From 4d superconformal indices to 3d partition functions

An exact formula for partition functions in 3d field theories was recently suggested by Jafferis, and Hama, Hosomichi, and Lee. These functions are expressed in terms of specific $q$-hypergeometric integrals whose key building block is the double sine function (or the hyperbolic gamma function). Elliptic hypergeometric integrals, discovered by the second author, define 4d superconformal indices. Using their reduction to the hyperbolic level, we describe a general scheme of reducing 4d superconformal indices to 3d partition functions which imply an efficient way of getting 3d $\mathcal{N}=2$ supersymmetric dualities for both SYM and CS theories from the "parent" 4d $\mathcal{N}=1$ dualities for SYM theories. As an example, we consider explicitly the duality pattern for 3d $\mathcal{N}=2$ SYM and CS theories with SP(2N) gauge group with the antisymmetric tensor matter.