An elliptic integrable deformation of the Principal Chiral Model

We introduce a new elliptic integrable σ-model in the form of a two-parameter deformation of the Principal Chiral Model on the group SL_ℝ(N), generalising a construction of Cherednik for N = 2 (up to reality conditions). We exhibit the Lax connection and $\mathcal{R}$ \( \mathcal{R} \) -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 σ-models, with the deformed Principal Chiral Model as the simplest example.