Physics Letters B 772 (2017) 791–799
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Physics Letters B
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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 we find a new family of the solutions to low energy string theory ϒ : −→ R, ϒ = 1, ..., dim(M) are obtained by the composi- with the BTZ black hole vacuum metric, constant dilaton field and tion ϒ = Xϒ ◦ 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 solution is stationary and is characterized by three parameters: the coordinates ϒ ={xμ, yα} on it, where xμ, 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. to the derivative of the asymptotic value of the dilaton field. In yα with α = 1, ···, d − dim(G) are the coordinates labeling the or- 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 yα (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- 2 ab aβ 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 and αβ may depend on all variables x and y . The back- vacuum metric. Moreover, the canonical equivalence of the original αb + ground matrix F (g, yα) 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. Investigation of the structure and asymptotic + α nature of the dual space–time including the horizon and singular- E (g, y ) AB ity are discussed at the end of Section 4. Some concluding remarks + −1 + − = A(g) + E (e, yα)B(g) C E (e, yα)(A 1) D (g), (2.6) are given in the last Section. A CD B + with e being the unit element of G. Here, the matrix E (e, yα) 2. A review of Poisson–Lie T-duality with spectators may be function of the spectator variables yα only and is defined + +(1) +(2) in terms of the new couplings E0 ab, Faβ , Fαb and Fαβ so that We start this section by recalling the main features of the it is read Poisson–Lie T-duality and introducing redefinitions to make the + α +(1) α duality transformations more symmetrical. According to [7], the + E (e, y ) F (e, y ) E (e, yα) = 0 ab aβ . (2.7) Poisson–Lie duality is based on the concepts of the Drinfeld dou- AB +(2) α α F (e, y ) Fαβ (y ) ble which is simply a Lie group D, such that the Lie algebra D of αb this Lie group as a vector space can be decomposed into a pair In addition, the matrices A(g) and B(g) appearing in relations ˜ of maximally isotropic subalgebras G and G of the Lie groups G (2.5) and (2.6) are given by1 ˜ ˜ a and G, respectively. We take the sets Ta, a = 1, ..., dim(G), and T G G˜ a(g) 0 b(g) 0 as the basis of the Lie algebras and , respectively. They satisfy A(g) = , B(g) = , (2.8) the commutation relations 0 Id 00 where the submatrices a(g) and b(g) associated with the Lie group [ ]= c [ ˜ a ˜ b]= ˜ ab ˜ c Ta, Tb f ab Tc, T , T f c T , G are constructed using b ˜ bc b c [T , T˜ ]= f T + f T˜ . (2.1) a a c ca −1 = c g Ta g aa (g)T c , In addition to (2.1), there is an inner product <., . > on D with − a − g 1 T˜ g = bac(g) T + (a 1) a(g)T˜ c, (2.9) the various generators obeying c c and for later use we also need to consider the following definition ˜ b b ˜ a ˜ b < Ta, T > = δa ,
1 Thus, using the relations (2.5)–(2.8) together with (2.10) one can Rϒ + ( − R) Gϒ = 0,<0, (3.1) obtain the backgrounds appearing in the action (2.4). They are then 2 given in matrix notation by where Rϒ and R are the respective Ricci tensor and scalar curva- ture. The line element for the black hole solutions is given by − −1 E+ α = + 1 α + (g, y ) E0 (e, y ) (g) , (2.11) 2 2 2 2 r 2 2 2 r J −1 2 (1) − (1) ds = (M − )dt − Jdtdφ + r dφ + − M + dr , + α = E+ α + 1 α + α l2 l2 4r2 (g, y ) (g, y ) E0 (e, y ) F (e, y ), (2.12) 0 ≤ φ<2π (3.2) +(2) α = +(2) α +−1 α E+ α (g, y ) F (e, y ) E0 (e, y ) (g, y ), (2.13) where the radius l is related to the cosmological constant by α α +(2) α + α − (y ) = F (y ) − F (e, y ) (g)E (g, y ) l = (−) 1/2. The constants of motion M and J are the mass +−1 +(1) and angular momentum of the BTZ black hole, respectively. They × E (e, yα)F (e, yα). (2.14) 0 are appeared due to the time translation symmetry and rotational As we shall see below, one can construct another sigma model symmetry of the metric, corresponding to the killing vectors ∂/∂t (denoted as usual with tilded symbols) which is said to be dual to and ∂/∂ϕ, respectively. The line element (3.2) describes a black (2.4) in the sense of the Poisson–Lie T-duality if the Lie algebras G hole solution with outer and inner horizons at r = r+ and r = r−, and G˜ form a pair of maximally isotropic subalgebras of the Lie al- respectively, + gebra D [7]. Similarly to (2.7) one can define the matrix E˜ (e˜, yα) + 1/2 1/2 and then obtain the relation between both the matrices E (e, yα) M 1/2 J 2 + r± = l 1 ± 1 − , (3.3) and E˜ (e˜, yα) as follows [7] 2 Ml + + −1 + where the mass and angular momentum are related to r± by E˜ (e˜, yα) = A + E (e, yα) B B + E (e, yα) A , (2.15) 2 2 r+ + r− 2r+r− in which M = , J = . l2 l 00 Id 0 A = , B = . (2.16) 1/2 0 Id 00 The region r+ < r < M l defines an ergosphere, in which the asymptotic timelike Killing field ∂/∂t becomes spacelike. The solu- Now, the dual sigma model on the manifold M˜ ≈ O × G˜ is, in the tions with −1 < M < 0, J = 0describe point particle sources with coordinate base {x˜μ, yα}, given by the following action naked conical singularities at r = 0. The metric with M =−1, J = 0may be recognized as that of ordinary anti-de Sitter space; it ˜ 1 + − ˜ +AB α ˜ ˜ is separated by a mass gap from the M = 0, J = 0. The vacuum S = dσ dσ F (g˜, y ) R+ R− 2 A B state which is regarded as empty space, is obtained by letting the a horizon size go to zero. This amounts to letting M → 0, which re- 1 + − +ab α +(1) α β = E˜ ˜ ˜ + ˜ − + ˜ ˜ ˜ + − dσ dσ (g, y )R a R b (g, y )R a ∂ y quires J → 0. We have to notice that the metric for the M = J = 0 2 β black hole is not the same as AdS metric which has negative mass b 3 +(2) α α α α β + ˜ ˜ + ˜ − + ˜ + − M =−1. Locally they are equivalent since there is locally only one α (g, y )∂ y R b αβ (y )∂ y ∂ y , (2.17) constant curvature metric in three dimensions. However they are ˜ = ˜μ ˜ inequivalent globally. where R±a ∂±x Rμa are the components of the right invariant In Ref. [4], first the BTZ black hole solutions have been consid- one-forms on the Lie group G˜ . The coupling matrices of the dual ered in the context of the low energy approximation, then as the sigma model are also determined in a completely analogous way as exact conformal field theory. In three dimensions, the low energy (2.7) and (2.11)–(2.14) such that by using (2.15) one relates them string effective action is given by to those of the original one by [7–9,20] √ − 3 −2φ + + 1 S = d −Ge E˜ (g˜, yα) = E (e, yα) + (˜ g˜) , (2.18) ef f 0 ˜ +(1) α ˜ + α +(1) α 2 1 ϒ (g˜, y ) = E (g˜, y ) F (e, y ), (2.19) × R + 4(∇φ) − Hϒ H − 4 , (3.4) 12 +(2) +(2) + ˜ g˜ yα =−F e yα E˜ g˜ yα (2.20) ( , ) ( , ) ( , ), where G = det Gϒ . Hϒ are the components of the torsion of ˜ α α +(2) α ˜ + α +(1) α the antisymmetric field B and are defined by H = B + (y ) = F (y ) − F (e, y ) E (g˜, y ) F (e, y ). ϒ ∂ϒ ∂ Bϒ + ∂ Bϒ , while φ is the dilaton field. The equations of (2.21) motion which follow from this action are [21] ˜ ˜ Notice that (g) is defined as (2.10) by replacing untilded quanti- 1 0 = Rϒ − HϒH + 2∇ϒ ∇ φ, ties by tilded ones. Hence, the actions (2.4) and (2.17) correspond 4 to Poisson–Lie T-dual sigma models [7]. 1 0 =− ∇ Hϒ + H ∇φ, (3.5) 2 ϒ 3. The Abelian T-dualization of the BTZ black hole solutions 2 2 1 ϒ 0 = R + 4∇ φ − 4(∇φ) − Hϒ H − 4. In this section we give the Abelian T-dualization of the BTZ 12 black hole solutions [1] by making use of the approach of the These equations are known as the vanishing of the one-loop beta- Poisson–Lie T-duality with the spectators reviewed in the previ- functions equations. Notice that the string effective action is con- ous section. Let us now begin this section by reviewing the BTZ nected to the two-dimensional non-linear sigma model through black hole solutions [1]. The BTZ black holes are 2 + 1-dimensional these equations [21]. solutions of Einstein’s equations with a negative cosmological con- In order to obtain an exact solution to string theory, one must stant, , modify the BTZ black hole solutions by adding an antisymmetric 794 A. Eghbali et al. / Physics Letters B 772 (2017) 791–799 tensor field Hϒ proportional to the volume form ϒ . It has Now one can use the above approach to obtain the Abelian T-dual been shown [4] that any solution to three-dimensional general rel- solutions with the BTZ black hole solutions. Here the target space ativity with negative cosmological constant is a solution to low M ≈ O × G is defined by the coordinates {ϕ, r, t}, where the Lie 2 energy string theory with φ = 0, Hϒ = 2ϒ /l and =−1/l . group G should be considered to be U (1). It is more convenient to In particular it was shown [4] that the two parameter family of write the action of sigma model corresponding to the BTZ metric black holes (3.2) along with (3.2) and the solutions (3.6). It is then read to be in the following form r2 = = Bϕt ,φ0, (3.6) 2 2 −1 l 1 + − 2 r J S = dσ dσ r ∂+ϕ∂−ϕ + − M + ∂+r∂−r satisfy the equations of motion (3.5). Then, it was obtained [4] 2 l2 4r2 the dual of this solution by Buscher’s duality transformations [10]. r2 J r2 Abelian duality is a well known symmetry of string theory that + M − ∂+t∂−t − − ∂+ϕ∂−t l2 2 l maps any solution of the low energy string equations of motion J r2 (3.5) with a translational symmetry to another solution. − + ∂+t∂−ϕ . (3.13) As a concluding remark for this section we first explicitly re- 2 l obtain Buscher’s duality transformations [10] using the approach On the other hand, since the background described by the action of the Poisson–Lie T-duality with the spectators. Then, in this way, (3.13) depend on the r coordinate only, from the Poisson–Lie T- we obtain the Abelian T-dualization of the BTZ black hole solutions ˜ duality standpoint the Lie group G can not be parametrized by given by (3.2). In this case, the Lie groups G and G are Abelian. this coordinate. However, we assume that the G is parametrized ˜ ab ˜ Therefore, both structures ab(g) and (g) are vanished. Conse- by the ϕ coordinate while the coordinates of orbit O are repre- F+ α F˜ +AB ˜ α quently, the background matrices AB(g, y ) and (g, y ) are sented by the r and t. Comparing the action (3.13) and (3.7) the + + α ˜ +AB ˜ α matrix E (e, yα) is derived as equaled with E AB(e, y ) and E (e, y ), respectively. Thus, the AB action (2.4) in the presence of the dilaton field φ is written as ⎛ ⎞ 2 r2 0 − J + r 1 + − ⎜ − 2 l ⎟ = 2 2 1 S dσ dσ + α = ⎜ r − + J ⎟ 2 E (e, y ) ⎝ 0 2 M 2 0 ⎠ . AB l 4r J r2 r2 + α A B ϒ 1 (2) − − 0 M − × E (e, y )δϒ δ ∂+ ∂− − R φ , (3.7) 2 l l2 AB 4 (3.14) where R(2) is two-dimensional scalar curvature in the world- Considering (3.14) in the form (2.7) and using the fact that G = sheet. Comparing the sigma model (3.7) and (2.3) we find that + + α A B = + U (1) we find that E (e, yα) = r2 and E AB(e, y )δϒ δ Gϒ Bϒ. Let us assume that the sigma 0 ab model (3.7) has an Abelian isometry represented by a translation 0 (1) (2) 0 in a coordinate x in the target space. Notice that in the coor- + J r2 + F = 0 − + , F = 2 , ϒ ={ 0 μ} = − aβ 2 l αb − J − r dinates x , x , μ 1, ..., d 1adapted to the isometry, 2 l 0 ⎛ ⎞ the metric, torsion and dilaton field are independent of x . If we −1 + r2 J 2 choose the matrix E (e, yα) as − M + 0 AB F = ⎝ l2 4r2 ⎠ . (3.15) αβ 2 + 0 M − r + α = G00 G0ν B0ν l2 E AB(e, y ) , (3.8) Gμ0 + Bμ0 Gμν + Bμν The dual target space M˜ ≈ O × G˜ is defined by the coordinates then from relations (2.15) and (2.16) one immediately gets the fol- {ϕ˜, r, t}. Obviously, in the standard Abelian case D = U (1) × U (1), lowing Buscher’s duality transformations [10] i.e., G˜ is also identical to U (1) and is parametrized by the ϕ˜ coor- dinate. Finally, utilizing the equations (2.18)–(2.21) together with G˜ G˜ + B˜ 00 0ν 0ν (2.17) the dual sigma model is read off to be of the form G˜ + B˜ G˜ + B˜ μ⎛0 μ0 μν μν ⎞ + 1 1 r2 J 2 −1 1 G0ν B0ν ˜ = + − ˜ ˜ + − + G G S dσ dσ ∂+ϕ ∂−ϕ M ∂+r ∂−r = ⎝ 00 00 ⎠ . (3.9) 2 r2 l2 4r2 Gμ0+Bμ0 (Gμ0+Bμ0)(G0ν +B0ν ) − Gμν + B − 2 G00 μν G00 J J 1 + M − ∂+t ∂−t + − + ∂+ϕ˜ ∂−t It has been shown [10,22] that for preserving conformal invariance, 4r2 2r2 l at least to one-loop order, the dilaton field has to transform as J 1 1 (2) ˜ + + ∂+t ∂−ϕ˜ − R φ , (3.16) 2r2 l 4 ˜ 1 φ = φ − ln G00. (3.10) 2 where the new dilaton φ˜ at the last term of the action can be This shift of the dilaton field is due to the duality transformation calculated by (3.10) to be of the form receiving corrections from the jacobian that comes from integrat- ˜ =− ing out the gauge fields [10,22]. In the Poisson–Lie T-duality case, φ ln r. (3.17) dilaton shifts in both models have been obtained by quantum con- In this way the dual geometry is given by siderations based on a regularization of a functional determinant in a path integral formulation of Poisson–Lie duality by incorpo- J 2 r2 J 2 −1 ds˜2 = M − dt2 + − M + dr2 rating spectator fields [8] (see, also, [23]) 4r2 l2 4r2 α 1 + α 1 + α 2 1 2 φ = φ0(y ) + ln det E (g, y ) − ln det E (e, y ) , (3.11) + dt dϕ˜ + dϕ˜ , (3.18) 2 2 0 l r2 ˜ α 1 ˜ + α ˜ J φ = φ0(y ) + ln det E (g˜, y ) . (3.12) B =− dϕ˜ ∧ dt. (3.19) 2 2r2 A. Eghbali et al. / Physics Letters B 772 (2017) 791–799 795
The solutions (3.17)–(3.19) have been obtained in Ref. [4] by using g = ex1 T1 ex2 T2 , (4.2) Buscher’s duality transformations. The three-dimensional charged where x1 and x2 are the coordinates of the Lie group A2. Then black string solutions can be obtained from the dual solutions a 2 R±’s are derived to be of the form (3.17)–(3.19) by making the coordinate transformation t = l(r+ − −1 −1 2 ˆ 2 2 2 ˆ 2 2 1 2 x1 r−) 2 (xˆ − t), ϕ˜ = l(r+ − r−) 2 (r+t − r−xˆ) and r = lrˆ as follows [4] R± = ∂±x1, R± = e ∂±x2. (4.3)
M Q2 We note that since the dual Lie group have been considered to be 2 ˆ2 2 ds˜ =− 1 − dt + 1 − dxˆ Abelian, hence, by using (2.9) and (2.10) it follows that the ab(g) rˆ Mrˆ is vanished. In addition to (4.3), we need to determine the cou- M Q2 −1 2 ˆ2 + +(1) +(2) −1 l dr plings E (g, yα), (g, yα) and (g, yα). By a convenient + 1 − 1 − , (3.20) ab aβ αb rˆ Mrˆ 4rˆ2 + +(1) +(2) choice of the matrices E (e, yα), F (e, yα) and F (e, yα) as Q 0 ab aβ αb ˜ = ˆ ∧ ˆ B dx dt, (3.21) 1 −2y rˆ + α 0 e E (e, y ) = − 2 , 1 0 ab 1 e 2y 0 φ˜ =− ln(lrˆ), (3.22) 2 2 +(1) α = 0 +(2) α = −2y 2 Faβ (e, y ) −2y , Fαb (e, y ) 0 e , (4.4) where M = r+/l and Q = J/2are the respective the mass and −e charge of the black string. For large rˆ the metric is asymptotically α and Fαβ (y ) = k (k is a non-zero real constant), and then with the flat. Thus, it is important to notice that the Abelian T-duality trans- help of relations (2.11)–(2.14) one can get the required couplings. formation changes the asymptotic behavior from AdS to flat. In 3 Finally by inserting these into action (2.4) the original sigma model addition, as it can be seen from the above results, for asymptoti- is found to be of the form cally flat solutions of three-dimensional low energy string theory, 1 1 the duality of conserved quantities defined on asymptotic region + − x1−2y S = dσ dσ k∂+ y∂− y + e (∂+x1∂−x2 + ∂+x2∂−x1) is given in such a way that a mass is unchanged, while an axion 2 2 charge and an angular momentum are interchanged each other [4]. − − ex1 2y(∂+x ∂− y − ∂+ y∂−x ) . (4.5) In the next section we give the main goal of the paper. We will 2 2 study the non-Abelian T-dualization of the BTZ black hole solution By identifying action (4.5) with the sigma model of the form (2.3) in such a way that we shall show that the dual model describes a one can read off the background matrix. The space–time metric new three-dimensional charged black string. and the antisymmetric tensor field corresponding to the action (4.5) can be written as 4. The BTZ black hole vacuum solution from Poisson–Lie T-dual 2 2 x1−2y sigma models with the spectators ds = kdy + e dx1 dx2, (4.6) − B =−ex1 2y dx ∧ dy. (4.7) In this section we explicitly construct a pair of Poisson–Lie T- 2 ˜ dual sigma models on 2 + 1-dimensional manifolds M and M For the metric (4.6) one can find that Rϒ =−(2/k)Gϒ and then as the target spaces. The original model is built on the manifold R =−6/k. This shows that the metric (4.6) with k > 0 describes M ≈ O × G, where G = A2 as a two-dimensional real non-Abelian an anti-de Sitter space while for k < 0we have a de Sitter space. Lie group acts freely on it while O is the orbit of G in M with Considering antisymmetric tensor field B of (4.7) one quickly finds − α ={ } =− x1 2y one spectator y y . The target space of the dual model is the that the only non-zero component of H is Hx1x2 y e , then, ˜ ˜ ϒ manifold M ≈ O × G with two-dimensional real Abelian Lie group it follows that Hϒ H =−24/k. Inserting the above results in ˜ G = 2A1 acting freely on it. We shall show the original model is the vanishing of the one-loop beta-functions equations (3.5), the canonically equivalent to the SL(2, R) WZW model for a given conformal invariance conditions up to one-loop order are satis- value of the background parameters. In particular, by a convenient fied with =−1/k and a constant dilaton field which satisfies coordinate transformation we show that the model describes a the equation (3.11). string propagating in a spacetime with the BTZ black hole vacuum On the one hand, action (4.5) can be simplified by making the metric. following new coordinates
x1 = = = 4.1. The original model as the SL(2, R) WZW model and BTZ black hole e θ+, x2 θ−, y γ , (4.8) vacuum solution then,2 by making use of the integrating by parts over the last two terms of the action (4.9) we get As mentioned above the original model is built on a 2 + 1-dimensional manifold M ≈ O ×G in which G is two-dimensional 2 Under the transformation (4.8) the action (4.5) turns into real non-Abelian Lie group whose Lie algebra is denoted by A2. Ac- cording to Section 2, having Drinfeld doubles we can construct the ¯ 1 + − 1 −2γ S = dσ dσ k∂+γ ∂−γ + e (∂+θ+∂−θ− Poisson–Lie T-dual sigma models on them. The Lie algebra of the 2 2 A A Drinfeld double ( 2, 2 1) is defined by the following non-zero Lie −2γ + ∂+θ−∂−θ+) − θ+e (∂+θ−∂−γ − ∂+γ ∂−θ−) . (4.9) brackets [24]
By calculating the momentums corresponding to the actions (4.5) and (4.9) (P ˜ 2 ˜ 2 ˜ 2 ˜ 1 ¯ [T1, T2]=T2, [T1 , T ]=−T , [T2 , T ]=T . (4.1) and P, respectively) and by making use of the transformation (4.8) we get the transformation between momentums. Then, by considering the basic equal-time ϒ We note that the Lie algebra (A2, 2A1) is isomorphic to the oscil- Poisson brackets for the pair of canonical variables ( , P) one can show that the equal-time Poisson brackets for the pair of canonical variables (¯ ϒ , P¯ ) are lator Lie algebra h4 [16]. According to action (2.4) to construct the also preserved. Furthermore, we can obtain the Hamiltonians corresponding to the original model we need the components of right invariant one- actions (4.5) and (4.9), and then show that they are identical together, that is, un- a ¯ forms R± on the Lie group A2. To this end, we parametrize an der the transformation (4.8) one goes from H()¯ to H() and vice versa, hence, element of A2 as proving that the transformation (4.8) is indeed a canonical transformation. 796 A. Eghbali et al. / Physics Letters B 772 (2017) 791–799 1 + − −2γ S = dσ dσ k∂+γ ∂−γ + e ∂+θ+∂−θ− . (4.10) Inserting (4.4) and (4.15) into equations (2.18)–(2.21) the dual cou- 2 plings are obtained to be of the form ⎛ ⎞ By setting k = 1the above action precisely becomes the SL(2, R) 0 1 ab ˜ 1 −2y WZW model [15]. On the other hand, to better understand of ac- + x2+ e E˜ = ⎝ 2 ⎠ , tion (4.5) we diagonalize the metric obtained by this action. Let 1 0 −x˜ + 1 e−2y 2 2 − x 1 y l −e 2y 1 = − =− + = (1)a (2)b − −2y e (t lϕ), x2 l(t lϕ), e , (4.11) ˜ + ˜ + 1 −2y ˜ + e = x2 e , = − 0 , (4.16) l r β 2 α −x˜ + 1 e 2y 0 2 2 where the radius l introduced in Section 3. Then the action (4.5) turns into and ˜ = k. Putting these pieces together into (2.17), the action αβ 2 2 of dual model is worked out to be 1 + − r 2 l S = dσ dσ − ∂+t∂−t + r ∂+ϕ∂−ϕ + ∂+r∂−r 2 l2 r2 ˜ 1 + − S = dσ dσ k∂+ y∂− y r 2 − (t − lϕ)(∂+t∂−r − ∂+r∂−t) 2 1 1 1 l + −2y − ˜ ˜ ˜ + −2y + ˜ ˜ ˜ r ( e x2)∂+x1 ∂−x2 ( e x2)∂+x2 ∂−x1 − (t − lϕ)(∂+ϕ∂−r − ∂+r∂−ϕ) , (4.12) 2 2 l −2y 1 −2y − e ( e − x˜2)∂+x˜1 ∂− y = 2 where we have, here, set k l . This describes a string propagating 2 in a space–time with the BTZ black hole vacuum metric with M = −2y 1 −2y − e ( e + x˜2)∂+ y ∂−x˜1 , (4.17) J = 0 2 − 2 2 where = 1 e 4y −x˜2. Comparing the above action with the sigma 2 r 2 2 2 l 2 4 2 ds =− dt + r dϕ + dr , (4.13) model action of the form (2.3), the corresponding metric and ten- l2 r2 sor field B˜ take the following forms and an antisymmetric tensor field −2y 2 2 e −2y r 1 ds˜ = kdy + dx˜1 dx˜2 − e dx˜1 dy , (4.18) B =− (t − lϕ) dϕ ∧ dr + dt ∧ dr . (4.14) l l x˜2 − = = B˜ =− dx˜ ∧ dx˜ − e 2y dx˜ ∧ dy . (4.19) Since the black hole metric with M J 0is locally equivalent 1 2 1 to the AdS3 metric, so, for the metric (4.13) one immediately 2 2 As mentioned in Section 2 the spectator fields do not participate finds that Rϒ =−(2/l )Gϒ and R =−6/l . The only non-zero component of antisymmetric field strength corresponding to the in the Poisson–Lie T-duality transformations. Therefore, as it can ϒ 2 be seen from the metrics (4.6) and (4.18) this duality changes the B-field (4.14) is Htrϕ = 2r/l; consequently Hϒ H =−24/l . Putting these pieces together, one verifies the equations (3.5) with group part of the metrics, while leaving the y component invari- =− 2 ant. The metric components (4.18) are ill defined at the regions 1/l and a constant dilaton field. Therefore, it follows that − − x˜ = 1 e 2y and x˜ =−1 e 2y . We can test whether there are true the conformal invariance of actions (4.5) and (4.12) is, under the 2 2 2 2 transformation (4.11), preserved. Notice that the metric (4.13) has singularities by calculating the scalar curvature, which is no horizon and no curvature singularity. Indeed, this solution is ev- −4y + ˜ −2y + ˜2 ˜ 2(11e 28x2e 12x2) erywhere regular including r = 0. In fact, this was expected since R =− . (4.20) k e−2y − 2x˜ 2 generally the (anti-)de Sitter space–time is a maximally symmet- ( 2) ric space with a constant curvature. The resulting space–time is ˜ = 1 −2y Thus x2 2 e is a true curvature singularity, while the dif- completely nonsingular. Consider now two killing vectors ∂/∂t and ˜ =−1 −2y ficult at the x2 2 e can be removed by an appropriate ∂/∂ϕ corresponding to the time translational and the rotational change of coordinates. As shown in subsection 4.1 the original isometries of the metric (4.13), respectively. The killing field ∂/∂t model (4.5) is canonically equivalent to the SL(2, R) WZW model. = becomes null at r 0 and it is timelike for the whole range r > 0, Therefore, it should be conformally invariant. To check the con- while the killing field ∂/∂ϕ is everywhere spacelike except for formal invariance of the dual model (4.17) we look at vanishing = r 0. The global structure associated with the metric (4.13) in- of the one-loop beta-functions equations (3.5). Given a B˜ -field cluding the Kruskal and Penrose diagrams has been discussed in with (4.19) we find that the only non-zero component of H˜ is ˜ −2y −2y 2 Ref. [25]. ˜ ˜ =− − ˜ Hx1x2 y 4e /(e 2x2) . Thus, the first two equations of (3.5) are satisfied by the new dilaton field 4.2. The dual model as a three-dimensional charged black string −2y 1 2x˜2 + e φ˜ = a + ln , (4.21) As mentioned at the first of this section, the dual model is con- −2y 2 2x˜2 − e structed on a 2 + 1-dimensional manifold M˜ ≈ O × G˜ with two- ˜ where a is an arbitrary constant. This additive constant plays an dimensional Abelian Lie group G = 2A1 acting freely on it. In the important role, as we will see later. Thus, the dilatonic contribution same way to construct out the dual sigma model we parametrize in (3.5) vanishes if the cosmological constant of the dual theory the corresponding Lie group (Abelian Lie group 2A1) with coor- ˜ μ does leave invariant, that is, =−1/k. It is also important to note dinates x˜ ={x˜1, x˜2} so that its elements are defined as (4.2) by that the dilaton field (4.21) is well behaved for the ranges x˜2 + replacing untilded quantities by tilded ones. Hence the compo- 1 −2y 1 −2y e < 0 and x˜2 − e > 0. nents of the right invariant one-forms on the Lie group 2A1 are 2 2 ˜ As shown, the metric (4.13) of the original model is the BTZ simply calculated to be R± = ∂±x˜a. Utilizing relation (2.10) for un- a black hole vacuum solution which is locally equivalent to the AdS tilded quantities we get 3 metric. To continue, we shall show that the dual solution repre- 0 −x˜ sents a three-dimensional charged black string which is station- ˜ ab(g˜) = 2 . (4.15) x˜2 0 ary and asymptotically flat. In order to enhancing and clarifying A. Eghbali et al. / Physics Letters B 772 (2017) 791–799 797
2 −W the structure of the dual space–time, horizon, singularity and also l W T e x˜1 = U − (e − W ), x˜2 = e (1 − ), determining the asymptotic nature of one, we first write the so- 2 2 { } lutions (4.18), (4.19) and (4.21) in the coordinate base t, x, r . 1 Furthermore, since we want to discuss the dual solution to the y˜ = (W − T ), (4.30) 2 BTZ black hole vacuum solution, we must here consider k = l2. In the following, we separately discuss the solutions for the ranges we obtain ˜ + 1 −2y ˜ − 1 −2y x2 2 e < 0 and x2 2 e > 0. 2 2 2 l 2 l 2 1 − ds˜ = dW + dT − dTdU, (4.31) • ˜ + 1 2y W The solution corresponding to the range x2 2 e < 0 4 4 e − 1 2 ˜ 1 −2y T ˜ 1 l W 1 In this case we consider x2 + e =−e for T ∈ . Then, we B = 1 + dU ∧ dT − (e − ) dW ∧ dT, (4.32) 2 W − introduce the following coordinate transformation 2(e 1) 2 2 W 2 −W ˜ 1 e l W T e φ = a + ln , (4.33) x˜1 = U + (W + e ), x˜2 =−e (1 + ), 2 eW − 1 2 2 1 where eW > 1, i.e., W > 0. We now define the transformation y˜ = (W − T ), (4.22) 2 eW = 1/(1 − rˆ) so that it requires that 0 < rˆ < 1. Applying this transformation and also using the linear transformation (4.26), one for U , W ∈ . Under the above transformation, the dual metric can show that the solutions (4.31)–(4.33) turn into the same form (4.18), the B˜ -field (4.19) and the dilaton field (4.21) are, respec- of the solution given by (4.27)–(4.29), that is, the solutions corre- tively, turn into ˜ + 1 −2y ˜ − 1 −2y sponding to both the valid ranges x2 2 e < 0 and x2 2 e > l2 l2 1 0can be expressed as the solution given by (4.27)–(4.29) only with ˜2 2 2 ds = dW + dT + dTdU, (4.23) 0 < rˆ < ∞. 4 4 eW + 1 One can simply check that the solution (4.27)–(4.29) does sat- 2 1 l 1 ˜ =− 2 B˜ = 1 − dU ∧ dT + (eW + ) dW ∧ dT, (4.24) isfy the equations (3.5) with 1/l . By considering this so- 2(eW + 1) 2 2 lution for the whole space–time, one sees that the metric com- W ponents (4.27) are ill behaved at rˆ = 0 and rˆ = 1. By looking at ˜ 1 e φ = a + ln , (4.25) the scalar curvature, which is R˜ = 2(4rˆ − 7)/l2rˆ2, we find that 2 eW + 1 rˆ = 0is a curvature singularity. Note that the singularity at rˆ = 0 Notice that there is no singularity for the metric (4.23). In fact, ˜ = 1 −2y corresponds to the same true singularity at the region x2 2 e this was expected since the solutions (4.18), (4.19) and (4.21) are, which mentioned above. We will see that rˆ = 1is also an event ˜ + 1 −2y in this case, defined only for the range x2 2 e < 0. As ex- horizon. The cross term appeared in the metric is constant and plained above, the true singularity of the metric (4.18) occurs at thus for large rˆ the metric is asymptotically flat. However, the so- ˜ = 1 −2y ˜ + x2 2 e , which this region is located out of the range x2 lution represents a stationary black string. We wish to express the 1 −2y W = ˆ − constant parameter a and the radius l in terms of the physical mass 2 e < 0. Let us now consider the transformation e 1/(r 1) so that it requires that 1 < rˆ < ∞. In addition to, we introduce the and axion charge per unit length of the black string. To calculate following linear transformation the mass we first need to have the asymptotic behavior of the so- lution given by (4.27)–(4.29). To this end, we set 2 xˆ xˆ T =− (tˆ + √ ), U = l(tˆ − √ ), (4.26) l 3 3 2l rˆ = e2ar, (4.34) 3 for tˆ, xˆ ∈ . By applying the above transformations to the solutions (4.23), (4.24) and (4.25), one obtains the forms of the dual space– in equations (4.27), (4.28) and (4.29) and only drop the hat sign time metric, antisymmetric field strength and dilaton field in new from on the tˆ and xˆ coordinates. Notice that it is not possible to ˆ coordinate base {t, xˆ, rˆ} as similarly to the t and x coordinates fix the overall scaling of the rˆ coordinate as rˆ goes to infinity, since the metric asymptotically ap- 2 2 ˜2 =− − ˆ2 + − ˆ2 proaches l2drˆ2/4rˆ2. Therefore, for large r the black string solution ds (1 ˆ )dt (1 ˆ )dx r 3r (4.27)–(4.29) approaches the following asymptotic solution 2 2 2 1 − l drˆ + √ dtdˆ xˆ + (1 − ) 2 , (4.27) ˆ ˆ2 2 2 2 2 3 r 4r ds˜η =−dt + dx + dρ , ˜ 2 Hˆˆˆ = √ , (4.28) ˜ ρ 1 3 ˜ rtx ˆ2 φ =− + ln( ), H = 0, (4.35) 3 r l 2 2l ˜ 1 ˆ φ = a − ln r. (4.29) where we have set r = e2ρ/l. The solution (4.35) is a flat solution 2 for equations (3.5). As it can be seen from (4.35), the radius l is re- ˜ + 1 −2y We note that this solution is valid only for the range x2 2 e < lated to the derivative of the asymptotic value of the dilaton field. 0or 1 < rˆ < ∞. Now, in order to calculate the mass per unit length of the string we use the Arnowitt–Deser–Misner (ADM) procedure which was been • ˜ − 1 −2y The solution corresponding to the range x2 2 e > 0 applied for obtaining the mass and charge of three-dimensional black strings [6,26] (see, also, [5]). In this way, to calculate the ˜ − 1 −2y = T − −2y = + For this case, we have assumed that x2 2 e e e for mass one needs to perturb the metric as Gϒ ηϒ γϒ. Then, which T + 2y > 0. Analogously, by introducing the following trans- by integrating the time–time component of the linearized form formation of the metric and dilaton contributions in equations (3.5) over a 798 A. Eghbali et al. / Physics Letters B 772 (2017) 791–799
3 spacelike surface, the total mass is derived to be of the form [6, asymptotic behavior from AdS3 to flat. One question that is in our 26]: mind is that although the black string metric (4.40) is asymptot- ically flat, its near horizon geometry is of the AdS3 type, that is, 1 −2φ j j i → ˜ →− 2 ˜ →− 2 ˜ Mtot = e ∂ γij − ∂iγ − 2γij∂ φ dS , (4.36) in the limit r M, one gets R 6/l and Rϒ (2/l )Gϒ. 2 The study of near horizon behavior is important, because accord- where i, j run over spatial indices and γ is the trace of the spatial ing to Strominger’s proposal [27] one asserts that the statistical i entropy of any black hole whose near horizon geometry contains components of γϒ, i.e., γ = γi . Finally, in three dimensions, the axion charge per unit length associated with the field strength H an AdS3 factor can be calculated by using the statistical counting is given by [6,26] of microstates of the BTZ black hole. Lastly, we have to note the fact that the charged black string solution (4.40)–(4.42) with l = 1 1 − R Q = e 2φ ∗ H, (4.37) is a dual solution for the SL(2, ) WZW model. 2 5. Discussion and conclusion where ∗ denotes the Hodge dual. Thus, by considering the metric ηϒ corresponding to the line element of equation (4.35) and by We have reviewed aspects of Poisson–Lie T-duality in the pres- using the specific form of γϒ for the solution (4.27) we carry out ence of spectator fields. We have reobtained Buscher’s duality the above procedure. For the black string solution (4.27)–(4.29), transformations from the Poisson–Lie T-duality transformations in the mass and axion charge per unit length measuring at the ρ =∞ the presence of spectators. Then using this approach we have stud- end are therefore ied the Abelian T-dualization of the BTZ black hole solutions given 3 − by (3.2) in such a way that our solutions are in agreement with M = e 2a, (4.38) 2l those of Ref. [4]. We have explicitly constructed a dual pair of 2 sigma models related by Poisson–Lie symmetry so that the orig- = √ −2a Q e . (4.39) inal model has built on a 2 + 1-dimensional manifold M ≈ O × G 3 l √ in which G is a two-dimensional real non-Abelian Lie group that = 3 3 acts freely on M and O is the orbit of G in M. The metric of Clearly, M 4 Q and thus M > Q . Using these results, the final expression for the black string solution is obtained to be the model depends on a non-zero real constant parameter k so that it describes an anti-de Sitter space with k > 0 while for k < 0 2M 2M ds˜2 =−(1 − )dt2 + (1 − )dx2 we have a de Sitter space. Then we have shown that the origi- r 3r nal model indeed is canonically equivalent to the SL(2, R) WZW 2 M l2dr2 model for a given value of the background parameters. Therefore, + √ + − −2 dtdx (1 ) , (4.40) we have shown that the Poisson–Lie T-duality relates the SL(2, R) 3 r 4r2 WZW model to a sigma model defined on a 2 + 1-dimensional ˜ 3Q manifold M. In addition, by a convenient coordinate transforma- Hrtx = , (4.41) 2r2 tion we have shown that the original model describes a string 1 1 3 propagating in a spacetime with the BTZ black hole vacuum met- φ˜ =− ln r + ln( ). (4.42) 2 2 2l ric. In this way we have found a new family of the solutions to low energy string theory with the BTZ black hole vacuum met- The metric (4.40) possesses a single horizon at r = M and a curva- ric, constant dilaton field and a new torsion potential. The dual ture singularity at r = 0, since the scalar curvature is R˜ =[2M(4r − model has built on a 2 + 1-dimensional target manifold M˜ with 7M)]/l2r2. The metric (4.40) appears to be somewhat analogous to two-dimensional real Abelian Lie group G˜ acting freely on it. We the extremal limit |Q| = M of the charged black string (3.20) [6]. have shown that the dual model yields a new three-dimensional However there is one important difference which that is the ap- charged black string which is stationary and asymptotically flat. In pearance of the cross term in the metric. The metric (4.40) also this way, the non-Abelian T-duality transformation has related a possesses two independent Killing vectors ∂/∂t and ∂/∂t + β∂/∂x solution with no horizon and no curvature singularity to a solu- for β = 0. tion with a single horizon and a curvature singularity. In addition to, it changes the asymptotic behavior of solutions from AdS to • The Killing vector ∂/∂t with the norm G˜ =−1 + 2M/r be- 3 tt flat. According to our findings, it seems that under the non-Abelian comes null at r = 2M, which lies outside the event horizon T-duality transformation the mass and charge have not restored r = M. It is a time translational at infinity and becomes time- from the dual model to the original one. We have thus found a like for r > 2M, while becomes spacelike for M ≤ r < 2M and new family of solutions in the abstract is rather too strong. We also inside the event horizon. have also investigated the effect of Poisson–Lie T-duality on the • The Killing vector √∂/∂t + β∂/∂x √becomes timelike for √the singularities of spaces of T-dual models and have shown that this ranges√ r < [2M(β − 3)]/[3(β −1/ 3)] when β ∈ (−∞, − 3) duality takes those singular regions to regular regions as was the ∪ ( 3, ∞√) which lies inside√ the event horizon,√ √ and r > case with the 2D black holes [28] and 3D black strings [6,26,29]. [2M(β − 3)]/[3(β − 1/ 3)] when β ∈√(− 3√, 1/ 3) −{0}. Nevertheless, in order to address the question of the non- It is also√ timelike at infinity when β ∈ (√− 3, 1/ 3) −{0}. For Abelian T-dualization of the BTZ black hole solutions one has to β =− 3it becomes null, and for β = 3it stays everywhere show that the BTZ black hole metric has sufficient number of inde- spacelike. pendent Killing vectors. Then the isometry subgroup of the metric can be taken as one of the subgroups of the Drinfeld double. In As mentioned above, our black string solution is asymptotically order to satisfy the dualizability conditions the other subgroup flat. Thus, the non-Abelian T-duality transformation changes the must be chosen Abelian. The isometry groups of metrics can be used for construction of non-Abelian T-dual backgrounds. In the 3 Note that the exponential contribution of dilaton field in action (3.4) has been present case, to construct the dualizable metrics by the Poisson– − appeared as e 2φ . Therefore, the formula of the total mass (4.36) differs from those Lie T-duality one needs a three-dimensional subalgebra of Killing in Ref. 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