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

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

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