On the Hamiltonian integrability of the bi-Yang-Baxter sigma-model

TL;DR

This paper proves the Hamiltonian integrability of the bi-Yang-Baxter sigma-model by constructing a Lax matrix with a standard r/s Poisson bracket and analyzing its symmetries and deformation properties.

Contribution

It demonstrates the Hamiltonian integrability of the bi-Yang-Baxter sigma-model through explicit Lax matrix construction and analysis of its Poisson structure and symmetries.

Findings

Existence of a Lax matrix with r/s-form Poisson bracket

Identification of Poisson commuting Kac-Moody currents

Explicit description of q-deformed symmetries

Abstract

The bi-Yang-Baxter sigma-model is a certain two-parameter deformation of the principal chiral model on a real Lie group G for which the left and right G-symmetries of the latter are both replaced by Poisson-Lie symmetries. It was introduced by C. Klimcik who also recently showed it admits a Lax pair, thereby proving it is integrable at the Lagrangian level. By working in the Hamiltonian formalism and starting from an equivalent description of the model as a two-parameter deformation of the coset sigma-model on G x G / G_diag, we show that it also admits a Lax matrix whose Poisson bracket is of the standard r/s-form characterised by a twist function which we determine. A number of results immediately follow from this, including the identification of certain complex Poisson commuting Kac-Moody currents as well as an explicit description of the q-deformed symmetries of the model. Moreover, the model is also shown to fit naturally in the general scheme recently developed for constructing integrable deformations of sigma-models. Finally, we show that although the Poisson bracket of the Lax matrix still takes the r/s-form after fixing the G_diag gauge symmetry, it is no longer characterised by a twist function.