T-duality/plurality of BTZ black hole metric coupled to two fermionic fields

We ask the question of classical super (non-)Abelian T-duality for BTZ black hole metric coupling to two fermionic fields. Our approach is based on super Poisson-Lie (PL) T-duality in the presence of spectator fields. In order to study the Abelian T-duality of the metric we dualize over the Abelian Lie supergroups of the types $(1|2)$ and $(2|2)$, in such a way that it is shown that both original and dual backgrounds of the models are conformally invariant up to one-loop order in the presence of field strength. Then, we study the non-Abelian T-duality of the BTZ vacuum metric coupling to two fermionic fields. The dualizing is performed on some non-Abelian Lie supergroups of the type $(2|2)$, in such a way that we are dealing with semi-Abelian superdoubles which are non-isomorphic as Lie superalgebras in each of the models. In the non-Abelian T-duality case, it is interesting to mention that the models can be conformally invariant up to one-loop order in both cases of the absence and presence of field strength. In addition, starting from the decomposition of semi-Abelian Drinfeld superdoubles generated by some of the ${\cal C}^3 \oplus {\cal A}_{1,1}$ Lie superbialgebras we study the super PL T-plurality of the BTZ vacuum metric coupled to two fermionic fields. However, our findings are interesting in themselves, but at a constructive level, can prompt many new insights into supergravity and manifestly have interesting mathematical relationships with double field theory.

The BTZ black hole, named after Banados, Teitelboim, and Zanelli (BTZ), is a black hole solution for 2 + 1-dimensional topological gravity with a negative cosmological constant, mass, angular momentum and charge [1].The BTZ black hole is asymptotically anti-de Sitter rather than asymptotically flat, and has no curvature singularity at the origin, because [2] it is a discrete quotient of SL(2, R).The line element for the black hole solutions is given by where the radius l is related to the cosmological constant by l = (−Λ) −1/2 .Here, the coordinate φ is periodic with period 2π.The constants of motion M and J are the mass and angular momentum of the BTZ black hole, respectively.They are appeared due to the time translation symmetry and rotational symmetry of the metric, corresponding to the killing vectors ∂/∂t and ∂/∂φ, respectively.The line element (1.1) describes a black hole solution with outer and inner horizons at r = r + and r = r − , respectively, where the mass and angular momentum are related to r ± by The region r + < r < M 1/2 l defines an ergosphere, in which the asymptotic timelike Killing field ∂/∂t becomes spacelike.The solutions with −1 < M < 0, J = 0 describe point particle sources with naked conical singularities at r = 0.The metric with M = −1, J = 0 may be recognized as that of ordinary anti-de Sitter space; it is separated by a mass gap from the M = 0, J = 0 "massless black hole", whose geometry is discussed in Ref. [2].
The vacuum state which is regarded as empty space, is obtained by letting the horizon size go to zero.This amounts to letting M → 0, which requires J → 0. We have to notice that the metric for the M = J = 0 black hole is not the same as AdS 3 metric which has negative mass M = −1.Locally they are equivalent since there is locally only one constant curvature metric in three dimensions.However they are inequivalent globally.Notice that the singularity at the center of the BTZ black hole is called a conical singularity.However, a conical singularity is not similar to a canonical singularity because it does not cause the spacetime curvature to diverge [3].
The BTZ black hole also arises in string theory as a near-horizon region of a non-extremal system of 1-and 5-branes.In Ref. [4] by using a presentation of the D-brane counting for extremal black holes, it has been shown that four-and five-dimensional non-extremal black holes can be mapped to the BTZ black hole (times a compact manifold) by means of dualities.A slight modification of this black hole solution yields an exact solution to string theory [5].There, the BTZ black hole solutions have been considered in the context of the low energy approximation, then as the exact conformal field theory.In order to obtain an exact solution to string theory, one must modify the BTZ black hole solutions by adding an antisymmetric tensor field H M N P (field strength) proportional to the volume form ε M N P .It has been shown that any solution to three-dimensional general relativity with negative cosmological constant is a solution to low energy string theory with H M N P = 2ϵ M N P /l, Λ = −1/l 2 and a dilaton field Φ = 0 [5].In particular, it was observed in [5] that, the two parameter family of black holes (1.1) along with satisfy the low energy string effective action equations.Then, it was obtained [5] the Abelian dual of this solution by Buscher's duality transformations [6].Indeed, T-duality symmetry is one of the most interesting properties of string theory that connects seemingly different backgrounds in which the strings can propagate.The duality is Abelian if it is constructed on an Abelian isometry group.We refer to Abelian T-duality for stressing the presence of global Abelian isometries in the target spaces of both the paired σ-models, while non-Abelian T-duality [7] refers to the existence of a non-Abelian isometry on the target space of one of the two σ-models.Now that we are discussing the non-Abelian Tduality, it is necessary to explain that the non-Abelian target space dual of BTZ vacuum solution was obtained in [8] by making use of the PL T-duality approach in the presence of spectator fields.There, it has been shown that the BTZ vacuum solution with no horizon and no curvature singularity is related to a solution with a single horizon and a curvature singularity.The PL T-duality was proposed by Klimcik and Severa [9,10] as a generalization of Abelian [6] and non-Abelian dualities [7].In this type of duality, the symmetry does not need to be realized as an isometry of the initial background; moreover, σ-models are formulated on a Drinfeld double group D [11] whose Lie algebra D can be decomposed into a pair of maximally isotropic sub-algebras with respect to a non-degenerate invariant bilinear form on D, so that the sub-algebras are duals of each other in the usual sense.In this case it is said that the σ-models are indeed dual in the sense of the PL T-duality.Generalization of T-duality to Lie supergroups and also supermanifolds have been explored in the context of PL T-duality in [12,13].When we deal with a semi-Abelian Drinfeld superdouble, that is, one of the sub-supergroups of Drinfeld superdouble is considered to be Abelian, the super PL T-duality reduces to the super non-Abelian T-duality.Recently, it has been considered the non-Abelian T-duality of σ-models on group manifolds, symmetric, and semi-symmetric spaces with the emphasis on the T-dualization of σ-models whose target spaces are supermanifolds [14] (see also [15]).There, it has been performed super non-Abelian T-dualization of the principal chiral model based on the OSP (1|2) Lie supergroup.
The first goal of this paper is to find (Abelian)non-Abelian target space duals of the BTZ metric when is coupled to two fermionic fields.In this regards, the target spaces of T-dual σ-models are considered to be the supermanifolds M ≈ O × G and M ≈ O × G where G and G are (Abelian)non-Abelian Lie supergroups of the type (2|2) acting freely on M and M , respectively, while O is the orbit of G in M .In the Abelian T-duality case, we also dualize over the Abelian Lie supergroup of the type (1|2).The indecomposable fourdimensional Lie superalgebras of the type (2|2) was first classified by Backhouse in [16].Unfortunately, there are some omissions and redundancies in that paper, probably because the author in some cases has not taken into account all isomorphisms to reduce the list of superalgebras.We have made some modifications to the Backhouse's classification; the detail of the calculations are given in Appendix A. In addition, one can find a classification of decomposable Lie superalgebras of the type (2|2) in [17].Based on the (anti-)commutation relations between bases of boson-boson (B-B), boson-fermion (B-F) and fermion-fermion (F-F) we have classified all decomposable and indecomposable Lie superalgebras of the type (2|2) into six disjoint families in Appendix A. Finally, we find super PL T-plurals of the BTZ vacuum metric coupled to two fermionic fields with respect to the C 3 ⊕ A 1,1 Lie supergroup.The general procedure that we shall apply is a straightforward generalization of the well-known prescription of purely bosonic PL T-plurality [18] to Lie supergroup case, and we shall refer to it as super PL T-plurality in this work.Notice that the generalization of PL T-plurality formulation to Lie supergroup case has been recently explored in [19].
The outline of this paper is as follows: in section 2 we introduce our notations and recall the origin of the super PL T-duality transformations in the presence of spectator fields at the level of the σ-model.Abelian and non-Abelian target space duals of the BTZ metric coupled to two fermionic fields are investigated in sections 3 and 4, respectively; the results of non-Abelian T-duality of the models are summarized in Table 1 at the end of section 4. In section 5, we study super PL T-plurality of the BTZ vacuum metric coupled to two fermionic fields, in such a way that we start from the decompositions of semi-Abelian Drinfeld superdoubles generated by the C 3 ⊕ A 1,1 Lie superbialgebras.Finally, in section 6, we present our conclusions and sketch possible developments of this work.The classification of decomposable and indecomposable Lie superalgebras of the type (2|2) is left to Appendix A. Appendix B is also devoted to solutions of the super Jacobi and mixed super Jacobi identities for the C 3 ⊕ A 1,1 Lie superalgebra.

A brief review of super PL T-duality with spectators
Before proceeding to review the super PL T-duality on supermanifolds, let us recall the properties of Z 2 -graded vector space and also some definitions related to Lie superalgebras [20].A super vector space V is a Z 2 -graded vector space, i.e., a vector space over a field K with a given decomposition of sub-spaces of grade 0 and grade 1, The parity of a nonzero homogeneous element, denoted by |x|, is 0 (even) or 1 (odd)1 according to whether it is in V 0 or V 1 , namely, |x| = 0 for any x ∈ V 0 , while for any x ∈ V 1 we have |x| = 1.A Lie superalgebra G is a Z 2 -graded vector space, thus admitting the decomposition G = G B ⊕ G F , equipped with a bilinear superbracket structure [., .]: G ⊗ G → G satisfying the requirements of anti-supersymmetry and super Jacobi identity.If G is finite-dimensional and the dimensions of G B and G F are m = #B and n = #F , respectively, then G is said to have dimension (m|n).We shall identify grading indices by the same indices in the power of (−1), i.e., we use (−1) x instead of (−1) |x| , where (−1) x equals 1 or -1 if the Lie sub-superalgebra element is even or odd, respectively2 .
Since our main discussion will be related to super PL T-duality, it is worth mentioning that super PL T-duality is based on the concepts of Drinfeld superdouble [20,22,23], so it is necessary to define Drinfeld superdouble supergroup D. Following [11], a Drinfeld superdouble is simply a Lie supergroup D whose Lie superalgebra D admits a decomposition D = G ⊕ G into a pair of sub-superalgebras maximally isotropic with respect to a supersymmetric ad-invariant non-degenerate bilinear form < ., .>.The dimension of subsuperalgebras have to be equal.We furthermore consider G and G as a pair of maximally isotropic sub-supergroups corresponding to the sub-superalgebras G and G , and choose a basis in each of the sub-superalgebras as T a ∈ G and The basis of the two sub-superalgebras satisfy the commutation relations where f c ab and f ab c are structure constants of G and G , respectively.Noted that the Lie superalgebra structure defined by relation (2.2) is called Drinfeld superdouble D. The super Jacobi identity on D relates the structure constants of the two Lie superalgebras as [22] To continue, let us now consider a two-dimensional non-linear σ-model for the d field variables X M = (y i , x µ ), where x µ , µ = 1, • • • , dim G stand for the coordinates of Lie supergroup G acting freely from right on the target supermanifold M ≈ O × G, and Here we work in the standard light-cone variables on the worldsheet Σ, It should be remarked that the coordinates y i do not participate in the PL T-duality transformations and are therefore called spectators [24].The corresponding action has the form3 [13] (see also [9,10,24,25]) where R a ± are the components of the right-invariant Maurer-Cartan super one-forms which are constructed by means of an element g of G as follows: ) ib and ϕ ij may depend on all variables x µ and y i .Similarly we introduce another σ-model for the d field variables XM = (x µ , y i ), where xµ 's, µ = 1, • • • , dim G parameterize an element g of the dual Lie supergroup G, whose dimension is, however, equal to that of G, and the rest of the variables are the same y i 's used in (2.4).Accordingly, we introduce a different set of bases T a of the Lie algebra G , with a = 1, 2, • • • , dim G.We furthermore consider the components of the right-invariant Maurer-Cartan super one-forms on G in the following form In this case, the corresponding action has the form Notice that here one does not require any isometry associated with the Lie supergroups G and G.The σ-models (2.4) and (2.7) will be dual to each other in the sense of super PL T-duality [12,13] if the associated Lie superalgebras G and G form a Drinfeld superdouble which can be decomposed.There remains to relate the couplings E ab , ϕ ib and i and φij in (2.6).It has been shown that [13] the various couplings in the σ-model action (2.4) are restricted to be where the new couplings E 0 , F (1) , F (2) and F maybe functions of the spectator variables y i ab (g) are defined in the following way Finally, the relation of the dual action couplings to those of the original one is given by [13] Ẽab = E 0 + Π −1 ab , φ(1) a j = (−1) b Ẽab F bj (1) , 1) . (2.10) Analogously, one can define Π(g) and sub-matrices ã(g), b(g) by just replacing the untilded symbols by tilded ones.Equivalently the action (2.4) can be expressed as where E M N (X) as a second order tensor field on the supermanifold M is a composition of the supersymmetric metric G M N (X) and the anti-supersymmetric torsion potential B M N (X) (B-field) 4 .In the absence of spectator fields, the E M N (X) reduces to E µν (x).The relationship between E µν (x) and E ab (g) is given by the formula 4 Note that G M N and B M N are the components of the supersymmetric metric G and the antisupersymmetric tensor field B, respectively, We will assume that the metric M G N is superinvertible and its superinverse is shown by where the superscript "st" stands for supertransposition of the matrix.The condition of super PL symmetry of σ-models on the level of the Lagrangian is given by the formula [12] L where f bc a are structure coefficients of the dual Lie superalgebra G and V a µ are the components of left-invariant supervector fields on the Lie supergroup G.The superalgebras G and G then define the Drinfeld superdouble that enables to construct tensor E µν satisfying (2.13).
3 Abelian T-duality of BTZ metric coupled to two fermionic fields where G is identical to G acting freely on M .In this case, both the Lie supergroups G and G are Abelian, so we are dealing with a Abelian Drinfeld superdoubles of the type (4|4).Accordingly, by using (2.9) we conclude that both the Poisson superbracket Π ab (g) and Πab (g) are vanished.In what follows we shall show that the original model describes a string propagating in a space with the BTZ black hole metric coupled to two fermionic fields.

The original model
In order to construct the original σ-model on the M ≈ O × G we assume that the Abelian Lie supergroup G is parameterized by an element where (t, φ) and (ψ, χ) are the bosonic and fermionic coordinates, respectively; moreover, {T 1 , T 2 } and {T 3 , T 4 } stand for the bosonic and fermionic basis of Lie superalgebra G of G, respectively.Here, the coordinate of orbit O is represented by one spectator field y i = {r}.So, the coordinates of M are represented by X M = (r, x a ) = (r, t, φ; ψ, χ).In this case, the components of the right-invariant super one-forms are simply obtained to be R a ± = ∂ ± x a .Now, we need to determine the coupling matrices of the model.Let us choose the spectatordependent matrices in the following form where k(r), h 1 (r), h 2 (r) and F (r) are some arbitrary functions that depend on the spectator field {r} only.As mentioned above, since the G is Abelian, the Poisson superbracket Π ab (g) on G is zero; consequently, the various couplings in the action (2.4) can be reduced to E = E 0 , ϕ (1) = F (1) , ϕ (2) = F (2) and ϕ = F .Thus, by making use of the (2.4) and also the above results, the action of original σ-model is worked out to be By considering the definition of line element and B-field on a supermanifold as and then by comparing the actions (3.3) and (2.11) one gets In the σ-model context, the ultraviolet finiteness of the quantum version of the model is guaranteed by the conformal invariance of the model.To achieve this invariance at the oneloop level we must add another term containing the so-called dilaton field to the Lagrangian of action (2.11) 5 .The dilaton field Φ can be understood as an additional function on M that defines the quantum non-linear σ-model and couples to scalar curvature of the worldsheet.The conformal invariance of the model is guaranteed by vanishing of the socalled beta-function.At the one-loop level the equations for vanishing of the beta-function on a supermanifold read [13] 6 (−1) ) 5 In the super Abelian T-duality case, the formula of dual dilaton transformation is given by where "sdet" stands for superdeterminant of the matrix; moreover, ϕ is the dilaton that makes the original σ-model conformal up to the one-loop order and may depend on both supergroup and spectator coordinates.
6 Notation: Suppose that f be a differentiable function on R m c × R n a (R m c are subset of all real numbers with dimension m while R n a are subset of all odd Grassmann variables with dimension n), then, the relation between the left partial differentiation and right one is given by where the covariant derivatives − → ∇ M , Ricci tensor R M N and scalar curvature R7 are calculated from the metric G M N that is also used for lowering and raising indices, and field strength (torsion) corresponding to the B-field is defined by Now we want to obtain conditions under which the background given by (3.6) and (3.7) satisfies the beta-function equations (3.9)- (3.11).These equations possess a solution with the metric (3.6) and B-field (3.7) and also a constant dilaton field Φ = ϕ 0 if the following conditions hold: for some constant C 0 .Notice that here the cosmological constant is obtained to be Λ = −1/l 2 .Taking into account the above results, the scalar curvature of the metric (3.6) is R = −6/l 2 , and this is exactly the same as the case where fermion fields are absent.Furthermore, for the B-field, the corresponding field strength is given by where (3.15)

The dual model
In the same way, to construct out the dual σ-model on the supermanifold M ≈ O × G we parameterize Abelian Lie supergroup G with the bosonic coordinates ( t, φ) and fermionic ones ( ψ, χ) so that its element is defined as (3.1) by replacing untilded quantities with tilded ones.In order to obtain the dual σ-model to (3.3), we use the action (2.7).By taking into account (3.13) for (3.2) and then by inserting the result into (2.10) one can obtain the dual coupling matrices.They are then read where h 1 (r) is given by relation (3.13).Finally, by using (3.16) and (2.7) one can construct out the action of dual model.The line element and anti-supersymmetric field corresponding to the dual action may be expressed as The components of metric are ill defined at the region r = J 2 M −1 2 .We can test whether there is true singularity by calculating the scalar curvature, which is 2 is a true curvature singularity, so that it cannot be removed by a change of coordinates.Moreover, the horizons of the dual metric are at the same location as the metric of original model.We also find that the field strength corresponding to the B-field is where H 144 is given by relation (3.15).The dual dilaton field that makes the model conformal is obtained by using equation (3.8) to be ). (3.20) Thus, the dual metric with the field strength (3.19) and dilaton field (3.20) satisfy oneloop beta-function equations (3.9)-(3.11) in a way that the cosmological constant must be Λ = −1/l 2 .

Abelian T-duality with Abelian Lie supergroup of the type (1|2)
In order to study Abelian T-duality of the BTZ black hole metric coupled to two fermionic fields, there is a possibility that we can perform the dualizing on Abelian Lie supergroups G and G of the type (1|2); moreover, the spectator fields are considered to be y i = (r, t).As we will see, the original model will be the same as (3.3).Here we find a new Abelian target space dual for the action (3.3).In fact, the result of duality will be different from those of (3.17) and also (3.18).In order to construct out the original σ-model, we parameterize an element of G as where φ is only the bosonic coordinate, while (ψ, χ) denote fermionic coordinates.Also, {T 1 } and {T 2 , T 3 } are bosonic and fermionic generators of G, respectively.In this case, the dualizing is performed on the coordinates (φ; ψ, χ), so it is more appropriate to choose the spectator-dependent matrices as follows: where h 1 (r) is given by relation (3.13).Then, using the fact that Π(g) = 0 and utilizing formulae (2.8) together with (2.4), the original σ-model is obtained to be the same as (3.3) provided that one also considers relation (3.13).
The corresponding coupling matrices to the dual model can be obtained by making use of relations (2.10) and (3.22).They are then read Thus, employing (2.7) the dual background can be cast in the form where h 1 (r) is given by relation (3.13).Let us now enhance and clarify the structure of the dual spacetime.As it is seen, the dual metric has an apparent singularity.Calculating the scalar curvature, which is one should test whether there is a true singularity, and we thus see that r = 0 is a true singularity.In addition to the above, one immediately verifies the field equations (3.9)- (3.11) for the dual background with the dilaton field Φ = ϕ 0 − log r; moreover, to satisfy the equations we must have a cosmological constant as in the original model, i.e., Λ = −1/l 2 .Notice that the dual dilaton field satisfies formula (3.8).
In summary, we obtained two Abelian duals for the BTZ metric coupled to two fermionic fields.When the dualizing was implemented by the Abelian Lie supergroup of the type (2|2), we found that the singularity of the dual metric has appeared at the point r = J 2 M −1 2 , whereas when we deal with a (1|2)-dimensional Abelian Lie supergroup, we encounter a singularity at the origin.Moreover, in both cases of the dual metrics, horizons are at the same location as the metric of original model.

Non-Abelian T-duality of BTZ vacuum metric coupled to two fermionic fields
Our goal in this section is that to calculate non-Abelian target space duals (here as super PL T-duality on a semi-Abelian superdouble) of the BTZ vacuum metric when is coupled to two fermionic fields.The Lie supergroup acting freely on the original target supermanifold of Tdual σ-models is considered to be one of the non-Abelian Lie supergroups of the type (2|2) whose Lie superalgebras were classified by Backhouse in [16] as mentioned in Introduction.
The classification of decomposable and indecomposable Lie superalgebras of the type (2|2) has been displayed into six disjoint families in Tables 2 to 7 of Appendix A. In this section we perform the duality on some Lie supergroups whose Lie superalgebras are considered to be the Family IV and (2A 1,1 + 2A ) 0 of Family V .These Lie superalgebras possess two bosonic generators {T 1 , T 2 } along with two fermionic ones {T 3 , T 4 }8 .Note that in order to satisfy the dualizability conditions, the Lie supergroup of dual target supermanifold must be chosen Abelian.In fact, we are dealing with semi-Abelian superdoubles which are non-isomorphic as Lie superalgebras in each of the models.
is defined by the following nonzero (anti-)commutation relations9 : where {T 1 , T 2 ; T 3 , T 4 } generate the (C 1 1 + A ), and { T 1 , T 2 } and { T 3 , T 4 } are the respective bosonic and fermionic generators of the I (2|2) .In order to calculate the components of the right-invariant super one-forms R a ± on the (C 1 1 + A) Lie supergroup we parameterize an element of the supergroup as where (x 1 , x 2 ) are bosonic fields, while (ψ, ξ) stand for fermionic ones.From now on, we will use this notation for bosonic and fermionic fields.Now, one may use equation (2.5) to get Let us now choose the spectator-dependent background matrices as where α 0 is an arbitrary constant.Also, h(y) and n(y) are functions of the variable {y} only.
In the case of the super non-Abelian T-duality, it can be deduced from formula (2.9) that the Poisson superbracket Π(g) vanishes provided that the dual Lie supergroup is Abelian.Thus, by using these and also employing (2.4) one can get the action of original model such that its corresponding line element and B-field are given by The scalar curvature of the metric is where prime denotes differentiation with respect to the argument {y}.It is straightforward to verify that the only non-zero component of the field strength corresponding to B-field (4.6) is H yχχ = α 0 n ′ (y).We are now interested in the satisfaction of the conformal invariance conditions of the model.In order to guarantee the conformal invariance of the model, at least at the one-loop level, one must show that the background of model including (4.5) and (4.6) satisfies the vanishing of the beta-function equations.Before we proceed to investigate these conditions further, let us briefly comment on the formula of dilaton transformation for the case of PL T-duality.In [18], von Unge showed that the duality transformation must be supplemented by a correction that comes from integrating out the fields on the dual group in path-integral formulation so that it can be absorbed at the one-loop level into the transformation of the dilaton field.Following [18], in the super PL T-duality case, these transformations may be expressed as where ϕ (0) may depend on both supergroup and spectator coordinates.Note that in the case of super non-Abelian T-duality, formula (4.8) becomes simpler, because we have E = E 0 .
In the present example, sub-matrix a b a (g) can be obtained from the first equation of (2.9), then, one can easily find that sdet a a b (g) = 1.Consequently, it follows from (4.8) that Φ = ϕ (0) such that the beta-function equations are satisfied with a constant dilaton field, It can be useful to comment on the fact that the metric (4.5) along with h(y) obtained in (4.10) is flat in the sense that its scalar curvature and Ricci tensor vanish.On the other hand, one may use the coordinate transformation then, the background of model, that is conformally invariant up to one-loop order, reduces to (4.12) Indeed, the above metric is nothing but the BTZ vacuum metric which has been coupled to two fermionic fields (ψ, χ).A remarkable point about the metric in (4.12) is that unlike 2 + 1-dimensional BTZ vacuum metric (the J = M = 0 case of (1.1)) which has a constant negative scalar curvature, it has vanishing scalar curvature.In fact, by coupling the fermionic fields to the metric, the scalar curvature undergoes a change.It is then straightforward to compute the corresponding dual background.By considering the coupling matrices in (4.4) and by employing (2.10) one can show that the dual couplings take the following form in which the Poisson superbracket Πab can be obtained from equation (2.9) by replacing untilded quantities with tilded ones.To this end, we parameterize the dual Lie supergroup I (2|2) with coordinates xa = (x 1 , x2 ; ψ, ξ) so that its elements are defined as in (4.2).We then find Putting (4.14) and E 0 from (4.4) together into (4.13) and using the fact that the components of the right-invariant super one-forms on I (2|2) are R±a = ∂ ± xa , the background of dual σ-model is obtained to be where ∆ = x2 2 − 1 4 e −4y .The metric (4.15) is flat in the sense that its scalar curvature vanishes.Looking at the field equations (3.9)-(3.11)one verifies the conformal invariance of dual model with a constant dilaton field and vanishing cosmological constant.Notice that the dilaton field making the dual model conformal can be also obtained from (4.9) which gives a constant value.Finally, it is seen that in this case, the super PL T-duality transforms rather extensive and complicated background given by equations (4.15) and (4.16) to much simpler form such as (4.12).
Before closing this subsection let us highlight some point concerning these models.As discussed in Introduction, the BTZ metric (1.1) (even for the vacuum state) satisfies the one-loop beta-function equations with a non-vanishing field strength, while here, as can be clearly seen, the original model with background (4.12) satisfies the one-loop beta-function equations in both cases of absence (α 0 = 0) and presence (α 0 ̸ = 0) of the field strength.

Non-Abelian target space dual starting from the (C 3 + A) Lie supergroup
The (C 3 + A ) Lie superalgebra10 is in turn interesting in the sense that its commutator of B-B is zero.Its (anti-)commutation relations are given in Table 3 of Appendix A. The Lie superalgebra of semie-Abelian Drinfeld superdouble (C 3 + A ) , I (2|2) obeys the following (anti-)commutation relations where T a 's generate the (C 3 + A ), while T a 's stand for the I (2|2) .The expression for a generic element of the (C 3 + A) Lie supergroup can be written as g = e ψT 3 e x 1 T 1 e x 2 T 2 e χT 4 .(4.18) In order to construct the original σ-model we need to calculate the right-invariant one-forms on the (C 3 + A).To this end, one should employ (4.17) and (4.18) and then use equation (2.5).The result is In what follows, the dualizing is performed on the (C 3 +A) Lie supergroup, and the spectator field is still displayed by {y}.Here we choose the spectator-dependent background matrices in the following form where a i (y), i = 1, 2, 3 and b(y) are some functions that may depend on the coordinate {y}.Now, by applying formulae (2.4) and (2.8) one can obtain the background of the original σ-model, giving us In order to obtain dilaton field that supports the background of the model one may apply formula (4.8).First, by using the first equation of (2.9) and (anti-)commutation relations of the (C 3 + A ) given in (4.17) one immediately concludes that sdet a a b (g) = 1.Hence, from (4.8) we find that Φ = ϕ (0) .As we will see, the dilaton field that makes the original σ-model conformal is found to be Φ = C 0 + y, so we choose ϕ for some constant β 0 .Inserting (4.23) into (4.21)one immediately finds that the scalar curvature of the metric is non-zero, and it has a negative constant value as R = −16/l 2 .It can be easily shown that under the coordinate transformation the background becomes Again, the above metric is nothing but the BTZ vacuum metric coupling to two fermionic fields.Note that here the terms related to the coupling of fermions to the metric are different from those of metric in (4.12).Analogously, we here can have a conformal background in both cases of absence (β 0 = 0) and presence (β 0 ̸ = 0) of the field strength.
To continue, we obtain a new dual background for the BTZ vacuum metric coupled to two fermionic fields.Employing (2.10), (4.20) and (4.23) one can get the dual couplings.In this case, the Poisson superbracket and non-zero background matrices read where we have here assumed that β 0 = 0. Finally, with the help of (2.7) the line element and the tensor field B take the following forms The scalar curvature of the dual metric is also constant and equal to the same value from the metric of original model.Before proceeding to investigate the conformal invariance conditions of the dual background, let us evaluate the dual dilaton.First we find that sdet( Ẽab ) = −e −4y and sdet(ã a b (g)) = 1, then it follows from (4.9) that Φ = ϕ (0) − 2y, and hence we get the dilaton by remembering that ϕ (0) = C 0 + y which gives the final result One can check that the field equations (3.9)- (3.11) are satisfied for the metric (4.27), the tensor field (4.28) and the dilaton field (4.29) together with a zero cosmological constant.This means that the dual background is also conformally invariant up to one-loop order.
In order to get more insight of the dual metric one may use the transformation then, the dual background takes the following form As can be clearly seen, the bosonic part of the metric remains unchanged under the non-Abelian T-duality.This means that the dual model metric also describes the BTZ vacuum metric coupled to two fermionic fields.
Starting with the superdouble (C 3 + A ), I (2|2) we thus illustrated a concrete example of the super PL T-duality and found another non-Abelian target space dual for the BTZ vacuum metric coupled to two fermionic fields in the form of equations (4.27)-(4.29).

Non-Abelian target space dual starting from the C
Here the Lie supergroup of target space supermanifold M ≈ O × G is considered to be C 3 ⊕ A 1,1 whose Lie superalgebra is decomposable and it belongs to Family III.According to Table 4 of Appendix A, only the commutator of B-F for this superalgebra is non-zero.The Lie superalgebra of semie-Abelian Drinfeld superdouble C 3 ⊕ A 1,1 , I (2|2) is defined by the following (anti-)commutation relations The parametrization of a general element of C 3 ⊕ A 1,1 we choose as in (4.18), giving us By a convenient choice of the spectator-dependent matrices in the form of and then with the help of relations (2.4) and (2.8) one can construct the action of original σ-model on the C 3 ⊕ A 1,1 .The corresponding background including the line element and zero B-field is given by The metric is flat in the sense that both its scalar curvature and Ricci tensor vanish.The bosonic part of the above metric (the first three terms) is similar to that of metric (4.21), therefore, one may use the transformation (4.24) to conclude that the metric in (4.35) is nothing but the BTZ vacuum metric coupled to two fermionic fields.In addition, the dilaton field that supports the background is found from (4.8) to be Φ = ϕ (0) = C 0 .Using these, one verifies the one-loop beta-function equations with zero cosmological constant.Similar to the construction of dual σ-models in the previous subsections, the Lie supergroup G of dual supermanifold is here assumed to be Abelian.With this in mind, one finds that the only non-zero component of the super Poisson structure on the dual Lie supergroup is Πx 1 χ = − ψ.Moreover, using the first equation of (2.10) one gets sdet( Ẽab ) = −1.We will use this in calculating the dual dilaton field in the next.The corresponding elements to the dual model can be obtained by making use of relations (4.34) and (2.10).They are then read By transforming coordinate y → −y (without changing the rest of the coordinates) it can be easily shown that the dual metric turns into the same metric as the original model in (4.35).Hence, it is said that the metric is self-dual.In order to investigate the conformal invariance conditions of the dual background one may use (4.9) to obtain Φ = C 0 .Using these, one verifies the field equations (3.9)-(3.11)for dual background (4.36) with zero cosmological constant.
4.4 Non-Abelian target space dual starting from the D 10

Lie supergroup
There is a possibility that we can also perform the dualization on the D 10 Lie supergroup to construct other non-Abelian target space dual for the BTZ vacuum metric coupling to two fermionic fields.The corresponding Lie superalgebra to this supergroup belongs to Familiy IV of Table 5, where only the commutator of F-F among all commutators is zero.In this subsection we deal with T-dual σ-models constructing out on the semie-Abelian Drinfeld superdouble D 10 p , I (2|2) whose Lie superalgebra is defined by the following non-zero Lie superbrackets To obtain the right-invariant super one-forms we parameterize an element of the D 10 p as in (4.18).We then find In this way, the BTZ vacuum metric coupled to two fermionic fields may be yielded from the original σ-model on the superdouble D 10 p , I (2|2) if one considers the spectator-dependent matrices as in (4.4) by setting α 0 = 0. Using these and employing (2.4) we then get Notice that the bosonic part of the metric is also similar to that of metric (4.5).Here the field strength is absent, and hence with a constant dilton field one guarantees the conformal invariance of the background (4.39) provided that we have h(y) = e −2y as well as p = ±1/2.It follows from this result that the metric is flat in the sense that its scalar curvature vanishes.In the following we will focus on the p = −1/2 case of the superdouble only.
Before proceeding to construct the dual background, let us calculate the dual couplings.Utilizing relation (2.9) for untilded quantities and also employing (4.37) we get then, the dual coupling matrices are worked out where ∆ ± = x2 ± 1 2 e −2y and ∆ = ∆ + ∆ − .We need to have the superdeterminant of matrix Ẽab .One immediately finds that sdet( Ẽab ) = e −4y /∆.Finally, the corresponding elements to the dual model can be obtained by making use of relations (4.40) and (4.41) together with (2.7), giving us The scalar curvature of the dual metric is also zero.It seems that under the non-Abelian T-duality the scalar curvature has been restored from the dual model to the original one.
From equation (4.9) and also from the superdeterminant of matrix Ẽab that found above, one gets the dual dilaton field, giving Φ = C 0 − 2y − 1 2 log |∆|.Unfortunately, the field equations (3.9)-(3.11)do not satisfy with the metric (4.42),B-field (4.43) and the above dilaton.In fact, unlike the original model, the dual model cannot be conformally invariant up to one-loop order.
Let us discuss below the reason for not preserving conformal invariance under non-Abelian T-duality for this example.In Ref. [24], Sfetsos proved the classical canonical equivalence to the σ-models related by PL T-duality.The canonical transformations are essentially classical and their equivalence can hold in some special cases but it fails in most cases.However, checking the equivalence by studying conformal invariance up to one-loop order is important.On the other hand, it has been shown that [28] a sufficient condition for the invariance of the reduced string effective action under PL T-duality is the vanishing of the traces of the structure constants of each Lie algebra constituting the Drinfeld double.Notice that the field equations (3.9)-(3.11)can be interpreted as field equations for G M N , B M N and Φ of the low energy string effective action.The string tree level effective action on a d-dimensional supermanifold M for these background fields is given by [13] where G stands for the superdeterminant of M G N .In the case of the above example, one first finds the supertrace (str) of adjoint representations , giving us, str(X 1 ) = −1, str(X i ) = 0, i = 2, 3, 4.Then, by calculating the effective Lagrangians corresponding to both original and dual models, we find that the dual Lagrangian is not invariant under super PL T-duality transformation and therefore both Lagrangians are not equal.Obviously, this anomaly is due to the non-vanishing traces of the structure constants of the superdouble D 10 p , I (2|2) .Moreover, in this example one can show that the integration weights √ −Ge −2Φ and − Ge −2 Φ are not equal.The reason behind this may be due to the particularity of our model.There is a possibility of absorbing the anomalous terms into dilaton shift which is the same as a diffeomorphism transformation.We note that the equations (4.8) and (4.9) are only transformations which lead to a proportionality between the integration weights √ −Ge −2Φ and − Ge −2 Φ [28].
Notice that under the transformation (4.24), the background of action (equation (4.35)) describes the BTZ vacuum metric coupling to two fermionic fields as was mentioned in subsection 4.3.In addition, it was shown there that the dilton field guaranteeing the conformal invariance of the model is The Lie superalgebra of the superdouble .i obeys the following set of non-trivial (anti-)commutation relations: where U a 's generate the C 3 ⊕ A 1,1 , while the bases Ũ a 's stand for the C 3 ⊕ A 1,1 .i.Also, β is a non-zero constant.Here and henceforth we denote the bosonic bases by and fermionic ones by (U 3 , U 4 , Ũ 3 , Ũ 4 ).The isomorphism between the superdoubles D = (C 3 ⊕A 1,1 , I (2|2) ) and D U = (C 3 ⊕A 1,1 , C 3 ⊕A 1,1 .i) is given by the following transformation: (5.12) Comparing relations (5.2) and (5.12) one can find the sub-matrices F a c , G ac , H bc and K b c .Then, using these and constant matrix E 0 of (4.34) together with formulae (5.5) and (5.6) we can find the matrices M ab and N a b leading to We use the parametrization of a general group element of C 3 ⊕ A 1,1 as in (4.18), but with T a 's replaced by U a 's.Then, one utilizes formula (2.9) to calculate the matrices a(g U ) and b(g U ) for this decomposition, giving us (5.17) By taking into consideration the scalar curvature of the metric which is one verifies the one-loop conformal invariance conditions of the background (5.17) with dilaton field (5.16) and a zero cosmological constant.Note that the metric of (5.17) is ill defined at the region y = − ln |β|.This singularity also appears in the scalar curvature, and hence, y = − ln |β| is a true singularity.In fact, we have shown that the solution of BTZ vacuum metric coupled to two fermionic fields with no curvature singularity is, under the T-plurality, related to a solution with a curvature singularity.This situation was not seen in the previous section regarding to super non-Abelian T-duality.
From the Manin supertriples of (5.9) we see that 2A ) 0 .ias Lie superalgebras are isomorphic, and so one can find an isomorphism transforming the Lie multiplication of D into that of D U and preserving the canonical form of the bilinear form < ., .> such that they belong to the same Drinfeld superdouble.The isomorphism transformation of Manin supertriples between (C 3 ⊕ A 1,1 , I (2|2) ) and C 3 ⊕ A 1,1 , (2A 1,1 + 2A ) 0 .i is given by Here, {U a , Ũ b } are elements of bases of the superdouble C 3 ⊕ A 1,1 , (2A 1,1 + 2A ) 0 .iwhose Lie superalgebra is defined by the following non-zero (anti-)commutation relations: where γ is a non-zero constant.In this case, the matrix a b a (g U ) is the same form as in (5.14).Calculating the matrices M ab , N a b and Π ab (g U ) for the decomposition (5.19) we then get leading to a background .22)In order to evaluate the total dilaton contribution we have to use ϕ (0) = C 0 which gives the final result Since the first sub-superalgebra is C 3 ⊕ A 1,1 , the right-invariant super one-forms are the same forms as in (4.33).Finally, using (5.3) one can construct the action of the model on the superdouble C 3 ⊕ A 1,1 , (2A 1,1 + 2A ) 0 .iwhose background is dψ ∧ dχ. (5.24) One immediately finds that the scalar curvature of the metric is Again we encounter a true singularity that appears here at the region y = − ln |γ/2|.Looking at the one-loop beta-function equations one verifies the conformal invariance conditions of the background (5.24) with dilaton field (5.23) and a zero cosmological constant.
.i : As the last example of this section we ask the question of super PL T-plurality of the BTZ vacuum metric coupled to two fermionic fields with respect to the superdouble C 3 ⊕ A 1,1 , (A 1,1 + 2A ) 2 ⊕ A 1,1 .iwhose Lie superalgebra is defined by the following non-zero Lie superbrackets: where γ is a non-zero constant.The isomorphism transforming the Lie multiplication of D = (C 3 ⊕A 1,1 , I (2|2) ) into that of D U = C 3 ⊕A 1,1 , (A 1,1 +2A ) 2 ⊕A 1,1 .iand preserving the canonical form of the bilinear form < ., .>, is given by (5.27) After calculating the sub-matrices F a c , G ac , H bc and K b c we employ formulae (5.5) and (5.6) to obtain the matrices M ab and N a b , in such a way that one must also use constant matrix E 0 of (4.34).In addition, the matrix Π(g U ) is obtained for this decomposition by using (2.9) and (5.7).It then follows from (5.4) that Unlike the previous two examples, in this case the metric is flat in the sense that both its scalar curvature and Ricci tensor vanish.In addition, it is interesting to note that the above metric is exactly the same as the metric of action (5.10) on the superdouble (C 3 ⊕ A 1,1 , I (2|2) ).Here, the dilaton obtained from equation (5.8) is Φ = ϕ (0) .Finally we get the dilaton by remembering that ϕ (0) = C 0 which gives the final result Φ = C 0 .

Conclusion
In summary, we have studied Abelian T-duality, non-Abelian T-duality and T-plurality of the BTZ metric coupled to two fermionic fields.In studying Abelian T-duality, we have obtained two duals for the BTZ metric when is coupled to two fermionic fields.When the dualizing was implemented by the Abelian Lie supergroup of the type (2|2), we found that the singularity of the dual metric is appeared at the region r = J 2 M −1 2 , whereas when we dealt with a (2|1)-dimensional Abelian Lie supergroup, we encountered a singularity at the origin.Moreover, in both cases of the dual metrics, horizons were at the same location as the metric of original model.
Using super PL T-duality approach in the presence of spectator fields we have constructed some non-Abelian T-dualizable σ-models on some of the semi-Abelian Drinfeld superdoubles, in such a way that we have dealt with (2|2)-dimensional non-Abelian Lie superalgebras (C 1 1 + A ), (C 3 + A ), C 3 ⊕ A 1,1 , D 10 p=±1/2 , and (2A 1,1 + 2A ) 0 as the first sub-superalgebras of the superdoubles.By a convenient choice of the spectator-dependent background matrices we showed that the background of original σ-models describes a string propagating in a target space with the BTZ vacuum metric coupling to two fermionic fields, in such a way that a new family of the solutions to supergravity equations was found in both cases of absence and presence of the field strength.Accordingly, we think that the choice of spectator-dependent matrices plays a key role in the structure of our models.In some cases we showed that the dual backgrounds also yields the BTZ vacuum metric coupled to two fermionic fields.

4 3 10 4
review of super PL T-duality with spectators Abelian T-duality of BTZ metric coupled to two fermionic fields 7 3.1 Abelian T-duality with Abelian Lie supergroup of the type (2|2) T-duality with Abelian Lie supergroup of the type (1|2) Non-Abelian T-duality of BTZ vacuum metric coupled to two fermionic fields 12 4.1 Non-Abelian target space dual starting from the (C 1 1 + A) Lie supergroup 12 4.2 Non-Abelian target space dual starting from the (C 3 + A) Lie supergroup 15 4.3 Non-Abelian target space dual starting from the C 3 ⊕ A 1,1 Lie supergroup 18 4.4 Non-Abelian target space dual starting from the D 10 only.In equation (2.8), the Poisson superbracket Π(g) is Π ab (g) = (−1) c b ac (g) (a −1 ) b c (g) so that sub-matrices a b a (g) and b

3. 1
Abelian T-duality with Abelian Lie supergroup of the type (2|2) In this subsection we construct explicitly a pair of Abelian T-dual σ-models on 4 + 1dimensional supermanifolds M and M as the target superspaces.The original model is built on the supermanifold M ≈ O × G, where G = I (2|2) is a four-dimensional Abelian Lie supergroup of the type (2|2) acting freely on M while O is the orbit of G in M .The target superspace of the dual model is the supermanifold M

4. 1
Non-Abelian target space dual starting from the (C 1 1 + A) Lie supergroupIn what follows we shall obtain the BTZ vacuum metric coupled to two fermionic fields from a T -dualizable σ-model constructing on a five-dimensional target supermanifold M ≈ O×G where G as the first sub-supergroup of Drinfeld superdouble is considered to be the (C1  1 +A) acting freely on M , while O as the orbit of G in M is a one-dimensional space with the coordinate y i = {y}.The second sub-supergroup, G = I (2|2) , acting freely on the dual supermanifold M ≈ O × G is assumed to be Abelian of the type (2|2).Hence, the super PL T-duality reduces to the super non-Abelian T-duality.One can find the (anti-)commutation relations of the (C 1 1 + A ) Lie superalgebra in Table 2 of Appendix A. As mentioned in Introduction section, having Drinfeld superdoubles one can construct super PL T-dual σmodels on them.The Lie superalgebra of the Drinfeld superdouble which we refer to as and vanishing cosmological constant if functions h(y) and n(y) (0) = C 0 + y.Looking at the one-loop conformal invariance conditions, the field equations (3.9)-(3.11)are then satisfied with the metric (4.21),B-field (4.22) and a zero cosmological constant together with the aforementioned dialton field if one considers a 1 (y) = −a 2 (y) = e y , a 3 (y) = e −y , b(y) = β 0 (1 − 1 2 e −2y ), (4.23) (4.46) 5.2 Super PL T-plurality with respect to the C 3 ⊕ A 1,1 Lie supergroup So far, there has been no classification of the Lie superbialgebras structures on the C 3 ⊕A 1,1 .
2|2) : In order to obtain a possible conformal duality chain on the isomorphic Drinfeld superdoubles of (5.9) we begin with D = (C 3 ⊕ A 1,1 , I (2|2) ) whose Lie superalgebra has been given in (4.32).The corresponding right-invariant super one-forms and constant matrix E 0 illustrated in subsection 4.3.Accordingly, the action of σ-model on the C 3 ⊕ A 1,1 in the form (5.1) is given by

. 14 )
Plugging (5.14) into (5.7)one gets the Poisson superbracket Π ab (g U ), which the result is exactly similar to sub-matrix b ab (g U ) of equation(5.14).These results give us the backgroundE ab (g U ) = y) = β 2 − e −2y .In order to calculate the corresponding dilaton field we findthat sdet( a E b (g U )) = e −2y /∆(y), sdet( a M b (g U )) = −1 and sdet(a b a (g U )) = 1.Then, the dilaton is obtained by making use of (5.8), and by remembering that ϕ (0) = C 0 one gets the final result in the formΦ U = C 0 − 1 2 log β 2 e 2y − 1 .(5.16) Using (5.15) and the right-invariant super one-forms on the C 3 ⊕ A 1,1 (equation (4.33)) one can write the action of model on the superdouble (C 3 ⊕ A 1,1 , C 3 ⊕ A 1,1 .i) in the form of (5.3).Finally, background including the line element and B-field is, in the coordinate basis, read off