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

TY - JOUR

T1 - On the Hamiltonian integrability of the bi-Yang-Baxter sigma-model

AU - Vicedo, Benoit

AU - Magro, Marc

AU - Delduc, Francois

AU - Lacroix, Sylvain

N1 - © The Authors, 2016 This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and re production in any medium, provided the original work is properly cited. https://creativecommons.org/licenses/by/3.0/ F. Deluc, S. Lacroix, M. Magro. And B. Vicedo, “On the Hamiltonian integrability of the bi-Yang-Baxter σ-model”, Journal of High Energy Physics, Vol. 2016 (3) March 2016.

PY - 2016/3/15

Y1 - 2016/3/15

N2 - 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/Gdiag, 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 Gdiag gauge symmetry, it is no longer characterised by a twist function.

AB - 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/Gdiag, 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 Gdiag gauge symmetry, it is no longer characterised by a twist function.

U2 - 10.1007/JHEP03(2016)104

DO - 10.1007/JHEP03(2016)104

M3 - Article

SN - 1029-8479

VL - 1603

JO - Journal of High Energy Physics

JF - Journal of High Energy Physics

IS - 104

ER -