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4-dimensional Chern-Simons theory and integrable field theories
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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.
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Cited by 4 Pith papers
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Time-Dependent Integrability from Gauge Theory, I
Spacetime-dependent 4d Chern-Simons theory generates time-dependent integrable field theories whose allowed time dependence coincides with one-loop RG flow while preserving Lax integrability.
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Twists of trigonometric sigma models
A new class of Z_N-twisted integrable sigma models is constructed from 4d Chern-Simons theory, and Z2 twisting by an outer automorphism of SU(n) produces models inequivalent to the untwisted ones.
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Higher-Spin Currents and Flows in Auxiliary Field Sigma Models
Auxiliary field sigma models admit local higher-spin conserved currents, and the corresponding Smirnov-Zamolodchikov flows can be solved recursively by reducing the problem to a free boson.
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Integrable black hole dynamics in the asymptotic structure of AdS$_{3}$
In AdS3, a relaxed class of boundary conditions makes Einstein's equations reduce to two AKNS integrable hierarchies, and stationary black holes in this class have constant temperatures fixed by hyperelliptic spectra,...
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