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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.

hep-th/9110068

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.

hep-th/9110069

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.

hep-th/9110070

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.

hep-th/9110071

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.

hep-th/9110072

Starting from SL(3,R) Chern-Simons theory we derive the covariant action for W_3 gravity.

hep-th/9110073

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.

hep-th/9110074

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.

hep-th/9110075

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.

hep-th/9110076

The author argues to Silicon Valley that the most important and powerful part of computer science is work that is simultaneously theoretical and practical. He particularly considers the intersection of the theory of algorithms and practical software development. He combines examples from the development of the TeX typesetting system with clever jokes, criticisms, and encouragements.

cs/9301114

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.

hep-th/9111001

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.

hep-th/9111002

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.

hep-th/9111003

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.

hep-th/9111005

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.

hep-th/9111007

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.

hep-th/9111006

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.

hep-th/9111008

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.

hep-th/9111009

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

hep-th/9111010

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.

hep-th/9111011

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.

hep-th/9111012

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.

hep-th/9111013

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.

hep-th/9111014

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.

hep-th/9111015

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.

hep-th/9111016

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.

hep-th/9111017

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.

hep-th/9111018

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.

hep-th/9111019

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.

hep-th/9111020

We prove the existence of at least $cl(M)$ periodic orbits for certain time dependant Hamiltonian systems on the cotangent bundle of an arbitrary compact manifold $M$. These Hamiltonians are not necessarily convex but they satisfy a certain boundary condition given by a Riemannian metric on $M$. We discretize the variational problem by decomposing the time 1 map into a product of ``symplectic twist maps''. A second theorem deals with homotopically non trivial orbits in manifolds of negative curvature.

math/9201297

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.

hep-th/9111021

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 ${\cal M}_{p/p+2}$.

hep-th/9111022

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.

hep-th/9111023

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.

hep-th/9111024

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.

hep-th/9111025

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.

hep-th/9111029

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.

hep-th/9111026

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.

hep-th/9111027

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.

hep-th/9111028

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.

hep-th/9111030

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.

hep-th/9111031

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.

hep-th/9111032

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.

hep-th/9111033

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.

hep-th/9111034

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.

hep-th/9111035

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.

hep-th/9111036

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 $c\leq 1.$

hep-th/9111037

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.

hep-th/9111038

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.

hep-th/9111040

We summarize some aspects of matrix models from the approaches directly based on their properties at finite N.

hep-th/9111039

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.

hep-th/9111041

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.

hep-th/9111042

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.

hep-th/9111044

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.

hep-th/9111046

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.

hep-th/9111047

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.

hep-th/9111048

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.

hep-th/9111049

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.

hep-th/9111050

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.

hep-th/9111052

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.

hep-th/9111053

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.

hep-th/9111054

Let $K$ be a compact subset of $\bar{\bold C} ={\bold R}^2$ and let $K^c$ denote its complement. We say $K\in HR$, $K$ is holomorphically removable, if whenever $F:\bar{\bold C} \to\bar{\bold C}$ is a homeomorphism and $F$ is holomorphic off $K$, then $F$ is a M\"obius transformation. By composing with a M\"obius transform, we may assume $F(\infty )=\infty$. The contribution of this paper is to show that a large class of sets are $HR$. Our motivation for these results is that these sets occur naturally (e.g. as certain Julia sets) in dynamical systems, and the property of being $HR$ plays an important role in the Douady-Hubbard description of their structure.

math/9201298

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.

hep-th/9111057

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.

hep-th/9111056

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 $M=N$.

hep-th/9111058

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.

hep-th/9111059

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.

hep-th/9111063

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.

hep-th/9111060

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.

hep-th/9111062

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 $\hat{B_{2}}$ .

hep-th/9111061

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.

hep-th/9111064

This article is a sketch of ideas that were once intended to appear in the author's famous series, "The Art of Computer Programming". He generalizes the notion of a context-free language from a set to a multiset of words over an alphabet. The idea is to keep track of the number of ways to parse a string. For example, "fruit flies like a banana" can famously be parsed in two ways; analogous examples in the setting of programming languages may yet be important in the future. The treatment is informal but essentially rigorous.

cs/9301115

A survey of ghost techniques in mathematical physics, which can be grouped under the rubric of `cohomological physics', particularly BRST cohomology.

hep-th/9112002

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.

hep-th/9112003

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.

hep-th/9112004

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.

hep-th/9112006

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.

hep-th/9112007

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.

hep-th/9112008

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 $k=9/4$.

hep-th/9112009

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.

hep-th/9112010

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.

hep-th/9112011

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.

hep-th/9112012

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.

hep-th/9112016

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

hep-th/9112013

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.

hep-th/9112014

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.

hep-th/9112015

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.

hep-th/9112017

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.

hep-th/9112018

Continuum and discrete approaches to 2d gravity coupled to $c<1$ matter are reviewed.

hep-th/9112019

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.

hep-th/9112021

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.

hep-th/9112023

In this paper, we assume that $G$ is a finitely generated torsion free non-elementary Kleinian group with $\Omega(G)$ nonempty. We show that the maximal number of elements of $G$ that can be pinched is precisely the maximal number of rank 1 parabolic subgroups that any group isomorphic to $G$ may contain. A group with this largest number of rank 1 maximal parabolic subgroups is called {\it maximally parabolic}. We show such groups exist. We state our main theorems concisely here. Theorem I. The limit set of a maximally parabolic group is a circle packing; that is, every component of its regular set is a round disc. Theorem II. A maximally parabolic group is geometrically finite. Theorem III. A maximally parabolic pinched function group is determined up to conjugacy in $PSL(2,{\bf C})$ by its abstract isomorphism class and its parabolic elements.

math/9201299

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.

hep-th/9112026

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.

hep-th/9112025

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.

hep-th/9112028

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.

hep-th/9112029

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.

hep-th/9112030

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.

hep-th/9112031

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'.

hep-th/9112032

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.

hep-th/9112033