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 $U_q(osp(1|2))$.
Highlights
We study the three-dimensional lens partition function for N 1⁄4 2 supersymmetric gauge dual theories on S3=Zr by using the gauge/Yang-Baxter equation correspondence
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 recent progress in gauge/Yang-Baxter equation (YBE) correspondence has lead to remarkable connections between supersymmetric gauge theories, integrable models of statistical mechanics, and special functions
Summary
The recent progress in gauge/Yang-Baxter equation (YBE) correspondence has lead to remarkable connections between supersymmetric gauge theories, integrable models of statistical mechanics, and special functions. The main idea of the correspondence is that the supersymmetric duality for gauge theories leads to the integrability for spin lattice models, see [1,2] for a review and references therein. We obtain the identity (1.1) from the equality of partition functions of supersymmetric dual theories on S3=Zr. The intriguing physical interpretation of this integral identity is that it can be written as the star-triangle relation for a certain two-dimensional Ising-type statistical model, as well as the pentagon identity for a certain triangulated 3-manifold. Once we know the group-theoretical data of three-dimensional supersymmetric theory on S3b=Zr, we can write down the partition function in terms of hyperbolic hypergeometric integral. Its role in integrable models of statistical mechanics was discovered in [9]
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