# On the Hamiltonian integrability of the bi-Yang-Baxter σ -model

Delduc, F. (Laboratoire de Physique, ENS de Lyon et CNRS UMR 5672, Université de Lyon, 46, allée d’Italie, 69364, LYON Cedex 07, France) ; Lacroix, S. (Laboratoire de Physique, ENS de Lyon et CNRS UMR 5672, Université de Lyon, 46, allée d’Italie, 69364, LYON Cedex 07, France) ; Magro, M. (Laboratoire de Physique, ENS de Lyon et CNRS UMR 5672, Université de Lyon, 46, allée d’Italie, 69364, LYON Cedex 07, France) ; Vicedo, B. (School of Physics, Astronomy and Mathematics, University of Hertfordshire, College Lane, Hatfield, AL10 9AB, U.K.)

16 March 2016

Abstract: The bi-Yang-Baxter σ -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. Klimčík 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 σ-model on G × 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 σ-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.

Published in: JHEP 1603 (2016) 104