Lens partition function, pentagon identity and star-triangle relation

TL;DR

This paper explores the connection between supersymmetric gauge theories and integrable models, deriving integral identities like the star-triangle relation and pentagon identity from the lens partition function, with implications for 3-manifold topology and quantum algebra.

Contribution

It establishes a new integral identity linking supersymmetric gauge dualities to star-triangle and pentagon relations, advancing the gauge/YBE correspondence.

Findings

Derived star-triangle relation for Ising type models.

Obtained pentagon identity representing Pachner moves.

Proved orthogonality and completeness of certain Clebsch-Gordan coefficients.

Abstract

We study the three-dimensional lens partition function for $\mathcal N=2$ supersymmetric gauge dual theories on $S^3/\mathbb{Z}_r$ by using the gauge/YBE correspondence. This correspondence relates supersymmetric gauge theories to exactly solvable models of statistical mechanics. The equality of partition functions for the three-dimensional supersymmetric dual theories can be written as an integral identity for hyperbolic hypergeometric functions. We obtain such an integral identity which can be written as the star-triangle relation for Ising type integrable models and as the integral pentagon identity. The latter represents the basic 2-3 Pachner move for triangulated 3-manifolds. A special case of our integral identity can be used for proving orthogonality and completeness relation of the Clebsch-Gordan coefficients for the self-dual continuous series of $U_q(osp(1|2))$.