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.
Categories: hep-th