# Elliptic hypergeometric sum/integral transformations and supersymmetric lens index

## TL;DR

This paper proves new transformation formulas for elliptic hypergeometric sums and integrals related to root systems, and applies these results to verify supersymmetric index equalities and integrable lattice models, advancing mathematical physics and gauge theory understanding.

## Contribution

It introduces generalized elliptic hypergeometric transformations involving the lens elliptic gamma function and applies them to supersymmetric gauge theories and integrable models.

## Key findings

- Established new transformation formulas for elliptic hypergeometric sums/integrals.
- Proved equality of supersymmetric indices for dual gauge theories, confirming Seiberg duality.
- Derived a star-star relation for an integrable lattice model.

## Abstract

We prove a pair of transformation formulas for multivariate elliptic hypergeometric sum/integrals associated to the $A_n$ and $BC_n$ 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 $S^1\times S^3/\mathbb{Z}_r$ supersymmetric indices, for a pair of four-dimensional $\mathcal{N}=1$ 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 $A_n$ integral, we prove a star-star relation for a two-dimensional integrable lattice model of statistical mechanics, previously given by the second author.