Integrable 𝜀-Models, 4d Chern-Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects

Abstract

We construct the actions of a very broad family of 2d integrable σ-models. Our starting point is a universal 2d action obtained in [arXiv:2008.01829] using the framework of Costello and Yamazaki based on 4d Chern-Simons theory. This 2d action depends on a pair of 2d fields 𝘩 and 𝓛, with 𝓛 depending rationally on an auxiliary complex parameter, which are tied together by a constraint. When the latter can be solved for 𝓛 in terms of 𝘩, this produces a 2d integrable field theory for the 2d field h whose Lax connection is given by 𝓛(𝘩). We construct a general class of solutions to this constraint and show that the resulting 2d integrable field theories can all naturally be described as 𝜀-models.