Integrable e-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 E-models.