Consistent string backgrounds and completely integrable 2D field theories

After reviewing the $\beta$-function equations for consistent string backgrounds in the $\sigma$-model approach, including metric and antisymmetric tensor, dilaton and tachyon potential, we apply this formalism to WZW models. We particularly emphasize the case where the WZW model is perturbed by an integrable marginal tachyon potential operator leading to the non-abelian Toda theories. Already in the simplest such theory, there is a large non-linear and non-local chiral algebra that extends the Virasoro algebra. This theory is shown to have two formulations, one being a classical reduction of the other. Only the non-reduced theory is shown to satisfy the $\beta$-function equations.Comment: 12 pages, uses PHYZZ

Small-time expansion of the Fokker-Planck kernel for space and time dependent diffusion and drift coefficients

We study the general solution of the Fokker-Planck equation in d dimensions with arbitrary space and time dependent diffusion matrix and drift term. We show how to construct the solution, for arbitrary initial distributions, as an asymptotic expansion for small time. This generalizes the well-known asymptotic expansion of the heat-kernel for the Laplace operator on a general Riemannian manifold. We explicitly work out the general solution to leading and next-to-leading order in this small-time expansion, as well as to next-to-next-to-leading order for vanishing drift. We illustrate our results on a several examples.Comment: 30 page

Aspects of Exactly Solvable Quantum-Corrected 2D Dilaton Gravity Theories

After reviewing the basic aspects of the exactly solvable quantum-corrected dilaton-gravity theories in two dimensions, we discuss a (subjective) selection of other aspects: a) supersymmetric extensions, b) canonical formalism, ADM-mass, and the functional integral measure, and c) a positive energy theorem and its application to the ADM- and Bondi-masses.Comment: 30 pages. Based on Talks given at Strings 93, Berkeley, May 1993, and at the Santa Barbara conference "Quantum Aspects of Black Holes". PUPT-141

Anomaly Cancellations on Lower-Dimensional Hypersurfaces by Inflow from the Bulk

Lower-dimensional (hyper)surfaces that can carry gauge or gauge/gravitational anomalies occur in many areas of physics: one-plus-one-dimensional boundaries or two-dimensional defect surfaces in condensed matter systems, four-dimensional brane-worlds in higher-dimensional cosmologies or various branes and orbifold planes in string or M-theory. In all cases we may have (quantum) anomalies localized on these hypersurfaces that are only cancelled by ``anomaly inflow'' from certain topological interactions in the bulk. Proper cancellation between these anomaly contributions of different origin requires a careful treatment of factors and signs. We review in some detail how these contributions occur and discuss applications in condensed matter (Quantum Hall Effect) and M-theory (five-branes and orbifold planes)Comment: 39 pages, to appear in: From Fields to Strings: Circumnavigating Theoretical Physics, Ian Kogan Memorial Volum

Does Coupling To Gravity Preserve Integrability ?

These are notes based on a lecture given at the Cargese summer school 1995. I describe evidence that the (two-dimensional) integrable chiral Gross-Neveu model might remain integrable when coupled to gravity. The results presented here were obtained in collaboration with Ian Kogan.Comment: 11 pages, uses PHYZZX, 7 figures (encapsulated postscript), (one reference added

NON-LOCAL EXTENSIONS OF THE CONFORMAL ALGEBRA : MATRIX WW-ALGEBRAS, MATRIX KdV-HIERARCHIES AND NON-ABELIAN TODA THEORIES,

In the present contribution, I report on certain {\it non-linear} and {\it non-local} extensions of the conformal (Virasoro) algebra. These so-called $V$-algebras are matrix generalizations of $W$-algebras. First, in the context of two-dimensional field theory, I discuss the non-abelian Toda model which possesses three conserved (chiral) ``currents". The Poisson brackets of these ``currents" give the simplest example of a $V$-algebra. The classical solutions of this model provide a free-field realization of the $V$-algebra. Then I show that this $V$-algebra is identical to the second Gelfand-Dikii symplectic structure on the manifold of $2\times 2$-matrix Schr\"odinger operators L=-\d^2+U (with \tr\sigma_3 U=0). This provides a relation with matrix KdV-hierarchies and allows me to obtain an infinite family of conserved charges (Hamiltonians in involution). Finally, I work out the general $V_{n,m}$-algebras as symplectic structures based on $n\times n$-matrix $m^{\rm th}$-order differential operators L=-\d^m +U_2\d^{m-2}+U_3 \d^{m-3}+\ldots +U_m. It is the absence of $U_1$, together with the non-commutativity of matrices that leads to the non-local terms in the $V_{n,m}$-algebras. I show that the conformal properties are similar to those of $W_m$-algebras, while the complete $V_{n,m}$-algebras are much more complicated, as is shown on the explicit example of $V_{n,3}$.Comment: 30 pages, uses phyzzx, lecture given at the "59. Rencontre de Strasbourg