Integrable deformations of coupled sigma-models

TL;DR

This paper develops a framework for creating integrability-preserving deformations of coupled sigma-models using affine Gaudin models, unifying various known deformations and exploring their relation to 4D Chern-Simons theory.

Contribution

It introduces a systematic method to deform coupled sigma-models via affine Gaudin models, encompassing Yang-Baxter and λ-deformations, and connects these to 4D Chern-Simons theory.

Findings

Constructed new integrable deformations of coupled sigma-models.

Expressed models explicitly in Hamiltonian and Lagrangian form.

Unified various known λ-deformed models within this framework.

Abstract

We construct integrability-preserving deformations of the integrable $\sigma$-model coupling together $N$ copies of the Principal Chiral Model. These deformed theories are obtained using the formalism of affine Gaudin models, by applying various combinations of Yang-Baxter and $\lambda$-deformations to the different copies of the undeformed model. We describe these models both in the Hamiltonian and Lagrangian formulation and give explicit expressions of their action and Lax pair. In particular, we recover through this construction various integrable $\lambda$-deformed models previously introduced in the literature. Finally, we discuss the relation of the present work with the semi-homolomorphic four-dimensional Chern-Simons theory.