Four-dimensional Chern–Simons theory and integrable field theories

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.

Introduction to classical and quantum integrability

“…• In order to learn about integrable deformations of sigma models see Ben Hoare's lecture notes [23] and references there in; • In order to learn about four-dimensional Chern-Simons and integrable field theories see Sylvain Lacroix's lecture notes [24] and references there in.…”

“…Integrable models play a role in many areas of physics ranging from condensed matter, to string theory, passing through Temperley-Lieb and Hecke algebras, quantum groups and Yangians, the bootstrap program, AdS/CFT, sigma models, quantum computing, statistical mechanics and many others . Three aspects of integrable models are explored in the Integrability, dualities and deformations school: the basic concepts and techniques on both classical and quantum integrability presented in these lecture notes, integrable deformations of sigma models in Ben Hoare's lecture notes [23] and four-dimensional Chern-Simons theory and integrable field theories in Sylvain Lacroix's lecture notes [24].…”

“…There is a particular class of models that has been widely studied and where a lot of progress has been achieved in this direction: the twodimensional sigma models and its deformations. The Lax connection appears very naturally in these models and in both [23,24] such cases are discussed using the principal chiral model and its deformations as examples. In [24] we learn that different integrable sigma models can appear depending on the choice of boundary conditions in a four-dimensional Chern-Simons theory [28][29][30][31].…”

“…The Lax connection appears very naturally in these models and in both [23,24] such cases are discussed using the principal chiral model and its deformations as examples. In [24] we learn that different integrable sigma models can appear depending on the choice of boundary conditions in a four-dimensional Chern-Simons theory [28][29][30][31]. So, this four-dimensional Chern-Simons model works as a method to construct integrable models, which given the usual difficulties already discussed in doing this, makes it even more valuable.…”

Integrability, dualities and deformations

“…This happens for example for the type IIB * Author to whom any correspondence should be addressed. 3 See the website https://indico.cern.ch/event/1001393/page/21966-school for more information.…”