An elliptic integrable deformation of the Principal Chiral Model

Abstract

We introduce a new elliptic integrable σ-model in the form of a two-parameter deformation of the Principal Chiral Model on the group SLℝ(N), generalising a construction of Cherednik for N = 2 (up to reality conditions). We exhibit the Lax connection and R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{R} $$\\end{document}-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.