Integrable $\mathcal{E}$-models, 4d Chern-Simons theory and affine Gaudin models, I -- Lagrangian aspects

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