On classical q-deformations of integrable sigma-models

TL;DR

This paper introduces a method for creating integrable deformations of sigma-models, leading to new models with q-deformed symmetries and interpolations between compact and non-compact geometries, expanding the landscape of integrable theories.

Contribution

A novel procedure for constructing classically integrable deformations of sigma-models, including new models with q-deformed symmetries and geometric interpolations.

Findings

Recovered the Yang-Baxter sigma-model for compact Lie groups.

Derived a new one-parameter family of integrable sigma-models for symmetric spaces.

Demonstrated interpolation between compact and non-compact target spaces in deformed models.

Abstract

A procedure is developed for constructing deformations of integrable sigma-models which are themselves classically integrable. When applied to the principal chiral model on any compact Lie group F, one recovers the Yang-Baxter sigma-model introduced a few years ago by C. Klimcik. In the case of the symmetric space sigma-model on F/G we obtain a new one-parameter family of integrable sigma-models. The actions of these models correspond to a deformation of the target space geometry and include a torsion term. An interesting feature of the construction is the q-deformation of the symmetry corresponding to left multiplication in the original models, which becomes replaced by a classical q-deformed Poisson-Hopf algebra. Another noteworthy aspect of the deformation in the coset sigma-model case is that it interpolates between a compact and a non-compact symmetric space. This is exemplified in the case of the SU(2)/U(1) coset sigma-model which interpolates all the way to the SU(1,1)/U(1) coset sigma-model.