BTZ black hole from Poisson-Lie T-dualizable sigma models with spectators

Abstract

The non-Abelian T-dualization of the BTZ black hole is discussed in detail by using the Poisson-Lie T-duality in the presence of spectators. We explicitly construct a dual pair of sigma models related by Poisson-Lie symmetry. The original model is built on a $2+1$-dimensional manifold $\mathcal{M}\approx O\times \mathbf{G}$, where $\mathbf{G}$ as a two-dimensional real non-Abelian Lie group acts freely on $\mathcal{M}$, while $O$ is the orbit of $\mathbf{G}$ in $\mathcal{M}$. The findings of our study show that the original model indeed is canonically equivalent to the $SL(2,\mathbb{R})$ Wess-Zumino-Witten (WZW) model for a given value of the background parameters. Moreover, by a convenient coordinate transformation we show that this model describes a string propagating in a spacetime with the BTZ black hole metric in such a way that a new family of the solutions to low energy string theory with the BTZ black hole vacuum metric, constant dilaton field and a new torsion potential is found. The dual model is built on a $2+1$-dimensional target manifold $\tilde{\mathcal{M}}$ with two-dimensional real Abelian Lie group $\tilde{\mathbf{G}}$ acting freely on it. We further show that the dual model yields a three-dimensional charged black string for which the mass $M$ and axion charge $Q$ per unit length are calculated. After that, the structure and asymptotic nature of the dual space-time including the horizon and singularity are determined.

20 pages

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• Poisson-Lie T-dual sigma models on supermanifolds
• Poisson Lie symmetry and D-branes in WZW model on the Heisenberg Lie group $H_4$

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