Yang-Baxter deformations of the $$GL(2,{\mathbb {R}})$$ WZW model and non-Abelian T-duality
Ali Eghbali (Department of Physics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, Tabriz, 53714-161, Iran); Tayebe Parvizi (Department of Physics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, Tabriz, 53714-161, Iran); Adel Rezaei-Aghdam (Department of Physics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, Tabriz, 53714-161, Iran)
By calculating inequivalent classical r-matrices for the $$gl(2,{\mathbb {R}})$$ Lie algebra as solutions of (modified) classical Yang-Baxter equation ((m)CYBE), we classify the YB deformations of Wess-Zumino-Witten (WZW) model on the $$GL(2,{\mathbb {R}})$$ Lie group in twelve inequivalent families. Most importantly, it is shown that each of these models can be obtained from a Poisson-Lie T-dual $$\sigma $$ -model in the presence of the spectator fields when the dual Lie group is considered to be Abelian, i.e. all deformed models have Poisson-Lie symmetry just as undeformed WZW model on the $$GL(2,{\mathbb {R}})$$ . In this way, all deformed models are specified via spectator-dependent background matrices. For one case, the dual background is clearly found.