Q-operators in the six-vertex model

TL;DR

This paper advances the understanding of Q-operators in the six-vertex model by constructing integral operator representations and connecting them with previous algebraic approaches, including extensions to higher and finite spins.

Contribution

It introduces a new integral operator construction for Q-operators and explicitly relates it to earlier algebraic methods, extending the framework to higher and finite spins.

Findings

Constructed Q-operators as integral operators with factorized kernels.

Established explicit connection between integral and algebraic constructions.

Discussed reduction to finite-dimensional representations with half-integer spins.

Abstract

In this paper we continue the study of $Q$-operators in the six-vertex model and its higher spin generalizations. In [1] we derived a new expression for the higher spin $R$-matrix associated with the affine quantum algebra $U_q(\widehat{sl(2)})$. Taking a special limit in this $R$-matrix we obtained new formulas for the $Q$-operators acting in the tensor product of representation spaces with arbitrary complex spin. Here we use a different strategy and construct $Q$-operators as integral operators with factorized kernels based on the original Baxter's method used in the solution of the eight-vertex model. We compare this approach with the method developed in [1] and find the explicit connection between two constructions. We also discuss a reduction to the case of finite-dimensional representations with (half-) integer spins.