Superconformal Indices of N=4 SYM Field Theories

Superconformal indices (SCIs) of 4d N = 4 SYM theories with simple gauge groups are described in terms of elliptic hypergeometric integrals. For F4, E6, E7, E8 gauge groups this yields first examples of integrals of such type. S-duality transformation for G(2) and F-4 SCIs is equivalent to a change of integration variables. Equality of SCIs for SP(2N) and SO(2N + 1) group theories is proved in several important special cases. Reduction of SCIs to partition functions of 3d N = 2 SYM theories with one matter field in the adjoint representation is investigated, corresponding 3d dual partners are found, and some new related hyperbolic beta integrals are conjectured.