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.


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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 Poisson-Lie T-plurality [14]. Examples of sigma models related by Poisson-Lie T-plurality were studied e.g. in [14,15], and decompositions of low-dimensional Drinfel'd doubles were classified in [16][17][18].
The goal of this paper is to show, using simple examples, that sigma models related by Poisson-Lie T-duality/plurality are given not only by the algebraic structure of decompositions of Lie algebra of the Drinfel'd double into Manin triples, but also by the particular embedding of Manin triples, i.e. maps that relate bases in various decompositions. For this purpose we shall consider the simplest possible case of Drinfel'd double accomodating plurality, i.e. a four-dimensional semi-Abelian Drinfel'd doubles.
After summarizing Poisson-Lie T-plurality in section 2 we identify transformations that yield equivalent sigma model backgrounds. In section 3 we develop examples of dual/plural models whose geometric properties depend on the choice of matrices transforming bases of Manin triples, and, in section 4, we show that nonequivalent models can be obtained even if we do not change the Manin triple at all. We study this "Poisson-Lie T-identity" further in section 5 trying to identify what the "canonical" duality/plurality should be.

Poisson-Lie T-plurality of sigma models
Let M be n-dimensional (pseudo-)Riemannian target manifold and consider sigma model on M given by Lagrangian L = ∂ − φ µ F µν (φ)∂ + φ ν , φ µ = φ µ (x + , x − ), µ = 1, . . . , n (2.1) where tensor F = G + B defines 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 field F defined by n × n matrix E(s) as

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Using non-Abelian T-duality one can find 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 Poisson-Lie T-plurality. In papers [21][22][23], non-Abelian T-duals of sigma model in flat torsionless four-dimensional background were constructed. The groups G were then subgroups of the Poincaré group [24].

Poisson-Lie 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 Poisson-Lie T-plurality [8,14] and find sigma model on N × G .
The 2d-dimensional Lie algebra d of the Drinfel'd double D is equipped with an adinvariant 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 C p a C q b F r pq = F c ab C r c . To preserve the bilinear form ., . and thus (2.4), C also has to satisfy where B ab are components of matrix B that can be written in block form 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 Poisson-Lie plural sigma models on matrices C.

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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 (2.5). For the following formulas it will be convenient to introduce d×d matrices P, Q, R, S as To accommodate the spectator fields 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 (2.3) via Poisson-Lie T-plurality is given by tensor F(s,ĝ) that is calculated as where E(s,ĝ) = (1 + E(s) · Π(ĝ)) −1 · E(s), , (2.10) matrices b(ĝ) and a(ĝ) are submatrices of the adjoint representation 11) and the matrix E(s) is obtained by formula Therefore, it is necessary that These formulas reduce to formulas for Poisson-Lie T-duality if we choose P = S = 0 d and Q = R = 1 d . Furthermore, for a semi-Abelian Drinfel'd double the well-known Buscher rules for non-Abelian T-duality are restored. 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 (2.3) and (2.9) 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 (2.5) of the form then preserve the algebraic structure (2.6), (2.7) and duality (2.4) of bases (T, T ) and ( T ,T ). Transformations (2.14) 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 (2.8), then it is easy to check that (2.16) Therefore, matrices define transformations between sigma model backgrounds F and F in various coordinates. That's why, from the perspective of Poisson-Lie T-plurality, they can be considered equivalent.

Sigma models with two-dimensional target space
In this section we shall consider atomic Poisson-Lie T-plurality of sigma models whose target space is a two-dimensional solvable Lie group G with generators T 1 , T 2 satisfying

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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 find that the background tensor F(g 1 , g 2 ) calculated according to the formula (2.3) is given by One can verify that F is invariant with respect to G and its symmetric part G is flat metric.
Since the target manifold is two-dimensional, the torsion H = dB of all the backgrounds discussed in this section vanishes. Therefore, the B-field can be eliminated by a gauge transformation and the only relevant part of F is the metric G.

Poisson-Lie T-plurality
In order to find Poisson-Lie T-dual or plural models associated to 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) Map relating both bases

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mentioned in the paper [17] preserves (2.4) and transforms Lie brackets (3.3) into (3.4). However, there are two different classes of linear maps (2.5) given by matrices that do the same and that define 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 (2.1) with F given by (3.2) one gets substantially different models. The first one, obtained from (2.9)-(2.12) and (3.6), is given by background 3 The other one, obtained using (3.7), is given by This is a background with flat metric, so indeed, using Poisson-Lie T-plurality with two different maps (3.6) and (3.7) we get two different sigma models. This essential difference 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 (3.6) and (3.7) are multiplied from the left by the exchange matrix One can simplify these matrices by (2.14), (2.16) and choose e.g. b1 = 1, b2 = b3 = 0 for C1 and b2 = 0 for C2. To get the map (3.5) one has to choose b1 = b2 = 1, b3 = 0. 3 We assume that elements of G are parametrized asĝ = eĝ 1 T 1 eĝ 2 T 2 .

Poisson-Lie T-duality
Maybe surprisingly, we observe the same phenomenon for Poisson-Lie T-duality as well. Dual sigma models can be obtained by exchange of Manin triples (d, g,g) and (d,g, g) mediated by the matrix (3.9). 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 (3.3) and meanwhile preserve the duality of the bases (2.4) are given by automorphisms (2.5) 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 (3.9) for b 1 = 1, b 2 = b 3 = 0. Inserting each of these matrices into (2.9)-(2.12) one gets again two different "dual" models. The first model, obtained using (3.10) and parametrizationg = eg 1 T 1 eg 2 T 2 , has flat background while background obtained using (3.11) is given by tensor with nonzero scalar curvature (3.14) One can see that once more we get two different Poisson-Lie 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

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that can be brought to the form (3.12) 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 (2.14), (2.16) 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 field V 1 = ∂ g 1 − g 2 ∂ g 2 that together with V 2 = ∂ g 2 satisfies (3.1).
To check our suspicion we have to find 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 (3.2) 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 (2.9)-(2.12) give tensor and subsequent change of coordinates Alternative formulation of duality given by matrix (3.15) follows from gauge invariant parent action Dualities of this form are called factorised in [25].

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where Integrating out gauge fields A + and A − we obtain sigma model with background tensor (3.18) that can be brought to the form (3.13) by coordinate transformation. One may also ask why Buscher duality with respect to V 2 is not included in (3.10) or (3.11). The reason is that change of bases is not an automorphism of the Manin triple given by (3.1) and [ T 1 , T 2 ] = 0.

Sigma models with four-dimensional target space
One can also ask if there are some "Poisson-Lie identities" preserving the structure of semi-Abelian Manin triple, i.e. Poisson-Lie T-pluralities generated by automorphisms of d = g ⊕g that preserve both Lie brackets (3.3) and duality of the basis (2.4). The answer is positive, and the mappings can have two possible forms given by matrices and Using (4.1) in atomic Poisson-Lie T-plurality transformation of the model given by (3.2) we get a sigma model in flat background. The example, however, is not particularly illuminating since the condition (2.13) is not satisfied for (4.2) and plural background cannot be calculated. For further investigation we focus on sigma models in four-dimensional target space and introduce spectator fields. In the papers [22,23] Poisson-Lie T-duality with spectators was used to study non-Abelian T-duals of sigma models in flat Minkowski space. Given the metric η = diag(−1, 1, 1, 1) in coordinates {t, x, y, z}, we consider Killing vectors

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satisfying [T 1 , T 2 ] = T 2 . 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 , from which one gets E(s) by setting g 1 = 0. The group G can be embedded into semi-Abelian Drinfel'd double with algebraic structure (3.3) allowing to find dual/plural backgrounds. 6 Inserting matrices (4.1) and (4.2) into (2.9)-(2.12) one gets two different sigma models on N × G related to the original model in background (4.5) by Poisson-Lie T-plurality. The first, obtained using (4.1), is given by background that is flat and torsionless for any values 7 of b 1 , b 2 , b 3 . The second background, obtained using (4.2), is given by tensor The action of G is not free and transitive for t + z = 0, i.e. for s1 = 0. We have to restrict our calculations to coordinate patches with s1 = 0. 6 For further details concerning the process of finding the adapted coordinates see [22] where this particular case was denoted S2,20. 7 Note that (4.1) reduces to identity matrix for b1 = 1, b2 = b3 = 0. Consequently, (4.6) reduces to (4.5).

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Similarly to the previous case, the torsion vanishes. However, the symmetric part of (4.7), 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 identified in [22,23] to be one of the gauged WZW models considered in [29,30]. We see that we again get two different sigma models, this time produced by Poisson-Lie Tpluralities that do not change the Manin triple.

Poisson-Lie T-dualities and pluralities generated by Poisson-Lie identities
Comparing transformation matrices (3.10), (3.11) that generate Poisson-Lie T-dualities and (4.1), (4.2) that generate Poisson-Lie identities one may notice that they are related by canonical duality matrix (3.9) that exchanges generators T a and T a . Indeed, it is easy to check that That means that all Poisson-Lie T-dualities described in subsection 3.2 can be obtained by canonical non-Abelian T-duality of sigma models generated by Poisson-Lie identities of the initial model. Writing (2.5) as we verify that this holds not only for the four-dimensional semi-Abelian Drinfel'd doubles, but for general Drinfel'd double (G | G ). One can try to obtain similar relation for Poisson-Lie T-pluralities described in subsection 3.1. However, question then is what is the "canonical Poisson-Lie T-plurality". Motivated by equations (5.1) we can define it for Manin triples (d, g,g) and (d,ĝ,ḡ) with (3.3) and (3.4) by

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i.e. by the matrix and all Poisson-Lie T-pluralities from subsection 3.1 can be obtained by Poisson-Lie Tplurality of sigma models generated by (5.2) and Poisson-Lie identities of the initial model. Unfortunately, the choice (5.2) 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 Poisson-Lie 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 Poisson-Lie 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 doubles.
This holds for the Poisson-Lie T-duality as well, so we may ask what should be considered as "true" T-duality. Is it only the procedure introduced in [1-3] 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 (2.3) and (2.9) 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). Poisson-Lie identity does not change the initial Manin triple (only its embedding in the Drinfel'd doubles) but changes the bilinear form E(s) according to the formula (2.12). Subsequent canonical duality or plurality then changes the Manin triple and produce Poisson-Lie 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.
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