Superconformal indices, matrix integrals, and duality

There is a direct connection between the Seiberg duality for four dimensional N=1 supersymmetric field theories and the theory of elliptic hypergeometric integrals formulated by the author around 10 years ago. Roemelsberger conjectured in 2007 that superconformal (topological) indices for dual field theories coincide. Dolan and Osborn in 2008 confirmed this for a number of simplest dualities by showing that the indices coincide with the particular elliptic hypergeometric integrals. Seiberg duality appears to be equivalent to discrete Weyl group symmetries in the parameter space of the latter integrals. In a joint work with G. Vartanov [arXiv:0910.5944 [4]] we have systematically analyzed all known dualities and suggested many new ones using known relations for integrlas. In this talk I will briefly outline the structure of indices and integrals and discuss known automorphic properties of these objects.