Four-dimensional Chern–Simons theory and integrable field theories

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.