4-dimensional Chern-Simons theory and integrable field theories

TL;DR

This paper explores the semi-holomorphic 4d Chern-Simons theory and its role in deriving integrable 2d field theories, including sigma-models, through boundary condition variations and the Lax formalism.

Contribution

It introduces the 4d analogue of Chern-Simons theory and demonstrates its application to classical integrable field theories, especially sigma-models like the Principal Chiral Model.

Findings

Relates 4d Chern-Simons theory to the Lax formalism of integrable models

Shows how boundary conditions yield different integrable sigma-models

Illustrates the approach with examples: Principal Chiral Model and Yang-Baxter deformation

Abstract

These lecture notes concern the semi-holomorphic 4d Chern-Simons theory and its applications to classical integrable field theories in 2d and in particular integrable sigma-models. After introducing the main properties of the Chern-Simons theory in 3d, we will define its 4d analogue and explain how it is naturally related to the Lax formalism of integrable 2d theories. Moreover, we will explain how varying the boundary conditions imposed on this 4d theory allows to recover various occurences of integrable sigma-models through this construction, in particular illustrating this on two simple examples: the Principal Chiral Model and its Yang-Baxter deformation. These notes were written for the lectures delivered at the school "Integrability, Dualities and Deformations", that ran from 23 to 27 August 2021 in Santiago de Compostela and virtually.