Non-Perturbative Effects in 2D Gravity and Matrix Models

Abstract

Two dimensional Euclidean quantum gravity may be formulated as a functional integral over 2-dimensional Riemannian manifolds. This infinite dimensional integral may be discretized in such a way that the topological expansion in terms of the genus of the manifold is mapped onto the 1/N expansion of some zero-dimensional matrix model [1]. The N = ∞ limit exhibits critical points which can be shown to describe the continuum limit of 2-dimensional gravity on a genus zero manifold, eventually coupled to some matter fields. Recently it was shown that a scaling limit can be constructed [2]. In this limit all the terms of the topological expansion survive and thus one obtains a fully non-perturbative solution for two dimensional gravity. However in the most interesting cases, in particular for pure gravity, the solution is defined as a solution of a non-linear differential equation of the Painlevé type and presents some non-perturbative ambiguities, related to the delicate issue of boundary conditions, which are usually attributed to some “non-perturbative effects” of the theory.