Non-Abelian T-duality of $AdS_{d\le3}$ families by Poisson-Lie T-duality

We proceed to investigate the non-Abelian T-duality of $AdS_{2}$, $AdS_{2}\times S^1$ and $AdS_{3}$ physical backgrounds, as well as the metric of the analytic continuation of $AdS_{2}$ from the point of view of Poisson-Lie (PL) T-duality. To this end, we reconstruct these metrics of the $AdS$ families as backgrounds of non-linear $\sigma$-models on two- and three-dimensional Lie groups. By considering the Killing vectors of these metrics and by taking into account the fact that the subgroups of isometry Lie group of the metrics can be taken as one of the subgroups of the Drinfeld double (with Abelian duals) we look up the PL T-duality. To construct the dualizable metrics by the PL T-duality we find all subalgebras of Killing vectors that generate subgroup of isometries which acts freely and transitively on the manifolds defined by aforementioned $AdS$ families. We then obtain the dual backgrounds for these families of $AdS$ in such a way that we apply the usual rules of PL T-duality without further corrections. We have also investigated the conformal invariance conditions of the original backgrounds ($AdS$ families) and their dual counterparts. Finally, by using the T-duality rules proposed by Kaloper and Meissner (KM) we calculate the Abelian T-duals of BTZ black hole up to two-loop by dualizing on the coordinates $ \varphi$ and $ t $. When the dualizing is implemented by the shift of direction $\varphi$, we show that the horizons and singularity of the dual spacetime are the same as in charged black string derived by Horne and Horowitz without $\alpha'$-corrections, whereas in dualizing on the coordinate $t$ we find a new three-dimensional black string whose structure and asymptotic nature are clearly determined. For this case, we show that the T-duality transformation changes the asymptotic behavior from $AdS_3$ to flat.


Introduction
Anti-de Sitter (AdS) space as a constant-negative-curvature spacetime is the maximally symmetric solution of Einstein's equation with a negative cosmological constant. AdS d space can be realized as a hyperboloid embedded in a d + 1-dimensional geometry whose metric may be, in d-dimensional half-space coordinate, expressed as where l is the radius of the AdS d metric. In this work we will deal with AdS 2 , AdS 2 × S 1 and AdS 3 backgrounds, and also the metric of the analytic continuation of AdS 2 in Poincaré coordinates. The AdS 3 metric is, in Poincaré coordinates, given by This metric is an exact solution of Einstein's equation so that it covers a subregion of AdS 3 called the Poincaré patch. A spacetime is called asymptotically AdS if it approaches (1.2) as r → ∞. AdS backgrounds have made numerous appearances in the context of the AdS/CFT correspondence, as well as the string theory. An example of an asymptotically AdS spacetime is the BTZ black hole. The BTZ as a 2 + 1-dimensional black hole solution with mass, charge, angular momentum, and negative cosmological constant was, first, found by Banados, Teitelboim and Zanelli [1]. A slight modification of this black hole solution yields an exact solution to string theory [2]. The BTZ black hole as an AdS 3 spacetime is a solution for the equations of motion of the low energy string effective action with antisymmetric tensor field B ϕt = r 2 /l and a zero dilaton field [2]. By studying the dual of this solution [2], it was shown that the BTZ black hole solution is, under the Abelian T-duality, equivalent to the charged black string solution discussed in [3]. Abelian T-duality [4] is a well known symmetry of string theory that maps any solution of the string equations with a translational symmetry to another solution. In fact, the duality symmetry is one of the most interesting properties of string theory that different spacetime geometries can correspond to equivalent classical solutions. In Ref. [5], it was discussed the Abelian T-duality for AdS 2 , as well as the two-sphere S 2 . These two examples are different in nature as T-duality performed along a compact direction for S 2 and along a non-compact direction for AdS 2 . In the case of the non-Abelian T-duality of AdS 3 spacetime, it was shown that [6] the non-Abelian T-duality transformation (as the PL T-duality on a semi-Abelian double) relates the AdS 3 spacetime 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). The PL T-duality is a generalization of Abelian [4] and non-Abelian dualities [7] that proposed by Klimcik and Severa [8].
They showed that the dual σ-models can be formulated on the Drinfeld double group D [9], which by definition has a pair of maximally isotropic subgroups G andG corresponding to the subalgebras G andG, such that the subalgebras are duals of each other in the usual sense, i.e.,G = G * . As mentioned above, the AdS 3 background is a solution for the low energy string effective action equations with a nonzero field strength H = dB. In the present work, we consider the AdS 3 background with zero B-field, i.e. H = 0. Indeed, an argument [10] claiming that AdS 3 solutions to the low energy string equations do not exist assumed that H = 0. Obviously, in the case of two-dimensional backgrounds such as AdS 2 , the field strength H is always absent.
The main goal of this paper is to investigate the non-Abelian T-duality of AdS 2 , AdS 2 × S 1 and AdS 3 backgrounds, as well as the metric of the analytic continuation of AdS 2 by using PL T-duality approach. For that purpose, we first obtain the isometry subgroups of these metrics acting freely and transitively on the corresponding target space manifolds. As we will address in the following, these isometry subgroups can be used for construction of their non-Abelian T-dual backgrounds. Sufficient condition for that is that the metrics have an isometry subgroup whose dimension is equal to the dimension of the manifold and its action on the manifold is free and transitive. This procedure was first applied for homogenous plane-parallel wave metric in [11] (see, also, [12]). On the other hand, we have lately studied the Abelian T-duality of Gödel string cosmologies up to α ′ -corrections [13] by applying the T-duality rules at two-loop order which were obtained by KM in [14]. Using the T-duality rules of KM, we shall study the Abelian T-duality of BTZ black hole up to α ′ -corrections by dualizing on both directions ϕ and t. When the dualizing is implemented by the shift of direction ϕ we show that the structure and asymptotic nature of the dual spacetime including the horizons and singularity are the same as in charged black string derived by Horne and Horowitz [3] without α ′ -corrections, whereas in performing the duality with respect to the coordinate t we find a new three-dimensional black string whose structure and asymptotic nature including the horizons and singularity are also determined.
Moreover, we show that the Abelian T-duality transformation of KM changes the asymptotic behavior of solutions from AdS 3 to flat. In the case of our black string, it is interesting to note that the true singularity lies outside the horizons. Similar to this, by using the version of the general relativity field equations produced in [15] it has been shown that [16] there is a singularity outside of a Schwarzschild black hole. Perhaps the most important feature of our black string solution is that its global structure is qualitatively different from previously discussed string solution, where dualizing is performed by the shift of direction ϕ.
This paper is organized as follows. After Introduction section, Sec. 2 reviews the construction of PL T-dual σ-models over the Lie groups, where necessary formulas are summarized. We furthermore review the conformal invariance conditions of the bosonic σ-model up to two-loop order at the end of this section. We start Sec. 3 by investigating the conformal invariance conditions of the AdS 2 metric up to α ′2 -corrections (three-loop order). Then, the non-Abelian target space dual of the metric is obtained by using the Lie subgroup of isometry group acting freely and transitively on the AdS 2 manifold; we also give a note on the non-Abelian T-duality of the metric of the analytic continuation of AdS 2 at the end of this section. Similar to the construction of T-dual σ-model for AdS 2 in Sec. 3, we obtain the non-Abelian T-dual background of the AdS 2 × S 1 metric in Sec. 4; the results of this section are summarized in Tables 1 and 2. The study of the non-Abelian T-dualization of the AdS 3 metric is given in Sec. 5. The results of this section including the constant matrix E 0 (e), the transformation between AdS 3 coordinates and group ones, and the metrics and B-fields corresponding to both original and dual backgrounds are clearly displayed in Table 4. The study of Abelian T-duality of BTZ black hole up to α ′ -corrections by using the KM approach, when the duality is implemented by a shift of the coordinates ϕ and t, is discussed in Sec. 6. Some concluding remarks are given in the last section.

Some review on PL T-duality and two-loop conformal invariance
In this section, we begin by reviewing the construction of PL T-dual σ-models over the Lie groups. In order to investigate the conformal invariance conditions of the models constructed out by the metrics AdS 2 , AdS 2 × S 1 , AdS 3 , and also the metric of the analytic continuation of AdS 2 up to the first order in α ′ we also review the two-loop beta function equations proposed by Hull and Townsend (HT) [17].

A brief review of PL T-duality
Here we review non-Abelian T-duality via the PL T-duality approach in the absence of spectators. According to [8] the PL duality is based on the concept of Drinfeld double [9]. The Drinfeld double D is a Lie group whose Lie algebra D as a vector space can be decomposed into direct sum of two Lie subalgebras G andG, such that D is maximal isotropic with respect to a non-degenerate invariant bilinear form , on D. Actually, by taking the sets {T a } and {T a } as the bases of the Lie algebras G andG, respectively, we have: where f c ab andf ab c are structure constants of the Lie algebras G andG, respectively. The isotropy of the subalgebras with respect to bilinear form means that Also, the Jacobi identity of Lie algebra D imposes the following mixed Jacobi relations over the structure constants of Lie algebras G andG In order to define σ-models with PL duality symmetry, we need to consider the following relations [8]: where g is an element of the Lie group G corresponding to the Lie algebra G. The invariance of inner product with respect to adjoint action of group together with (2.2) and (2.4) requires the above matrices to possess the following properties [8,18] 4 : Now, one may define below the σ-model with d-dimensional target manifold M where the action of group G on M is free and transitive so that M ≈ G [8] where σ ± stand for the standard light-cone variables which are defined by means of the coordinates of worldsheet Σ, giving σ ± = (τ ± σ)/2 together with ∂ ± = ∂ τ ± ∂ σ , and R a ± are the components of right invariant one-forms on G which are defined in the following way where X M , M = 1, ..., d are the coordinates of manifold M. The background matrix E is given in matrix notation by where E 0 (e) is an invertible constant matrix in which e is the unit element of the group G. The target space of dual model is the d-dimensional manifoldM with coordinatesX M . Analogously, the group G (whose dimension is, however, equal to that of G) corresponding toG acts freely and transitively on the manifoldM so thatM ≈G. Then, the corresponding dual action can be written as [8] whereg is an element of the Lie groupG, andR ±a are the components of right invariant one-forms oñ G which are defined as in (2.7). The background appearing in this action are given in matrix notation byẼ whereΠ(g) is defined as in (2.5) by replacing untilded quantities by tilded ones and vice versa. It should be noted that the actions (2.6) and (2.9) correspond to PL dual σ-models [8]. If the group G becomes the isometry group of the manifold M with dual Abelian groupG, then one can obtain the standard non-Abelian duality [7]. Let us now compare the σ-model (2.6) with the following standard two-dimensional non-linear σ-model defining on the worldsheet in d spacetime dimensions Notice that for the non-Abelian duality case,f ab c = 0, we find that b(g) = 0, then, Π(g) = 0; consequently, E(g) = E 0 (e). In this case, if E 0 (e) is chosen to be symmetric, then one concludes that the B-field vanishes. In general E 0 (e) of (2.8) can have an antisymmetric part, and in that case the B-field would be non-vanishing.
Before closing this subsection, we shall give a short summary of the dualization procedure that we will apply in the next sections. First of all, we obtain the subalgebras of d(d + 1)/2-dimensional isometry Lie algebras that generate subgroups of isometry Lie groups that act freely and transitively on d-dimensional target space M where the metrics of AdS families are defined. Since the dualizable metrics can be constructed by virtue of Drinfeld double, the isometry subgroups of the metrics are taken as one of the subgroups of the Drinfeld double. In order to satisfy the dualizability conditions the other subgroup is considered to be Abelian. In other words, since we are dealing with non-Abelian T-duality, the dual Lie algebra must be chosen Abelian. In order to obtain the metrics of AdS families by the Drinfeld double construction we need to find the transformation between group (isometry subgroup) coordinates and geometrical coordinates by choosing a convenient element of group. In this case, we have to get the left-invariant vector fields on the group. Accordingly, the metrics can be transformed into the group coordinates and finally one can write the corresponding actions to the transformed metrics in the from of (2.11). Thus, one may use (2.12) to obtain the original σ-model from (2.9) and then dual one from (2.6).

Two-loop conformal invariance conditions of the bosonic string σ-model
In order to study the conformal invariance conditions of the bosonic string σ-model, Fradkin and Tseytlin [19] have suggested that one should add to action (2.11) the renormalizable, but not Weyl invariant, term where R (2) is the curvature scalar for the worldsheet metric, and Φ(X) is the background dilaton field in the manifold M. The term (2.13) breaks Weyl invariance on a classical level as do the one-loop corrections to G and B. It is essential for string consistency that, as a quantum field theory, the σ-model be locally scale invariant. This is equivalent to the requirement that the two-dimensional worldsheet stress-energy tensor of the theory be traceless. In the σ-model S nlsm + S dil , local scale invariance is broken explicitly by the term (2.13). Consistency of the string theory requires that the action S nlsm + S dil defines a conformally invariant quantum field theory. In the σ-model context, the conformal invariance conditions of the σ-model S nlsm + S dil are provided by the vanishing of the three functions β G M N , β B M N and β Φ (beta function equations) [20]. In the HT scheme, these equations are, at the two-loop level (first order in α ′ ) 5 , given by [17,21] where β (i) 's stand for the i-loop beta function equations. Moreover, the covariant derivatives ∇ M , Ricci tensor R M N and Riemann tensor field R M NP Q are calculated from the metric G M N that is also used for lowering and raising indices, and H M NP is the field strength corresponding to the B-field which is defined by We have moreover introduced the conventional notations Λ is a cosmological constant. In the following we shall investigate the conformal invariance conditions of the σ-models with metrics AdS 2 , AdS 2 × S 1 , AdS 3 and the metric of the analytic continuation of AdS 2 up to the second order in α ′ and also BTZ black hole up to the first order. Then, we will study the non-Abelian T-duality of these metrics (except for BTZ) using the procedure mentioned in subsection 2.1. As mentioned in Introduction section, the Abelian T-duality of AdS 2 metric has been discussed in Ref. [5]. Here we are going to investigate the non-Abelian T-duality of this metric by PL T-duality procedure. Let us now start with AdS 2 metric.

Non-Abelian T-duality of the AdS 2 background
The simplest space of AdS family is AdS 2 manifold whose metric may be expressed as One can easily deduce that the field equations (2.14)-(2.16) to zeroth order in α ′ don't possess a AdS 2 solution with metric (3.1). Notice that in the case of two-dimensional backgrounds such as AdS 2 , the field strength H is always zero. Now we consider the α ′ terms to the equations (2.14)-(2.16), but neglect the B-field. In this case, the B-field equation of motion (2.15) is fulfilled. Considering a constant dilaton field the resulting equations are then reduced to two polynomial constraints: As it can be seen from the above equations, the solutions we obtain are not under control. Namely, from equation (3.2) one takes the AdS radius to be the string length or α ′ = l 2 . In this way, we can cancel one-loop terms against two-loop terms in the beta function equations. Technically one can do this but it is not physically admissible. Similarly to our work, in the case of the Gödel spacetimes in string theory for the full O(α ′ ) action, Barrow and Dabrowski [22] found a simple relation between the angular velocity of the Gödel universe, Ω, and the inverse string tension of the form α ′ = 1/Ω 2 in the absence of the B-field. Any way, it seems that the α ′ expansion is uncontrollable, since all orders now contribute equally. In order to show that the expansion in the higher orders is uncontrollable, we calculate three-loop beta functions for the AdS 2 metric. By following [23], in the absence of the field strength, the three-loop beta functions for a general theory are given by Now we add the three-loop beta functions (3.4) and (3.5) to the field equations (2.14)-(2.16). Using the fact that the field strength is zero, the beta function equations to second order in α ′ for the AdS 2 metric with a constant dilaton reduce to the following polynomials, The above results render the α ′ expansion is uncontrollable, and thus one can't guarantee the conformal invariance of the AdS 2 metric. To continue, in order to study the non-Abelian T-duality of the AdS 2 background one may obtain Lie algebra generated by Killing vectors of (3.1). The corresponding Killing vectors k a of AdS 2 can be derived by solving Killing equations, L ka G M N = 0. They are then read off such that the vectors k 1 and k 3 become everywhere timelike except for x = 0. The Killing vectors (3.8) also satisfy in the sl(2, R) algebra ( ∼ = V III Bianchi Lie algebra) with the following commutation relations Note that this Lie algebra has only a two-dimensional non-Abelian subalgebra, A 2 , which can be defined by bases T 1 = k 2 and T 2 = k 1 (or T 1 = −k 2 and T 2 = k 3 ). As we will show below the Lie group corresponding to A 2 acts freely and transitively on the AdS 2 manifold. In the next subsection we shall study the non-Abelian T-duality of the AdS 2 background by constructing the PL T-dual σ-models stating from the Lie bialgebra (A 2 , 2A 1 ).

Non-Abelian T-duality as PL T-duality on the semi-Abelian double
As mentioned above, the two-dimensional Lie subalgebra of (3.9) is A 2 with the following commutation relation: where we have considered T 1 = k 2 , T 2 = k 1 . In order to construct dualizable backgrounds we need to first investigate whether the action of the Lie group A 2 A 2 A 2 on AdS 2 manifold is free and transitive. Before proceeding further, let us have some review on free and transitive actions [24]: Free action: The free action of the Lie group G G G on a manifold M is given by for any g = e α a Ta ∈ G G G and x M ∈ M in which T a 's are the bases of Lie algebra corresponding to G G G.
Considering the infinitesimal form of this action one easily finds that ξ In order to have the free action one must expand the bases T a in terms of the Killing vectors of the manifold metric. Then using ξ • x M = 0 it should be concluded that α a = 0, that is, g = e.
Transitive action: The transitive action of the Lie group G G G on the manifold M means that for every two points In order to turn the above definition into a computational method for determining the transitive action of G G G on M one may expand the bases of Lie algebra G of G G G in terms of the Killing vectors k a of M, Notice that from the linear independence of the bases T a one concludes that the matrix A M a must be invertible. Let g = e α a Ta be an element of G G G. By considering infinitesimal form of g and by using g • x M = x ′ M and also the invertibility condition on A M a we then find that α a = 0. That is, g is a non-trivial element of G G G. Thus, in order to have the transitive action of G G G on M, the matrix A M a must be invertible.
In what follows we consider the action of Lie group such that detA = 0. Therefore, the action of A 2 A 2 A 2 on AdS 2 is also transitive. Now we are ready to find the non-Abelian target space dual of AdS 2 background. Consider the Drinfeld double (A 2 , 2A 1 ) [25] (see, also, [26]), where 2A 1 is the two-dimensional Abelian Lie algebra. Making use of (2.1) and (3.10) we obtain the corresponding non-zero Lie brackets (3.14) where (T 1 ,T 2 ) are the bases of the dual Lie algebra 2A 1 . Choosing a convenient element of A 2 A 2 A 2 as g = e x 1 T 1 e x 2 T 2 and then using (2.7) we get the components of the right invariant one-forms where (x 1 , x 2 ) stand for the coordinates of A 2 A 2 A 2 . It can be inferred from (2.4) and (3.14) that Π = 0. Thus by setting E 0 (e) as the σ-model (2.6) can be written in the following form Comparing (3.17) and the general form of σ-model (2.11) we obtain that B M N = 0 and (3.18) Indeed the above metric can be transform to the AdS 2 metric (3.1) by an appropriate coordinate transformation. To find relation between coordinates of AdS 2 , (x, t), and group coordinates, (x 1 , x 2 ), one must find the left invariant vector fields of A 2 A 2 A 2 , giving us and then consider the transformation between these vector fields and ∂ M of AdS 2 (3.20) Finally, one can find the following transformation To construct the dual σ-model of (3.17), in other words, the dual space to AdS 2 we choose an element Then by using relation (2.4) for the dual group and by applying (3.14) one can obtain the Poisson structure on 2A 1 Noting the fact that the dual Lie algebra is Abelian we find thatR M a = δ M a . Then using (2.10), (3.16), (3.22) together with (2.9) the dual σ-model corresponding to (3.17) is worked out One may compare the dual model (3.23) with dual version of general σ-model (2.11) to obtain It is worthwhile to mention some specific and important properties of this dual solution. The metric (3.24) is ill defined at the regionsx 2 = ±l 2 . We can test whether there are true singularities by calculating the scalar curvature. To be more specific, one may use the coordinate transformatioñ to write (3.24) and (3.25) in the following form Thus, r = 0 is a true singularity; moreover, one can show that this singularity also appears in the Kretschmann scalar, which is,K =R 2 . On the other hand, only the Killing vector of (3.27) is −∂ t whose norm is −l 2 / sinh 2 (r), hence, after the dualization only a timelike isometry is preserved. Note that in the case of the dual metric (3.24) the α ′ expansion is also uncontrollable. Namely, the dual background can't be conformally invariant.
A note on the non-Abelian T-duality of the metric of the analytic continuation of AdS 2 .
The metric of the analytic continuation of AdS 2 in Poincaré coordinates can be derived by doing the Wick rotation t → it on the metric (3.1), giving us The Killing vectors of this metric are Unlike AdS 2 , both Killing vectors k 1 and k 3 of (3.30) are spacelike. One can easily check that the Lie algebra spanned by these vectors is IX Bianchi Lie algebra. Analogously, only two-dimensional non-Abelian subalgebra of IX Bianchi is A 2 which is defined by bases T 1 = k 2 and T 2 = k 1 . In this case, the A 2 Lie group also acts freely and transitively on the manifold defined by the metric (3.30).
To find the non-Abelian target space dual of (3.30) we first construct the original σ-model on the semi-Abelian double (A 2 , 2A 1 ) such that the metric of model can be turned into (3.30). To this end, one must choose the constant matrix E 0 (e) as E 0 (e) = l 2 I, where I is the 2 × 2 identity matrix, to obtain the metric of the model in the form of Under the coordinate transformation the metric (3.32) can be turned into (3.30). The corresponding dual model can be constructed out by means of the procedure applied for AdS 2 . Finally, the dual background for the metric (3.30) reads Unlike the dual metric of AdS 2 , there is no singularity for the dual metric of (3.30) as expected, because these two examples are different in nature as Abelian T-duality performed in [5] for both AdS 2 and S 2 . There is only a spacelike Killing vector ∂x 1 for the metric (3.34). So, after dualization only a spacelike isometry is preserved. We also note that due to the uncontrollable α ′ expansion, the conformal invariance of both original and dual backgrounds fail here as in AdS 2 case.

Non-Abelian T-duality of the AdS 2 × S 1 background
The AdS 2 × S 1 metric can be, in the coordinates (x, z, t), written as Looking at the equations (2.14)-(2.16) together with (3.4) and (3.5), one can check the conformal invariance conditions of the metric (4.1). Hence, the vanishing of the beta function equations up to three-loop order for the AdS 2 × S 1 metric with a zero B-field and dilaton field Φ = c 1 x + c 2 , where c i 's are some constant parameters, reduce to the two polynomials, From the above equations one concludes that α ′ expansion is uncontrollable as in the AdS 2 metric. Therefore the AdS 2 × S 1 background fails to satisfy the beta function equations which indicates that the corresponding σ-model is not Weyl invariant, i.e. does not define a critical string theory in the usual sense.
In the following, in order to investigate the non-Abelian T-duality of the AdS 2 × S 1 metric we need to obtain the Lie algebra generated by the Killing vectors of (4.1). The metric (4.1) admits the following Killing vectors The Lie algebra generated by these Killing vectors is According to the norms of the Killing vectors |k 1 | 2 = l 2 , |k 2 | 2 = −l 2 t 2 − z 2 2 /(4z 2 ) and |k 4 | 2 = −l 2 /z 2 we find that the vector k 1 is everywhere spacelike while k 2 and k 4 are everywhere timelike except for z = 0. Indeed, the Lie algebra given by (4.5) is isomorphic to the gl(2, R). One can show that there are two classes of three-dimensional subalgebras of the isometry algebra of the AdS 2 ×S 1 metric isomorphic to the Bianchi Lie algebras III and V III which are denoted by III .i and V III .i , respectively.
On the other hand, one easily shows that the Lie group corresponding to III .i , III .i III .i III .i , acts freely and transitively on AdS 2 × S 1 space, while for the Lie group V III .i V III .i V III .i we have a free action only. The results are summarized in Table 1.
Similar to the T-dual σ-models construction for the AdS 2 metric, which was represented in preceding section, we find the non-Abelian target space dual including the metric and B-field of the AdS 2 ×S 1 Original background Dual background transformation background. As shown in Table 2, we have constructed T-dual σ-models 8 based on the semi-Abelian double (III .i , 3A 1 ) [27,28] by a convenient choice of the constant matrix E 0 (e). It has been shown that there is an appropriate coordinate transformation that turns original σ-model into the AdS 2 × S 1 background. As it is seen from the dual solution, we don't expect to see any dramatic change in physical properties of AdS 2 × S 1 versus AdS 2 alone.
For the metric (5.1) with zero B-field and a constant dilaton field, the vanishing of the three-loop beta function equations are reduced to two polynomial constraints: Note that we can also obtain the original background from the PL T-duality in the presence of spectator fields. To this end, one may use two-dimensional Lie group A2 A2 A2 with the coordinates (x1, x2) and choose a spectator with the coordinate x3. So far, our findings show that the Lie group A2 A2 A2 is wealthy. Recently, in order to study the non-Abelian T-duality of the Gödel spacetimes [13] we have constructed the T-dual σ-models on the manifold M ≈ O × A2 A2 A2 where A2 A2 A2 acts freely on M while O is the two-dimensional orbit of G in M (see, also, [6,29]). 9 It should be remarked that the non-Abelian T-dualization of AdS3 background has been already discussed in Ref. [30]. There, the AdS3 metric has been applied in Poincaré coordinates as in (1.2), hence, the forms of their Killing vectors and Lie algebra spanned by them are different from ours (Eqs. (5.4) and (5.5)). Although subalgebras of the isometry subgroups of those are isomorphic to our results, the form of the resulting dual backgrounds is different from ours (Table  4). Most importantly, we have, here, investigated the spacetime structure of T-dual findings for AdS3 background and also some their physical interpretations by introducing some convenient coordinate transformations, while these cases are not seen in [30].
In this case, the α ′ expansion can't be also controlled in the same way as in the previous cases. Nevertheless, as mentioned in [2] the metric (5.1) in the presence of the B-field B tx = −l 2 /y 2 with a constant dilaton field satisfy the one-loop beta function equations provided that Λ = 2/l 2 . It can be also useful to comment on the fact that one can verify the field equations (2.14)-(2.16) up to twoloop order for the aforementioned solution provided that Λ = (2l 2 + 4α ′ )/l 4 . It's worth mentioning AdS 3 inherits the isometries of the embedding space that preserve the hyperboloid. The group of rotations+boosts in a 4D geometry with signature (+, +, −, −) is SO(2, 2), so we expect this to be the isometry group of AdS 3 10 . In this section we will confirm this. We are interested in metrics that admit at least three independent Killing vectors because they can be interpreted as T-dualizable backgrounds for σ-models in three dimensions. In order to investigate the non-Abelian T-duality of the AdS 3 we need all three-dimensional subalgebras of Killing vectors that generate group of isometries acting freely and transitively on the AdS 3 manifold. Metric (5.1) has a number of symmetries important for the construction of the dualizable σ-models. It admits the following Killing vectors It is easily shown that the vectors k 1 and k 5 are everywhere spacelike except for y = 0, while k 2 and k 6 stay everywhere timelike except for y = 0. The Lie algebra spanned by Killing vectors (5.4) is isomorphic to the so(2, 2) Lie algebra with nonzero commutation relations One can check that the three-dimensional Bianchi Lie algebras III .i , V, V I 0 , V I q , and V III .i are Lie subalgebras of (5.5) such that all the corresponding Lie subgroups (except for the V I 0 V I 0 V I 0 Lie group) act freely and transitively on the AdS 3 space. The results are summarized in Table 3. 11 Similar to the construction of T-dual σ-models for AdS 2 and AdS 2 × S 1 backgrounds which were represented in preceding sections and also by applying the results of Table 3 we find the non-Abelian 10 In general the AdS d isometry group is SO(d − 1, 2) [5]. 11 To obtain the commutation relations of the semi-Abelian doubles generated by the Bianchi Lie algebras [27,28] of Table 3, one must use (2.1) and (5.5) together with the basis represented in terms of the linear combination of Killing vectors.
target space duals of AdS 3 . For the sake of clarity the results obtained in this section are summarized in Table 4; we display the metrics and B-fields corresponding to both original and dual backgrounds, together with the transformation between AdS 3 coordinates and group ones, as well as the form of constant matrix E 0 (e). Note: In order to better understand of the spacetime structure of T-dual findings for AdS 3 background and also some their physical interpretations we use some coordinate transformations that make the metrics simpler. Below we discuss the dual backgrounds for each case separately.
• The dual background on the double (III .i , 3A 1 ): As shown in Table 4, the dual metric obtained by this double has apparent singularities at the regionsx 2 = ±l 2 . Indeed, these are the coordinate singularities in the metric. To remove them one may use the change of coordinates x 1 = l 2 y,x 2 = −l 2 tanh z,x 3 = l 2 x, (5.6) to rewrite the dual background in the form The scalar curvature and Kretschmann scalar corresponding to the metric are, respectively, given bỹ As it is seen from equations (5.7), (5.9) and (5.10), the singular points are not true points. In addition, note that the metric (5.7) also possesses two independent Killing vectors ∂ x and ∂ y − ∂ x which have the norm l 2 , thus, they are everywhere spacelike. Therefore, the duality has not here involved the timelike directions. In the case of the conformal invariance conditions of background given by (5.7) and (5.8) we have checked that this background does not satisfy the field equations (2.14)-(2.16) up to two-loop order.
• The dual background on the double (V, 3A 1 ): In this case, it is simply shown that the field strength corresponding to the B-field represented in Table 4 is zero. Hence, if we introduce the coordinate transformationx then, the dual background turns into ds 2 = l 2 dx 2 + dy 2 y 2 + 1 − y 2 dt 2 , (5.12) Here we have ignored the total derivative terms that appeared in the B-field part. As it is seen from (5.12), the singularities appeared in dual metric have been removed by coordinate transformation (5.11). Solving the field equations (2.14)-(2.16) for the metric (5.12) with zero field strength one concludes that there is no suitable dilaton field to satisfy these equations. It is also interesting to see that the metric (5.12) possesses two independent Killing vectors −∂ t and ∂ x which are timelike and spacelike, respectively. Thus, in this case, the duality involves the timelike directions.
• The dual background on the double (V I q , 3A 1 ): In this case one may use the coordinate transformationx then, the dual background related to this double yields where △ = 4 q 2 − 1 e u − q 2 . For the metric (5.15) one finds that the scalar curvature is Therefore, here the singularity is true and represents itself as the curve u = − ln 4[1 − q −2 ] .
• The dual background on the double (V III .i , 3A 1 ): In this case, the from of the metric represented in Table 4 is a bit complicated. In order to get the simpler form of the metric one may consider the following coordinate transformatioñ Under the above transformation, the background becomes ds 2 = l 2 2(e 2y + e 2z − 1) e 2y dx 2 + e 2(y+z) dydz + e 3y sinh(y)dy 2 + e 3z sinh(z)dz 2 , By calculating the scalar curvature corresponding to the metric (5.19) one concludes that the singularity appeared in the metric, which represents itself as the curve e 2y + e 2z = 1, is true. In this case, the metric just has one Killing vector which is 2∂x with the norm 2l 2 e 2y /(e 2y + e 2z − 1). Accordingly, the behavior of Killing vector changes between spacelike and timelike regions as we pass through the singularity curve.
6 Abelian T-duality of the BTZ black hole up to two-loop order As announced in Introduction section, the BTZ black hole metric [1] with the following B-field and dilaton can be considered as a solution for the equations of motion of the low energy string effective action [2] for some constant b. By studying Buscher-duality of this solution [2], it was shown that the BTZ black hole solution is, under the Abelian T-duality, equivalent to the charged black string solution discussed in [3]. Then, in [6] by investigating the non-Abelian T-duality of the BTZ vacuum solution it was shown that the non-Abelian T-duality transformation relates the BTZ vacuum solution with no horizon and no curvature singularity to a solution with a single horizon and a curvature singularity.
In this section, we study the Abelian T-duality of BTZ background up to α ′ -corrections when the dualizing is implemented by the shift of directions ϕ and t. To this end, we use the T-duality rules at two-loop order derived by KM [14]. Before proceeding further, let us review the α ′ -corrected T-duality rules of KM in the next subsection.

A review of Abelian T-duality up to α ′ -corrections
The two-loop σ-model corrections to the T-duality map in string theory by using the effective action approach were obtained by KM [14] 12 . They had found the explicit form for the O(α ′ ) modifications of the lowest order duality transformations by focusing on backgrounds that have a single Abelian isometry. Following Ref. [14], here we consider the reduced metric g µν , antisymmetric field b µν and dilaton Φ of the d-dimensional spacetime as where V µ is the standard Kaluza-Klein gauge field coupled to the momentum modes of the theory, and W µ is the other gauge field coupling to the winding modes. It has been assumed that the isometry direction we want to dualize is implemented by a shift of a coordinate x. Furthermore, σ andφ in equation (6.4) are some scalars fields. The relations to identify the fields of the dimensional reduction are given by As first demonstrated in Ref. [32], for applying rules of KM, one first needs to implement the field redefinitions to go from HT scheme to that of KM. The field redefinitions are given by [32] G (HT ) The equations of two-loop T-duality transformation in the KM scheme that we will use are [14] 12) where also, All the lowering and raising of the indices will be done with respect to the reduced metric g µν . Note that after perform two-loop T-dulaity transformation one must return to HT scheme by use of (6.8)-(6.10). In the next subsections we will apply this method in order to study the Abelian T-duality of BTZ background up to two-loop order.

Dualizing with respect to the coordinate ϕ
As we mentioned earlier, background (6.1) is a solution for the field equations (2.14)-(2.16) up to zeroth order in α ′ . In addition, one can show that (6.1) satisfies (2.14)-(2.16) up to first order in α ′ if the following relation holds between the constants l, α ′ and Λ: We note that solution (6.1) has been obtained in the HT scheme. In order to use the field redefinitions in equations (6.8)-(6.10) we need to write (6.1) in the KM scheme, giving us In this way we are ready to obtain the shifting coordinates and then perform the KM T-duality transformations. Here, the isometry we want to dualize is that the shift of the ϕ coordinate, i.e., x = ϕ. It should be remarked that the rules of T-duality are derived assuming that the coordinate to be dualized is spacelike. Fortunately, for the metric (6.1), G xx = G ϕϕ is positive. In fact, we are faced with a spacelike case. Comparing (6.21) with equations (6.2)-(6.4) or using (6.5)-(6.7) one concludes that only non-zero components of the fields V µ and W µ are Then, one may use (6.15), (6.16) together with (6.22) to obtain only non-zero components of V µν and W µν as follows In this case one also gets that h µνρ = 0. Thus, by using (6.18), (6.19) together with (6.24), functions Z and T are, up to first order of α ′ , obtained to be One can check that the dual background (6.35) is conformally invariant up to the first order in α ′ Let us investigate the asymptotic behavior of the metric. Note that for larger it is not possible to similarly to the t and φ coordinates fix the overall scaling of ther asr goes to infinity, since the metric asymptotically approaches (l 2 − 2α ′ )dr 2 /4r 2 . Therefore, for large r the black string solution (6.37) approaches the following asymptotic solution for some constantb. Here we have set Now, one verifies the field equations (2.14)-(2.16) up to two-loop order for the asymptotic solution (6.40) with the same condition of the conformal invariance of background (6.35). The above result shows that the Abelian T-duality transformation of KM changes the asymptotic behavior of solutions from AdS 3 to flat.

Conclusions
In order to study the non-Abelian T-duality of the metrics of Riemannian manifolds one may use the isometry subgroups of the metrics. Sufficient condition for that is that the dimension of the isometry subgroups of the metric is equal to the dimension of the Riemannian manifold and its action on the manifold is transitive and free. Using this fact we have shown that for the metrics of AdS families such isometry subgroups exist and the metrics can be dualized by the PL T-duality transformation.
We have shown that the Lie subgroup A 2 corresponding to non-Abelian two-dimensional subalgebra A 2 acts freely and transitively on AdS 2 manifold. In this way, it has been found the dual of the AdS 2 background stating from the Lie bialgebra (A 2 , 2A 1 ). Our results show that the dual metric of the AdS 2 has a true singularity. In fact, T-duality takes that singular region to regular region as was the case with the 2D black holes [33]. As we have shown the Lie algebra generated by Killing vectors of AdS 2 × S 1 is isomorphic to the gl(2, R). Then we have found that only the Lie group corresponding to the III .i Bianchi Lie algebra (as subalgebra of the gl(2, R)) acts freely and transitively on AdS 2 × S 1 space. Accordingly, we have determined the metric and B-field dual to the AdS 2 × S 1 . In the case of AdS 3 , there are five classes of three-dimensional subalgebras of the isometry algebra of the AdS 3 metric isomorphic to the Bianchi Lie algebras III, V, V I 0 , V I q and V III. All Lie subgroups corresponding to these subalgebras act freely and transitively on the AdS 3 space, except for the V I 0 V I 0 V I 0 Lie group. We have investigated the spacetime structure of T-dual findings for AdS 3 background and also some their physical interpretations by introducing some convenient coordinate transformations. Among the four dual backgrounds to AdS 3 , only the dual metrics constructed out on the semi-Abelian Drinfeld doubles (V I q , 3A 1 ) and (V III .i , 3A 1 ) have true singularities.
Most importantly, in the absence of B-field, for all metrics of AdS 2 , AdS 2 × S 1 , AdS 3 , and also the metric of the analytic continuation of AdS 2 we have investigated the conformal invariance conditions of the backgrounds up to three-loop order. The results render the α ′ expansion is uncontrollable, and thus one can't guarantee the conformal invariance of the backgrounds. Notice that in the case of these spacetimes we have applied the usual rules of non-Abelian T-duality without further corrections. We have obtained the non-Abelian T-duals of the metrics of AdS families by using the PL T-duality approach and have then checked the conformal invariance conditions of the duals up to two-loop order. Unfortunately, all of the dual backgrounds corresponding to these metrics do not remain conformally invariant up to two-loop order. Indeed, this was expected, because we did not use the α ′ -corrected rules of non-abelian T-duality that are necessary to have conformal invariance at two-loop order [31]. It has been shown that the PL duality can be extended to order α ′ , i.e. two loops in the σ-model perturbation theory, provided that the map is corrected [31] (see, also, [34,35]). It is possible that one applies the usual rules of non-Abelian T-duality without further corrections, and still be able to obtain two-loop solutions (e.g. [13]). However, in general, further corrections to the rules are necessary. With the modified rules one will be able to find the right α ′ -corrections to the non-Abelian dual backgrounds, so that the two-loop equations are satisfied. We intend to address this problem in the future.
Finally, we have studied the Abelian T-duality of BTZ background up to α ′ -corrections by using the T-duality rules of KM, when the dualizing is implemented by the shift of directions ϕ and t. In dualizing on the direction ϕ, we have shown that the structure and asymptotic nature of the dual spacetime including the horizons and singularity are the same as in charged black string derived in [3] without α ′ -corrections, whereas in performing the duality with respect to the coordinate t it has been found a new three-dimensional black string for which we have determined the horizons and singularity. For this case, we have also shown that the Abelian T-duality transformation of KM changes the asymptotic behavior of solutions from AdS 3 to flat.