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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.

hep-th/9112034

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.

hep-th/9112035

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 ($\hat{c} = 1$).

hep-th/9112036

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.

hep-th/9112037

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.

hep-th/9112038

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.

hep-th/9112039

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.

hep-th/9112040

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.

hep-th/9112041

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.

hep-th/9112042

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.

hep-th/9112043

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.

hep-th/9112044

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.

hep-th/9112047

$O(N)$ 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.

hep-th/9112048

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.

hep-th/9112049

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 $d=1-3Q_m^2$.

hep-th/9112055

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.

hep-th/9112067

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.

hep-th/9112050

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.

hep-th/9112051

We generalize Toda--like integrable lattice systems to non--symmetric case. We show that they possess the bi--Hamiltonian structure.

hep-th/9112053

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.

hep-th/9112054

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.

hep-th/9112052

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.

hep-th/9112056

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.

hep-th/9112058

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.

hep-th/9112057

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.

hep-th/9112060

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.

hep-th/9112062

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.

hep-th/9112063

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.

hep-th/9112064

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.

hep-th/9112065

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.

hep-th/9112066

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.

hep-th/9112068

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.

hep-th/9112069

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.

hep-th/9112070

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.

hep-th/9112072

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 $L_{p,1}$.

hep-th/9112071

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.

hep-th/9112073

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.

hep-th/9112074

Certain subclasses of $B_1(K)$, the Baire-1 functions on a compact metric space $K$, are defined and characterized. Some applications to Banach spaces are given.

math/9201236

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.

hep-th/9112076

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

hep-th/9201001

We present here "the" cartesian closed theory for real analytic mappings. It is based on the concept of real analytic curves in locally convex vector spaces. A mapping is real analytic, if it maps smooth curves to smooth curves and real analytic curves to real analytic curves. Under mild completeness conditions the second requirement can be replaced by: real analytic along affine lines. Enclosed and necessary is a careful study of locally convex topologies on spaces of real analytic mappings. As an application we also present the theory of manifolds of real analytic mappings: the group of real analytic diffeomorphisms of a compact real analytic manifold is a real analytic Lie group.

math/9201254

A rather simple natural outer derivation of the graded Lie algebra of all vector valued differential forms with the Fr\"olicher-Nijenhuis bracket turns out to be a differential and gives rise to a cohomology of the manifold, which is functorial under local diffeomorphisms. This cohomology is determined as the direct product of the de Rham cohomology space and the graded Lie algebra of "traceless" vector valued differential forms, equipped with a new natural differential concomitant as graded Lie bracket. We find two graded Lie algebra structures on the space of differential forms. Some consequences and related results are also discussed.

math/9201255

For any unitary representation of an arbitrary Lie group I construct a moment mapping from the space of smooth vectors of the representation into the dual of the Lie algebra. This moment mapping is equivariant and smooth. For the space of analytic vectors the same construction is possible and leads to a real analytic moment mapping.

math/9201256

We define two $(n+1)$ graded Lie brackets on spaces of multilinear mappings. The first one is able to recognize $n$-graded associative algebras and their modules and gives immediately the correct differential for Hochschild cohomology. The second one recognizes $n$-graded Lie algebra structures and their modules and gives rise to the notion of Chevalley cohomology.

math/9201257

The space of all non degenerate bilinear structures on a manifold $M$ carries a one parameter family of pseudo Riemannian metrics. We determine the geodesic equation, covariant derivative, curvature, and we solve the geodesic equation explicitly. Each space of pseudo Riemannian metrics with fixed signature is a geodesically closed submanifold. The space of non degenerate 2-forms is also a geodesically closed submanifold. Then we show that, if we fix a distribution on $M$, the space of all Riemannia metrics splits as the product of three spaces which are everywhere mutually orthogonal, for the usual metric. We investigate this situation in detail.

math/9201258

The space of all Riemannian metrics on a smooth second countable finite dimensional manifold is itself a smooth manifold modeled on the space of symmetric (0,2)-tensor fields with compact support. It carries a canonical Riemannian metric which is invariant under the action of the diffeomorphism group. We determine its geodesics, exponential mapping, curvature, and Jacobi fields in a very explicit manner.

math/9201259

Certain second-order partial differential operators, which are expressed as sums of squares of real-analytic vector fields in $\Bbb R^3$ and which are well known to be $C^\infty$ hypoelliptic, fail to be analytic hypoelliptic.

math/9201260

In this announcement we present a general and new approach to analyzing the asymptotics of oscillatory Riemann-Hilbert problems. Such problems arise, in particular, in evaluating the long-time behavior of nonlinear wave equations solvable by the inverse scattering method. We will restrict ourselves here exclusively to the modified Korteweg de Vries (MKdV) equation, $$y_t-6y^2y_x+y_{xxx}=0,\qquad -\infty<x<\infty,\ t\ge0, y(x,t=0)=y_0(x),$$ but it will be clear immediately to the reader with some experience in the field, that the method extends naturally and easily to the general class of wave equations solvable by the inverse scattering method, such as the KdV, nonlinear Schr\"odinger (NLS), and Boussinesq equations, etc., and also to ``integrable'' ordinary differential equations such as the Painlev\'e transcendents.

math/9201261

The authors discuss the role of controversy in mathematics as a preface to two opposing articles on computational complexity theory: "Some basic information on information-based complexity theory" by Beresford Parlett [math.NA/9201266] and "Perspectives on information-based complexity" by J. F. Traub and Henryk Wo\'zniakowski [math.NA/9201269].

math/9201262

We construct new coordinates for the Teichm\"uller space Teich of a punctured torus into $\bold{R} \times\bold{R}^+$. The coordinates depend on the representation of Teich as a space of marked Kleinian groups $G_\mu$ that depend holomorphically on a parameter $\mu$ varying in a simply connected domain in $\bold{C}$. They describe the geometry of the hyperbolic manifold $\bold{H}^3/G_\mu$; they reflect exactly the visual patterns one sees in the limit sets of the groups $G_\mu$; and they are directly computable from the generators of $G_\mu$.

math/9201263

The authors announce the following theorem. Theorem 1. If $G=A*_H B$ is an amalgamated product where $A$ and $B$ are finitely presented and semistable at infinity, and $H$ is finitely generated, then $G$ is semistable at infinity. If $G=A*_H$ is an HNN-extension where $A$ is finitely presented and semistable at infinity, and $H$ is finitely generated, then $G$ is semistable at infinity.

math/9201264

To most mathematicians and computer scientists the word ``tree'' conjures up, in addition to the usual image, the image of a connected graph with no circuits. In the last few years various types of trees have been the subject of much investigation, but this activity has not been exposed much to the wider mathematical community. This article attempts to fill this gap and explain various aspects of the recent work on generalized trees. The subject is very appealing for it mixes very na\"{\i}ve geometric considerations with the very sophisticated geometric and algebraic structures. In fact, part of the drama of the subject is guessing what type of techniques will be appropriate for a given investigation: Will it be direct and simple notions related to schematic drawings of trees or will it be notions from the deepest parts of algebraic group theory, ergodic theory, or commutative algebra which must be brought to bear? Part of the beauty of the subject is that the na\"{\i}ve tree considerations have an impact on these more sophisticated topics and that in addition, trees form a bridge between these disparate subjects.

math/9201265

Numerical analysts might be expected to pay close attention to a branch of complexity theory called information-based complexity theory (IBCT), which produces an abundance of impressive results about the quest for approximate solutions to mathematical problems. Why then do most numerical analysts turn a cold shoulder to IBCT? Close analysis of two representative papers reveals a mixture of nice new observations, error bounds repackaged in new language, misdirected examples, and misleading theorems. Some elements in the framework of IBCT, erected to support a rigorous yet flexible theory, make it difficult to judge whether a model is off-target or reasonably realistic. For instance, a sharp distinction is made between information and algorithms restricted to this information. Yet the information itself usually comes from an algorithm, so the distinction clouds the issues and can lead to true but misleading inferences. Another troublesome aspect of IBCT is a free parameter $F$, the class of admissible problem instances. By overlooking $F$'s membership fee, the theory sometimes distorts the economics of problem solving in a way reminiscent of agricultural subsidies. The current theory's surprising results pertain only to unnatural situations, and its genuinely new insights might serve us better if expressed in the conventional modes of error analysis and approximation theory.

math/9201266

Let $f$ be a holomorphic mapping between compact complex manifolds. We give a criterion for $f$ to have {\it unobstructed deformations}, i.e. for the local moduli space of $f$ to be smooth: this says, roughly speaking, that the group of infinitesimal deformations of $f$, when viewed as a functor, itself satisfies a natural lifting property with respect to infinitesimal deformations. This lifting property is satisfied e.g. whenever the group in question admits a `topological' or Hodge-theoretic interpretation, and we give a number of examples, mainly involving Calabi-Yau manifolds, where that is the case.

math/9201267

We survey existence and regularity results for semi-linear wave equations. In particular, we review the recent regularity results for the $u^5$-Klein Gordon equation by Grillakis and this author and give a self-contained, slightly simplified proof.

math/9201268

The authors discuss information-based complexity theory, which is a model of finite-precision computations with real numbers, and its applications to numerical analysis.

math/9201269

Shoen and Uhlenbeck showed that ``tangent maps'' can be defined at singular points of energy minimizing maps. Unfortunately these are not unique, even for generic boundary conditions. Examples are discussed which have isolated singularities with a continuum of distinct tangent maps.

math/9201270

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.

hep-th/9212156

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.

hep-th/9201002

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.

hep-th/9201003

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.

hep-th/9201004

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 $w_\infty$.

hep-th/9201005

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.

hep-th/9201007

This is a continuation of the paper [FJS] with a similar title. Several results from there are strengthened, in particular: 1. If T is a "natural" embedding of l_2^n into L_1 then, for any well-bounded factorization of T through an L_1 space in the form T=uv with v of norm one, u well-preserves a copy of l_1^k with k exponential in n. 2. Any norm one operator from a C(K) space which well-preserves a copy of l_2^n also well-preserves a copy of l_{\infty}^k with k exponential in n. As an application of these and other results we show the existence, for any n, of an n-dimensional space which well-embeds into a space with an unconditional basis only if the latter contains a copy of l_{\infty}^k with k exponential in n.

math/9201202

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.

hep-th/9201006

If E is a nonempty closed subset of the locally finite Hausdorff (2n-2)-measure on an n-dimensional complex manifold M and all points of E are nonremovable for a meromorphic mapping of M \ E into a compact K\"ahler manifold, then E is a pure (n-1)-dimensional complex analytic subset of M.

math/9201201

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.

hep-th/9201008

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.

hep-th/9201015

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.

hep-th/9201009

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.

hep-th/9201010

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.

hep-th/9201011

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.

hep-th/9201013

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.

hep-th/9201012

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.

hep-th/9201014

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.

hep-th/9201019

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.

hep-th/9201020

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.

hep-th/9201022

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.

hep-th/9201023

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.

hep-th/9201024

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.

hep-th/9201025

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.

hep-th/9201027

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.

hep-th/9201026

It is shown that the weak $L^p$ spaces $\ell^{p,\infty}, L^{p,\infty}[0,1]$, and $L^{p,\infty}[0,\infty)$ are isomorphic as Banach spaces.

math/9201237

In [Sh:89] we, answering a question of Monk, have explicated the notion of ``a Boolean algebra with no endomorphisms except the ones induced by ultrafilters on it'' (see section 2 here) and proved the existence of one with character density aleph_0, assuming first diamondsuit_{aleph_1} and then only CH. The idea was that if h is an endomorphism of B, not among the ``trivial'' ones, then there are pairwise disjoint D_n in B with h(d_n) not subset d_n. Then we can, for some S subset omega, add an element x such that d <= x for n in S, x cap d_n=0 for n not in S while forbidding a solution for {y cap h(d_n):n in S} cup {y cap h(d_n)=0:n not in S}. Further analysis showed that the point is that we are omitting positive quantifier free types. Continuing this Monk succeeded to prove in ZFC, the existence of such Boolean algebras of cardinality 2^{aleph_0}. We prove (in ZFC) the existence of such B of density character lambda and cardinality lambda^{aleph_0} whenever lambda > aleph_0. We can conclude answers to some questions from Monk's list. We use a combinatorial method from [Sh:45],[Sh:172], that is represented in Section 1.

math/9201238

Assuming 0^sharp does not exist, kappa is an uncountable cardinal and for all cardinals lambda with kappa <= lambda < kappa^{+ omega}, 2^lambda = lambda^+, we present a ``mini-coding'' between kappa and kappa^{+ omega}. This allows us to prove that any subset of kappa^{+ omega} can be coded into a subset, W of kappa^+ which, further, ``reshapes'' the interval [kappa, kappa^+), i.e., for all kappa < delta < kappa^+, kappa = (card delta)^{L[W cap delta]}. We sketch two applications of this result, assuming 0^sharp does not exist. First, we point out that this shows that any set can be coded by a real, via a set forcing. The second application involves a notion of abstract condensation, due to Woodin. Our methods can be used to show that for any cardinal mu, condensation for mu holds in a generic extension by a set forcing.

math/9201249

We prove that assuming suitable cardinal arithmetic, if B is a Boolean algebra every homomorphic image of which is isomorphic to a factor, then B has locally small density. We also prove that for an (infinite) Boolean algebra B, the number of subalgebras is not smaller than the number of endomorphisms, and other related inequalities. Lastly we deal with the obtainment of the supremum of the cardinalities of sets of pairwise incomparable elements of a Boolean algebra.

math/9201250

We present a survey of some results of the pcf-theory and their applications to cardinal arithmetic. We review basics notions (in section 1), briefly look at history in section 2 (and some personal history in section 3). We present main results on pcf in section 5 and describe applications to cardinal arithmetic in section 6. The limitations on independence proofs are discussed in section 7, and in section 8 we discuss the status of two axioms that arise in the new setting. Applications to other areas are found in section 9.

math/9201251

We show that it is consistent with ZFC that L^infty (Y,B, nu) has no linear lifting for many non-complete probability spaces (Y,B, nu), in particular for Y=[0,1]^A, B= Borel subsets of Y, nu = usual Radon measure on B .

math/9201252

It is shown that if T is stable unsuperstable, and aleph_1< lambda =cf(lambda)< 2^{aleph_0}, or 2^{aleph_0} < mu^+< lambda =cf(lambda)< mu^{aleph_0} then T has no universal model in cardinality lambda, and if e.g. aleph_omega < 2^{aleph_0} then T has no universal model in aleph_omega. These results are generalized to kappa =cf(kappa) < kappa (T) in the place of aleph_0. Also: if there is a universal model in lambda >|T|, T stable and kappa < kappa (T) then there is a universal tree of height kappa +1 in cardinality lambda .

math/9201253

A general construction for $\sigma-$finite absolutely continuous invariant measure will be presented. It will be shown that the local bounded distortion of the Radon-Nykodym derivatives of $f^n_*(\lambda)$ will imply the existence of a $\sigma-$finite invariant measure for the map $f$ which is absolutely continuous with respect to $\lambda$, a measure on the phase space describing the sets of measure zero. Furthermore we will discuss sufficient conditions for the existence of $\sigma-$finite invariant absolutely continuous measures for real 1-dimensional dynamical systems.

math/9201300

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.

hep-th/9201029

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.

hep-th/9201035

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.

hep-th/9201030

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.

hep-th/9201031

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.

hep-th/9201032

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.

hep-th/9201033

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.

hep-th/9201034

$G/G$ 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.

hep-th/9201036

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.

hep-th/9201037

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.

hep-th/9201039