The master T-operator for vertex models with trigonometric $R$-matrices as classical tau-function

Abstract

The construction of the master T-operator recently suggested in Alexandrov et al. (arXiv:1112.3310) is applied to integrable vertex models and associated quantum spin chains with trigonometric R-matrices. The master T-operator is a generating function for commuting transfer matrices of integrable vertex models depending on infinitely many parameters. At the same time it turns out to be the tau-function of an integrable hierarchy of classical soliton equations in the sense that it satisfies the the same bilinear Hirota equations. The class of solutions of the Hirota equations that correspond to eigenvalues of the master T-operator is characterized and its relation to the classical Ruijsenaars-Schneider system of particles is discussed.