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

TL;DR

This paper investigates non-perturbative effects in matrix models related to two-dimensional quantum gravity, revealing complex eigenvalue distributions, the role of theta-parameters, and connections to instanton methods and loop equations.

Contribution

It introduces a novel description of large N eigenvalue distributions using complex contours and identifies the non-perturbative theta-parameter as key to solving the string equation.

Findings

Eigenvalues form tree-like structures in the complex plane.

The theta-parameter determines the non-perturbative vacuum structure.

Non-perturbative effects are linked to poles in loop operators.

Abstract

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 split solution describes the strong coupling phase of 2d quantum gravity (c=0 non-critical 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 theta-parameter that fixes uniquely the solution of the string equation (Painleve 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 theta-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.