Complex Curve of the Two Matrix Model and its Tau-function

TL;DR

This paper explores the complex algebraic curves arising in two matrix models, revealing their structure, expressing key quantities via periods of differentials, and linking the partition function to tau-functions, with implications for supersymmetric theories.

Contribution

It introduces a new understanding of the complex curves in two matrix models and relates the partition function to tau-functions, extending previous hyperelliptic curve analysis.

Findings

The complex curve in two matrix models differs from the hyperelliptic case.

Partition functions are expressed as quasiclassical tau-functions.

The study connects matrix models to supersymmetric Yang-Mills theories.

Abstract

We study the hermitean and normal two matrix models in planar approximation for an arbitrary number of eigenvalue supports. Its planar graph interpretation is given. The study reveals a general structure of the underlying analytic complex curve, different from the hyperelliptic curve of the one matrix model. The matrix model quantities are expressed through the periods of meromorphic generating differential on this curve and the partition function of the multiple support solution, as a function of filling numbers and coefficients of the matrix potential, is shown to be the quasiclassical tau-function. The relation to softly broken N=1 supersymmetric Yang-Mills theories is discussed. A general class of solvable multimatrix models with tree-like interactions is considered.