### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 14 (2018), 013, 29 pages      arXiv:1704.03159      https://doi.org/10.3842/SIGMA.2018.013
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

### Elliptic Hypergeometric Sum/Integral Transformations and Supersymmetric Lens Index

Andrew P. Kels a and Masahito Yamazaki b
a) Institute of Physics, University of Tokyo, Komaba, Tokyo 153-8902, Japan
b) Kavli Institute for the Physics and Mathematics of the Universe (WPI), University of Tokyo, Chiba 277-8583, Japan

Received April 24, 2017, in final form February 02, 2018; Published online February 16, 2018

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 ${\rm SU}(n+1)$ and ${\rm Sp}(2n)$. 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.

Key words: elliptic hypergeometric; elliptic gamma; supersymmetric; Seiberg duality; integrable; exactly solvable; Yang-Baxter; star-star.

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