aa r X i v : . [ h e p - t h ] F e b Poisson–Lie T -plurality for WZW backgrounds Yuho Sakatani
Department of Physics, Kyoto Prefectural University of Medicine,1-5 Shimogamohangi-cho, Sakyo-ku, Kyoto, Japan [email protected]
Abstract
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 solutionswith H -flux, which are not Poisson–Lie symmetric but admit non-Abelian isometries, £ v a g mn = 0 and £ v a H = 0 with £ v a B = 0. After introducing the general procedure,we study the Poisson–Lie T -plurality for two WZW backgrounds, the AdS with H -fluxand the Nappi–Witten background. ontents T -plurality for DD2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1.1 Manin triple ( .i | |
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.1.2 Manin triple ( | .i |
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.1.3 Manin triple ( .iii | |
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.1.4 Manin triple ( | .iii |
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 PL T -plurality for DD7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2.1 Manin triple ( | .iii |
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 133.2.2 Manin triple ( .iii | |
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 153.2.3 Manin triples ( | |
1) and ( .i | | −
1) . . . . . . . . . . . . . . . . . . 153.2.4 Manin triples ( | |
1) and ( | .i | −
1) . . . . . . . . . . . . . . . . . . 16 E c | A ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.2 Semi-Abelian double ( A | E c ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194.3 Manin triple ( G | G
2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204.4 Manin triple ( G | G
1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214.5 Yang–Baxter deformation based on modified CYBE . . . . . . . . . . . . . . . 22 Introduction
Abelian T -duality, which is a symmetry of string theory, is based on the existence of a set ofcommuting vector fields v ma satisfying the Killing equations £ v a g mn = 0 , £ v a H = 0 . (1.1)At least in supergravity, this symmetry continues to hold even when the Killing vector fieldsform a non-Abelian algebra [ v a , v b ] = f abc 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 vectorfields v ma satisfying the condition for the PL symmetry, £ v a E mn = − f abc E mp v pb v qc E qn (cid:0) E mn ≡ g mn + B mn (cid:1) . (1.2)Here f abc (= − f acb ) are called the dual structure constants, and if they are absent, this condi-tion reduces to the Killing equations, £ v a g mn = £ v a B = 0 . Then the PL T -duality reducesto the traditional non-Abelian T -duality. However, the condition £ v a B = 0 is stronger than £ v a H = 0 of Eq. (1.1), and naively, not all of the traditional non-Abelian T -duality can berealized as a PL T -duality. In this short paper, we discuss how to perform the PL T -pluralityin backgrounds with non-Abelian isometries, £ v a g mn = 0 and £ v a H = 0 but £ v a B = 0 .For the sake of clarity, let us comment on some of the related earlier works. There are atleast two approaches to perform the PL T -duality in a H -fluxed background. The first is takenin [5, 6], where Wess–Zumino–Witten (WZW) backgrounds are constructed as PL symmetricbackgrounds satisfying the condition (1.2). For a given WZW background, how to find theset of vector fields v ma and the dual structure constants f abc is quite non-trivial, but once wefind these, we can perform the PL T -duality by following the standard procedure. The secondapproach is taken in [7, 8], where the non-trivial H -flux is produced by the spectator fieldsand the internal part satisfies the usual Killing equations £ v a E mn = 0 . Namely, the WZWbackgrounds are realized as PL symmetric backgrounds associated with semi-Abelian doubles(i.e., f abc = 0). Again, once we found such a realization, we can perform the PL T -duality byusing 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) [9–12], whose flux formulation [13] is especially useful here. In DFT, the supergravityfields in the NS–NS sector are packaged into the generalized metric and the DFT dilaton H IJ ≡ g mn ( B g − ) mn − ( g − B ) mn g mn , e − d ≡ e − p | det g mn | , (1.3)2here I, J = 1 , . . . , D 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 H IJ ( x ) = E I A ( x ) E J B ( x ) ˆ H AB , (1.4)where ˆ H AB is constant and E I A is the inverse of the generalized frame fields E AI satisfying[ E A , E B ] D = −F ABC E C . (1.5)Here, [ · , · ] D denotes the D-bracket or the generalized Lie derivative in DFT and F ABC are thestructure constants with components F abc = H abc , F abc = F bca = F cab = f abc , F abc = F cab = F bca = 0 = F abc , (1.6)under the decompositions, { A } = { a , a } and { A } = { a , a } ( A = 1 , . . . , D and a = 1 , . . . , D ).For the DFT dilaton, for simplicity, we assume the absence of the dilaton flux, F A ≡ E AI D B E I B + 2 D A d = 0 (cid:0) D A ≡ E AI ∂ I (cid:1) . (1.7)In the usual setup of the PL T -duality/plurality in DFT [14–16], the components of thestructure constants are supposed to have the form F abc = −F acb = F cab = f abc , F abc = F cab = F bca = f abc , F abc = 0 = F abc , (1.8)which are the structure constants of the Drinfel’d double. However, as a solution generatingtechnique, Eqs. (1.4), (1.5), and (1.7) are sufficient, and instead of (1.8), we rather suppose(1.6) in the initial background. As usual, the equations of motion of DFT reduce toˆ H AD (cid:0) η BE η CF − ˆ H BE ˆ H CF (cid:1) F ABC F DEF = 0 , (cid:0) η CE η DF − ˆ H CE ˆ H DF (cid:1) ˆ H G [ A F CDB ] F EF T = 0 , (1.9)and they are covariant under the O( D, D ) transformationˆ H AB → ˆ H ′ AB ≡ C AC C BD ˆ H CD , F ABC → F ′ ABC ≡ C AD C BE ( C − ) F C F DEF . (1.10)Then, by finding a new set of generalized frame fields E ′ AI satisfying [ E ′ A , E ′ B ] D = −F ′ ABC E ′ C (but E ′ A = C AB E B ), we can generate a new DFT solution H ′ IJ ≡ E ′ I A E ′ J B ˆ H AB and d ′ ,which satisfies F ′ A ≡ E ′ AI D ′ B E ′ I B + 2 D ′ A d ′ = 0 .As concrete backgrounds satisfying Eqs. (1.4), (1.5), and (1.7), we consider the targetspace of an (ungauged) WZW model. As we explain in section 2, for any WZW model, wecan explicitly construct the generalized frame fields E AI and the DFT dilaton d satisfyingEqs. (1.4)–(1.7). There, we find a set of (generalized) Killing vector fields ( v ma , ˜ v am ) satisfying £ v a g mn = 0 , £ v a B + d˜ v a = 0 , [ v a , v b ] = f abc v c , (1.11) We thank Chris Blair for correspondence regarding this point. T -duality. Due to the presenceof the 1-form fields ˜ v a , this is different from the usual condition for the PL symmetry (1.2)and the standard procedure of the PL T -duality cannot be applied. However, if the 2 D -dimensional Lie algebra [ T A , T B ] = F ABC T C with F ABC given in (1.6) can be decomposedinto two maximally isotropic subalgebras, we can construct the dual geometry by means ofthe PL T -plurality. Namely, if we find an O( D, D ) transformation (1.10) such that the newstructure constants F ′ ABC have the form of Eq. (1.8), the 2 D -dimensional algebra becomes aDrinfel’d double. For a Drinfel’d double, we can systematically construct the dual fields E ′ AI and d ′ satisfying Eqs. (1.5) and (1.7). Then the resulting background should be a solution ofDFT as long as the original WZW background is a solution of DFT. This is the idea of thesolution generation method proposed in this paper.As concrete examples of the WZW model, we consider the SL(2) WZW model and theNappi–Witten (NW) model [17]. For the SL(2) WZW model, we consider two approaches forthe PL T -plurality: the one studied in [5] and our approach. We find that the two approachesare based on different six-dimensional Drinfel’d doubles. For each of the Drinfel’d doubles,there are several inequivalent decompositions into the pair of algebras { f abc , f cab } , called theManin triples, and we construct a background for each of the Manin triples. In general, eachManin triple corresponds to a different target geometry but it turns out that many of theManin triples correspond to the same AdS solution with H -flux. This kind of self-dualityunder the PL T -duality has been observed in [8,18] and this may represent the high symmetryof the AdS background. We also find that some Manin triples correspond to a flat spacewith a linear dilaton and some Manin triples correspond to a known AdS solution of thegeneralized supergravity equations of motion [19, 20].For the NW model, the Drinfel’d double is eight-dimensional and the Manin triples havenot been classified. We find several inequivalent Manin triples and construct the correspondingbackgrounds. 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 descriptionin 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 modifiedclassical YB equations (CYBE). As a result, we find a one-parameter family of solutions whichcontains 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 generalizedframe fields E AI 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 -pluralityof the SL(2) WZW model. In section 3, we study the PL T -plurality and the YB deformationof the NW background. Section 5 is devoted to conclusions and discussions.4 WZW background
Let us consider a group G associated with a Lie algebra, [ T a , T b ] = f abc T c . We define theleft-/right-invariant 1-forms and their duals as ℓ ≡ g − d g ≡ ℓ am d x m T a , r ≡ d g g − ≡ r am d x m T a , v ma ℓ bm = δ ba , e ma r bm = δ ba . (2.1)Then the WZW model can be defined as S = 14 πα ′ Z Σ g mn d x m ∧ ∗ d x n + 12 πα ′ Z B H ( ∂B = Σ) , (2.2)where the metric and the 3-form field strength are g mn ≡ ˆ g ab ℓ am ℓ bn = ˆ g ab r am r bn ,H ≡ f abc ℓ a ∧ ℓ b ∧ ℓ c = 13! f abc r a ∧ r b ∧ r c . (2.3)Here ˆ g ab is a non-degenerate invariant metric and f abc ≡ f abd ˆ g dc is totally antisymmetric.If we introduce a 2-form field B satisfying d B = H , we can check the generalized Killingvector fields V aI ≡ ( v ma , ˜ v am ≡ E mn v na d x m ) satisfy Eq. (1.11) for any choice of B . In general˜ v am does not vanish and this background is not PL symmetric, i.e., we cannot perform thePL T -duality. Now we define the generalized frame fields and a constant matrix as E AI ≡ e ma − e na B nm r am , ˆ H AB ≡ ˆ g ab
00 ˆ g ab , (2.4)and then the generalized metric for the WZW background (2.3) is expressed as H IJ ( x ) = E I A ( x ) E J B ( x ) ˆ H AB . (2.5)Using e ma e nb e pc H mnp = f abc , we can easily check that the frame fields { E A } = { E a , E a } satisfythe algebra (1.5) with the fluxes F ABC given by Eq. (1.6) and H abc = f abc , [ E a , E b ] D = − f abc E c − f abc E c , [ E a , E b ] D = 0 , [ E a , E b ] D = − [ E b , E a ] D = f acb E c . (2.6)Here the D-bracket or the generalized Lie derivative in DFT is defined as[ V, W ] ID = V J ∂ J W I − (cid:0) ∂ J V I − ∂ I V J (cid:1) W J , (2.7) If a B -field satisfies d B = H abc r a ∧ r b ∧ r c for a general skew-symmetric constants H abc , the E AI definedin Eq. (2.4) satisfy the algebra (2.6) with f abc replaced by H abc . In such a general case, H IJ = E IA E J B ˆ H AB does not describe the target space of a WZW model, but we can still perform the PL T -plurality. For the PL T -plurality, ˆ H AB also can be an arbitrary constant O( D, D ) matrix and is not restricted to have the form (2.4). ∂ I ) = ( ∂∂x m , ∂∂ ˜ x m ) and the indices { I } = { m , m } and { I } = { m , m } are raised or loweredwith the O( D, D )-invariant metric η IJ ≡ δ nm δ mn , η IJ ≡ δ mn δ nm . (2.8)Regarding the DFT dilaton, the requirement (1.7) reduces to − ∂ m d = e nb ∂ n r bm = − ∂ n e nb r bm , (2.9)under our assumption ∂∂ ˜ x m = 0 . By using ∂ n e nb = ℓ am e nb ∂ n v ma = − e nb ∂ n ln | det( ℓ am ) | , whichfollows from ℓ am £ v a e mb = 0 , the DFT dilaton can be found to have the forme − d = e − d | det( ℓ am ) | , (2.10)where d is a constant. In our examples, we simply choose d = 0 because the constant is notimportant in the equations of motion.Now let us perform the O( D, D ) transformation (1.10) such that the new algebra F ′ ABC isthe Lie algebra of a Drinfel’d double. Then we parameterize the group element, for example,as g ( x ) = e x a T a and define the right-/left-invariant 1-forms and their duals. We also define g − T A g ≡ a ab π a ) ab ( a − ) ba (cid:0) π ab = − π ba (cid:1) , (2.11)and construct new generalized frame fields and a DFT dilaton as E AI ( x ) ≡ e ma π ab e mb r am , e − d ( x ) = e − d | det( ℓ am ) | . (2.12)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. In order to make a solution of a ten-dimensional supergravity, we may consider a productof 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 H ˆ I ˆ J ( x m , y µ ) = H IJ ( x ) 00 H sMN ( y ) , (2.13)where ˆ I = 1 , . . . ,
20 and { M } = { µ , µ } with µ = 1 , . . . , − n . The DFT dilaton is given bya product e − d ( x m , y µ ) = e − d ( x ) e − d s ( y ) . (2.14)The fields H sMN ( 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. When the dual algebra is non-unimodular ( f aab = 0), a PL symmetric background is known to satisfy thegeneralized supergravity equations of motion [19, 20] with the vector field I = f bba v a [15]. SL(2) WZW model
Here we consider the SL(2) WZW model, where the target geometry can be identified withthe AdS with H -fluxd s = l d z − d t + d x z , H = ± l d z ∧ d t ∧ d xz = ± l − ∗ . (3.1)The Riemann curvature tensors is R mnpq = − l − ( g mp g nq − g mq g np ) and the Ricci scalar is R = − /l . This becomes a ten-dimensional supergravity solution by adding spectator fields(which are not affected by T -dualities)d s × T = l (cid:2) d θ + sin θ d φ + (cid:0) d ψ + cos θ d φ (cid:1) (cid:3) + d y + d y + d y + d y ,H (S × T )3 = l sin θ d θ ∧ d φ ∧ d ψ . e − d s ( y ) = ( l/ sin θ . (3.2)Since e − d s ( y ) is just the volume element, the dilaton Φ can be found as e − = e − d ( x ) √ | det g mn | .In [5], this WZW background was reproduced as a PL symmetric background by usinga six-dimensional Drinfel’d double. Six-dimensional Drinfel’d doubles and the Manin tripleshave been classified in [23] and it is known that there are 22 Drinfel’d doubles. According tothis 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: ( .i | | b ) ∼ = ( | .i | b ) ∼ = ( | .iii | b ) ∼ = ( .iii | | b ) ( b > , (3.3)and it was found that the SL(2) WZW background is associated with ( .i | | b ) [5]. Thenthe PL T -dual model, whose Manin triple is ( | .i | b ), was found to be a constrained sigmamodel [5]. In this section, we find that the target geometry of this dual model is a kind ofnon-Riemannian backgrounds (see [25–27] for details of non-Riemannian backgrounds). Thenwe further perform PL T -plurarities and obtain the backgrounds associated with the othertwo Manin triples, ( | .iii | b ) and ( .iii | | 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 abc and the H -flux H abc = 0 . Then weidentify 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: ( | | b ) ∼ = ( | | b ) ∼ = ( | .iii | b ) ∼ = ( .iii | | b ) ∼ = ( | .i | − b ) ∼ = ( .i | | − b ) , (3.4)and we identify the corresponding backgrounds. T -plurality for DD2 Here we review the result of [5] and then complete the orbit of DD2 described in Eq. (3.3).7 .1.1 Manin triple ( .i | | f = 1 , f = 1 , f = 1 , f = − , f = 2 . (3.5)If we consider a redefinition, T a → Λ ab T b , T a → (Λ − ) ba T b , Λ ab = − − , (3.6)the structure constants become f = − , f = − , f = − , f = 1 , f = − . (3.7)Therefore, this Manin triple is isomorphic to ( .i | | b = 1) of [23].Using the algebra (3.5) and a parameterization g = e x T e y T + z T , we obtain r ma = x
00 0 e x , ℓ ma = y z , (3.8) π ab = − e x y e x z e x y − e x ( y z + 1) − e x z e x ( y z + 1) − . (3.9)By following [5], we introduce the constant matrix asˆ H AB = − − , (3.10)and then compute the generalized metric H IJ ( x ) = E I A E J B ˆ H AB , where E AI is defined inEq. (2.12). Then the three-dimensional part of the supergravity fields are found as g mn = y z z y z y , B mn = z − y − z − y . (3.11)The DFT dilaton is trivial e − d ( x ) = e − d | det( ℓ am ) | = e − d = 1 , and the dilaton Φ is just aconstant. This background is a conformally flat Einstein space with R = − H -fluxis H = d x ∧ d y ∧ d z = ∗ H -fluxed AdS background (3.1) with unit AdS radius l = 1 . 8e can easily check that the condition for the PL symmetry (1.2) is satisfied by the left-invariant vector fields v ma and the structure constants (3.5). We can also check that these vectorfields v ma do not generate the isometries of the target space, £ v a g mn = 0 and £ v a H = 0 ,unlike the usual setup of the traditional (non-)Abelian T -duality. ( | .i | T -duality T a ↔ T a , namely, the O(3 ,
3) transformation (1.10) with C AB = , (3.12)the algebra is mapped to a Manin triple ( | .i |
1) , where f abcc and f cab are swapped. Theconstant metric becomes ˆ H AB = − − . (3.13)By using a parameterization g = e z T e y T e x T , we find r ma = y z − y y z + z − z − z , ℓ ma = − x − y − e − x y e x , (3.14) π ab = − y − z (1 + y z ) y y zz (1 + y z ) − y z , (3.15)and the generalized metric and the DFT dilaton become H IJ ( x ) = − y − − y y y
10 0 2 y − y , d ( x ) = 0 . (3.16)Since the dual algebra is non-unimodular, this background is a solution of the generalizedsupergravity equations of motion with the vector field I = f bba v a = − ∂ x . In DFT, theconstant vector field I can be included into the DFT dilaton as d ( x ) = I m ˜ x m = − ˜ x [28], andthen we find that this background is a solution of DFT (by adding the spectator fields).9ince the matrix H mn is degenerate, the standard fields { g mn , B mn , Φ } are not definedand this is an example of non-Riemannian backgrounds [25]. If we parameterize H IJ ( x ) as H IJ ( x ) = G mn G mq β qn − β mq G qn G mn − β mp G pq β qn , (3.17)we find G mn = − y − y , β mn = − y y − , d ( x ) = − ˜ x . (3.18)It turns out that the (open-string) metric G mn is the AdS with the radius l = 2 , and the β -field produces a constant Q -flux Q yxy ≡ ∂ y β xy = − We can remove the ˜ x -dependence of the dilaton, by performing an Abelian T -duality alongthe x -direction. We then find G mn ( x ) = y y , β mn ( x ) = y − y − , d ( x ) = − x , (3.19)which is again the AdS geometry with a constant Q -flux. Moreover, in order to describethis background in the usual supergravity fields, let us consider a further constant O(3 , B -shift) H IJ → O I K O J L H KL , O I J = δ nm b mn δ mn , b mn = η − η . (3.20)In terms of { G mn , β mn } , we find G mn ( x ) = y y , β mn ( x ) = y − η − y η − , d ( x ) = − x , (3.21)and only the β -field gets a constant shift. In terms of the usual (closed-string) metric g mn andthe B -field, the non-zero η is crucial and we find a solution of the usual supergravity g mn = η y η y y η y − η y η y − η y η , B mn = η y − η − η y − η y η y η η y − η y , e − = 2 η y e x . (3.22)Interestingly, this is a flat metric without H -flux for an arbitrary value of η ( = 0), and thedilaton satisfies ∇ m ∂ n Φ = 0 and g mn ∂ m Φ ∂ n Φ = 1 . Thus, we can find a certain coordinatetransformation which makes this solution a flat space with a (spacelike) linear dilaton Φ = ± x . In [25], the string sigma model in a general non-Riemannian background is shown to be constrained by acertain self-duality relation (or a chirality constraint), and this seems to be consistent with the result of [5]. .1.3 Manin triple ( .iii | | T A → C AB T B with C AB = − − −
11 0 0 0 − − − − −
18 12 −
14 12 18 12 − − . (3.23)We then obtain a new Manin triple with f = − , f = − , f = 1 , f = 1 , (3.24)which is called ( .iii | |
1) . Under the transformation, the constant matrix is transformed asˆ H AB = . (3.25)Considering a parameterization g = e x T + y T e z T , we find r ma = − x − y , ℓ ma = e − z − z
00 0 1 , (3.26) π ab = − x − y − y x − y − xy x . (3.27)The generalized metric and the DFT dilaton can be found as H IJ ( x ) = , e − d ( x ) = | det ℓ am | = e − z . (3.28)This is again a non-Riemannian background, but if we perform a factorized T -duality alongthe x -direction (which is a symmetry of string theory), we get a Riemannian backgroundd s = 2 d x d y + d z , B = 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 background with H -flux is related to the flat space witha linear dilaton through a PL T -plurality (and the usual T -duality).11 .1.4 Manin triple ( | .iii | | .iii |
1) by considering the PL T -duality from the previousexample. Again we consider a parameterization g = e x T + y T e z T and then obtain r ma = y x , ℓ ma = cosh z sinh z z cosh z
00 0 1 , π ab = x − y x − x − y y − x − y . (3.30)The generalized metric and the DFT dilaton become H IJ ( x ) = , d ( x ) = 0 . (3.31)Since the dual algebra is non-unimodular, the vector field I = − ∂ z is needed to satisfy thegeneralized supergravity equations of motion. Again, under a factorized T -duality along the x -direction, this non-Riemannian background is mapped to a flat Riemannian backgroundd s = 2 d x d y + d z , B = 0 , Φ = 0 , I = − ∂ z . (3.32)Moreover, we can include I into the DFT dilaton as d ( x ) = − ˜ z , and the by performing theformal T -duality along the z -direction (which interchanges z and ˜ z ), we getd s = 2 d x d y + d z , B = 0 , Φ = − z . (3.33)Then, again, we obtained the flat space with a linear dilaton, although the sign of the dilatonis changed. T -plurality for DD7 Now we consider the PL T -plurality based on our approach. We consider the SL(2) algebra f = − , f = − , f = 1 , (3.34)and denote an invariant metric as ˆ g ab = − − . (3.35)Then we find the non-vanishing components of f abc ≡ f abd ˆ g dc as f = − g = e √ t − x ) T e − z T e √ t + x ) T , (3.36)we obtain r ma = − √ t − x ) z − z √ z √ t − x ) + z ] z t − x ) z √ z √ t − x ) − z ] z t − x ) z , ℓ ma = − √ t + x ) z − z √ t + x ) + z ] z √ z t + x ) z − √ t + x ) − z ] z − √ z − t + x ) z . (3.37)Then the metric and the H -flux of the WZW background are found asd s = 4 d z − d x + d x z , H = 13! f abc r a ∧ r b ∧ r c = − z ∧ d t ∧ d xz . (3.38)Here £ v a g mn = 0 and £ v a H = 0 are satisfied, but we find £ v B = 0 for B = z d t ∧ d x .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 AI as given in Eq. (2.4). Then we cancheck that they satisfy the algebra [ E A , E B ] D = −F ABC E C with the fluxes F = − , F = − , F = 1 , F = − . (3.39)We also obtain the constant metric ˆ H AB by substituting Eq. (3.35) into Eq. (2.4). In thefollowing, we perform the PL T -plurality by rotating the pair of the constants {F ABC , ˆ H AB } . ( | .iii | ,
3) transformation T A → C AB T B with C AB = . (3.40)Then we arrive at the Manin triple known as ( | .iii | b = 1), f = − , f = 1 , f = − , f = − . (3.41)This shows that the flux algebra (3.39) corresponds to the Drinfel’d double, called DD7 [23],which admits six inequivalent Manin triples described in Eq. (3.4).For ( | .iii | g = e x T e y T + z T , (3.42)13nd then by using r ma = − x e − x x − x , ℓ ma = − y y − z , (3.43) π ab = − x y y e − x − e − x y − y e − x , (3.44)the generalized metric and the DFT dilaton become H IJ ( x ) = y − x x −
10 0 0 0 − x y − xy − x − y y − xy y x − x − xy − xy x y , d ( x ) = 0 . (3.45)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, letus we perform a factorized T -duality along the z -direction. Then we get g mn = y − y − , B mn = x − x , Φ = 0 . (3.46)This background is (at least locally) the original AdS background. Indeed, under a coordinatetransformation w ≡ − − x/ , x + ≡ − e − x y , x − ≡ z , (3.47)this three-dimensional background becomesd s = 4 d w + 2 d x + d x − w , H = 8 d w ∧ d x + ∧ d x − w . (3.48)The above result shows that the AdS with H -flux has the ( | .iii |
1) symmetry. One canexplicitly construct the generators of the ( | .iii |
1) symmetry as E = ∂ x , E = e x ∂ y − e x x d z , E = e x d z ,E = d x + y d z , E = e − x d y + e − x y z ,E = − e − x y ∂ x − e − x y ∂ y + e − x ∂ z + e − x x d y + e − x xy z . (3.49)We can easily check that they satisfy[ E A , E B ] D = −F ABC E C , (3.50)where F ABC is the structure constant of the algebra ( | .iii | .2.2 Manin triple ( .iii | | T -dual of the previous background. By using the parameterization g = e z T e ( y + z ) T e x T , (3.51)and r ma = z
00 1 00 1 1 , ℓ ma = x + 1 1 , π ab = y z − y z − z − z , (3.52)the dual background is found as g mn = y y y y y , B mn = y y − y y − y − y , e − = y , I = ∂ x . (3.53)This satisfies the generalized supergravity equations of motion.Again we see that the three-dimensional geometry is AdS . Indeed, assuming y >
0, acoordinate transformation w = √ y , x + = x + ln y , x − = z , (3.54)and a B -field gauge transformation gives a ten-dimensional backgroundd s = 4 d w + 2 d x + d x − w + d s × T , e − = w ,B = 4 d x + ∧ d x − w + 2 d x + ∧ d ww − cos θ d φ ∧ d ψ , I = ∂ + . (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). ( | | and ( .i | | − ,
3) transformation C AB = σ σ − σ σ , σ = ± , (3.56)of the algebra (3.39). Then we arrive at the algebra F = − σ , F = σ , F = − σ , F = 1 , F = − σ . (3.57)15his corresponds to the Manin triple ( | |
1) or ( .i | | −
1) for σ = +1 or − H AB = −
10 0 0 0 − − σ − − σ . (3.58)By using g = e x T e y T + z T , (3.59)we obtain r ma = − σx σ x e − σx − σx , ℓ ma = y − y + z , (3.60) π ab = σ y e − σx e − σx ( σ + y ) − σ − σ y e − σ x − e − σx ( σ + y ) − σ . (3.61)Then the supergravity fields are g mn = σ y − y − y σ (1 − x ) − x − x − σ , B mn = − σ x y − yσ x y σy − σ , Φ = 0 . (3.62)By computing the curvature tensor and the H -flux, we can identify that this is again theoriginal AdS background with the AdS radius l = 2 . ( | | and ( | .i | − T -dual of the previous example, which corresponds to the Manintriple ( | |
1) or ( | .i | −
1) for σ = +1 or − g = e y T e x T +( y − z ) T , (3.63)we find r ma = cos y sin y − sin y cos y
10 0 1 , ℓ ma = z − y x + 1 1 z − y x , (3.64) π ab = − z sin yz − cos y − sin y cos y − , (3.65)16or σ = +1 and r ma = cosh y − sinh y − sinh y cosh y
10 0 1 , ℓ ma = z − y − x z − y − x , (3.66) π ab = z − sinh y − z y − y − cosh y , (3.67)for σ = − g mn = − σz − σz − σz − σ − σ − z − σz − σ − z − σ − z , B mn = − σz σz z − z , e − = z , I = σ ∂ x . (3.68)If we perform a coordinate transformation, x ≡ σ ( x + − w ) , y ≡ − x − − w , z ≡ w , (3.69)we obtain d s = 4 d w + 2 d x + d x − w − σ (d x − ) , e − = w ,B = 4 d x + ∧ d x − w + 2 d x + ∧ d ww , I = ∂ + . (3.70)This AdS solution will be the same as the solution given in Eq. (3.55) up to a furthercoordinate transformation. The NW model [17] is the WZW model based on a central extension of the two-dimensionalEuclidean group E c [ J, P i ] = ǫ ij P j , [ P i , P j ] = ǫ ij T . (4.1)We denote the generators collectively as { T a } = { P , P , J, T } , (4.2)and parameterize the group element as [17] g = e x T + y T e u T + v T . (4.3)17he right- and left-invariant 1-forms are r ma = − y x y − x − x + y , ℓ ma = cos u − sin u y sin u cos u − x . (4.4)Using the non-degenerate invariant metricˆ g ab = b
10 0 1 0 , (4.5)we obtain the NW background g mn = y
00 1 − x y − x b
10 0 1 0 , H = 13! f abc r a ∧ r b ∧ r c = d x ∧ d y ∧ d u . (4.6)Choosing the B -field, for example, as B = u d x ∧ d y , (4.7)we can construct the generalized frame fields E AI (2.4) satisfying the algebra (1.5) with F = 1 , F = − , F = 1 , F = 1 . (4.8)Using the constant metric ˆ H AB , defined by Eq. (2.4), we obtain the generalized metric H IJ = E I A E J B ˆ H AB , (4.9)which describes the NW background (4.6).The fluxes F ABC and ˆ H AB satisfy the equations of motion (1.9), and in the followingsubsections, we consider several O(4 ,
4) rotations T A → C AB T B and ˆ H AB → C AC C BD ˆ H CD to find the dual solutions. ( E c | A ) As the first example, let us consider an O(4 ,
4) transformation C AB = −
10 0 0 1 0 0 1 00 0 0 0 1 0 0 00 0 0 0 0 1 0 00 0 0 0 0 0 1 00 0 0 0 0 0 0 1 . (4.10)18he original fluxes (4.8) are mapped to the fluxes F = 1 , F = − , F = 1 . (4.11)The (geometric) fluxes F abc are precisely the original ones and the H -flux disappeared underthe O(4 ,
4) transformation. This is a an algebra of the Drinfel’d double (especially a semi-Abelian double F abc = 0) and we denote the Manin triple as ( E c | A ) , where A denotes thefour-dimensional Abelian algebra. We can easily construct the generalized frame fields byusing the group element (4.3). In this frame, the constant metric becomesˆ H AB = − b − b − b . (4.12)Then we find the supergravity fields as g mn = y
00 1 − x y − x b
10 0 1 0 , B mn = − y
00 0 x y − x −
10 0 1 0 , Φ = 0 . (4.13)The H -flux is H = d x ∧ d y ∧ d u , (4.14)and it turns out that this background is precisely the original NW background. Here, thecondition £ v a E mn = 0 for the PL symmetry is satisfied, and we can perform the PL T -duality/plurality as usual. ( A | E c ) Here we consider the PL T -duality of the previous example, where the Manin triple can bedenoted as ( A | E c ). By using the parameterization g = e x T + y T + u T + v T , we find r am = ℓ am = δ am , π ab = v − y − v x y − x . (4.15)The generalized metric cannot be parameterized by g mn and B mn , but we find a solution ofDFT, H IJ ( x ) = G mn G mp β pn − β mp G pn G mn − β mp G pq β qn , d ( x ) = 0 , (4.16)19here G mn ( x ) = − b , β mn ( x ) = − v y v − x − y x − . (4.17)The open-string metric is flat and there is the constant Q -flux Q xyu = − , Q yxu = 1 , Q vxy = − . (4.18)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 whichbrings this non-Riemannian background into a Riemannian one. ( G | G From the algebra (4.11), by performing an O(4 ,
4) transformation, C AB = , (4.19)we obtain another Manin triple F = 1 , F = − , F = 1 . (4.20)For convenience, we denote the four-dimensional algebras, characterized by F abc and F cab , by G G H AB = − − b b − b . (4.21)20s one can easily expect, we get a non-Riemannian background when b = 0 . Assuming b = 0and using a parameterization g = e x T + y T − ( b u +2 v ) T + v T , we find r ma = − y x − b
00 0 − , ℓ ma = y − x − b
00 0 − , (4.22) π ab = y
00 0 − x − y x x + y − x + y , (4.23)and the dual background is precisely the original one g mn = y
00 1 − x y − x b
10 0 1 0 , B mn = − y
00 0 x y − x − , e − = b − , (4.24)up to a B -field gauge transformation and a constant shift of the dilaton. Then, we find theinvariance of the NW background under the PL T -plurality. ( G | G The PL T -duality of the previous example gives a Manin triple ( G | G
1) . There, again as-suming b = 0 and using g = e ub T e x T + y T + v T , we obtain r ma = cos ub sin ub − sin ub cos ub b
00 0 0 1 , ℓ ma = − yb xb b
00 0 0 1 , π ab = − v v . (4.25)The dual geometry is non-Riemannian and we find G mn ( x ) = − b , β mn ( x ) = − v v − , e − d ( x ) = b − . (4.26)In fact, we find that G mn ( x ) = − b , β mn ( x ) = − v − c y v c x c y − c x − , e − d ( x ) = const. , (4.27)solves the equations of motion for an arbitrary parameter c . Both (4.17) and (4.27) arecontained in this one-parameter family of solutions.21 .5 Yang–Baxter deformation based on modified CYBE Here we consider the YB deformation of the NW background (4.13). YB deformations of theNW model were studied in [29] by following the prescription of [30]. There, the general solutionof the (modified) CYBE was found, but the deformation can be removed by a coordinatetransformation 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 YBdeformation.When the r -matrix solves the homogeneous CYBE, the YB deformation is a particularcase of the PL T -plurality. In the case of the NW background, the general solution of thehomogeneous CYBE is Abelian [29] and then the deformation is just an Abelian T -dualitytransformation. The only non-Abelian solution can be found by considering a solution of themodified CYBE. Here, we consider the YB deformation by using a solution of the modifiedCYBE. It seems that our deformation is different from the one studied in [29], and we find asolution which connects the NW background and a flat solution.We consider a Lie algebra of the Drinfel’d double where the dual structure constants satisfythe coboundary ansatz, f abc = f adb r dc − f adc r db (cid:0) r ab = − r ba (cid:1) . (4.28)This Drinfel’d double satisfies the Jacobi identity if the following (modified) CYBE is satisfied: f d d a r d a r d a + f d d a r d a r d a + f d d a r d a r d a = c f d d a ˆ g d a ˆ g d a , (4.29)where ˆ g ab is the inverse matrix of an invariant metric ˆ g ab satisfying f cad ˆ g db + f cbd ˆ g ad = 0 .Here, we consider the Lie algebra f abc given in Eq. (4.11) and the invariant metricˆ g ab = b
10 0 1 0 . (4.30)The general solution of (4.29) was found in [29], and the only non-Abelian solution is r = c , (4.31)which gives the structure constants f = 1 , f = − , f = 1 , f = η , f = η ( η ≡ c/ . (4.32)The YB deformation can be understood as the deformation of the algebra from η = 0 to η = c/ r -matrix satisfies the homogeneous CYBE ( c = 0), a YB deformation is22recisely an O(4 ,
4) transformation T A → C AB T B , C AB = δ ba r ab δ ab . (4.33)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 backto the original one (4.11). Rather, the algebra becomes F = 1 , F = − , F = 1 , F = − η , (4.34)and this includes the R -flux F abc (= F [ abc ] ) in addition to the original geometric flux (see, forexample, [31] for more details of the generalized fluxes).Fortunately, our constant metric (4.12),ˆ H AB = − b − b − b , (4.35)satisfies the equations of motion of DFT (1.9) for an arbitrary value of the R -flux F . Thus,the fluxes (4.32) together with the O(4 , H AB = − η η − b η η + 1 0 0 0 − η η + 1 0 00 0 − b − b , (4.36)satisfies the equations of motion of DFT.Now let us explicitly construct the YB-deformed background. Using the group element g = e x T + y T e u T + v T , (4.37) r am and ℓ am are obtained as in Eq. (4.4) and we also find π ab = − η x − η y η x η y . (4.38)23ince the dual structure constants are non-unimodular f bba = 0, we find a solution of thegeneralized supergravity equations of motion g mn = η y − η x η η − x − ηy η y − ηx η − x − ηy η b + η ( x + y ) η +1
10 0 1 0 , e − = 1 + η ,B mn = − η η y − yη +1 η η x η − x y η − y x − x η −
10 0 1 0 , I = f bba v a = η v = η ∂ v . (4.39)Interestingly, due to the degeneracy ( g + B ) mn I n = 0 , this vector field I disappears from theequations of motion and can be removed. Consequently we obtain a one-parameter family ofsupergravity 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 = ± g mn = y − η x − x + η y y − η x − x + η y b + x + y
10 0 1 0 , B mn = − η η −
10 0 1 0 , e − = 2 , (4.40)where the curvature tensor and the H -flux vanish. Therefore, we found a one-parameter familyof solutions which contains the NW background ( η = 0) and the four-dimensional Minkowskispacetime ( η = ±
1) as specific cases.
The main purpose of this paper is to point out that the target space of a WZW modelcan be used as a seed solution to generate a chain of solutions through the PL T -plurality.The generalized frame fields E AI and the constant metric ˆ H AB given in Eq. (2.4) constructthe WZW background, and additionally E AI satisfy the algebra [ E A , E B ] D = −F ABC E C associated with the 2 D -dimensional algebra F abc = H abc , F abc = F bca = F cab = f abc , F abc = F cab = F bca = 0 = F abc . (5.1)If we find an O( D, D ) transformation which transforms this algebra into a Drinfel’d double, weobtain a PL symmetric background, and we can construct further PL symmetric backgroundsby following the standard procedure of the PL T -plurality. As demonstrations, we studied thePL T -plurality for two WZW backgrounds, AdS with H -flux and the NW background. A similar situation has been observed in [32] and studied in more detail in [33].
24n the case of the AdS 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 foursolutions through a coordinate transformation or the standard Abelian T -duality:Sol1: d s = l d z − d t + d x z , H = ± l − ∗ , Φ = 0 , (5.2)Sol2: d s = d x + 4 (cid:0) d y + y d x (cid:1) d z , β = (cid:0) y ∂ x − ∂ z (cid:1) ∧ ∂ y , d ( x ) = − x , (5.3)Sol3: d s = 2 d x d y + d z , B = 0 , Φ = ± z , (5.4)Sol4: d s = l d z + 2 d x + d x − z + λ (d x − ) , B = d x + ∧ ( l d x − + 2 z d z ) z , e − = z , I = ∂ + ( λ ∈ R ) . (5.5)The first and the third solutions are the familiar solution and the last one is the solution of thegeneralized supergravity equations of motion known in [16]. The second one is an interestingAdS solution with a traceful constant Q -flux, which is non-Riemannian. This seems to be anew solution, but as we explained around Eq. (3.22), a B -shift makes this solution to a flatsolution with a linear dilaton, and this is essentially the same as the third solution.The correspondence between the Manin triples and the solutions can be summarized asDD2: ( .i | | | {z } Sol1 ∼ = ( | .i | | {z } Sol2 ∗ ∼ = ( | .iii | | {z } Sol3 ∗ ∼ = ( .iii | | | {z } Sol3 ∗ , DD7: SL(2) WZW | {z }
Sol1 ∼ = ( | | | {z } Sol4 ∼ = ( | | | {z } Sol1 ∼ = ( | .iii | | {z } Sol1 ∗ ∼ = ( .iii | | | {z } Sol4 ∼ = ( | .i | − | {z } Sol4 ∼ = ( .i | | − | {z } Sol1 , (5.6)where SL(2) WZW represents the algebra (3.39) and ∗ denotes that Abelian T -duality wasrequired in order to bring the non-Riemannian backgrounds into the Riemannian frame or tomake solutions of the generalized supergravity to the standard solutions.In the case of the NW background, we considered four Manin triples,NW ∼ = ( E c | A ) ∼ = ( A | E c ) ∼ = ( G | G ∼ = ( G | G , (5.7)where NW denotes the algebra (4.8). We found that NW, ( E c | A ), and ( G | G
2) correspondto the same NW background while ( A | E c ) and ( G | G
1) correspond to a non-Riemannianbackground of the form (4.27). This non-Riemannian background (4.27) can be regarded as aflat space with a constant Q -flux, and is a kind of T -fold if we make some periodic identificationof the spatial direction.The most interesting solution will be the one obtained by the YB deformation based onthe modified CYBE. In this case, we found a one-parameter family of solutions which containsthe NW background and the flat Minkowski space as particular cases.25e can apply our procedure to other WZW models, such as the WZW model based onthe Heisenberg group H f = 1 , f = − , f = − . (5.8)In [6], this WZW background was realized as a PL symmetric background by considering aDrinfel’d double f = 1 , f = − , f = − , f = 1 . (5.9)On the other hand, in our approach, the flux algebra is given by F = 1 , F = − , F = − , F = 1 . (5.10)Since the two eight-dimensional algebras (5.9) and (5.10) are inequivalent, our procedure willprovide another way to construct the H WZW background based on a new Drinfel’d algebra,and will give a new family of dual geometries. Of course, our procedure is not applicable toall of the WZW model. For example, if we consider the SU(2) WZW model, we may startwith the algebra F = 1 , F = 1 , F = 1 , F = 1 . (5.11)However, it seems to be impossible to map this algebra into any Lie algebra of a Drinfel’ddouble 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 cantake advantage of this situation [18]. For example, one may exploit the self-duality under PL T -plurality in order to search for a new D-brane configuration in a WZW background. Anembedding of a D-brane can be characterized by the boundary condition of the open string,and the boundary condition on the endpoints of the open string can be characterized by thegluing matrix R mn which relates the left-/right-moving derivatives on the worldsheet ∂ − x m = R mn ∂ + x n . (5.12)The gluing matrix for the NW model was studied in [35] and a similar analysis was done forthe AdS with H -flux in [36], and several D-brane configurations were found in these WZWbackgrounds. Subsequently, the transformation rule of the gluing matrix transforms under thePL T -duality was found in [37] and its extension to the PL T -plurality was found in [38]. As wefound in this paper, the WZW backgrounds are self-dual under several PL T -pluralities, and anaive expectation is that we can find new D-brane configurations by mapping the known gluing26atrices through the PL T -plurality transformations. Moreover, since the AdS backgroundis related to the flat space with the linear dilaton, it is also interesting to map the D-braneconfigurations in these spaces onto each other. In addition, our PL T -plurality produced manynon-Rimannian backgrounds, but D-branes in these backgrounds have not been studied. Thenit will be interesting to study various D-brane configurations in non-Riemannian backgroundsusing 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 alsobe applied to the non-Abelian U -duality [39–44]. For example, in the E n ( n ) exceptional fieldtheory (EFT) with n ≤
4, we may consider a solution of 11D supergravity where the internalparts of the supergravity fields are given by g ij = r ai r bj ˆ g ab , F = F abcd r a ∧ · · · ∧ r d ( F ≡ d C ) , (5.13)where F abcd is constant. In this case, we can construct the (weightful) generalized metric as M IJ = E I A E J B ˆ M AB by using the generalized frame fields E AI = E aIE a a I √ = e ia − e ka C ki i √ r a [ i r a i ] , (5.14)and a constant matrix ˆ M AB ∈ SL(5) . We can easily check that this set of the generalizedframe fields satisfies the algebra [ E A , E B ] E = − X ABC E C , (5.15)where [ · , · ] E is the generalized Lie derivative in EFT and the structure constants are given by X abc = f abc , X abc c = F abc c , X ab b c c = 4 f ad [ b δ b ] dc c ,X a a bc c = 6 f [ c c [ a δ a ] b ] , X ab b c = X a a bc = X a a b b C = 0 . (5.16)Under an SL(5) rotationˆ M AB → ˆ M ′ AB ≡ C AC C BD ˆ H CD , X ABC → X ′ ABC ≡ C AD C BE ( C − ) F C X DEF , (5.17)the components X abcd may vanish and the new algebra can be regarded as an exceptionalDrinfel’d algebra [39, 40]. In that case, we can construct the new generalized frame fields E ′ AI that satisfy the algebra (5.16) for the structure constants X ′ ABC . Then we obtain the dualsolution M ′ IJ = E ′ I A E ′ J B ˆ M ′ AB , similar to the PL T -plurality discussed in this paper. Acknowledgments
This work is supported by JSPS Grant-in-Aids for Scientific Research (C) 18K13540 and (B)18H01214. 27 eferences [1] X. C. de la Ossa and F. Quevedo, “Duality symmetries from nonAbelian isometries instring theory,” Nucl. Phys. B , 377 (1993) [hep-th/9210021].[2] C. Klimcik and P. Severa, “Dual nonAbelian duality and the Drinfeld double,” Phys.Lett. B , 455 (1995) [hep-th/9502122].[3] C. Klimcik, “Poisson-Lie T duality,” Nucl. Phys. Proc. Suppl. , 116 (1996) [hep-th/9509095].[4] R. Von Unge, “Poisson Lie T plurality,” JHEP , 014 (2002) [hep-th/0205245].[5] A. Y. Alekseev, C. Klimcik and A. A. Tseytlin, “Quantum Poisson-Lie T duality andWZNW model,” Nucl. Phys. B , 430 (1996) [hep-th/9509123].[6] A. Eghbali and A. Rezaei-Aghdam, “Poisson Lie symmetry and D-branes in WZW modelon the Heisenberg Lie group H ,” Nucl. Phys. B , 165 (2015) [arXiv:1506.06233 [hep-th]].[7] A. Eghbali, L. Mehran-nia and A. Rezaei-Aghdam, “BTZ black hole from Poisson–Lie T-dualizable sigma models with spectators,” Phys. Lett. B , 791 (2017)[arXiv:1705.00458 [hep-th]].[8] A. Eghbali, “Exact conformal field theories from mutually T-dualizable σ -models,” Phys.Rev. D , no. 2, 026001 (2019) [arXiv:1812.07664 [hep-th]].[9] W. Siegel, “Two vierbein formalism for string inspired axionic gravity,” Phys. Rev. D ,5453 (1993) [hep-th/9302036].[10] W. Siegel, “Superspace duality in low-energy superstrings,” Phys. Rev. D , 2826 (1993)[hep-th/9305073].[11] W. Siegel, “Manifest duality in low-energy superstrings,” hep-th/9308133.[12] C. Hull and B. Zwiebach, “Double Field Theory,” JHEP , 099 (2009)[arXiv:0904.4664 [hep-th]].[13] D. Geissbuhler, D. Marques, C. Nunez and V. Penas, “Exploring Double Field Theory,”JHEP , 101 (2013) [arXiv:1304.1472 [hep-th]].[14] F. Hassler, “Poisson-Lie T-Duality in Double Field Theory,” Phys. Lett. B , 135455(2020) [arXiv:1707.08624 [hep-th]]. 2815] S. Demulder, F. Hassler and D. C. Thompson, “Doubled aspects of generalised dualitiesand integrable deformations,” JHEP , 189 (2019) [arXiv:1810.11446 [hep-th]].[16] Y. Sakatani, “Type II DFT solutions from Poisson-Lie T-duality/plurality,” PTEP,073B04 (2019) [arXiv:1903.12175 [hep-th]].[17] C. R. Nappi and E. Witten, “A WZW model based on a nonsemisimple group,” Phys.Rev. Lett. , 3751 (1993) [hep-th/9310112].[18] C. Klimcik and P. Severa, “Open strings and D-branes in WZNW model,” Nucl. Phys.B , 653 (1997) [hep-th/9609112].[19] G. Arutyunov, S. Frolov, B. Hoare, R. Roiban and A. A. Tseytlin, “Scale invariance ofthe η -deformed AdS × S superstring, T-duality and modified type II equations,” Nucl.Phys. B , 262 (2016) [arXiv:1511.05795 [hep-th]].[20] A. A. Tseytlin and L. Wulff, “Kappa-symmetry of superstring sigma model and general-ized 10d supergravity equations,” JHEP , 174 (2016) [arXiv:1605.04884 [hep-th]].[21] C. M. Hull, “A Geometry for non-geometric string backgrounds,” JHEP , 065 (2005)[hep-th/0406102].[22] C. Klimcik, “Yang-Baxter sigma models and dS/AdS T duality,” JHEP , 051 (2002)[hep-th/0210095].[23] L. Snobl and L. Hlavaty, “Classification of six-dimensional real Drinfeld doubles,” Int. J.Mod. Phys. A , 4043 (2002) [math/0202210 [math-qa]].[24] L. Hlavaty and L. Snobl, “Poisson-Lie T plurality of three-dimensional conformally in-variant sigma models,” JHEP , 010 (2004) [hep-th/0403164].[25] K. Lee and J. H. Park, “Covariant action for a string in ”doubled yet gauged” spacetime,”Nucl. Phys. B , 134 (2014) [arXiv:1307.8377 [hep-th]].[26] K. Morand and J. H. Park, “Classification of non-Riemannian doubled-yet-gauged space-time,” Eur. Phys. J. C , no. 10, 685 (2017) Erratum: [Eur. Phys. J. C , no. 11, 901(2018)] [arXiv:1707.03713 [hep-th]].[27] K. Cho and J. H. Park, “Remarks on the non-Riemannian sector in Double Field Theory,”Eur. Phys. J. C , no. 2, 101 (2020) [arXiv:1909.10711 [hep-th]].[28] J. Sakamoto, Y. Sakatani and K. Yoshida, “Weyl invariance for generalized super-gravity backgrounds from the doubled formalism,” PTEP , no. 5, 053B07 (2017)[arXiv:1703.09213 [hep-th]]. 2929] H. Kyono and K. Yoshida, “Yang–Baxter invariance of the Nappi–Witten model,” Nucl.Phys. B , 242 (2016) [arXiv:1511.00404 [hep-th]].[30] F. Delduc, M. Magro and B. Vicedo, “Integrable double deformation of the principalchiral model,” Nucl. Phys. B , 312 (2015) [arXiv:1410.8066 [hep-th]].[31] D. L¨ust and D. Osten, “Generalised fluxes, Yang-Baxter deformations and the O(d,d)structure of non-abelian T-duality,” JHEP , 165 (2018) [arXiv:1803.03971 [hep-th]].[32] J. Sakamoto and Y. Sakatani, “Local β -deformations and Yang-Baxter sigma model,”JHEP , 147 (2018) [arXiv:1803.05903 [hep-th]].[33] L. Wulff, “Trivial solutions of generalized supergravity vs non-abelian T-duality anomaly,”Phys. Lett. B , 417 (2018) [arXiv:1803.07391 [hep-th]].[34] F. Bascone, F. Pezzella and P. Vitale, “Poisson-Lie T-Duality of WZW Model via CurrentAlgebra Deformation,” JHEP , 060 (2020) [arXiv:2004.12858 [hep-th]].[35] S. Stanciu and A. A. Tseytlin, “D-branes in curved space-time: Nappi-Witten back-ground,” JHEP , 010 (1998) [hep-th/9805006].[36] S. Stanciu, “D-branes in an AdS(3) background,” JHEP , 028 (1999) [hep-th/9901122].[37] C. Albertsson and R. A. Reid-Edwards, “Worldsheet boundary conditions in Poisson-LieT-duality,” JHEP , 004 (2007) [hep-th/0606024].[38] C. Albertsson, L. Hlavaty and L. Snobl, “On the Poisson-Lie T-plurality of boundaryconditions,” J. Math. Phys. , 032301 (2008) [arXiv:0706.0820 [hep-th]].[39] Y. Sakatani, “ U -duality extension of Drinfel’d double,” PTEP , no. 2, 023B08 (2020)[arXiv:1911.06320 [hep-th]].[40] E. Malek and D. C. Thompson, “Poisson-Lie U-duality in Exceptional Field Theory,”JHEP , 058 (2020) [arXiv:1911.07833 [hep-th]].[41] C. D. A. Blair, D. C. Thompson and S. Zhidkova, “Exploring Exceptional Drinfeld Ge-ometries,” JHEP , 151 (2020) [arXiv:2006.12452 [hep-th]].[42] E. Malek, Y. Sakatani and D. C. Thompson, “E exceptional Drinfel’d algebras,” JHEP2101