Matrix Models vs. Seiberg–witten/whitham Theories

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. 1. It is well known that partition functions of matrix models are τ-functions of integrable hierarchies of the Toda type [1]. In the specific double scaling limit, these τ-functions become τ-functions of various reduction of the KP hierarchy [2]. If one makes the simplest, large-N (planar) limit, the partition function becomes the τ-function of the dispersionless Toda hierarchy, which in turn becomes the τ-function of the dispersionless (reductions of) KP hierarchy [3] after performing the continuum limit (which basically means working nearby a singularity of the partition function). All these dispersionless hierarchies are just Whitham equations over trivial solutions to integrable (Toda, KP) hierarchies. When solving matrix models, most attention was paid to one-cut solutions where the limiting eigenvalue distribution spans one interval on the real axis [4]. The results on multi-cut solutions were few [5, 6, 7]. Recently, Dijkgraaf and Vafa proposed [8] the new insight on the multi-cut large-N limit of matrix models. Namely, they associated this limit with a Riemann surface and some related SW system. Its prepotential, which we prove here to be the logarithm of the large-N partition function, is typically associated with the logarithm of some Whitham τ-functions [9, 10]. This hints that the matrix matrix model in the large N limit of multi-cut type describes the Whitham system over a non-trivial, finite-gap solution to integrable (Toda, KP) hierarchy. In particular, this solution passes to a finite-gap solution of (reductions of) the KP hierarchy in the continuum limit. In this paper, we restrict ourselves with the simplest example of the Hermitean one-matrix model. We show that coefficients of the potential of the model gives rise to Whitham flows and manifestly construct this Whitham system. In fact, the authors of [8] associated the N = 1 SUSY gauge theory studied in [11] with the SW system related to the multi-cut planar limit of matrix models. From the point of view of N = 1 SUSY theory, these coefficients must be identified with couplings in the tree …