T-folds as Poisson–Lie plurals

Eur. Phys. J. C (2020) 80:892 https://doi.org/10.1140/epjc/s10052-020-08446-1 Regular Article - Theoretical Physics T-folds as Poisson–Lie plurals Ladislav Hlavatý1,a , Ivo Petr2,b 1 Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Br̆ehová 7, Prague 1 115 19, Czech Republic 2 Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, 160 00 Prague 6, Czech Republic Received: 28 July 2020 / Accepted: 5 September 2020 © The Author(s) 2020 Abstract 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 confirm that all non-Abelian T-duals can be compactified as T-folds. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . 2.1 Poisson–Lie T-duality/plurality . . . . . . . . . . 2.2 Generalized supergravity equations . . . . . . . . 2.3 Short review of double field theory and T-folds . 3 Transformations of flat metric . . . . . . . . . . . . . 3.1 T-folds obtained by Poisson–Lie transformations given by Bianchi V isometry . . . . . . . . 3.1.1 Transformation of bV  a to bV I−1  bV ii 3.1.2 Transformation of bV  a to b I I  bV . . 3.2 T-folds obtained by Poisson–Lie transformations given by Bianchi I V isometry . . . . . . . 3.2.1 Transformation of b I V  a to bV I−1  b I I 3.2.2 Transformation of b I V  a to b I I  bV I−1 4 Transformations of curved cosmologies . . . . . . . . 4.1 T-folds obtained by transformation of Bianchi V I−1 cosmology . . . . . . . . . . . . . . . . . 4.1.1 Transformation of bV I−1  a to a  bV . . 4.2 T-folds obtained by transformation of Bianchi V Iκ cosmology . . . . . . . . . . . . . . . . . . a e-mail: b e-mail: (corresponding author) 0123456789().: V,-vol 4.2.1 Transformation of bV Iκ  a to bV Iκ  bV I−κ .iii . . . . . . . . . . . . . . . . . . . 5 Conclusions . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . 1 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 [5–8] and generalized supergravity [9–11]. It also often contributes to the study of integrable models [12,13]. However, most of the papers deal with local aspects of non-Abelian duals, and global properties remain unclear, see e.g. [14]. The same holds for Poisson–Lie T-duality [15,16] or plurality [17] 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 [18,19] appeared that describe compactifications of Yang–Baxter deformations of Minkowski and AdS5 × S5 backgrounds in terms of Tfolds. T-folds represent a special class of nongeometric backgrounds that appeared in string theory in an attempt to accommodate T-duality as symmetry of some models [20–25]. Tfolds 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 [26–29]. It turns out that Double Field Theory can describe not only Abelian T-duality but also Poisson–Lie T-duality [30–33] and may help investigate quantum aspects of Poisson–Lie T-duality [34] or its extension to U-duality [35,36]. 123 892 Page 2 of 12 In [37,38] 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 [39,40]. 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  in the open string picture [41]. 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 one finds that all non-Abelian T-duals can be compactified as T-folds (as noticed e.g. in [42]). In the case of general plurality transformation additional conditions have to be satisfied. The paper is organized as follows. In Sects. 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 Sect. 2.3. In Sect. 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 Sect. 4. 2 Preliminaries In this Section we will summarize main features of Poisson– Lie T-duality and plurality [15–17,43], Generalized Supergravity Equations [9–11] and T-folds [19,21,27–29,44]. For detailed information see the original papers. 2.1 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. [38], 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 coordinates1 {sα , x a }, α = 1, . . . , n = dim N , a = 1, . . . , d = dim G 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 1 For a thorough description of the process of finding adapted coordi- nates see [45]. 123 Eur. Phys. J. C (2020) 80:892 of group G . Such F is defined by spectator-dependent (n + d) × (n + d) matrix E(s) and group dependent E(x) as   1n 0 T . E(x) = F(s, x) = E(x) · E(s) · E (x), 0 e(x) (1) 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 L = ∂− φ μ Fμν (φ)∂+ φ ν , φ μ = φ μ (σ+ , σ− ), μ = 1, (...truncated)