The symmetric six-vertex model and the Segre cubic threefold

TL;DR

This paper explores the integrability of the symmetric six-vertex model using algebraic geometry, revealing a birational isomorphism to the Segre cubic primal and providing a general solution for the Yang-Baxter triple.

Contribution

It establishes a novel connection between the symmetric six-vertex model's algebraic variety and the Segre cubic primal, enabling a comprehensive parametrization of the R-matrix and Lax operators.

Findings

Algebraic variety from Baxter's method is birationally isomorphic to Segre cubic primal.

Provides the most general solution for the Yang-Baxter triple in this model.

Parametrizes R-matrix and Lax operators with three independent spectral variables.

Abstract

In this paper we investigate the mathematical properties of the integrability of the symmetric six-vertex model towards the view of Algebraic Geometry. We show that the algebraic variety originated from Baxter's commuting transfer method is birationally isomorphic to a ubiquitous threefold known as Segre cubic primal. This relation makes it possible to present the most generic solution for the Yang-Baxter triple associated to this lattice model. The respective $\mathrm{R}$-matrix and Lax operators are parametrized by three independent affine spectral variables.