An elliptic integrable deformation of the Principal Chiral Model

Abstract

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.

53 pages. v2: published version; added references and a paragraph on quantum integrability in the conclusion

References in corpus (8)

• Non-local charges, Zm gradings and coset space actions
• Assembling integrable sigma-models as affine Gaudin models
• Combining the bi-Yang-Baxter deformation, the Wess-Zumino term and TsT transformations in one integrable sigma-model
• 4-dimensional Chern-Simons theory and integrable field theories
• Yang Baxter and Anisotropic Sigma and Lambda Models, Cyclic RG and Exact S-Matrices
• Universal 1-loop divergences for integrable sigma models
• Quartic Hamiltonians, and higher Hamiltonians at next-to-leading order, for the affine ${\mathfrak{sl}}_2$ Gaudin model
• The Magic Renormalisability of Affine Gaudin Models

Cited by in corpus (2)

• Higher Gauge Theory and Integrability
• Integrability and renormalizability for the fully anisotropic $\mathrm{SU}(2)$ principal chiral field and its deformations