Root-of-unity limit of the $A_{n-1}$ lens elliptic gamma function solution of the star-star relation

Abstract

We study the root-of-unity limit of the multi-component spin solution of the star-star relation whose Boltzmann weights are products of lens elliptic gamma functions, and whose spins take values in $\mathbb{R}^{n-1}\times(\mathbb{Z}_r)^{n-1}$. When one of the elliptic nomes tends to a primitive $2rN$-th root of unity, the quasi-classical expansion gives at leading order the classical star-star relation of the Bazhanov-Sergeev multi-spin model; at the next order, under the standard assumptions on the saddle points, a $(\mathbb{Z}_{rN})^{n-1}$-state solution of the star-star relation, which depends on $r$ and $N$ only through $rN$. Real $\mathbb{Z}_n$-symmetric configurations of the continuous spins give elliptic $\mathfrak{sl}(n)$ Kashiwara-Miwa type models labeled by $n$-torsion points. Extended to general primitive torsion points, these include the broken $\mathbb{Z}_{rN}$ model of Quano and Fujii, for which the star-star relation follows from their Yang-Baxter equation when $\gcd(n,rN)=1$. In the trigonometric limit the saddle point equations single out the spectral curve of the $\mathfrak{sl}(n)$ chiral Potts model of Bazhanov, Kashaev, Mangazeev and Stroganov, and the Boltzmann weights become those of that model.