4-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.

v2: 64 pages, published version, references added and minor corrections

References in corpus (23)

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• Integrable Deformations of Sigma Models
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• Branes and integrable lattice models
• Q-operators are 't Hooft lines
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Cited by in corpus (18)

• Duality-Invariant Non-linear Electrodynamics and Stress Tensor Flows
• Introduction to classical and quantum integrability
• Infinite Family of Integrable Sigma Models Using Auxiliary Fields
• 2d Integrable systems, 4d Chern-Simons theory and Affine Higgs bundles
• Generalized Dualities and Supergroups
• Integrable Deformations from Twistor Space
• Geometry of the spectral parameter and renormalisation of integrable $\sigma$-models
• An elliptic integrable deformation of the Principal Chiral Model
• Higher Gauge Theory and Integrability
• 4D Chern-Simons theory with auxiliary fields
• $T\bar{T}$ and root-$T\bar{T}$ deformations in four-dimensional Chern-Simons theory
• Gauging The Diamond: Integrable Coset Models from Twistor Space
• The Courant-Hilbert construction in 4D Chern-Simons theory
• Integrability in Gravity from Chern-Simons Theory
• A generalized 4d Chern-Simons theory
• Topological 4D gravity and gravitational defects
• Integrable models from 4d holomorphic BF theory
• Point particle E-models