Matrix models vs. Seiberg-Witten/Whitham theories

TL;DR

This paper establishes a precise mathematical connection between matrix models and Seiberg-Witten theories, showing that the matrix model partition function matches the SW prepotential and is related to Whitham hierarchies.

Contribution

It proves the equivalence of the Hermitean one-matrix model's partition function with the Seiberg-Witten prepotential in the planar limit, linking matrix models to integrable systems.

Findings

Partition function equals SW prepotential

Partition function is a Whitham τ-function

Explicit construction of the Whitham hierarchy

Abstract

We discuss the relation between matrix models and the Seiberg--Witten type (SW) theories, recently proposed by Dijkgraaf and Vafa. In particular, we prove that the partition function of the Hermitean one-matrix model in the planar (large $N$) limit coincides with the prepotential of the corresponding SW theory. This partition function is the logarithm of a Whitham $\tau$-function. The corresponding Whitham hierarchy is explicitly constructed. The double-point problem is solved.