Physics Letters B 772 (2017) 791–799 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb BTZ black hole from Poisson–Lie T-dualizable sigma models with spectators ∗ A. Eghbali , L. Mehran-nia, A. Rezaei-Aghdam Department of Physics, Faculty of Basic Sciences, Azarbaijan Shahid Madani University, 53714-161, Tabriz, Iran a r t i c l e i n f o a b s t r a c t Article history: The non-Abelian T-dualization of the BTZ black hole is discussed in detail by using the Poisson–Lie Received 1 May 2017 T-duality in the presence of spectators. We explicitly construct a dual pair of sigma models related Received in revised form 18 July 2017 by Poisson–Lie symmetry. The original model is built on a 2 + 1-dimensional manifold M ≈ O × G, Accepted 21 July 2017 where G as a two-dimensional real non-Abelian Lie group acts freely on M, while O is the orbit of G Available online 26 July 2017 in M. The findings of our study show that the original model indeed is canonically equivalent to the Editor: N. Lambert SL(2, R) Wess–Zumino–Witten (WZW) model for a given value of the background parameters. Moreover, Keywords: by a convenient coordinate transformation we show that this model describes a string propagating in a BTZ black hole spacetime with the BTZ black hole metric in such a way that a new family of the solutions to low energy Sigma model string theory with the BTZ black hole vacuum metric, constant dilaton field and a new torsion potential String duality is found. The dual model is built on a 2 + 1-dimensional target manifold M˜ with two-dimensional real ˜ Poisson–Lie symmetry Abelian Lie group G acting freely on it. We further show that the dual model yields a three-dimensional Charged black string 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. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 1. Introduction Then, it was shown [6] that a simple extension of Witten’s con- struction yields a three-dimensional charged black string such that 2 + 1-dimensional black hole solution with a negative cosmo- this solution is characterized by three parameters the mass, ax- logical constant, mass, angular momentum and charge was, first, ion charge per unit length, and a constant k [6]. After all, it was found by Banados, Teitelboim and Zanelli (BTZ) [1]. The study of shown [4] that the BTZ black hole is, under the standard Abelian 2 + 1-dimensional solutions have received a lot of attention, since T-duality, equivalent to the charged black string solution discussed the near horizon geometry of these solutions serves as a worth- in Ref. [6]. In the present paper, we obtain this result by making while model to investigate some conceptual questions of AdS/CFT use of the approach of the Poisson–Lie T-duality with spectators correspondence [2,3]. The BTZ black hole is asymptotically anti-de [7–9]. The Poisson–Lie T-duality is a generalization of Abelian [10] Sitter rather than asymptotically flat, and has no curvature singu- and traditional non-Abelian dualities [11] that proposed by Klimcik larity at the origin. A slight modification of this black hole solution and Severa [7]. It deals with sigma models based on two Lie groups yields an exact solution to string theory [4]. which form a Drinfeld double and the duality transformation ex- One of the most interesting properties of string theories, or changes their roles. This duality is a canonical transformation and more specially, two-dimensional non-linear sigma models is that two sigma models related by Poisson–Lie duality are equivalent a certain class of these models admits the action of duality trans- at the classical level [12]. A generalization of the Poisson–Lie formations which change the different spacetime geometries but T-duality transformations from manifolds to supermanifolds has leave unchanged the physics of such theories at the classical level. been also carried out in Ref. [13] (see, also, [14]). One of the Witten had shown [5] that two-dimensional black hole could most interesting applications of the Poisson–Lie T-duality trans- be obtained by gauging a one-dimensional subgroup of SL(2, R). formations is to the WZW models [15,16]. Up to now, only few examples of the Poisson–Lie symmetric sigma models have been treated at the quantum level [15,17]. Furthermore, the Poisson–Lie * Corresponding author. E-mail addresses: [email protected] (A. Eghbali), [email protected] symmetry in the WZW models based on the Lie supergroups have (A. Rezaei-Aghdam). recently studied in Refs. [18] and [19]. We also refer the reader http://dx.doi.org/10.1016/j.physletb.2017.07.044 0370-2693/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3. 792 A. Eghbali et al. / Physics Letters B 772 (2017) 791–799 1 to the literatures [20]. In [15] it has been shown that the dual- + − ϒ S = dσ dσ Gϒ() + Bϒ() ∂+ ∂− , (2.3) ity relates the SL(2, R) WZW model to a constrained sigma model 2 defined on the SL(2, R) group space. Here in this paper we ob- ± tain a new non-Abelian T-dual background for the SL(2, R) WZW where σ = τ ± σ are the standard light-cone variables on the + model so that it is constructed out on a 2 1-dimensional mani- worldsheet . Gϒ and Bϒ are components of the metric and fold. Moreover, by using the Poisson–Lie T-duality with spectators antisymmetric tensor field B on a manifold M. The functions ϒ we find a new family of the solutions to low energy string theory : −→ R, ϒ = 1, ..., dim(M) are obtained by the composi- ϒ ϒ with the BTZ black hole vacuum metric, constant dilaton field and tion = X ◦ of a map : −→ M and components of a a new torsion potential. The findings of our study show that the coordinate map X on a chart of M. dual model yields a three-dimensional charged black string. This Let us now consider a d-dimensional manifold M and some ϒ μ α μ solution is stationary and is characterized by three parameters: the coordinates ={x , y } on it, where x , μ = 1, ···, dim(G) are mass M and axion charge Q per unit length, and a radius l related the coordinates of Lie group G that act freely from right on M. α to the derivative of the asymptotic value of the dilaton field. In y with α = 1, ···, d − dim(G) are the coordinates labeling the or- α this way, we show that the non-Abelian T-duality transformation bit O of G in the target space M. We note that the coordinates y (here as the Poisson–Lie T-duality on a semi-Abelian double) re- do not participate in the Poisson–Lie T-duality transformations and lates a solution with no horizon and no curvature singularity (the are therefore called spectators [9]. We also introduce the compo- −1 a a μ a BTZ vacuum solution) to a solution with a single horizon and a nents of the right invariant one-forms (∂± gg ) = R± = ∂±x Rμ α α curvature singularity (the charged black string). and for notational convenience we will also use R± = ∂± y . Then To set up conventions and notations, as well as to make the pa- the original sigma model on the manifold M ≈ O × G is defined per self-contained, we review the Poisson–Lie T-dual sigma models by the following action construction in the presence of spectators in Section 2. In Section 3 1 + − + α A B we discuss in detail the Abelian T-dualization of the BTZ black hole S = dσ dσ F (g, y ) R+ R−, 2 AB solutions by using the approach of the Poisson–Lie T-duality with spectators. Our main result is derived in Section 4: we explicitly 1 + − + α a b +(1) α a β = dσ dσ E (g, y ) R R + (g, y )R ∂− y construct a dual pair of sigma models related by Poisson–Lie sym- ab + − aβ + ˜ 2 metry on 2 + 1-dimensional manifolds M and M such that the +(2) α α b α α β ˜ + + + + − group parts of M and M are two-dimensional real non-Abelian αb (g, y )∂ y R− αβ (y )∂ y ∂ y , (2.4) and Abelian Lie groups, respectively. In subsection 4.1, by a conve- ={ } E+ +(1) nient coordinate transformation we show that the original model where the index A a, α . As it is seen the couplings ab, aβ , describes a string propagating in a spacetime with BTZ black hole +(2) μ α αb and αβ may depend on all variables x and y . The back- vacuum metric. Moreover, the canonical equivalence of the original + α ground matrix F (g, y ) is defined as model to the SL(2, R) WZW model is discussed in subsection 4.1. AB In subsection 4.2, first the dual model construction is given. Then it + α C + α D F (g, y ) = A A (g)E (g, y ) A B (g), (2.5) is shown that the dual model yields a three-dimensional charged AB CD black string for which the mass M and axion charge Q per unit where [7] length are calculated.