Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index

We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the A(n) and BCn root systems, generalising the formulas previously obtained by Rains. The sum/integrals are expressed in terms of the lens elliptic gamma function, a generalisation of the elliptic gamma function that depends on an additional integer variable, as well as a complex variable and two elliptic nomes. As an application of our results, we prove an equality between S-1 x S-3 = Z(r) supersymmetric indices, for a pair of four-dimensional N = 1 supersymmetric gauge theories related by Seiberg duality, with gauge groups SU(n + 1) and Sp(2 ). This provides one of the most elaborate checks of the Seiberg duality known to date. As another application of the A(n) integral, we prove a star-star relation for a two-dimensional integrable lattice model of statistical mechanics, previously given by the second author.