Abelian and non-Abelian T-dualization of the BTZ black hole solution

Using the approach of the Poisson-Lie T-duality in the presence of spectators we first study the Abelian T-dualization of the BTZ black hole solution in such a way that our solution is in agreement with the solution obtained by Horowitz et al. Then, in order to study the non-Abelian T-duality of the BTZ black hole solution, we construct out a dual pair of sigma models related by Poisson-Lie symmetry so that the original model is built on a 2 + 1-dimensional manifold M≈ O ×G in which G is a two-dimensional real non-Abelian Lie group that acts freely on M and O is the orbit of G in M. The findings of our study show that the original model indeed is canonically equivalent to the SL(2,R) WZW model for a special value of the background parameters. In addition, by a convenient coordinate transformation we show that the original model describes a string propagating in a space-time with the BTZ black hole vacuum metric. The dual model is built on a 2 + 1-dimensional target manifold M˜ with two-dimensional real Abelian Lie group ˜G acting freely on it. We investigate the effect of Poisson-Lie T-duality on the singularities of spaces of those models. Finally, the conformal invariance conditions of the dual model are checked by looking at the vanishing of one-loop beta-functions equations.