A unifying 2d action for integrable $\sigma$-models from 4d Chern-Simons theory

TL;DR

This paper derives a unified 2D action framework for various integrable sigma-models using 4D Chern-Simons theory, encompassing deformations and dualities, thus providing a comprehensive geometric understanding.

Contribution

It introduces a simple, unifying 2D action for integrable sigma-models derived from 4D Chern-Simons theory, including deformations and dualities, advancing the geometric and algebraic understanding.

Findings

Unified 2D action for integrable sigma-models derived from 4D Chern-Simons theory

Inclusion of Yang-Baxter and lambda deformations within the framework

Interpretation of Poisson-Lie T-duality and derivation of the E-model action

Abstract

In the approach recently proposed by K. Costello and M. Yamazaki, which is based on a four-dimensional variant of Chern-Simons theory, we derive a simple and unifying two-dimensional form for the action of many integrable $\sigma$-models which are known to admit descriptions as affine Gaudin models. This includes both the Yang-Baxter deformation and the $\lambda$-deformation of the principal chiral model. We also give an interpretation of Poisson-Lie $T$-duality in this setting and derive the action of the $\mathsf{E}$-model.