An elliptic integrable deformation of the Principal Chiral Model

We introduce a new elliptic integrable $\sigma$-model in the form of a two-parameter deformation of the Principal Chiral Model on the group ${\text{SL}}_{\mathbb{R}}(N)$, generalising a construction of Cherednik for $N=2$ (up to reality conditions). We exhibit the Lax connection and $\mathcal{R}$-matrix of this theory, which depend meromorphically on a spectral parameter valued in the torus. Furthermore, we explain the origin of this model from an equivariant semi-holomorphic 4-dimensional Chern-Simons theory on the torus. This approach opens the way for the construction of a large class of elliptic integrable $\sigma$-models, with the deformed Principal Chiral Model as the simplest example.