Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index

Abstract

We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the An 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 S¹×S³/Zr 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(2n). This provides one of the most elaborate checks of the Seiberg duality known to date. As another application of the An integral, we prove a star-star relation for a two-dimensional integrable lattice model of statistical mechanics, previously given by the second author.