Lens elliptic gamma function solution of the Yang-Baxter equation at roots of unity

Abstract

We study the root of unity limit of the lens elliptic gamma function solution of the star-triangle relation, for an integrable model with continuous and discrete spin variables. This limit involves taking an elliptic nome to a primitive $rN$-th root of unity, where $r$ is an existing integer parameter of the lens elliptic gamma function, and $N$ is an additional integer parameter. This is a singular limit of the star-triangle relation, and at subleading order of an asymptotic expansion, another star-triangle relation is obtained for a model with discrete spin variables in ${\mathbb{Z}}_{rN}$. Some special choices of solutions of equation of motion are shown to result in well-known discrete spin solutions of the star-triangle relation. The saddle point equations themselves are identified with three-leg forms of "3D-consistent" classical discrete integrable equations, known as $Q4$ and $Q{3}_{(\delta =0)}$. We also comment on the implications for supersymmetric gauge theories, and in particular comment on a close parallel with the works of Nekrasov and Shatashvili.

32 pages; v2: published version

References in corpus (6)

• Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry
• Quivers, YBE and 3-manifolds
• Exact solution of the Faddeev-Volkov model
• Emergent 3-manifolds from 4d Superconformal Indices
• The star-triangle relation, lens partition function, and hypergeometric sum/integrals
• T-Reflection

Cited by in corpus (4)

• Integrable quad equations derived from the quantum Yang-Baxter equation
• Extended Z-invariance for integrable vector and face models and multi-component integrable quad equations
• Hyperbolic and trigonometric hypergeometric solutions to the star-star equation
• Two-component Yang-Baxter maps and star-triangle relations