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

SIGMA 14 (2018), 043, 30 pages      arXiv:1712.09933
Contribution to the Special Issue on Elliptic Hypergeometric Functions and Their Applications

The Hyperbolic Asymptotics of Elliptic Hypergeometric Integrals Arising in Supersymmetric Gauge Theory

Arash Arabi Ardehali
School of Physics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5531, Tehran, Iran

Received January 09, 2018, in final form April 29, 2018; Published online May 06, 2018

The purpose of this article is to demonstrate that $i)$ the framework of elliptic hypergeometric integrals (EHIs) can be extended by input from supersymmetric gauge theory, and $ii)$ analyzing the hyperbolic limit of the EHIs in the extended framework leads to a rich structure containing sharp mathematical problems of interest to supersymmetric quantum field theorists. Both of the above items have already been discussed in the theoretical physics literature. Item $i$ was demonstrated by Dolan and Osborn in 2008. Item $ii$ was discussed in the present author's Ph.D. Thesis in 2016, wherein crucial elements were borrowed from the 2006 work of Rains on the hyperbolic limit of certain classes of EHIs. This article contains a concise review of these developments, along with minor refinements and clarifying remarks, written mainly for mathematicians interested in EHIs. In particular, we work with a representation-theoretic definition of a supersymmetric gauge theory, so that readers without any background in gauge theory - but familiar with the representation theory of semi-simple Lie algebras - can follow the discussion.

Key words: elliptic hypergeometric integrals; supersymmetric gauge theory; hyperbolic asymptotics.

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