2d Integrable systems, 4d Chern-Simons theory and Affine Higgs bundles

TL;DR

This paper compares two gauge-theoretic approaches to constructing 2d integrable models, highlighting their connections through examples like elliptic Calogero-Moser and Gaudin models.

Contribution

It demonstrates the relationship between 4d Chern-Simons theory and affine Higgs bundles in formulating 2d integrable systems.

Findings

Establishes correspondence between 4d-CS and AHB constructions.

Provides explicit examples of elliptic integrable field theories.

Shows how these approaches unify different integrable models.

Abstract

In this short review we compare constructions of 2d integrable models by means of two gauge field theories. The first one is the 4d Chern-Simons (4d-CS) theory proposed by Costello and Yamazaki. The second one is the 2d generalization of the Hitchin integrable systems constructed by means the Affine Higgs bundles (AHB). We illustrate this approach by considering 1+1 field versions of elliptic integrable systems including the Calogero-Moser field theory, the Landau-Lifshitz model and the field theory generalization of the elliptic Gaudin model.