T-folds as Poisson-Lie plurals

In previous papers we have presented many purely bosonic solutions of Generalized Supergravity Equations obtained by Poisson-Lie T-duality and plurality of flat and Bianchi cosmologies. In this paper we focus on their compactifications and identify solutions that can be interpreted as T-folds. To recognize T-folds we adopt the language of Double Field Theory and discuss how Poisson-Lie T-duality/plurality fits into this framework. As a special case we find that all non-Abelian T-duals can be compactified as T-folds.


Introduction
Dualities in string theory relate apparently different physical models and allow to address issues that are otherwise hard to tackle. T-duality [1] connects models in backgrounds with distinct curvature properties. Together with its non-Abelian generalization [2] it was extended to RR fields [3,4] and can be used as solution generating technique in supergravity [5,6] and generalized supergravity [7][8][9]. It also often contributes to the study of integrable models [10,11]. However, most of the papers deal with local aspects of non-Abelian duals, and global properties remain unclear, see e.g. [12]. The same holds for Poisson-Lie T-duality [13,14] or plurality [15] that introduce Drinfel'd double as the underlying algebraic structure of T-duality allowing us to treat both original and dual/plural models equally.
Recently several papers [16,17] appeared that describe compactifications of Yang-Baxter deformations of Minkowski and AdS 5 × S 5 backgrounds in terms of T-folds. T-folds represent a special class of nongeometric backgrounds that appeared in string theory in an attempt to accommodate Tduality as symmetry of some models [18][19][20][21][22][23]. T-folds generalize the notion of manifold by allowing not only diffeomorphisms but also T-duality transformations as transition functions between local charts. Natural language for their description has become Double Field Theory [24][25][26][27].
In [28,29] we have presented many purely bosonic solutions of Generalized Supergravity Equations that were obtained by Poisson-Lie T-duality or plurality transformation of flat and Bianchi cosmologies [30,31]. The purpose of this paper is to present solutions that can be interpreted as T-folds. We follow the idea that T-folds can be identified using non-commutative structure Θ in the open string picture [32]. We give the argument both in terms of Poisson-Lie T-plurality and Double Field Theory to show the interplay between these two formalisms. From the structure of Drinfel'd double underlying non-Abelian T-duality we conclude that all non-Abelian T-duals can be compactified as T-folds, whereas additional conditions have to be satisfied in the case of general plurality transformation. Discussion of the connection of Poisson-Lie T-plurality and Double Field Theory can be found also in [33][34][35][36].
The paper is organized as follows. In Sections 2.1 and 2.2 we briefly recapitulate Poisson-Lie T-plurality and Generalized Supergravity Equations. Elements of Double Field Theory, T-folds, and the method that we use to identify T-folds are explained in Section 2.3. In Section 3 we present backgrounds obtained as Poisson-Lie plurals of flat background that can be interpreted as T-folds. Examples of T-folds obtained as Poisson-Lie plurals of curved Bianchi cosmologies are presented in Section 4.

Poisson-Lie T-duality/plurality
A convenient way to describe Poisson-Lie T-plurality is in terms of a Drinfel'd double. As this has been done in many preceeding papers,e.g. [29], we shall not go into details and restrict to a summary of necessary formulas.
Let G be a d-dimensional Lie group with free action on manifold M of dimension M = n+d. Since the action of G is transitive on its orbits, we may locally consider M ≈ (M /G ) × G = N × G . This allows us to introduce the so-called adapted coordinates 1 where x a denote group coordinates while s α label the orbits of G and will be treated as spectators in the duality/plurality transformation. We shall consider sigma models on N × G given by covariant tensor field F invariant with respect to the action of group G . Such F is defined by spectator-dependent (n + d) × (n + d) matrix E(s) and group dependent E(x) as .
The d×d matrix e(x) contains components of right-invariant Maurer-Cartan form (dg)g −1 on G . The dynamics of sigma model on M follows from Lagrangian where tensor field F = G + B on M contains metric G and torsion potential (Kalb-Ramond field) B.
To find Poisson-Lie dual model [13] on N × G we embed group G into Drinfel'd double, i.e. 2d-dimensional Lie group D = (G | G ) formed by a pair of Lie subgroups G and G . The Lie algebra d of the Drinfel'd double is endowed with an ad-invariant non-degenerate symmetric bilinear form ., . . d can be written as a double cross sum g ⊲⊳g of subalgebras g andg [41] corresponding to G and G . g andg are maximally isotropic with respect to the form ., . . The resulting algebraic structure (d, g,g) is called Manin triple. As noted already in [13], for a particular Drinfel'd double D = (G | G ) there may exist various Manin triples. If there is another pair of subgroups G andḠ (with corresponding Manin triple (d,ĝ,ḡ)) that form the same Drinfel'd double D, we can find Poisson-Lie plural models [15] on N × G or N ×Ḡ . The Poisson-Lie T-plural sigma model on N × G is specified by tensor field where e(x) contains components of (dĝ)ĝ −1 on G , , , a(x) denote submatrices of the adjoint representation of G on algebra d given by The matrix E(s) is obtained as follows. Let C be an invertible 2d×2d matrix relating bases of Manin triples g ⊲⊳g and g ⊲⊳ḡ as T a ∈ g, T a ∈g, T a ∈ĝ,T a ∈ḡ, a = 1, . . . , d. (5) We denote the d × d blocks of C −1 as P, Q, R, S, i.e.
To account for the spectator fields we form (n + d) × (n + d) matrices E(s) then reads This procedure was used in [28,29] to construct new solutions of Generalized Supergravity Equations from Bianchi cosmologies. Matrices C relating Manin triples g ⊲⊳g and g ⊲⊳ḡ are not unique and different choices may lead to backgrounds with different curvature or torsion properties [42]. However, in [28,29] we have shown that many parameters appearing in general C are irrelevant as the resulting backgrounds differ only by a coordinate or gauge transformation. In this paper we deal with particular matrices C leading to backgrounds that can be interpreted as Tfolds. Important special cases include P = S = 0 d , Q = R = 1 d in which case plurality reduces to Poisson-Lie T-duality. For P = S = 1 d , R = 0 d , Q = B where B = −B T we obtain the so-called B-shifts, while the so-called β-shifts are given by P = S = 1 d , Q = 0 d and R = β, β = −β T .
Throughout this paper we deal with non-semisimple Bianchi groups G , while G is three-dimensional Abelian group A . We parametrize group elements 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 normal subgroups of G . Poisson-Lie T-plurality transformation between models on groups G , G and G ,Ḡ is specified by mapping (5), but the relation can be also formulated in terms of group elements as l = g(y)h(ỹ) = g(x)h(x), l ∈ D, g ∈ G ,h ∈G , g ∈ G ,h ∈Ḡ .
In this paper we consider Bianchi cosmologies on four-dimensional manifolds, thus dim N = 1. For simplicity we denote the spectator coordinate s 1 by t.

Generalized Supergravity Equations
Adopting the convention used in [31] we write the Generalized Supergravity Equations of Motion [7,8,38] H ρµν are components of torsion 3 We restrict to bosonic fields in the backgrounds.
∇ µ are covariant derivatives with respect to metric G, and X is given by where Φ is the dilaton. When vector field J vanishes, the usual beta function equations are recovered.
In the papers [9,43] the authors discuss Yang-Baxter deformations and find components of vector J as and D ν is covariant derivative with respect to metric These two tensors form the so-called open background [32] and the form of bivector Θ is important for identification of T-folds. Div Θ, however, fails to provide correct J for Poisson-Lie T-plurality. A formula for J that takes into account possible non-locality of Φ has been presented in [29]. We shall use it in this paper.

Short review of Double Filed Theory and T-folds
The presence of dualities in string theory suggests that it might be convenient to generalize the standard concept of manifold. T-folds do so by allowing Tdualities beside diffeomorphisms as transition functions between charts. To describe T-folds, we shall use the framework of Double Field Theory (DFT).
While T-folds are examples of the so-called nongeometric backgrounds [39], in DFT T-duality transformations become diffeomorphisms of a manifold with doubled dimension. In this construction all the local patches are geometric.
The central concepts in DFT are the generalized metric H ∈ O(M, M) and DFT dilaton D that are defined using the initial background fields G, B, Φ as and The possibility to compactify both initial and extended 2M-dimensional manifold in direction α µ is conditioned by existence of a monodromy matrix Ω α whose action is equivalent to the shift of coordinates in direction α µ , i.e.
and invariance of DFT dilaton with respect to the shift If the transformation by Ω α can be reinterpreted as transformation of the background F = G + B induced by coordinate or gauge transformation, then F can be considered as background on compactified manifold M . However, monodromy matrix of the form where β is constant antisymmetric M ×M matrix cannot be obtained neither by transformation of coordinates on M nor gauge transformation of B and tensor F must be interpreted as background on a T-fold. Authors of paper [17] give simple procedure for T-fold identification via open 4 background [32]. To understand the action of Ω β we rewrite the generalized metric (13) in terms of open fields G and Θ as It is then easy to verify that action of Ω β does not change G, but shifts Θ by β. Necessary condition (15) then is It can be satisfied only for Θ linear in a coordinate x µ and suitable matrices β can be obtained from Θ as linear combinations of the so-called Q-fluxes [35] However, the sum (20) runs only over λ for which the open metric G is invariant with respect to x λ → x λ + α λ .
In terms of the background on M the condition (15) is then equivalent to The right-hand side can be considered as In [28,29] we investigated Poisson-Lie duals and plurals of Bianchi cosmologies. To find backgrounds that can be interpreted as T-folds we will present in the rest of this paper dual or plural backgrounds whose bivector Θ is linear in coordinates x µ . As the plural backgrounds are given by (3), (4), (8), bivector Θ can be expressed as where v(x) is d × d matrix of components of right-invariant vector fields of the group G . From the formula (22) it is clear that bivectors Θ linear inx occur beside others for backgrounds on Abelian groups G = A . For such groups V(x) = 1 M and Π is linear inx since Moreover, the open metric G is completely independent of the group coordinatesx. It means that all backgrounds obtained by non-Abelian T-duality, i.e. on a semi-Abelian Drinfel'd double D = (G |A ), can be compactified as T-folds. In a different way this has been shown also in [44].

Transformations of flat metric
In this section we investigate T-folds obtained from Poisson-Lie T-pluralities of the Minkowski metric following from its invariance with respect to Bianchi groups [45].

T-folds obtained by Poisson-Lie transformations given by Bianchi V isometry
The flat Minkowski metric is invariant with respect to the action of Bianchi V group. In adapted coordinates it has the form 5 Together with vanishing dilaton Φ = 0 the background satisfies beta function equations, i.e. equations (9)-(11) with J = 0. Its non-Abelian dual with respect to non-semisimple Bianchi V group appears repeatedly in the literature as it is not conformal [46,47] but satisfies Generalized Supergravity Equations instead [28,31]. The flat background in the form (24) can be ob- The algebra d = b V ⊲⊳ a allows several other decompositions into Manin triples [48], such as d = b V I −1 ⊲⊳ b V ii . Interested reader may find its commutation relations in [29] where matrices C transforming Manin triple b V ⊲⊳ a to b V I −1 ⊲⊳ b V ii were found as . 5 To get matrix E(s) one has to set y 1 = 0 in F . 6 We use B V and b V to denote the group Bianchi V and its Lie algebra. Other Bianchi groups and algebras are denoted similarly. The three-dimensional Abelian group and its Lie algebra are written as A and a.
Using C 1 we get background tensor whose metric is curved and torsion does not vanish. Supported by dilaton the background satisfies Generalized Supergravity Equations with constant vector J = (0, 0, 2, 0). DFT metric then has the form and DFT dilaton is Bivector Θ for the background above is of the form so that the Q-flux is constant. Its nonvanishing components are and β-shift matrix is β-shift (21) of F is then equivalent to the coordinate shiftx 3 →x 3 + α. DFT dilaton is independent ofx 3 so the condition (16) is satisfied. Therefore, the background (25) can be compactified as a T -fold as Using C 2 we obtain plural background that reads (29) Its metric is curved but torsion vanishes. The beta function equations are satisfied for dilaton Bivector Θ is the same as in the preceding case and DFT dilaton is D(t,x 1 ) = − 1 4 ln(t 6 ) −x 1 . The background (29) can be again compactified as T -fold by (t,x 1 ,x 2 ,x 3 ) ∼ (t,x 1 ,x 2 ,x 3 + α).

Examples of mappings that transform the algebraic structure of Manin triple
Background obtained by C 1 is given by tensor with curved metric and nonvanishing torsion. Together with the background satisfies Generalized Supergravity Equations. Bivector Θ has the form and is clearly linear in bothx 2 andx 3 . However, components of Q-flux corresponding to shift inx 2 depend on t and the open metric G depends onx 2 so the above given procedure does not apply. Constant nonvanishing components of Q-flux are and β-shift (21) of F given by is equivalent to the coordinate shiftx 3 →x 3 + α. DFT dilaton is independent ofx 3 . Therefore, the background (30) can be compactified as T -fold by Background obtained by C 2 is given by tensor with curved metric and vanishing torsion. Together with dilaton they satisfy beta function equations. Bivector Θ for this background is of the form It is slightly different from the previous case, however, the constant nonvanishing components of Q-flux, β-shift matrix and DFT dilaton are the same as in (31), (32) and (33), and the background (34) can be compactified as

T-folds obtained by Poisson-Lie transformations given by Bianchi IV isometry
Next we investigate backgrounds that follow from Poisson-Lie T-pluralities obtained from the invariance of Minkowski metric with respect to the action of Bianchi IV group. The group B IV is not semisimple and trace of its structure constants does not vanish. The metric in adapted coordinates reads Since the background is flat and torsionless, the beta function equations are satisfied if we choose vanishing dilaton Φ = 0. In this form the background can be obtained by formula (1) The algebra d allows several other decompositions into Manin triples.

Transformation of
Mapping that transforms the algebraic structure of Manin triple b IV ⊲⊳ a to b V I −1 ⊲⊳ b II is given by matrix Resulting background tensor reads The background has vanishing torsion. Its metric is curved with vanishing scalar curvature. Simple coordinate transformation brings the metric to the Brinkmann form of a plane-parallel wave Together with the dilaton Φ(x 1 ) = −x 1 , the background satisfies beta function equations. Bivector Θ for this background is of the form Nonvanishing components of constant Q-flux are so the β-shift matrix is Unfortunately, DFT dilaton D and the open metric depend onx 1 . Therefore, background (39) cannot be compactified as T -fold in the directionx 1 .

Transformation of
Manin triple b II ⊲⊳ b V I −1 is dual to that of preceding section. Background given by matrix that is dual to (38) has the form Formulas for transformation of dilaton and supplementary Killing vectorJ presented in [29] givē Together with this nontrivial dilaton andJ the background satisfies Generalized Supergravity Equations. The background has flat metric and vanishing torsion so it also satisfies beta function equations with vanishing dilaton. This is rather interesting situation. DilatonΦ andJ enter Generalized Supergravity Equations combined into one-form X and there might be more suitable combinations. BivectorΘ is of the form and DFT dilaton D vanishes. Nonvanishing components of Q-flux are and it may seem that β-shift matrices can be arbitrary linear combinations However, the open metricḠ depends linearly onx 2 . Therefore, background (43) can be compactified as T -fold by

T-folds obtained by transformation of Bianchi V I −1 cosmology
Let us now focus on the curved background with metric [31] F (t, that is invariant with respect to the action of Bianchi V I −1 group. Dilaton Φ(t) = βt and functions a i (t) are given as in [30] by The beta function equations are satisfied if parameters p 1 , p 2 and β fulfill condition To obtain metric (47) via (1) we shall consider Manin triple d = b V I −1 ⊲⊳ a that corresponds to the same Drinfel'd double that we discussed in Section 3.1. Although the Drinfel'd double is the same, the background in this Section is different as we use different matrix E(t) = F (t, y 1 = 0).

Transformation of
Example of mappings that transform the algebraic structure of Manin triple and background obtained by Poisson-Lie T-plurality with this matrix is given by tensor where ∆ = a 1 (t) 2 a 2 (t) 2 +x 2 2 a 2 (t) 4 +x 2 3 . The background is curved and has nontrivial torsion. Together with dilaton Φ(t,x 2 ,x 3 ) = βt − 1 2 ln ∆ they satisfy Generalized Supergravity Equations with J = (0, 1, 0, 0). Bivector Θ for the background (49) is of the form and the Q-flux is constant. Its nonvanishing components are and β-shift matrices are arbitrary linear combinations DFT dilaton is . Therefore, the background (49) can be compactified as For α 2 , α 3 ∈ Z we have Ω ∈ O(M, M, Z).

T-folds obtained by transformation of Bianchi V I κ cosmology
Finally we are going discuss Poisson-Lie T-pluralities of curved cosmology invariant with respect to Bianchi V I κ group. Its metric is [31] F (t, (52) and dilaton reads Φ(t) = βt.
The functions a i (t) are given as in [30] by The beta function equations are satisfied provided constants p 1 , p 2 , β and κ fulfill condition The background (52) can be obtained from (1) if we consider six-dimensional semi-Abelian Drinfel'd double D = (B V Iκ |A ). Among the decompositions of d = b V Iκ ⊲⊳ a given in [48] is also Manin triple iii . For its commutation relations see [29] where general forms of mappings C relating b V Iκ ⊲⊳ a and b V Iκ ⊲⊳ b V I −κ .iii were found as well. In particular we shall consider matrices Since the corresponding DFT dilatons D(t,x 1 ) = − ln(a 1 (t)a 2 (t)a 3 (t)) + 2βt ± (κ + 1)x 1 2 do not depend onx 2 , β-shifts (21) of F are equivalent to coordinate shift x 2 →x 2 + α and backgrounds (55), (56) can be compactified as T -folds by (t,x 1 ,x 2 ,x 3 ) ∼ (t,x 1 ,x 2 + α,x 3 ).

Conclusions
In this paper we discussed the connection between Poisson-Lie T-plurality and Double Field Theory. Using tensors Θ and G constituting the open background we have derived conditions (19) and (21) that need to be satisfied in order to identify background as T-fold. If Θ is linear in a coordinate x µ , the shift x µ → x µ + α µ is equivalent to a β-shift (21) provided G and DFT dilaton D are independent of x µ . We have shown that due to the formula (22) all backgrounds obtained by non-Abelian T-duality can be compactified as T-folds. In Sections 3 and 4 we have tested conditions (19) and (21) for Poisson-Lie plurals of flat and Bianchi cosmologies obtained in [29]. We have shown that in spite of their rather complicated forms, many Poisson-Lie plurals can be considered as T-folds.