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

A Eghbali,L Mehran-Nia,A Rezaei-Aghdam

doi:10.1016/j.physletb.2017.07.044

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 M≈O×G, where G as a two-dimensional real non-Abelian Lie group acts freely on M, while 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) 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 M˜ with two-dimensional real Abelian Lie group 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.

Highlights

The non-Abelian T-dualization of the BTZ black hole is discussed in detail by using the PoissonLie T-duality in the presence of spectators
We have explicitly constructed a dual pair of sigma models related by Poisson-Lie symmetry so that the original model has 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
We have shown that the Poisson-Lie T-duality relates the SL(2, R) WZW model to a sigma model defined on a 2+1-dimensional manifold M

Summary

Introduction

2+1-dimensional black hole solution with a negative cosmological constant, mass, angular momentum and charge was, first, found by Banados, Teitelboim and Zanelli (BTZ) [1]. The findings of our study show that the dual model yields a three-dimensional charged black string This solution is stationary and is characterized by three parameters: the mass M and axion charge Q per unit length, and a radius l related to the derivative of the asymptotic value of the dilaton field. In this way, we show that the non-Abelian T-duality transformation (here as the Poisson-Lie T-duality on a semi-Abelian double) relates a solution with no horizon and no curvature singularity (the BTZ vacuum solution) to a solution with a single horizon and a curvature singularity (the charged black string).

A review of Poisson-Lie T-duality with spectators

The Abelian T-dualization of the BTZ black hole solutions

The dual model as a three-dimensional charged black string

Discussion and Conclusion