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arXiv · 1704.03159

Elliptic hypergeometric sum/integral transformations and supersymmetric lens index

Andrew P. Kels·Masahito Yamazaki
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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.

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Andrew P. Kels, Masahito Yamazaki. 2017-04-11. Elliptic hypergeometric sum/integral transformations and supersymmetric lens index. https://doi.org/10.3842/sigma.2018.013

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