@MISC{Marshakov05complexgeometry, author = {A. Marshakov and A. Mironov}, title = {Complex Geometry of Matrix Models L. Chekhov a}, year = {2005} }

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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. Recent interest to matrix models and especially to their so-called multisupport (multicut) solutions was inspired by the studies in N = 1 SUSY gauge theories due to Cachazo, Intrilligator and Vafa [1], [2] and by the proposal of Dijkgraaf and Vafa [3] to calculate the low energy superpotentials, using the partition function of multicut solutions. The solutions themselves are well-known already for a long time (see, e.g., [4, 5]) with a new vim due to the paper by Bonnet, David and Eynard [6]. The Dijkgraaf–Vafa proposal was to consider the nonperturbative superpotentials of N = 1 SUSY gauge theories in four dimensions (possibly coming as the softly broken N = 2 Seiberg–Witten (SW) theories [7, 8]) arising from the partition functions of the one-matrix model (1MM) in the leading order in 1/N, N being the matrix size. The leading order (of the ’t Hooft 1/N-expansion) of the matrix model is described by the quasiclassical tau-function of the so-called universal Whitham hierarchy [9]