Root-of-unity limit of the An−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 ℝn−1×(ℤr)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 (ℤrN)n−1-state solution of the star-star relation, which depends on r and N only through rN. Real ℤn-symmetric configurations of the continuous spins give elliptic sl(n) Kashiwara-Miwa type models labeled by n-torsion points. Extended to general primitive torsion points, these include the broken ℤrN 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 sl(n) chiral Potts model of Bazhanov, Kashaev, Mangazeev and Stroganov, and the Boltzmann weights become those of that model.
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BibTeX
@misc{yamazaki2026rootofunity,
title = {{Root-of-unity limit of the $A_n-1$ lens elliptic gamma function solution of the star-star relation}},
author = {Masahito Yamazaki},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41438},
eprint = {2610.41438},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41438},
}
RIS
TY - UNPB
TI - Root-of-unity limit of the A_n−1 lens elliptic gamma function solution of the star-star relation
AU - Masahito Yamazaki
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41438
UR - https://arxiv.org/abs/2610.41438
AB - 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 ℝ^n−1×(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 (ℤ_rN)^n−1-state solution of the star-star relation, which depends on r and N only through rN. Real ℤ_n-symmetric configurations of the continuous spins give elliptic sl(n) Kashiwara-Miwa type models labeled by n-torsion points. Extended to general primitive torsion points, these include the broken 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 sl(n) chiral Potts model of Bazhanov, Kashaev, Mangazeev and Stroganov, and the Boltzmann weights become those of that model.
ER -