Lens partition function, pentagon identity and star-triangle relation

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 $U_q(osp(1|2))$.

22 pages, v2: minor corrections and comments, v3: minor corrections

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Cited by in corpus (2)

• Hyperbolic and trigonometric hypergeometric solutions to the star-star equation
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$