On Bailey pairs for $\mathcal N=2$ supersymmetric gauge theories on $S_b^3/\mathbb{Z}_r$

TL;DR

This paper constructs new Bailey pairs related to hyperbolic hypergeometric identities, revealing connections between supersymmetric gauge theories, integrable models, and 3-manifold topology.

Contribution

It introduces novel Bailey pairs for key identities, linking supersymmetric dualities with integrability and topological moves in 3-manifolds.

Findings

Bailey pairs for star-triangle and star-star relations

Representation of Pachner move via Bailey pairs

Connections between gauge theories and integrable models

Abstract

We study Bailey pairs construction for hyperbolic hypergeometric integral identities acquired via the duality of lens partitions functions for the three-dimensional $\mathcal N=2$ supersymmetric gauge theories on $S_b^3/\mathbb{Z}_r$. The novel Bailey pairs are constructed for the star-triangle relation, the star-star relation and the pentagon identity. The first two of them are integrability conditions for the Ising-type integrable lattice models. The last one corresponds to the representation of the basic $2-3$ Pachner move for triangulated 3-manifolds.