Auxiliary Field Sigma Models and Yang-Baxter Deformations

TL;DR

This paper develops multi-parameter integrable deformations of the principal chiral model on Lie groups by combining Yang-Baxter and auxiliary field methods, establishing their classical integrability via Lax pairs.

Contribution

It introduces new multi-parameter families of integrable models combining Yang-Baxter and auxiliary field deformations, with explicit Lax representations.

Findings

Constructs integrable deformations for each pair (R, E) and triplet (R, R~, E).

Demonstrates classical integrability through Lax pairs.

Extends the framework of Yang-Baxter deformations to include auxiliary fields.

Abstract

We combine the Yang-Baxter (YB) and bi-Yang-Baxter (bi-YB) deformations with higher-spin auxiliary field deformations to construct multi-parameter families of integrable deformations of the principal chiral model on a Lie group $G$ with semi-simple Lie algebra $\mathfrak{g}$. In the YB case, our construction produces one integrable deformation for each pair $(\mathcal{R}, E)$, where $\mathcal{R}$ is an antisymmetric bilinear operator on $\mathfrak{g}$ obeying the modified classical Yang-Baxter equation and $E$ is a function of several variables. In the bi-YB case, the pair becomes a triplet $(\mathcal{R},\tilde{\mathcal{R}}, E)$, where $\tilde{\mathcal{R}}$ is another antisymmetric bilinear operator on $\mathfrak{g}$ and obeys the non-split inhomogeneous modified classical Yang-Baxter equation. We show that every model in these families is (weakly) classically integrable by exhibiting a Lax representation for their equations of motion.