Matrix Models, Complex Geometry and Integrable Systems. I

TL;DR

This paper explores the geometric and integrable structures underlying one- and two-matrix models, revealing their universal description via complex curves and tau-functions, and suggesting potential generalizations beyond one complex dimension.

Contribution

It provides a detailed analysis of the geometric properties of matrix models and links them to integrable systems and complex geometry, extending the understanding of their universal structures.

Findings

Matrix models are described by complex curves and tau-functions.

The geometric picture can be generalized beyond one complex dimension.

Planar matrix models are universally related to integrable systems.

Abstract

We consider the simplest gauge theories given by one- and two- matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. We study the main ingredients of this geometric picture, suggesting that it can be generalized beyond one complex dimension, and formulate them in terms of the quasiclassical integrable systems, solved by construction of tau-functions or prepotentials. The complex curves and tau-functions of one- and two- matrix models are discussed in detail.