  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.
