The hyperbolic asymptotics of elliptic hypergeometric integrals arising in supersymmetric gauge theory

摘要

The purpose of this article is to demonstrate that $i)$ the framework ofElliptic Hypergeometric Integrals (EHIs) can be extended by input fromsupersymmetric (SUSY) gauge theory, and $ii)$ analyzing the hyperbolic limit ofthe EHIs in the extended framework leads to a rich structure containing sharpmathematical problems of interest to supersymmetric quantum field theorists. Both of the above items have already been discussed in the theoreticalphysics literature. Item $i$ was demonstrated by Dolan and Osborn in 2008. Item$ii$ was discussed in the present author's phd thesis in 2016, wherein crucialelements were borrowed from the 2006 work of Eric Rains on the hyperbolic limitof certain classes of EHIs. This article is a concise review of thesedevelopments, written for mathematicians interested in EHIs. In particular, wework with a representation-theoretic definition of a SUSY gauge theory, so thatreaders without any background in gauge theory---but familiar with therepresentation theory of semi-simple Lie algebras---can follow the discussion.