Matrix models, complex geometry, and integrable systems: I

Matrix Models , Complex Geometry and Integrable Systems . I ∗

We consider the simplest gauge theories given by one-and two-matrix integrals and concentrate on their stringy and geometric properties. We remind general integrable structure behind the matrix integrals and turn to the geometric properties of planar matrix models, demonstrating that they are universally described in terms of integrable systems directly related to the theory of complex curves. ...

Matrix Models , Complex Geometry and Integrable Systems . II ∗

We consider certain examples of applications of the general methods, based on geometry and in-tegrability of matrix models, described in [1]. In particular, the nonlinear differential equations, satisfied by quasiclassical tau-functions are investigated. We also discuss a similar quasiclassical geometric picture, arising in the context of multidimensional supersymmetric gauge theories and the A...

Complex Geometry of Matrix Models

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 derivativ...

Discrete Integrable Systems and Geometry

Integrable Discrete Linear Systems and One - Matrix Random Models

In this paper we analyze one–matrix models by means of the associated discrete linear systems. We see that the consistency conditions of the discrete linear system lead to the Virasoro constraints. The linear system is endowed with gauge invariances. We show that invariance under time–independent gauge transformations entails the integrability of the model, while the double scaling limit is con...