Matrix models vs. Seiberg-Witten/Whitham theories

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 $τ$-function. The corresponding Whitham hierarchy is explicitly constructed. The double-point problem is solved.