New solutions of the star-triangle relation with discrete and continuous spin variables

TL;DR

This paper presents a novel solution to the star-triangle relation involving spins with integer and real components, connecting integrable models with supersymmetric gauge theories and elliptic hypergeometric integrals.

Contribution

It introduces a new star-triangle relation solution based on lens elliptic-gamma functions, linking it to Seiberg duality and elliptic hypergeometric identities, extending previous models.

Findings

Provides a proof of the star-triangle relation.

Derives a new elliptic hypergeometric integral identity.

Includes limiting cases leading to new relations involving q-products.

Abstract

A new solution to the star-triangle relation is given, for an Ising type model that involves interacting spins, that contain integer and real valued components. Boltzmann weights of the model are given in terms of the lens elliptic-gamma function, and are based on Yamazaki's recently obtained solution of the star-star relation. The star-triangle given here, implies Seiberg duality for the $4\!-\!d$ $\mathcal{N}=1$ $S_1\times S_3/\mathbb{Z}_r$ index of the $SU(2)$ quiver gauge theory, and the corresponding two component spin case of the star-star relation of Yamazaki. A proof of the star-triangle relation is given, resulting in a new elliptic hypergeometric integral identity. The star-triangle relation in this paper contains the master solution of Bazhanov and Sergeev as a special case. Two other limiting cases are considered one of which gives a new star-triangle relation in terms of ratios of infinite $q$-products, while the other case gives a new way of deriving a star-triangle relation previously obtained by the author.