Poisson-Lie T-plurality for WZW backgrounds

Poisson-Lie T-plurality constructs a chain of supergravity solutions from a Poisson-Lie symmetric solution. We study the Poisson-Lie T-plurality for supergravity solutions with $H$-flux, which are not Poisson-Lie symmetric but admit non-Abelian isometries, $\mathcal{L}_{v_a}g_{mn}=0$ and $\mathcal{L}_{v_a}H_3=0$ with $\mathcal{L}_{v_a}B_2\neq 0$. After introducing the general procedure, we study the Poisson-Lie T-plurality for two WZW backgrounds, the AdS$_3$ with $H$-flux and the Nappi-Witten background.


Introduction
Abelian T -duality, which is a symmetry of string theory, is based on the existence of a set of commuting vector fields v m a satisfying the Killing equations At least in supergravity, this symmetry continues to hold even when the Killing vector fields form a non-Abelian algebra [v a , v b ] = f ab c v c , and this is known as non-Abelian T -duality [1].
There is a further extension of the T -duality, called the Poisson-Lie (PL) T -duality [2,3] or T -plurality [4]. We can perform the PL T -duality when the background admits a set of vector fields v m a satisfying the condition for the PL symmetry, Here f a bc (= −f a cb ) are called the dual structure constants, and if they are absent, this condition reduces to the Killing equations, £ va g mn = £ va B 2 = 0 . Then the PL T -duality reduces to the traditional non-Abelian T -duality. However, the condition £ va B 2 = 0 is stronger than £ va H 3 = 0 of Eq. (1.1), and naively, not all of the traditional non-Abelian T -duality can be realized as a PL T -duality. In this short paper, we discuss how to perform the PL T -plurality in backgrounds with non-Abelian isometries, £ va g mn = 0 and £ va H 3 = 0 but £ va B 2 = 0 .
For the sake of clarity, let us comment on some of the related earlier works. There are at least two approaches to perform the PL T -duality in a H-fluxed background. The first is taken in [5,6], where Wess-Zumino-Witten (WZW) backgrounds are constructed as PL symmetric backgrounds satisfying the condition (1.2). For a given WZW background, how to find the set of vector fields v m a and the dual structure constants f a bc is quite non-trivial, but once we find these, we can perform the PL T -duality by following the standard procedure. The second approach is taken in [7,8], where the non-trivial H-flux is produced by the spectator fields and the internal part satisfies the usual Killing equations £ va E mn = 0 . Namely, the WZW backgrounds are realized as PL symmetric backgrounds associated with semi-Abelian doubles (i.e., f a bc = 0). Again, once we found such a realization, we can perform the PL T -duality by using the standard procedure of the PL T -duality with spectator fields. Now we explain our approach. For convenience, we use the language of double field theory (DFT) [9][10][11][12], whose flux formulation [13] is especially useful here. In DFT, the supergravity fields in the NS-NS sector are packaged into the generalized metric and the DFT dilaton where I, J = 1, . . . , 2D and m, n = 1, . . . , D . Similar to the standard setup to study the PL T -duality in DFT [14], we assume that the generalized metric has the form (1.4) whereĤ AB is constant and E I A is the inverse of the generalized frame fields E A I satisfying Here, [·, ·] D denotes the D-bracket or the generalized Lie derivative in DFT and F AB C are the structure constants with components For the DFT dilaton, for simplicity, we assume the absence of the dilaton flux, In the usual setup of the PL T -duality/plurality in DFT [14][15][16], the components of the structure constants are supposed to have the form Then, by finding a new set of generalized frame fields E ′  Manin triple corresponds to a different target geometry but it turns out that many of the Manin triples correspond to the same AdS 3 solution with H-flux. This kind of self-duality under the PL T -duality has been observed in [8,18] and this may represent the high symmetry of the AdS 3 background. We also find that some Manin triples correspond to a flat space with a linear dilaton and some Manin triples correspond to a known AdS 3 solution of the generalized supergravity equations of motion [19,20].
For the NW model, the Drinfel'd double is eight-dimensional and the Manin triples have not been classified. We find several inequivalent Manin triples and construct the corresponding backgrounds. Again, we find that two Manin triples correspond to the same NW background.
Other Manin triples correspond to a kind of T -fold [21] which does not allow for the description in terms of the standard fields in the NS-NS sector. We also perform the Yang-Baxter (YB) deformation [22] of the NW background by using a classical r-matrix satisfying the modified classical YB equations (CYBE). As a result, we find a one-parameter family of solutions which contains the NW background and the flat Minkowski spacetime as specific cases.
This paper is organized as follows. In section 2, we explain how to construct the generalized frame fields E A I and the DFT dilaton d satisfying Eqs. (1.5) and (1.7) in WZW backgrounds.
We also explain the procedure of the PL T -plurality. In section 3, we study the PL T -plurality of the SL(2) WZW model. In section 3, we study the PL T -plurality and the YB deformation of the NW background. Section 5 is devoted to conclusions and discussions.

WZW background
Let us consider a group G associated with a Lie algebra, [T a , T b ] = f ab c T c . We define the left-/right-invariant 1-forms and their duals as Then the WZW model can be defined as where the metric and the 3-form field strength are g mn ≡ĝ ab ℓ a m ℓ b n =ĝ ab r a m r b n , Hereĝ ab is a non-degenerate invariant metric and f abc ≡ f ab dĝ dc is totally antisymmetric.
Using e m a e n b e p c H mnp = f abc , we can easily check that the frame fields {E A } = {E a , E a } satisfy the algebra (1.5) with the fluxes F AB C given by Eq. (1.6) and Here the D-bracket or the generalized Lie derivative in DFT is defined as

7)
2 If a B-field satisfies dB2 = 1 3! H abc r a ∧r b ∧r c for a general skew-symmetric constants H abc , the EA I defined in Eq. Regarding the DFT dilaton, the requirement (1.7) reduces to under our assumption ∂ ∂xm = 0 . By using ∂ n e n b = ℓ a m e n b ∂ n v m a = −e n b ∂ n ln|det(ℓ a m )| , which follows from ℓ a m £ va e m b = 0 , the DFT dilaton can be found to have the form the Lie algebra of a Drinfel'd double. Then we parameterize the group element, for example, as g(x) = e x a Ta and define the right-/left-invariant 1-forms and their duals. We also define They are known to satisfy the desired properties (1.5) and (1.7) [14,15], and using these fields, we obtain the dual background, in the same way as the standard PL T -plurality. 3 In order to make a solution of a ten-dimensional supergravity, we may consider a product of the WZW background and a certain space with coordinates y µ , called the spectator fields.
In the examples to be studied in this paper, the generalized metric is given by a direct sum (2.14) The fields H s M N (y) and d s (y) are invariant under the PL T -pluralities and only H IJ (x) and d(x) are transformed in the following discussion.

SL(2) WZW model
Here we consider the SL(2) WZW model, where the target geometry can be identified with the AdS 3 with H-flux The Riemann curvature tensors is R mnpq = −l −2 (g mp g nq − g mq g np ) and the Ricci scalar is R = −6/l 2 . This becomes a ten-dimensional supergravity solution by adding spectator fields (which are not affected by T -dualities) ds 2 S 3 ×T 4 = l 2 4 dθ 2 + sin 2 θ dφ 2 + dψ + cos θ dφ 2 + dy 2 1 + dy 2 2 + dy 2 3 + dy 2 4 , Since e −2 ds(y) is just the volume element, the dilaton Φ can be found as In [5], this WZW background was reproduced as a PL symmetric background by using a six-dimensional Drinfel'd double. Six-dimensional Drinfel'd doubles and the Manin triples have been classified in [23] and it is known that there are 22 Drinfel'd doubles. According to this classification, the Drinfel'd double used in [5] is called DD2 (see [24] for the notation).
This Drinfel'd double can be decomposed into four Manin triples [23] DD2: and it was found that the SL(2) WZW background is associated with (5.i|8|b) [5]. Then the PL T -dual model, whose Manin triple is (8|5.i|b), was found to be a constrained sigma model [5]. In this section, we find that the target geometry of this dual model is a kind of non-Riemannian backgrounds (see [25][26][27] for details of non-Riemannian backgrounds). Then we further perform PL T -plurarities and obtain the backgrounds associated with the other two Manin triples, (6 0 |5.iii|b) and (5.iii|6 0 |b).
After completing the orbit of DD2, we consider the PL T -pluralities based on our approach.
We start with a flux algebra with the SL(2) algebra f ab c and the H-flux H abc = 0 . Then we identify that this six-dimensional Lie algebra corresponds to a Drinfel'd double called DD7.
The DD7 can be decomposed into six Manin-triples [5] DD7: and we identify the corresponding backgrounds.

PL T -plurality for DD2
Here we review the result of [5] and then complete the orbit of DD2 described in Eq. (3.3).

Manin triple (5.i|8|1)
The Lie algebra of the Drinfel'd double considered in [5] has the following structure constants If we consider a redefinition, the structure constants become Therefore, this Manin triple is isomorphic to (5.i|8|b = 1) of [23].
Using the algebra (3.5) and a parameterization g = e x T 1 e y T 2 +z T 3 , we obtain By following [5], we introduce the constant matrix aŝ and then compute the generalized metric Eq. (2.12). Then the three-dimensional part of the supergravity fields are found as The DFT dilaton is trivial e −2 d(x) = e −2 d 0 |det(ℓ a m )| = e −2 d 0 = 1 , and the dilaton Φ is just a constant. This background is a conformally flat Einstein space with R = −6 and the H-flux is H 3 = dx ∧ dy ∧ dz = * 2 . Namely, at least locally, this is precisely the H-fluxed AdS 3 background (3.1) with unit AdS radius l = 1 .
We can easily check that the condition for the PL symmetry (1.2) is satisfied by the leftinvariant vector fields v m a and the structure constants (3.5). We can also check that these vector fields v m a do not generate the isometries of the target space, £ va g mn = 0 and £ va H 3 = 0 , unlike the usual setup of the traditional (non-)Abelian T -duality.
the algebra is mapped to a Manin triple (8|5.i|1) , where f abc c and f c ab are swapped. The By using a parameterization g = e z T 3 e y T 2 e x T 1 , we find 15) and the generalized metric and the DFT dilaton become Since the dual algebra is non-unimodular, this background is a solution of the generalized supergravity equations of motion with the vector field In DFT, the constant vector field I can be included into the DFT dilaton as d(x) = I mx m = −x [28], and then we find that this background is a solution of DFT (by adding the spectator fields).
Since the matrix H mn is degenerate, the standard fields {g mn , B mn , Φ} are not defined and this is an example of non-Riemannian backgrounds [25]. If we parameterize H IJ (x) as we find It turns out that the (open-string) metric G mn is the AdS 3 with the radius l = 2 , and the β-field produces a constant Q-flux Q y xy ≡ ∂ y β xy = −1 with a non-zero trace. This will be the target geometry of the constrained sigma model studied in [5]. 4 We can remove thex-dependence of the dilaton, by performing an Abelian T -duality along the x-direction. We then find which is again the AdS 3 geometry with a constant Q-flux. Moreover, in order to describe this background in the usual supergravity fields, let us consider a further constant O (3, 3) transformation (or a constant B-shift) In terms of {G mn , β mn }, we find and only the β-field gets a constant shift. In terms of the usual (closed-string) metric g mn and the B-field, the non-zero η is crucial and we find a solution of the usual supergravity Interestingly, this is a flat metric without H-flux for an arbitrary value of η ( = 0), and the dilaton satisfies ∇ m ∂ n Φ = 0 and g mn ∂ m Φ ∂ n Φ = 1 . Thus, we can find a certain coordinate transformation which makes this solution a flat space with a (spacelike) linear dilaton Φ = ±x .

Manin triple (5.iii|6 0 |1)
From the algebra (3.5), we consider a redefinition of generators We then obtain a new Manin triple with which is called (5.iii|6 0 |1) . Under the transformation, the constant matrix is transformed aŝ Considering a parameterization g = e x T 1 +y T 2 e z T 3 , we find The generalized metric and the DFT dilaton can be found as This is again a non-Riemannian background, but if we perform a factorized T -duality along the x-direction (which is a symmetry of string theory), we get a Riemannian background ds 2 = 2 dx dy + dz 2 , B 2 = 0 , Φ = z . (3.29) This (together with the spectator fields) is a solution of the ten-dimensional supergravity.
Then, we again found that the AdS 3 background with H-flux is related to the flat space with a linear dilaton through a PL T -plurality (and the usual T -duality).

Manin triple (6 0 |5.iii|1)
We can realize the Manin triple (6 0 |5.iii|1) by considering the PL T -duality from the previous example. Again we consider a parameterization g = e x T 1 +y T 2 e z T 3 and then obtain The generalized metric and the DFT dilaton become Then, again, we obtained the flat space with a linear dilaton, although the sign of the dilaton is changed.

PL T -plurality for DD7
Now we consider the PL T -plurality based on our approach. We consider the SL(2) algebra and denote an invariant metric asĝ Then we find the non-vanishing components of f abc ≡ f ab dĝ dc as f 123 = −1 .
Using the parameterization (3.37) Then the metric and the H-flux of the WZW background are found as Here £ va g mn = 0 and £ va H 3 = 0 are satisfied, but we find £ v 2 B 2 = 0 for B 2 = 4 z 2 dt ∧ dx . The AdS radius is l = 2 and we add spectator fields (3.2) with l = 2 in this subsection.
We can construct the generalized frame fields E A I as given in Eq. (2.4). Then we can check that they satisfy the algebra We also obtain the constant metricĤ AB by substituting Eq. Then we arrive at the Manin triple known as (4|2.iii|b = 1), For (4|2.iii|1), we parameterize the group element as and then by using the generalized metric and the DFT dilaton become This (together with the spectator fields) satisfies the equations of motion of DFT.
Again, this is a non-Riemannian background. In order to get the standard description, let us we perform a factorized T -duality along the z-direction. Then we get The above result shows that the AdS 3 with H-flux has the (4|2.iii|1) symmetry. One can explicitly construct the generators of the (4|2.iii|1) symmetry as (3.49) We can easily check that they satisfy where F AB C is the structure constant of the algebra (4|2.iii|1).

Manin triple (2.iii|4|1)
Here we consider the PL T -dual of the previous background. By using the parameterization and the dual background is found as This satisfies the generalized supergravity equations of motion.
Again we see that the three-dimensional geometry is AdS 3 . Indeed, assuming y > 0, a coordinate transformation and a B-field gauge transformation gives a ten-dimensional background (3.55) This is precisely the solution obtained in [16] [see Eq. (4.11)] via the traditional non-Abelian T -duality (which is not based on the Drinfel'd double).

The constant matrix becomesĤ
(3.58) By using we obtain Then the supergravity fields are Let us also consider the PL T -dual of the previous example, which corresponds to the Manin triple (7 0 |4|1) or (6 0 |4.i| − 1) for σ = +1 or −1 , respectively. Using a parameterization, for σ = −1 . Considering that the dual structure constants have non-vanishing trace, we find a solution of the generalized supergravity equations of motion (3.68) If we perform a coordinate transformation, we obtain

Nappi-Witten model
The NW model [17] is the WZW model based on a central extension of the two-dimensional We denote the generators collectively as and parameterize the group element as [17] g = e x T 1 +y T 2 e u T 3 +v T 4 . The right-and left-invariant 1-forms are Using the non-degenerate invariant metriĉ we obtain the NW background Choosing the B-field, for example, as we can construct the generalized frame fields E A I (2.4) satisfying the algebra (1.5) with Using the constant metricĤ AB , defined by Eq. (2.4), we obtain the generalized metric which describes the NW background (4.6).
The fluxes F AB C andĤ AB satisfy the equations of motion (1.9), and in the following subsections, we consider several O(4, 4) rotations to find the dual solutions.

Semi-Abelian double (E c 2 |A 4 )
As the first example, let us consider an O(4, 4) transformation (4.10) The original fluxes (4.8) are mapped to the fluxes The (geometric) fluxes F ab c are precisely the original ones and the H-flux disappeared under the O (4, 4) transformation. This is a an algebra of the Drinfel'd double (especially a semi-Abelian double F a bc = 0) and we denote the Manin triple as (E c 2 |A 4 ) , where A 4 denotes the four-dimensional Abelian algebra. We can easily construct the generalized frame fields by using the group element (4.3). In this frame, the constant metric becomeŝ (4.12) Then we find the supergravity fields as The H-flux is 14) and it turns out that this background is precisely the original NW background. Here, the condition £ va E mn = 0 for the PL symmetry is satisfied, and we can perform the PL Tduality/plurality as usual.

Semi-Abelian double (A 4 |E c 2 )
Here we consider the PL T -duality of the previous example, where the Manin triple can be denoted as (A 4 |E c 2 ). By using the parameterization g = e x T 1 +y T 2 +u T 3 +v T 4 , we find The generalized metric cannot be parameterized by g mn and B mn , but we find a solution of DFT, where The open-string metric is flat and there is the constant Q-flux If we make a periodic identification, such as x ∼ x + 1 , this spacetime can be regarded a T -fold [21]. In this example, it is not easy to find an Abelian O(4, 4) transformation which brings this non-Riemannian background into a Riemannian one.

Manin triple (G1|G2)
From the algebra (4.11), by performing an O(4, 4) transformation, we obtain another Manin triple For convenience, we denote the four-dimensional algebras, characterized by F ab c and F c ab , by G1 and G2 , respectively. The constant matrix becomeŝ As one can easily expect, we get a non-Riemannian background when b = 0 . Assuming b = 0 and using a parameterization g = e x T 1 +y T 2 −(b u+2 v) T 3 +v T 4 , we find 23) and the dual background is precisely the original one The dual geometry is non-Riemannian and we find In fact, we find that

Yang-Baxter deformation based on modified CYBE
Here we consider the YB deformation of the NW background (4.13). YB deformations of the NW model were studied in [29] by following the prescription of [30]. There, the general solution of the (modified) CYBE was found, but the deformation can be removed by a coordinate transformation and a B-field gauge transformation and a new background was not found.
In other words, the NW background was found to be invariant (or self-dual) under the YB deformation.
When the r-matrix solves the homogeneous CYBE, the YB deformation is a particular case of the PL T -plurality. In the case of the NW background, the general solution of the homogeneous CYBE is Abelian [29] and then the deformation is just an Abelian T -duality transformation. The only non-Abelian solution can be found by considering a solution of the modified CYBE. Here, we consider the YB deformation by using a solution of the modified CYBE. It seems that our deformation is different from the one studied in [29], and we find a solution which connects the NW background and a flat solution.
We consider a Lie algebra of the Drinfel'd double where the dual structure constants satisfy the coboundary ansatz, This Drinfel'd double satisfies the Jacobi identity if the following (modified) CYBE is satisfied: The general solution of (4.29) was found in [29], and the only non-Abelian solution is which gives the structure constants The YB deformation can be understood as the deformation of the algebra from η = 0 to η = c/2 . When the r-matrix satisfies the homogeneous CYBE (c = 0), a YB deformation is precisely an O(4, 4) transformation However, when the r-matrix satisfies the modified CYBE, it is not an O(4, 4) transformation.
Indeed, if we perform the inverse transformation of (4.33), the algebra (4.32) does not go back to the original one (4.11). Rather, the algebra becomes 34) and this includes the R-flux F abc (= F [abc] ) in addition to the original geometric flux (see, for example, [31] for more details of the generalized fluxes).
Fortunately, our constant metric (4.12),   Since the dual structure constants are non-unimodular f b ba = 0, we find a solution of the generalized supergravity equations of motion (4.39) Interestingly, due to the degeneracy (g + B) mn I n = 0 , this vector field I disappears from the equations of motion and can be removed. 5 Consequently we obtain a one-parameter family of supergravity solutions (4.39) without I .
By the construction, this solution reduces to the NW background by choosing η = 0 .
Moreover, we find an interesting special case η = ±1 where the curvature tensor and the H-flux vanish. Therefore, we found a one-parameter family of solutions which contains the NW background (η = 0) and the four-dimensional Minkowski spacetime (η = ±1) as specific cases.

Conclusions
The main purpose of this paper is to point out that the target space of a WZW model can be used as a seed solution to generate a chain of solutions through the PL T -plurality.
If we find an O(D, D) transformation which transforms this algebra into a Drinfel'd double, we obtain a PL symmetric background, and we can construct further PL symmetric backgrounds by following the standard procedure of the PL T -plurality. As demonstrations, we studied the PL T -plurality for two WZW backgrounds, AdS 3 with H-flux and the NW background. 5 A similar situation has been observed in [32] and studied in more detail in [33].
In the case of the AdS 3 with H-flux, we considered two Drinfel'd doubles DD2 and DD7.
There are 4 + 6 inequivalent Manin triples, but all of them are related to the following four solutions through a coordinate transformation or the standard Abelian T -duality: Sol4: The first and the third solutions are the familiar solution and the last one is the solution of the generalized supergravity equations of motion known in [16]. The second one is an interesting AdS 3 solution with a traceful constant Q-flux, which is non-Riemannian. This seems to be a new solution, but as we explained around Eq. The correspondence between the Manin triples and the solutions can be summarized as DD2: where SL(2) WZW represents the algebra (3.39) and * denotes that Abelian T -duality was required in order to bring the non-Riemannian backgrounds into the Riemannian frame or to make solutions of the generalized supergravity to the standard solutions.
In the case of the NW background, we considered four Manin triples, where NW denotes the algebra (4.8). We found that NW, (E c 2 |A 4 ), and (G1|G2) correspond to the same NW background while (A 4 |E c 2 ) and (G2|G1) correspond to a non-Riemannian background of the form (4.27). This non-Riemannian background (4.27) can be regarded as a flat space with a constant Q-flux, and is a kind of T -fold if we make some periodic identification of the spatial direction.
The most interesting solution will be the one obtained by the YB deformation based on the modified CYBE. In this case, we found a one-parameter family of solutions which contains the NW background and the flat Minkowski space as particular cases.
We can apply our procedure to other WZW models, such as the WZW model based on the Heisenberg group H 4 In [6], this WZW background was realized as a PL symmetric background by considering a Drinfel'd double On the other hand, in our approach, the flux algebra is given by However, it seems to be impossible to map this algebra into any Lie algebra of a Drinfel'd double through an O (3,3) transformation. In such cases, our procedure does not work (see [34] for some discussion on the PL T -duality for SU(2) WZW model).
In our examples, we found that many Manin triples correspond to a single background.
As a solution generating technique, this may not seem like a desirable situation, but one can take advantage of this situation [18]. The gluing matrix for the NW model was studied in [35] and a similar analysis was done for the AdS 3 with H-flux in [36], and several D-brane configurations were found in these WZW backgrounds. Subsequently, the transformation rule of the gluing matrix transforms under the PL T -duality was found in [37] and its extension to the PL T -plurality was found in [38]. As we found in this paper, the WZW backgrounds are self-dual under several PL T -pluralities, and a naive expectation is that we can find new D-brane configurations by mapping the known gluing matrices through the PL T -plurality transformations. Moreover, since the AdS 3 background is related to the flat space with the linear dilaton, it is also interesting to map the D-brane configurations in these spaces onto each other. In addition, our PL T -plurality produced many non-Rimannian backgrounds, but D-branes in these backgrounds have not been studied. Then it will be interesting to study various D-brane configurations in non-Riemannian backgrounds using the procedure of the PL T -plurality studied in [38].
In this paper, we restricted ourselves to the PL T -plurality, but the same idea can also be applied to the non-Abelian U -duality [39][40][41][42][43][44]. For example, in the E n(n) exceptional field theory (EFT) with n ≤ 4, we may consider a solution of 11D supergravity where the internal parts of the supergravity fields are given by where F abcd is constant. In this case, we can construct the (weightful) generalized metric as AB by using the generalized frame fields and a constant matrixM AB ∈ SL(5) . We can easily check that this set of the generalized frame fields satisfies the algebra where [·, ·] E is the generalized Lie derivative in EFT and the structure constants are given by Under an SL (5) rotation the components X abcd may vanish and the new algebra can be regarded as an exceptional Drinfel'd algebra [39,40]. In that case, we can construct the new generalized frame fields E ′ A I that satisfy the algebra (5.16) for the structure constants X ′