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On classical q-deformations of integrable sigma-models

Francois Delduc, Marc Magro, Benoit Vicedo

TL;DR

The paper develops a general, Hamiltonian-based procedure to deform integrable sigma-models while preserving classical integrability by combining the original Poisson bracket with a compatible second bracket. Applied to the principal chiral model on a compact group F, the construction reproduces Klimčík’s Yang–Baxter sigma-model and reveals a classical q-deformed symmetry U^P_q(𝔣) that arises as a semiclassical limit of the quantum group Û_q(𝔣). For symmetric space coset models F/G, the method yields a new one-parameter family of integrable deformations with a deformed target geometry and torsion, interpolating between compact and non-compact cosets; in SU(2)/U(1) this interpolates towards SU(1,1)/U(1). The deformation hinges on a non-split R-matrix and the poles of the deformed twist function, which drive the definition of the deformed group-valued field g and the conserved charges. The work also connects to the semiclassical limits of quantum groups and sets the stage for potential extensions to AdS/CFT-related string backgrounds such as the AdS_5 × S^5 superstring.

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.

On classical q-deformations of integrable sigma-models

TL;DR

The paper develops a general, Hamiltonian-based procedure to deform integrable sigma-models while preserving classical integrability by combining the original Poisson bracket with a compatible second bracket. Applied to the principal chiral model on a compact group F, the construction reproduces Klimčík’s Yang–Baxter sigma-model and reveals a classical q-deformed symmetry U^P_q(𝔣) that arises as a semiclassical limit of the quantum group Û_q(𝔣). For symmetric space coset models F/G, the method yields a new one-parameter family of integrable deformations with a deformed target geometry and torsion, interpolating between compact and non-compact cosets; in SU(2)/U(1) this interpolates towards SU(1,1)/U(1). The deformation hinges on a non-split R-matrix and the poles of the deformed twist function, which drive the definition of the deformed group-valued field g and the conserved charges. The work also connects to the semiclassical limits of quantum groups and sets the stage for potential extensions to AdS/CFT-related string backgrounds such as the AdS_5 × S^5 superstring.

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.

Paper Structure

This paper contains 55 sections, 174 equations, 1 figure.

Table of Contents

  1. Abstract.
  2. Introduction
  3. Deforming the principal chiral model
  4. Principal chiral model
  5. Hamiltonian, equations of motion and Lax matrix.
  6. Symmetry algebra and group valued field.
  7. Setting up the deformation
  8. Deformed Poisson bracket.
  9. Lax matrix and Hamiltonian.
  10. Deformed twist function.
  11. Poles of the deformed twist function.
  12. Defining the group valued field
  13. Definition of $g$.
  14. Singularity at $\epsilon = 1$.
  15. Definition of the conjugate momentum.
  16. ...and 40 more sections

Figures (1)

  • Figure 1: The four poles $\lambda_{\pm}$, $\lambda_{\pm}^{-1}$ of the twist function $\varphi_{\epsilon}(\lambda)$ for $\epsilon \in [0, 1]$.