A new pentagon identity for the tetrahedron index

Recently Kashaev, Luo and Vartanov, using the reduction from a four-dimensional superconformal index to a three-dimensional partition function, found a pentagon identity for a special combination of hyperbolic Gamma functions. Following their idea we have obtained a new pentagon identity for a certain combination of so-called tetrahedron indices arising from the equality of superconformal indices of dual three-dimensional N=2 supersymmetric theories and give a mathematical proof of it.Comment: 13 pages, v2: we added a new section with the proof of the identity, misprints correcte

Comments on the multi-spin solution to the Yang-Baxter equation and basic hypergeometric sum/integral identity

We present a multi-spin solution to the Yang-Baxter equation. The solution corresponds to the integrable lattice spin model of statistical mechanics with positive Boltzmann weights and parameterized in terms of the basic hypergeometric functions. We obtain this solution from a non-trivial basic hypergeometric sum-integral identity which originates from the equality of supersymmetric indices for certain three-dimensional N=2 Seiberg dual theories.Comment: 8 pp, based on a talk given at the XXVth International Conference on Integrable Systems and Quantum symmetries (ISQS-25), Prague, Czech Republic, 06-10 June, 2017; v2: minor change

Integral pentagon relations for 3d superconformal indices

The superconformal index of a three-dimensional supersymmetric field theory can be expressed in terms of basic hypergeometric integrals. By comparing the indices of dual theories, one can find new integral identities for basic hypergeometric integrals. Some of these integral identities have the form of the pentagon identity which can be interpreted as the 2-3 Pachner move for triangulated 3-manifolds.Comment: 9 pages. Based on arXiv:1309.2195 with new results and comments. Presented at String-Math conference, Edmonton, Canada, June 9-13, 2014; v2: minor corrections and comments adde

Basic hypergeometry of supersymmetric dualities

We introduce several new identities combining basic hypergeometric sums and integrals. Such identities appear in the context of superconformal index computations for three-dimensional supersymmetric dual theories. We give both analytic proofs and physical interpretations of the presented identities.Comment: 25 pages, v2: minor corrections and comment

Integrable lattice spin models from supersymmetric dualities

Recently, there has been observed an interesting correspondence between supersymmetric quiver gauge theories with four supercharges and integrable lattice models of statistical mechanics such that the two-dimensional spin lattice is the quiver diagram, the partition function of the lattice model is the partition function of the gauge theory and the Yang-Baxter equation expresses the identity of partition functions for dual pairs. This correspondence is a powerful tool which enables us to generate new integrable models. The aim of the present paper is to give a short account on a progress in integrable lattice models which has been made due to the relationship with supersymmetric gauge theories.Comment: 35 pages, preliminary versio

A resurgence analysis for cubic and quartic anharmonic potentials

In this work, we explicitly show resurgence relations between perturbative and one instanton sectors of the resonance energy levels for cubic and quartic anharmonic potentials in one-dimensional quantum mechanics. Both systems satisfy the Dunne–Unsal relation ¨ and hence we are able to derive one-instanton nonperturbative contributions with the fluctuation terms to the energy merely from the perturbative data. We confirm our results with previous results obtained in the literature

The star-triangle relation, lens partition function, and hypergeometric sum/integrals

The aim of the present paper is to consider the hyperbolic limit of an elliptic hypergeometric sum/integral identity, and associated lattice model of statistical mechanics previously obtained by the second author. The hyperbolic sum/integral identity obtained from this limit, has two important physical applications in the context of the so-called gauge/YBE correspondence. For statistical mechanics, this identity is equivalent to a new solution of the star-triangle relation form of the Yang-Baxter equation, that directly generalises the Faddeev-Volkov models to the case of discrete and continuous spin variables. On the gauge theory side, this identity represents the duality of lens (S 3 b /Zr) partition functions, for certain three-dimensional N = 2 supersymmetric gauge theories

Hyperbolic and trigonometric hypergeometric solutions to the star-star equation

We construct the hyperbolic and trigonometric solutions to the star-star relation via the gauge/YBE correspondence by using the three-dimensional lens partition function and superconformal index for a certain N=2 supersymmetric gauge dual theories. 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 and basic hypergeometric functions.Comment: 14 pages, v2: minor corrections and comments, v3: minor correction

Notes on the lens integral pentagon identity

We obtain the lens integral pentagon identity for three-dimensional mirror dual theories in terms of hyperbolic hypergeometric functions via reduction of equality for $\mathcal N=2$ lens supersymmetric partition functions of a certain supersymmetric IR duality.Comment: 9 page