The star-triangle relation, lens partition function, and hypergeometric sum/integrals

TL;DR

This paper explores the hyperbolic limit of elliptic hypergeometric identities, revealing new solutions to the star-triangle relation in statistical mechanics and dualities in supersymmetric gauge theories, bridging mathematical physics and quantum field theory.

Contribution

It introduces a hyperbolic sum/integral identity derived from elliptic hypergeometric functions, generalizing the Faddeev-Volkov models and connecting to gauge theory dualities.

Findings

New solution to the star-triangle relation in lattice models

Duality of lens partition functions in 3D supersymmetric gauge theories

Hyperbolic sum/integral identity generalizing previous models

Abstract

The aim of the present paper is to consider the hyperbolic limit of an elliptic hypergeometric sum/integral identity, and associated lattice model of statistical mechanics previously obtained by the second author. The hyperbolic sum/integral identity obtained from this limit, has two important physical applications in the context of the so-called gauge/YBE correspondence. For statistical mechanics, this identity is equivalent to a new solution of the star-triangle relation form of the Yang-Baxter equation, that directly generalises the Faddeev-Volkov models to the case of discrete and continuous spin variables. On the gauge theory side, this identity represents the duality of lens ($S_b^3/\mathbb{Z}_r$) partition functions, for certain three-dimensional $\mathcal N = 2$ supersymmetric gauge theories.