Poisson-Lie plurals of Bianchi cosmologies and Generalized Supergravity Equations

Poisson-Lie T-duality and plurality are important solution generating techniques in string theory and (generalized) supergravity. Since duality/plurality does not preserve conformal invariance, the usual beta function equations are replaced by Generalized Supergravity Equations containing vector J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{J} $$\end{document}. In this paper we apply Poisson-Lie T-plurality on Bianchi cosmologies. We present a formula for the vector J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{J} $$\end{document} as well as transformation rule for dilaton, and show that plural backgrounds together with this dilaton and J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{J} $$\end{document} satisfy the Generalized Supergravity Equations. The procedure is valid also for non-local dilaton and non-constant J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{J} $$\end{document}. We also show that Div Θ of the non-commutative structure Θ used for non-Abelian T-duality or integrable deformations does not give correct J\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathcal{J} $$\end{document} for Poisson-Lie T-plurality.


Introduction
In recent years we have seen renewed interest in (non-)Abelian T-duality [1,2] of sigma models. Its generalization [3,4] includes RR fields and is often used to find new supergravity solutions [5,6]. Duality transformation can be performed whenever there is an isometry of the sigma model background. However, dualization with respect to non-semisimple Lie groups G does not preserve conformal invariance [7] and dual models exhibit mixed gauge and gravitational anomaly proportional to the trace of structure constants of Lie algebra g corresponding to G [8].
Non-Abelian T-duality also contributes to the study of integrable deformations [9,10] where similar problem has been dealt with [11]. After publication of [12] it became clear that integrable deformations should satisfy the so-called Generalized Supergavity Equations JHEP04(2020)068 of the flat background, and in sections 4.1, 4.2 we present metrics, B-fields and dilatons obtained by Poisson-Lie T-plurality of curved Bianchi cosmologies with nontrivial dilaton. It turns out that for appropriately defined vector field J all of them satisfy Generalized Supergravity Equations.

Basics of Poisson-Lie T-plurality
In the first two subsections we recapitulate Poisson-Lie T-plurality with spectators [18,19,27].

Sigma models
Let M be (n + d)-dimensional (pseudo-)Riemannian target manifold and consider sigma model on M given by Lagrangian where s α label the orbits of G and are treated as spectators, and x a are group coordinates. Dualizable sigma model on N ×G is given by tensor field F defined by spectator-dependent (n + d) × (n + d) matrix E(s) and group-dependent E(x) as F (s, x) = E(x) · E(s) · E T (x), E(x) = 1 n 0 0 e(x) (2.3) where e(x) is d × d matrix of components of right-invariant Maurer-Cartan form (dg)g −1 on G .
Using non-Abelian T-duality one can find dual sigma model on N × A , where A is Abelian subgroup of semi-Abelian Drinfel'd double D = (G |A ). The necessary formulas will be given in the following subsection as a special case of Poisson-Lie T-plurality.
In this paper the groups G will be non-semisimple Bianchi groups. Their elements will be parametrized as g = e x 1 T 1 e x 2 T 2 e x 3 T 3 where e x 2 T 2 e x 3 T 3 and e x 3 T 3 parametrize their normal subgroups. Bianchi cosmologies are defined on four-dimensional manifolds, hence dim N = 1 and we denote the spectator s 1 as t. JHEP04(2020)068

Formulas for Poisson-Lie T-plurality with spectators
Drinfel'd double D = (G | G ) is a 2d-dimensional Lie group whose Lie algebra d is equipped with an ad-invariant non-degenerate symmetric bilinear form ., . . The Lie algebra d decomposes into double cross sum g g of subalgebras g andg [29] that are maximally isotropic with respect to ., . giving rise to Manin triple (d, g,g). This means that mutually dual bases T a ∈ g, T a ∈g, a = 1, . . . , d, satisfy relations and due to ad-invariance of the bilinear form ., . the algebraic structure of Manin triple is given by For many Drinfel'd doubles D = (G | G ) several Manin triples may exist. Suppose that we have sigma model on N × G and the Drinfel'd double is formed by another pair of subgroups G andḠ corresponding to Manin tripleĝ ḡ. Then we can apply the full framework of Poisson-Lie T-plurality [18,19] and find backgrounds for sigma models on N × G and N ×Ḡ . Relation between groups G , G and G ,Ḡ is given by relation between the Manin triples. In terms of group elements it reads The mutually dual bases T a ∈ĝ,T a ∈ĝ, a = 1, . . . , d, satisfying then must be related to T a ∈ g and T a ∈g by linear transformation where C is an invertible 2d × 2d matrix. Given the structure constants F k ij of d = g g and F k ij of d =ĝ ḡ, the matrix C has to satisfy equation To preserve the bilinear form ., . and thus (2.4) and (2.6), C also has to satisfy where (D 0 ) ab are components of matrix D 0 that can be written in block form as

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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 (2.8).
For the following formulas it will be convenient to introduce d×d matrices P, Q, R, S as and extend these to (n + d) × (n + d) matrices to accommodate the spectator fields. 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 and matrices b(x) and a(x) are submatrices of the adjoint representation The matrix E(s) is obtained from E(s) in (2.3) by formula Analogously we get tensor dual to (2.13) on N ×Ḡ . Formulas (2.13)-(2.15) reduce to those for full Poisson-Lie duality if we choose P = S = 0 d and Q = R = 1 d . Furthermore, for a semi-Abelian Drinfel'd double the wellknown Buscher rules for (non-)Abelian T-duality are restored. If there are no spectators the plurality is called atomic.

Generalized Supergravity Equations and transformation of dilaton
Main goal of this paper is to verify whether backgrounds obtained by Poisson-Lie Tpluralities satisfy Generalized Supergravity Equations of Motion [11,12,30,31]. They can be written in different forms. We adopt convention used in [14] so the equations read 3 where ∇ µ are covariant derivatives with respect to metric G, Φ is the dilaton and vector field J will be defined bellow. For vanishing J the usual beta function equations are recovered. Formula for transformation of dilaton under Poisson-Lie T-plurality was given in [19] as where y represent coordinates of G , Problem with the formula (2.20) is that in general it is non-local in the sense that y = y(x,x) obtained by decompositions of Drinfel'd double elements This is the cause of the puzzle mentioned in [23], namely that Φ in formula (2.20) for dilaton on G depends not only on coordinatesx of the group G but also on coordinatesx ofḠ . For coordinates y depending linearly onx andx which holds for all cases bellow, the problem can be solved in the following way. We set

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and vectors J for backgrounds on N × G J α = 0, α = 1, . . . , dim N , For the dual model on N ×ḠΦ where V a ,V a are left-invariant fields of the groups G ,Ḡ andf ba c andf ba c are structure constants of their Lie algebras. Note that in case of Poisson-Lie T-plurality linear combinations of left-invariant vector fields need not be Killing vectors of plural backgrounds as they satisfy condition [18] of pluralizability with generally nonvanishing right-hand side. Nevertheless, it turns out that all nontrivial vectors J andJ found below are Killing vectors of corresponding backgrounds. For non-Abelian T-duality, authors of [24,32] calculate components J µ of vector J using G and B are symmetric and antisymmetric part of tensor F and D ν is covariant derivative with respect to Unfortunately, as we shall show on many examples below, this formula does not work in general for Poisson-Lie T-plurality.
3 Poisson-Lie T-plurality of flat background

Bianchi III cosmology
First we shall study Poisson-Lie T-pluralities that follow from the Bianchi III invariance of Minkowski metric. Consider six-dimensional semi-Abelian Drinfel'd double 5 D = (B III |A ) whose Lie algebra d = b III a is spanned by basis (T 1 , T 2 , T 3 , T 1 , T 2 , T 3 ) with nonvanishing commutation relations

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The group B III is not semisimple and trace of its structure constants does not vanish. Lie algebra d of this Drinfel'd double admits several other decompositions into Manin triples found in [22]. In adapted coordinates the Bianchi III flat cosmology is given by metric 6 (3. 2) The corresponding matrix E(s) can be found by setting y 1 = 0 in F . The background is invariant with respect to the action of Bianchi III group generated by left-invariant vector fields The metric is flat and there is no torsion so the equations (2.17)- (2.19) are satisfied if we choose dilaton Φ(y) = 0 and vector J = 0. This background was mentioned in [33] where the authors noted that its non-Abelian dual does not satisfy standard beta function equations.
3.1.1 Transformation of b III a to b III b II and to its dual As was found in [22], the algebra of the Drinfel'd double D = (B III |A ) can be decomposed into There is rather large number of solutions of equations (2.9) and (2.10) giving linear mappings (2.8) between generators of b III a and b III b II (cf. the table of Poisson-Lie identities of b III a in [16]). However, up to a change of coordinates and gauge shift of the B field, all the resulting plural backgrounds F on G are equivalent. Since the parameters 6 For typographic reasons from now on we shall use subscripts to label group coordinates.

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coming from mappings C can be eliminated in plural backgrounds by transformations of coordinates or gauge shifts, it is sufficient to consider matrix which gives rise to tensor F on G calculated from (2.13), (2.14) as The metric is not flat but its scalar curvature vanishes. It is possible to find coordinate transformation that brings the metric to the Brinkmann form of a plane parallel wave

The background is accompanied by a non-trivial torsion
This background was repeatedly found in [35][36][37] as non-Abelian T-dual of flat metric.
Let us now focus on the dilaton. Change of Manin triples given by mapping (3.6) induces transformation of coordinates of the Drinfel'd double. From (2.22) we find that As Φ (0) (y) = y 1 2 , we need only y 1 and that gives us Φ (0) =x 1 2 . From the formula (2.20) we obtain transformed dilaton Vector J given by both (2.23) and (2.26) vanishes, Generalized Supergravity Equations become standard beta function equations, and one can check that they are satisfied. For sigma model onḠ different C matrices give different backgrounds. For instance, using with non-trivial curvature and torsion. To find the dilaton and vectorJ we express y 1 from (2.22) as found by Poisson-Lie T-plurality generated by is calculated via (2.24). Formula (2.26) fails to provide correctJ as From (2.22) we find that y 1 = −x 1 and formulas (2.20) and (2.24) givē The standard beta function equations are thus satisfied. Since D ν Θ νµ = 0, − t 2 1+t 2 , 0, 0 , we see that formula (2.26) is again not applicable.

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3.1.2 Transformation of b III a to b III b IIIiii and to its dual The left-invariant fields of G andḠ are There is again rather large number of linear mappings (2.8) between b III a and b III b IIIiii depending on many parameters. The backgrounds obtained by plurality are quite complicated, hence, we shall restrict to two simple ones.
First of them is given by matrix for which tensor F has the form The background is curved and torsionless. From the change of decompositions (2.22) we express y 1 as satisfies Generalized Supergravity Equations with J = (0, 0, 0, 0) but Div Θ = (0, 0, −2, 0). Torsionless background is obtained by Poisson-Lie T-plurality given by that satisfies beta function equations, i.e. Generalized Supergravity Equations withJ = (0, 0, 0, 0). That disagrees with formula (2.26) giving Div Θ = (0, −4e −2x 2 , 0, 0). Another Poisson-Lie T-plurality is given by the matrix with nontrivial torsion. For C 2 we find that   [14,16] dealt only with semi-Abelian Drinfel'd doubles whereV a,m = δ a,m , vectorsJ were constant and proportional to the trace of structure constants so the full form of (2.24) was not necessary.

Bianchi V cosmology
Next we shall study Poisson-Lie T-pluralities that follow from the well-known Bianchi V invariance of the flat metric that in adapted coordinates reads (3.8) Authors of [7] first noticed that non-Abelian dual of this background is not conformal. Further study of the emerging gravitational-gauge anomaly was carried out in [34]. Metric The group B V is not semisimple and trace of its structure constants does not vanish. The algebra d allows several other decompositions into Manin triplesĝ ḡ. Corresponding dilaton can be again chosen Φ = 0 and transformation formulas (2.20), (2.21) now give so it is again necessary to find how y 1 depends onx,x using (2.22).

Transformation
a and to its dual Left-invariant vector fields of G andḠ are

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Mappings that transform commutation relations of Manin triple b V a to those of b V I −1 a are in general given by matrices Although backgrounds calculated from C 1 and C 2 seem rather complicated at first sight, dependence on constants c ij can be eliminated in the resulting metrics and c ij appear only in B. Moreover, the torsion vanishes in both cases, hence, up to coordinate or gauge transformations, tensors F are equivalent to i.e. Buscher duality 7 in coordinate y 3 . Tensor (3.13) together with dilaton Φ(t,x 1 ) = − ln t +x 1 found from (3.2) and y 1 = −x 1 obtained using (2.22) satisfy standard beta function equations. 7 Duality is accompanied by a change of sign inx1 that is necessary to get (3.10).

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Let us note that metric (3.13) can be brought to the Brinkmann form of a plane-parallel wave found earlier in [35][36][37] as non-Abelian T-dual of flat metric. BackgroundsF onḠ found from mappings D 0 · C 1 and D 0 · C 2 can be simplified substantially since after suitable coordinate transformation they differ from background

Transformation of
Left-invariant vector fields of G andḠ are Mappings C that transform the algebraic relations of Manin triple b V a to b V b II are given by matrices Dual of (3.16) calculated using D 0 · C 10 reads (3.20) Background obtained by transformation (3.20) is given by tensor that again gives plane parallel wave (3.17) with torsion (3.18). Dilaton that together with (3.21) satisfy beta function equations is Dual of (3.21) found from D 0 · C 20 is Left-invariant vector fields of G andḠ are Examples of mappings C that transform the algebraic structure of Manin triple b V I −1 a to b V I −1 b V ii are given by matrices Background obtained by C 1 is given by tensor (3.24) Together with the dilaton they satisfy Generalized Supergravity Equations with J = (0, 0, 2, 0), but Div Θ = (0, 0, 1, 0).
The metric is invariant with respect to the action of Bianchi V I −1 group and can be constructed by virtue of Manin triple d = b V I −1 a spanned by basis (T 1 , T 2 , T 3 , T 1 , T 2 , T 3 ) with algebraic relations Structure constants of b V I −1 are traceless. Drinfel'd double is the same as in section 3.2, where we have seen that beside ii and their duals. In this section, however, the backgrounds are different as we use different matrix E(s).
Formula (2.20) for new dilatons does not depend on coordinates y and is, therefore, applicable for any Manin triple of this Drinfel'd double.

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4.1.1 Transformation of b V I −1 a to b V a and to its dual Left-invariant vector fields of G andḠ are Mappings C that transform the algebraic structure of Manin triple b V I −1 a to b V a are given by matrices inverse to (3.11) and (3.12). After a suitable change of coordinates we find that backgrounds obtained from these general solutions differ from background obtained using factorized duality (3.14) only by a torsionless B field. Note that F is again invariant with respect to group B V . This metric is not flat and dilaton that together with (4.3) satisfy beta function equations is Φ(t,x 1 ) = βt +x 1 − ln a 2 (t).
Dual of (4.3) is given by tensor Together with dilatonΦ Now we are ready to study Poisson-Lie T-pluralities of the most complicated curved cosmology invariant with respect to Bianchi V I κ . Its Lie algebra is contained in semi-Abelian six-dimensional Manin triple 8 d = b V Iκ a spanned by basis (T 1 , T 2 , T 3 , T 1 , T 2 , T 3 ) with non-trivial algebraic relations Note that for κ = 0, or κ = 1, these are comutation relations of b III , or b V respectively. The case κ = −1 was treated separately in section 4.1. The group B V Iκ is not semisimple and trace of its structure constants does not vanish. Lie algebra of the Drinfel'd double (B V Iκ |A ) admits several other Manin triples [22]. Metric of Bianchi V I κ cosmology reads [14] F (t, where the functions a i (t) have the form and dilaton is Φ(t) = βt. The background (4.8) is invariant with respect to symmetry generated by left-invariant vector fields satisfying (4.6). The beta function equations reduce to condition The background is torsionless and for β = 0 also Ricci flat. 8 Linear transformation between basis of bV Iκ a used in this paper and vectors Xi, Xj that span Lie algebra of Drinfel'd double (6a|1) in [22] is

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Formula (2.20) for new dilatons then reads so that, it is again necessary to solve y 1 from (2.22) and find how it depends onx andx.
4.2.1 Transformation of b V Iκ a to b V Iκ b II and to its dual , which belongs to decompositions of the Lie algebra of (B V Iκ |A ), is spanned by ( T 1 , T 2 , T 3 ,T 1 ,T 2 ,T 3 ) with algebraic relations Left-invariant fields of G andḠ are There are two different linear mappings (2.8) between b V Iκ a and b V Iκ b II . One of them is given by the matrix Tensors F on G = G = B V Iκ calculated from these general forms are rather extensive. Nevertheless, up to change of coordinates and gauge shift the backgrounds coming from C 1 are equivalent to  and J = (0, 0, 0, 0) = Div Θ. Dilaton together with (4.12) satisfies beta function equations if the condition (4.10) holds. Coordinate transformation brings background onḠ = B II calculated from D 0 · C 1 to the form With the corresponding dilaton the backgroundF satisfies Generalized Supergravity Equations withJ = (0, 0, 0, 0) obtained from y 1 = −x 1 and (2.24). However, Div Θ is nontrivial and depends on a 2 (t)a 3 (t).
Transformation of Manin triple given by C 2 gives similar results. All constants c ij except c 53 and c 55 can be set to zero, and the only relevant difference is that in this case y 1 =x 1 , so the Generalized Supergravity Equations are satisfied with J = (0, 0, 0, 0) and J = (0, κ + 1, 0, 0).

Transformation of
iii and to its dual iii is another Manin triple of the Drinfel'd double (B V Iκ |A ). It corresponds to (6 a |6 1/a .iii) in [22] by the transformation (4.5). It is spanned by Left-invariant fields of G andḠ are

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There are two different Poisson-Lie T-pluralities between b V Iκ a and b V Iκ b V I −κ .iii . One of them is given by the matrix gives again rather extensive tensors and dilatons. The only relevant difference is that now y 1 =x 1 −x 3 and the Generalized Supergravity Equations are satisfied with J = (0, 0, 0, −1) andJ = (0, (κ + 1)e −x 3 , 0, 0). VectorJ does not agree with Div Θ that depends on the product a 1 (t)a 3 (t).

Conclusions
We have presented many examples of Poisson-Lie T-plurality transformation acting on flat or curved backgrounds invariant with respect to Bianchi groups B III , B V , B V Iκ and B V I −1 . Coresponding dilatons were found using formulas (2.20), (2.21). In many cases the dilatons were non-local in the sense that they were both functions of coordinatesx on G andx onḠ . We have shown how to deal with this issue if the dependency is linear. This partially resolves the puzzle explained in 2.3. It turns out that plural backgrounds and dilatons often do not satisfy the usual beta function equations but Generalized Supergravity Equations provided the supplementary vector field J is computed using formulas (2.23) or (2.24) presented in section 2.3. The formulas were repeatedly checked not only for the examples presented here, but also for other Manin triples given in [22] and their embedings into d. All tested backgrounds and dilatons obtained by Poisson-Lie transformations (2.13), (2.14), (2.20), (2.21) satisfy Generalized Supergravity Equations.
As we noted in the Introduction, complete classification of plural models is beyond the scope of the paper as there are too many cases to discuss. Therefore, we present Manin triples that demonstrate important properties of plural sigma models, dilatons and vectors J . The examples show that vector fields J need not be constant as it turns out in cases of plurality b III a to b iii . Let us note that vector fields J are Killing vectors of corresponding backgrounds in spite of the fact that they are linear combinations of left-invariant fields of corresponding groups that satisfy condition (2.25) with generally nonvanishing right-hand side. Beside that, we have shown that the alternative formula (2.26) for the supplementary vector J developed for the (non-)Abelian T-duality does not work in general for Poisson-Lie T-plurality.
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