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

## TL;DR

This paper explores the Poisson-Lie T-duality of the BTZ black hole, constructing dual sigma models that reveal new string solutions and black string configurations with detailed geometric and physical properties.

## Contribution

It explicitly constructs a dual pair of sigma models related by Poisson-Lie symmetry, linking the BTZ black hole to new string backgrounds and black string solutions.

## Key findings

- Original model is equivalent to the SL(2,R) WZW model.
- Derived a family of solutions with BTZ black hole metric, constant dilaton, and torsion.
- Obtained a dual model describing a charged black string with computed mass and charge.

## 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 ${\cal M} \approx O \times \bf G$, where $\bf G$ as a two-dimensional real non-Abelian Lie group acts freely on ${\cal M}$, while $O$ is the orbit of $\bf G$ in ${\cal M}$. 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 $\tilde {\cal M}$ with two-dimensional real Abelian Lie group ${\tilde {\bf G}}$ 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.