Superconformal indices and integrable systems

Seiberg duality (1995) assumes an equivalence of certain 4d N=1 supersymmetrie nonabelian gauge field theories. Superconformal indices of such theories (2006) are described by elliptic hypergeometrie integrals introduced by the speaker in 2000. Symmetry transformations of these integrals give rigorous confirmations of the duality hypotheses. Many of such symmetries are found from the Bailey lemma yielding the star-triangle relation. The latter leads to solutions of the Yang-Baxter equation in terms of the integral operators with elliptic hypergeometrie kernels and relates the Seiberg duality to integrability of 2d lattice systems.