Elliptic hypergeometry of supersymmetric dualities

We give a full list of known supersymmetric quantum field theories related by the Seiberg duality conjectures for the SU(N), SP(2N) and G_2 gauge groups. Many of the presented dualities are new, not considered earlier in the literature. For all these theories we construct superconformal indices and express them in terms of elliptic hypergeometric integrals. This gives a systematic extension of the related Romelsberger and Dolan-Osborn results. Equality of indices in dual theories leads to various identities for elliptic hypergeometric integrals. About half of them was proven earlier, and another half represents new challenging conjectures. In particular, we conjecture a dozen of new elliptic beta integrals on root systems, extending the univariate elliptic beta integral discovered by the first author