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

Published for SISSA by Springer Received: December 14, Revised: February 28, Accepted: April 4, Published: April 29, 2018 2019 2019 2019 Ladislav Hlavatýa and Ivo Petrb a Department of Physics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University in Prague, Břehová 7, Prague 1, 115 19 Czech Republic b Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Thákurova 9, Prague 6, 160 00 Czech Republic E-mail: , Abstract: 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 Tduality is defined and what should be the “canonical” duality or plurality transformation. Keywords: Sigma Models, String Duality, Integrable Field Theories ArXiv ePrint: 1811.12235 Open Access, c The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP04(2019)157 JHEP04(2019)157 Poisson-Lie T-plurality revisited. Is T-duality unique? Contents 1 2 Poisson-Lie T-plurality of sigma models 2.1 Poisson-Lie T-plurality with spectators 2.2 Equivalence of transformation matrices 2 3 5 3 Sigma models with two-dimensional target space 3.1 Poisson-Lie T-plurality 3.2 Poisson-Lie T-duality 5 6 8 4 Sigma models with four-dimensional target space 10 5 Poisson-Lie T-dualities and pluralities generated by Poisson-Lie iden12 tities 6 Conclusion 1 13 Introduction The notion of (non-)Abelian T-duality [1–3] 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 fields 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. Poisson-Lie T-duality, introduced in the seminal paper [8] by Klimčı́k and Ševera, treats both models equally and offers a remedy to this issue. The algebraic structure underlying Poisson-Lie T-duality is the Drinfel’d double, a Lie group D that decomposes into two Lie subgroups G and Ge 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 Ge are non-Abelian. In such a case the symmetry of the initial model is replaced by the so-called Poisson-Lie symmetry (or generalized symmetry), see [9], and the full Poisson-Lie 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 [11–13]. –1– JHEP04(2019)157 1 Introduction 2 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 = ∂− φµ Fµν (φ)∂+ φν , φµ = φµ (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 xµ = {sδ , ga }, δ = 1, . . . , n − d, a = 1, . . . , d (2.2) where sδ label the orbits of G and are treated as spectators and ga 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 ! 1 0 T F(s, g) = E(g) · E(s) · E (g), E(g) = (2.3) 0 e(g) where e(g) is d × d matrix of components of right-invariant Maurer-Cartan form (dg)g −1 on G . –2– JHEP04(2019)157 Since duality exchanges roles of G and Ge, we may understand it in terms of Drinfel’d double as a switch between decompositions (G |Ge) and (Ge|G ) of D. The authors of [8] mene (K|K), e tion the fact that a Drinfel’d double D can have other decompositions (K|K), ... beside (G |Ge) and (Ge|G ). All these decompositions can be used to construct mutually related sigma models. The transformation of the initial model constructed by decomposition e was later denoted Poisson-Lie T-plurality [14]. (G |Ge) to a model constructed by (K|K) 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 [16–18]. 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. Using non-Abelian T-duality one can find dual sigma model on N × Ge, where Ge is Abelian subgroup of semi-Abelian Drinfel’d double D that splits into subgroups G and Ge. The necessary formulas will be given in the following subsection as a special case of PoissonLie T-plurality. In papers [21–23], 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]. 2.1 Poisson-Lie T-plurality with spectators hTa , Tb i = 0, hTba , Tbb i = 0, hTea , Teb i = 0, hT̄ a , T̄ b i = 0, are related by transformation Tb T̄ ! =C· T Te ! hTa , Teb i = δab , hTba , T̄ b i = δab (2.4) (2.5) where C is an invertible 2d × 2d matrix. Due to ad-invariance of the bilinear form h., .i the algebraic structure of d is given both by [Ti , Tj ] = fijk Tk , and [Tbi , Tbj ] = fˆijk Tbk , [Tei , Tej ] = f˜kij Tek , [T̄ i , T̄ j ] = f¯kij T̄ k , j ek [Ti , Tej ] = fki T + f˜ijk Tk j k [Tbi , T̄ j ] = fˆki T̄ + f¯ (...truncated)