Pentagon Identities Arising in Supersymmetric Gauge Theory Computations

Abstract

The partition functions of three-dimensional N = 2 supersymmetric gauge theories on different manifolds can be expressed as q-hypergeometric integrals. By comparing the partition functions of three-dimensional mirror dual theories, one finds complicated integral identities. In some cases, these identities can be written in the form of pentagon relations. Such identities often have an interpretation as the Pachner's 3-2 move for triangulated manifolds via the so-called 3d-3d correspondence. From the physics perspective, another important application of pentagon identities is that they may be used to construct new solutions to the quantum Yang-Baxter equation.

New Pentagon Identities Revisited

“…See[7,10,16] for pentagon identities from gauge theories 4. Special case of this identity corresponds to the pentagon identity related to the tetrahedron index[9].…”

“…See[6,12,15] for the relation between supersymmetric dualities and star-triangle relation. More information about pentagon identities and star-triangle relations see[19].…”

Lens partition function, pentagon identity and star-triangle relation

“…There are several examples of integral pentagon relations computed via three-dimensional supersymmetric dualities, see, e.g. [24,[40][41][42][43][45][46][47]. Here we present a new pentagon identity in terms of hyperbolic gamma functions.…”

Lens Partition Functions and Integrability Properties

“…In this section, we explain the work [36] and give some details of the new result by applying gauge symmetry breaking. The idea is first done in [8] (see also [37]) for the duality of partition functions on S 3 b and used in [4] for the dual theories on S 3 b /Z r so for details both studies represent the way of doing gauge symmetry breaking step by step.…”