Integrable $\mathcal{E}$-Models, $4\mathrm{d}$ Chern–Simons Theory and Affine Gaudin Models. I. Lagrangian Aspects

We construct the actions of a very broad family of $2\mathrm{d}$ integrable $\sigma$-models. Our starting point is a universal $2\mathrm{d}$ action obtained in [arXiv:2008.01829] using the framework of Costello and Yamazaki based on $4\mathrm{d}$ Chern–Simons theory. This $2\mathrm{d}$ action depends on a pair of $2\mathrm{d}$ 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 $2\mathrm{d}$ integrable field theory for the $2\mathrm{d}$ 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 $2\mathrm{d}$ integrable field theories can all naturally be described as $\mathcal{E}$-models.