Non-perturbative Effects in Matrix Models and Vacua of Two Dimensional Gravity

The most general large N eigenvalues distribution for the one matrix model is shown to consist of tree-like structures in the complex plane. For the m = 2 critical point, such a solution describes the strong coupling phase of 2d quantum gravity (c = 0 noncritical string). It is obtained by taking combinations of complex contours in the matrix integral, and the relative weight of the contours is identified with the non-perturbative “θ-parameter” that fixes uniquely the solution of the string equation (Painlevé I). This allows to recover by instanton methods results on the non-perturbative effects obtained by the Isomonodromic Deformation Method, and to construct for each θ-vacuum the observables (the loop correlation functions) which satisfy the loop equations. The breakdown of analyticity of the large N solution is related to the existence of poles for the loop operators. December 1992 † Physique Théorique CNRS ⋆ Laboratoire de la Direction des Sciences de la Matière du Commissariat à l’Energie Atomique The discovery of the “double scaling solutions” of the matrix models [1] [2] [3] led to important progress in the understanding of string theories in d ≤ 2 backgrounds and of 2d gravity (see [4] [5] for reviews). However, the important issue of the non-perturbative status of some of these theories remains unclear, in particular for 2d gravity coupled to unitary matter for c ≤ 1. In this letter, we discuss some of these questions in the framework of the Hermitian one matrix models. We shall show that a simple generalization of the complex integration contour prescription [6] [7] , which allows to construct non-perturbative — but in general complex — solutions of the string equations and of the continuous loop equations, leads to real non-perturbative solutions of these equations. This generalization, which consists in taking combinations of inequivalent integration contours, has been already discussed by Fokas, Its and Kitaev [8] in the framework of the Isomonodromic Deformation Method (IDM) approach to the string equations [9] , but does not seem to have attracted much attention. Our treatment is based on the BIPZ solution of the one matrix model [10], and follows our previous analysis of [11]. We shall show that in the limit N → ∞, new solutions for the eigenvalues (e.v.) distribution exist, which have not been discussed before. They correspond to a distribution of e.v. along tree-like structures in the complex plane. Moreover, these solutions depend non-analytically of the coupling constant of the matrix model, and will be associated with the sectors with an infinite number of poles of the string equation solutions. The non-perturbative parameter which characterizes the non-perturbative solutions is simply related to the different weights chosen for the contours, and our treatment allows to recover easily by instanton methods some results of [7][8]. In addition, we show that to each real solution of the string equation is associated a prescription for the asymptotics of the loop operators which defines uniquely observables (i.e. macroscopic loop v.e.v.) which obey the loop equations. Finally we shall show that these new solutions allow to explain the properties of the solutions for the double well matrix models recently discussed by Brower, Deo, Jain and Tan [12]. In the matrix model formulation of 2d gravity, the partition function F (sum over orientable connected 2-dimensional Riemannian spaces) is discretized into a sum over triangulations, and is written as the logarithm of the partition function Z for the Hermitian one-matrix model (F = lnZ), which after diagonalization of the matrix Φ can be written as an e.v. integral