Poisson-Lie T-plurality revisited. Is T-duality unique?

We investigate (non-)Abelian T-duality from the perspective of Poisson-Lie T-plurality. We show that sigma models related by duality/plurality are given not only by Manin triples obtained from decompositions of Drinfel'd double, but also by their particular embeddings, i.e. maps that relate bases of these decompositions. This allows us to get richer set of dual or plural sigma models than previously thought. That's why we ask how T-duality is defined and what should be the `canonical' duality or plurality transformation.


Introduction
The notion of (non-)Abelian T-duality [1,2,3] of sigma models relies on the presence of symmetries of the sigma model backgrounds. Whenever there is such a symmetry, one may gauge it to arrive at a model related to the original one by T-duality. This technique, extended to RR elds in [4,5], is used frequently to generate new supergravity solutions, see e.g. [6,7] and references therein. Non-Abelian T-duality, however, does not preserve the symmetries, and it may not be possible to return back to the initial model.
PoissonLie T-duality, introduced in the seminal paper [8] by Klim£ík and evera, treats both models equally and oers a remedy to this issue.
The algebraic structure underlying PoissonLie T-duality is the Drinfel'd double, a Lie group D that decomposes into two Lie subgroups G and G of equal dimension. In case of (non-)Abelian T-duality the former represents group of symmetries of the initial sigma model, while the latter is Abelian. There are also Drinfel'd doubles where both G and G are non-Abelian. In such a case the symmetry of the initial model is replaced by the so-called PoissonLie symmetry (or generalized symmetry), see [9], and the full PoissonLie T-duality transformation applies. Nevertheless, the presence of symmetries remains crucial if one wants to dualize a particular background [10]. Recently (PoissonLie) T-duality also appears as an important tool in the study of integrable models and their deformations [11,12,13].
Since duality exchanges roles of G and G , we may understand it in terms of Drinfel'd double as a switch between decompositions (G | G ) and ( G |G ) of D. The authors of [8] mention the fact that a Drinfel'd double D can have other decompositions (K| K), ( K|K), . . . beside (G | G ) and ( G |G ). All these decompositions can be used to construct mutually related sigma models. The transformation of the initial model constructed by decomposition (G | G ) to a model constructed by (K| K) was later denoted PoissonLie T-plurality [14].
where tensor F = G + B denes metric and torsion potential of the target manifold. Assume that there is a d-dimensional Lie group G with free action on M that leaves the tensor invariant. The action of G is transitive on its orbits, hence we may locally consider M ≈ (M /G ) × G = N × G , and introduce adapted coordinates where s δ label the orbits of G and are treated as spectators and g a are group coordinates [19,20]. Dualizable sigma model on N × G is given by tensor eld F dened by n × n matrix E(s) as where e(g) is d × d matrix of components of right-invariant MaurerCartan form (dg)g −1 on G .
Using non-Abelian T-duality one can nd dual sigma model on N × G , where G is Abelian subgroup of semi-Abelian Drinfel'd double D that splits into subgroups G and G . The necessary formulas will be given in the following subsection as a special case of PoissonLie T-plurality. In papers [21,22,23], non-Abelian T-duals of sigma model in at torsionless four-dimensional background were constructed. The groups G were then subgroups of the Poincaré group [24].

PoissonLie T-plurality with spectators
For certain Drinfel'd doubles several decompositions may exist. Suppose that D = (G | G ) splits into another pair of subgroups G andḠ . Then we can apply the full framework of PoissonLie T-plurality [8,14] and nd sigma model on N × G .
The 2d-dimensional Lie algebra d of the Drinfel'd double D is equipped with an ad-invariant non-degenerate symmetric bilinear form ., . . Let d = g ⊕g and d = g ⊕ḡ be two decompositions (Manin triples (d, g,g) and (d, g,ḡ)) of d into subalgebras that are maximally isotropic with respect to ., . . The pairs of mutually dual bases T a ∈ g, T a ∈g and T a ∈ g,T a ∈ḡ, a = 1, . . . , d, satisfying are related by transformation where C is an invertible 2d × 2d matrix. Due to ad-invariance of the bilinear form ., . the algebraic structure of d is given both by and Given the structure constants F k ij of d = g ⊕g and F k ij of d = g ⊕ḡ, the matrix C has to satisfy equation 1 To preserve the bilinear form ., . and thus (4), C also has to satisfy where B ab are components of matrix B that can be written in block form as In other words, C is an element of O(d, d) but, unlike the case of Abelian T-duality, not every element of O(d, d) is allowed in (5). For the following formulas it will be convenient to introduce d×d matrices P, Q, R, S as To accommodate the spectator elds we have to extend these to n×n matrices It is also advantageous to introduce block form of E(s) as The sigma model on N × G related to (3) via PoissonLie T-plurality is given by tensor F(s,ĝ) that is calculated as 1 Conditions on C's are more restrictive than those for NATD group investigated in [25] (We are grateful to D. Osten for bringing our attention to this paper) but as said in the Introduction, our main goal is to present dependence of geometrical properties of the PoissonLie plural sigma models on matrices C.
matrices b(ĝ) and a(ĝ) are submatrices of the adjoint representation Therefore, it is necessary that These formulas reduce to formulas for PoissonLie T-duality if we choose If there are no spectators, i.e. if n = d, the plurality is called atomic.

Equivalence of transformation matrices
Tensors F, F are expressed by formulas (3) and (9) in particular bases of subalgebras g, g of Manin triples (d, g,g) and (d, g,ḡ). However, both initial and dual/plural tensor do not depend on the choice of bases in g or g. Their geometric properties are thus independent as well.
Let us consider automorphisms of both Manin triples given by linear transformations of g and g that preserve their algebraic structure. Let A and B be d × d matrices that transform bases T a and T a . Transformations (5) of the form then preserve the algebraic structure (6), (7) and duality (4) of bases (T, T ) and ( T ,T ). Transformations (13) induce changes in matrices E(s) and E(s) that are used in construction of background tensors. We have If the relation of bases in Manin triples is written as in (8), then it is easy to check that 3 Sigma models with two-dimensional target space In this section we shall consider atomic PoissonLie T-plurality of sigma models whose target space is a two-dimensional solvable Lie group G with generators T 1 , The trace f k ik of the structure constants is not zero since f 2 12 = 1 and it is known that this leads to mixed gauge and gravitational anomaly [26] in the dual model. Yet, it is worth considering such groups in the context of integrable models [27] and generalized supergravity [28]. We parametrize the elements g ∈ G as g = e g 1 T 1 e g 2 T 2 . Since there are no spectators, the matrix E(s) is constant. Choosing it in the form we nd that the background tensor F(g 1 , g 2 ) calculated according to the formula (3) is given by One can verify that F is invariant with respect to G and its symmetric part G is at metric. Since the target manifold is two-dimensional, the torsion H = dB of all the backgrounds discussed in this section vanishes. Therefore, the B-eld can be eliminated by a gauge transformation and the only relevant part of F is the metric G.

PoissonLie T-plurality
In order to nd PoissonLie T-dual or plural models associated to (16), we The algebraic structure of four-dimensional Drinfel'd doubles was studied in [17], where it was shown that for such Drinfel'd double there are two nonequivalent Manin triples: • Semi-Abelian triple d = g ⊕g with dual basis (T 1 , T 2 , T 1 , T 2 ) and Lie brackets (only nontrivial brackets are displayed) or • Type B non-Abelian triple d = g ⊕ḡ with dual basis ( T 1 , T 2 ,T 1 ,T 2 ) and Lie brackets Map relating both bases mentioned in the paper [17] preserves (4) and transforms Lie brackets (17) into (18). However, there are two dierent classes of linear maps (5) given by matrices that do the same and that dene much richer set of decompositions 2 . Note that det C 1 = −1 and det C 2 = 1. We will show that using these two maps to generate models plural to (1) with F given by (16) one gets substantially dierent models.
3 We assume that elements of G are parametrized asĝ = eĝ 1 T1 eĝ 2 T2 . This is a background with at metric, so indeed, using PoissonLie Tplurality with two dierent maps (20) and (21) we get two dierent sigma models. This essential dierence of curvature properties of the backgrounds remains true for any choice of b 1 , b 2 , b 3 .
Similar results are obtained if we consider plural sigma models onḠ .
In that case we use transformations between bases of semi-Abelian Manin triple and a dual to type-B Manin triple, i.e. the matrices (20) and (21) are multiplied from the left by the exchange matrix

PoissonLie T-duality
Maybe surprisingly, we observe the same phenomenon for PoissonLie Tduality as well. Dual sigma models can be obtained by exchange of Manin triples (d, g,g) and (d,g, g) mediated by the matrix (22). On the other hand, there are more general maps between bases of the semi-Abelian Manin triple and its dual. Linear maps on d that switch the roles of T a and T a in (17) and meanwhile preserve the duality of the bases (4) are given by automorphisms (5) where the matrix C is either with determinants of D 1 and D 2 equal to ±1. Note that D 1 is equal to the exchange matrix (22) for b 1 = 1, b 2 = b 3 = 0. Inserting each of these matrices into (9)(11) one gets again two dierent dual models.
The rst model, obtained using (23) and parametrizationg = eg 1 T 1 eg 2 T 2 , has at background F(g 1 ,g 2 ) = 0 , (25) while background obtained using (24) is given by tensor (26) with nonzero scalar curvature One can see that once more we get two dierent PoissonLie T-dual sigma models no matter what the parameters b 1 , b 2 and b 3 are. It is well known that (non-)Abelian T-duality is induced by matrix D 0 that is a special case of D 1 . In fact, dualizing F using D 0 we get tensor F (g 1 ,g 2 ) = 0 that can be brought to the form (25) by coordinate transformatioñ Alternatively, it can be obtained by gauging the (non-)Abelian isometry and introduction of Abelian Lagrange multipliers. We shall investigate whether the duality induced by matrix D 2 can be obtained in a similar way. Matrix D 2 can be transformed by automorpisms (13), (14) to the form Up to the change of sign necessary for being an automorphism of semi-Abelian Manin triple, matrix D 2 acts by switching T 1 ↔ T 1 . One may suspect that this can actually be Buscher duality with respect to one-dimensional subgroup 4 of isometry group G . This subgroup is generated by left-invariant vector eld V 1 = ∂ g 1 − g 2 ∂ g 2 that together with V 2 = ∂ g 2 satises (15).
To check our suspicion we have to nd adapted coordinates {s 1 , h 1 } such that V 1 becomes V 1 = ∂ h 1 and F becomes independent of h 1 . Suitable transformation of coordinates is given by Tensor (16) is then transformed to the form Treating s 1 as spectator we may dualize F with respect to h 1 . Buscher rules that follow from (9)(11) give tensor and subsequent change of coordinates Alternative formulation of duality given by matrix (27) follows from gauge invariant parent action Integrating out gauge elds A + and A − we obtain sigma model with background tensor (30) that can be brought to the form (26) by coordinate transformation.
One may also ask why Buscher duality with respect to V 2 is not included in (23) or (24). The reason is that change of bases is not an automorphism of the Manin triple given by (15) and [ T 1 , T 2 ] = 0.
Using (31) in atomic PoissonLie T-plurality transformation of the model given by (16) we get a sigma model in at background. The example, however, is not particularly illuminating since the condition (12) is not satised for (32) and plural background cannot be calculated. For further investigation we focus on sigma models in four-dimensional target space and introduce spectator elds.
In the papers [22], [23] PoissonLie T-duality with spectators was used to study non-Abelian T-duals of sigma models in at Minkowski space. Given the metric η = diag(−1, 1, 1, 1) in coordinates {t, x, y, z}, we consider Killing vectors T 1 := K 3 = z∂ t + t∂ z , These vectors generate a solvable two-dimensional group G of symmetries of the background η. The coordinates {s 1 , s 2 , g 1 , g 2 } given by 5 t = 1 2 e −g 1 |s 1 | sgn(s 1 ) + e 2g 1 g 2 2 + 1 , PoissonLie T-plurality. The rst, obtained using (31), is given by background F(s 1 , s 2 , g 1 , that is at and torsionless for any values The action of G is not free and transitive for t + z = 0, i.e. for s 1 = 0. We have to restrict our calculations to coordinate patches with s 1 = 0. 6 For further details concerning the process of nding the adapted coordinates see [22] where this particular case was denoted S 2,20 . 7 Note that (31) reduces to identity matrix for b 1 = 1, b 2 = b 3 = 0. Consequently, (35) reduces to (34).
ground, obtained using (32), is given by tensor Similarly to the previous case, the torsion vanishes. However, the symmetric part of (36), i.e. the metric, has vanishing scalar curvature but nontrivial Ricci tensor. Using transformation of coordinates one can bring this background to a pp-wave in the Brinkmann form identied in [22,23] to be one of the gauged WZW models considered in [29,30]. We see that we again get two dierent sigma models, this time produced by PoissonLie T-pluralities that do not change the Manin triple.
That means that all PoissonLie T-dualities described in subsection 3.2 can be obtained by canonical non-Abelian T-duality of sigma models generated by PoissonLie identities of the initial model. Writing (5) as  (17) and (18) by i.e. by the matrix C 1 we then have C 1 = C 0 · I 1 , C 2 = C 0 · I 2 and all PoissonLie T-pluralities from subsection 3.1 can be obtained by PoissonLie T-plurality of sigma models generated by (38) and PoissonLie identities of the initial model. Unfortunately, the choice (38) is by far not unique, and, as opposed to duality, we can hardly call it canonical Poisson Lie T-plurality.
We have obtained similar results for PoissonLie T-pluralities of sigma models embedded in six-dimensional Drinfel'd doubles.

Conclusion
The examples presented in sections 3 and 4 prove that families of sigma models related by PoissonLie T-plurality may depend not only on the algebraic structure of Manin triple but also on the way how the given Manin triple is embedded in the Drinfel'd double.
This holds for the PoissonLie T-duality as well, so we may ask what should be considered as true T-duality. Is it only the procedure introduced in [1, 2, 3] that uses gauging of initial sigma model and is alternatively described in [8] as an exchange of dual bases T a and T a of the isotropic subalgebras of Manin triple? Or can we admit any linear transformation of bases of the Drinfel'd double that give decompositions isomorphic to the Manin triple obtained by the exchange?
Possible answer to the question above follows from expressions (3) and (9) of both initial and dual/plural tensor. Namely, from these expressions we can see that tensors F and F depend both on algebraic structure of Manin triple and bilinear forms E(s), E(s). PoissonLie identity does not change the initial Manin triple (only its embedding in the Drinfel'd double) but changes the bilinear form E(s) according to the formula (11). Subsequent canonical duality or plurality then changes the Manin triple and produce PoissonLie T-dual/plural tensor. Moreover, it turns out that non-canonical dualities may be hidden canonical dualities with respect to subgroups of the initial group of isometries.