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

Abstract

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 ($S_b^3/{\mathbb{Z}}_r$) partition functions, for certain three-dimensional $\mathcal{N}=2$ supersymmetric gauge theories.

42 pages, 4 figures

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Cited by in corpus (22)

• Elliptic hypergeometric sum/integral transformations and supersymmetric lens index
• Basic hypergeometry of supersymmetric dualities
• Integrable quad equations derived from the quantum Yang-Baxter equation
• Integrable lattice spin models from supersymmetric dualities
• From rarefied elliptic beta integral to parafermionic star-triangle relation
• Lens elliptic gamma function solution of the Yang-Baxter equation at roots of unity
• A parafermionic hypergeometric function and supersymmetric 6j-symbols
• A New Integrable Ising-type Model from 2d $\mathcal{N}$=(2,2) Dualities
• Star-triangle type relations from $2d$ $\mathcal{N}=(0,2)$ $USp(2N)$ dualities
• Lens partition function, pentagon identity and star-triangle relation
• The rarefied elliptic Bailey lemma and the Yang-Baxter equation
• Hyperbolic and trigonometric hypergeometric solutions to the star-star equation
• General modular quantum dilogarithm and beta integrals
• Pentagon identities arising in supersymmetric gauge theory computations
• Exactly solved models on planar graphs with vertices in ${\mathbb{Z}}^3$
• Two-component Yang-Baxter maps and star-triangle relations
• On Bailey pairs for $\mathcal{N}=2$ supersymmetric gauge theories on $S_b^3/{\mathbb{Z}}_r$
• Lens Generalisation of $\tau$-functions for the Elliptic Discrete Painlevé Equation
• Integrability As Duality: The Gauge/YBE Correspondence
• Cluster-Enriched Yang-Baxter Equation from SUSY Gauge Theories
• Rahman's biorthogonal functions and superconformal indices
• Quasi-classical expansion of a hyperbolic solution to the star-star relation and multicomponent 5-point difference equations