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 3 b /Z r ) partition functions, for certain three-dimensional N = 2 supersymmetric gauge theories.

The rarefied elliptic Bailey lemma and the Yang–Baxter equation

“…Following the suggestion of [20] to use analytical dependence on the discrete variables, in [12] the following normalized rarefied elliptic gamma function was introduced Γ(u, m; τ, σ) := e πi m(m−r)…”

Lens Generalisation of τ-Functions for the Elliptic Discrete Painlevé Equation

“…In another direction, the lens elliptic gamma function, which is a central function for this paper, first appeared in the study of four-dimensional N = 1 supersymmetric gauge theories on a circle times the lens space S 3 /Z r [7]. This connection suggests that there exists an interpretation of the results of this paper in terms of supersymmetric gauge theories and associated integrable lattice models [8,[11][12][13][14][15][16][17][18][19][20].…”

“…These identities can be verified by direct computation. A proof of the r-periodicity in (25) previously appeared in Appendix C of [19]. For (27), (28), the normalisation of the lens theta functions (5) are in fact chosen to satisfy…”

Rahman’s Biorthogonal Rational Functions and Superconformal Indices

“…We then have the biorthogonality relations J(Q (i, j) n Q ( j,i) m ) = 0 for m = n. We may assume that i = 5, j = 6. Then, Q (5,6) n is given by the function (our notation differs from that of Rahman) More precisely, Rahman proved that if…”

“…As an example, a top level integral of this type is [7] ∞ x=−∞ 4) valid for generic parameters b j and integer parameters N j subject to .5) and N 1 + · · · + N 6 = 0. The identity (1.4) and some related results can also be interpreted as star-triangle relations for solvable lattice models [5,8,15,16,40]. The main purpose of the present work is to investigate the "classical orthogonal polynomials" corresponding to the integral (1.4) and another integral from [7] [see (4.2) below].…”