Lens Hyperbolic Modular Double

TL;DR

The paper introduces the lens hyperbolic modular double, an extension of Faddeev’s hyperbolic modular double motivated by an Ising‑like lens space lattice model. It constructs an intertwining operator that yields a lens hyperbolic hypergeometric solution to the Yang–Baxter equation, realized as an integral operator on infinite‑dimensional representations, and shows that in the limit $r\to1$ the structure reduces to the standard hyperbolic modular double. Central to the framework are the lens hyperbolic Sklyanin‑like algebra, its two dual finite‑difference realizations on $f(z,m)$, and the two‑copy modular double generated by $A,B,C,D$ and $\tilde{A},\tilde{B},\tilde{C},\tilde{D}$ tied together by an intertwining kernel $M(t,p)$. The work forges connections between integrable lattice models, quantum groups, and SUSY gauge theories on lens spaces, with potential extensions to higher rank and broader physical contexts such as knot theory and Liouville gravity.

Abstract

We construct the lens hyperbolic modular double, a new algebraic structure whose intertwining operator produces a lens hyperbolic hypergeometric solution of the Yang--Baxter equation.