Yang-Baxter deformations of the $$GL(2,{\mathbb {R}})$$ G L ( 2 , R ) WZW model and non-Abelian T-duality

Abstract

Abstract By calculating inequivalent classical r-matrices for the $$gl(2,{\mathbb {R}})$$ g l ( 2 , 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 $$GL(2,{\mathbb {R}})$$ G L ( 2 , 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 $$\sigma $$ σ -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 $$GL(2,{\mathbb {R}})$$ G L ( 2 , R ) . In this way, all deformed models are specified via spectator-dependent background matrices. For one case, the dual background is clearly found.