Complex Geometry of Matrix Models

TL;DR

This paper explores the complex geometric structures underlying matrix models, focusing on multisupport solutions, their relation to supersymmetric gauge theories, and the mathematical properties of associated tau-functions and free energy.

Contribution

It introduces new results on the geometry of matrix models, including derivatives of tau-functions, relations from complex geometry, and subleading order free energy analysis.

Findings

Derivatives of tau-functions relate to Riemann surface geometry.

Free energy satisfies determinant relations.

Solutions described by quasiclassical hierarchies and WDVV equations.

Abstract

The paper contains some new results and a review of recent achievements, concerning the multisupport solutions to matrix models. In the leading order of the 't Hooft expansion for matrix integral, these solutions are described by quasiclassical or generalized Whitham hierarchies and are directly related to the superpotentials of four-dimensional N=1 SUSY gauge theories. We study the derivatives of tau-functions for these solutions, associated with the families of Riemann surfaces (with possible double points), and relations for these derivatives imposed by complex geometry, including the WDVV equations. We also find the free energy in subleading order of the 't Hooft expansion and prove that it satisfies certain determinant relations.