Hyperbolic and trigonometric hypergeometric solutions to the star-star equation

Abstract

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 $$\mathcal N=2$$ 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.

On Bailey pairs for $$ \mathcal{N} $$ = 2 supersymmetric gauge theories on $$ {S}_b^3/{\mathbb{Z}}_r $$

“…the gauge symmetry to U(1) via the gauge symmetry breaking gives another solution to the star-triangle relation which is realized as the generalized Faddeev-Volkov model [25,28], since the r = 1 case gives the Faddeev-Volkov model [51,52] corresponding to N = 2 supersymmetric dual theories on S 3 b [53]. The star-star relations [54] of these two models are constructed, and for the factorized interaction-round-a-face (IRF) spin models, IRFtype YBE are presented in [31,32]. The integral identity obtained by supersymmetric duality with the gauge group U(1) can also be written as a pentagon identity [28].…”

“…3 The three-dimensional N = 2 partition functions on the squashed lens space S 3 b /Z r have been computed via dimensional reduction of the four-dimensional lens superconformal index [2,7,26] and via the supersymmetric localization technique [3,4]. Such theories have been studied in [5,24,28,31,32].…”

“…of lattice spin models (Ising-like and IRF-type) in statistical mechanics and pentagon identity as a 2 − 3 Pachner move [28,31,32,41]. In this work, we have constructed Bailey pairs that generate these integral identities.…”

“…In recent years, the remarkable concept of hypergeometric identities sits at the intersection of diverse studies such as exact results in supersymmetric gauge theories [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and their mathematical structures interacting with various fields in mathematics, see, e.g., [15][16][17][18], star-triangle relation (Yang-Baxter equation) [19][20][21][22][23][24][25][26][27][28] or star-star relation [29][30][31][32] for spin lattice models, knot theory [33], pentagon identities [33][34][35][36][37][38][39][40][41], Bailey pairs [23,[42][43][44][45], quantum algebras [28,…”

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“…Note that the basic hypergeometric integral identity (7) is new in the literature and needs detailed study and classification [35]. Such identities (containing summation and integration) recently appear in supersymmetric gauge theory computations [14,19,20,33,36,37,38,39,40,41] and integrable models of statistical mechanics [16,34,42,43,44,45].…”