Non-Abelian T-duality as a transformation in Double Field Theory

Abstract

Non-Abelian T-duality (NATD) is a solution generating transformation for supergravity backgrounds with non-Abelian isometries. We show that NATD can be described as a coordinate dependent O(d,d) transformation, where the dependence on the coordinates is determined by the structure constants of the Lie algebra associated with the isometry group. Besides making calculations significantly easier, this approach gives a natural embedding of NATD in Double Field Theory (DFT), a framework which provides an O(d,d) covariant formulation for effective string actions. As a result of this embedding, it becomes easy to prove that the NATD transformed backgrounds solve supergravity equations, when the isometry algebra is unimodular. If the isometry algebra is nonunimodular, the generalized dilaton field is forced to have a linear dependence on the dual coordinates, which implies that the resulting background solves generalized supergravity equations. IntroductionNon-Abelian T-duality (NATD) is a generalization of T-duality for strings on backgrounds with non-Abelian isometries [1]- [5]. Although it is not as well established as T-duality is as a string duality symmetry, it works well as a solution generating transformation for supergravity. The rules for the transformation of the fields in the NS-NS sector, namely the metric, the B-field and the dilaton field has been known for a long time. Recently, NATD has gained a new interest, as the rules for the transformation of the fields in the RR sector of Type II strings has also been found [6]. This has been applied to many supergravity backgrounds by various groups, especially to backgrounds that are relevant for AdS-CFT correspondence, see for example [7]-[16].Recently, a compact formula for the transformation of the supergravity fields for a generic Green-Schwarz string with isometry G has been obtained in [17], where they also showed that the sigma model after NATD has kappa symmetry. This means that the resulting background is a solution of the generalized supergravity equations (GSE), which have recently been introduced in [18] as a generalization of supergravity equations, see also [19]. To be more precise, when the isometry group G is unimodular, the dualized sigma model is Weyl invariant and the target space is a solution of standard Type II supergravity equations. If G is non-unimodular so that the structure constants of the Lie algebra of G is not traceless, the trace components give rise to a deformation of the equations to be satisfied by the target space fields to GSE 1 .The purpose of this paper is to describe the NATD transformation rules obtained in [17] as a coordinate dependent O(10, 10) transformation 2 . In Abelian T-duality with d commuting isometries, the transformation rules for the supergravity fields in the NS-NS sector can be neatly described through the action of a constant O(d, d) matrix embedded in O(10, 10) [23]. The RR fields are then packaged in a differential form, which can be a regarded as a spinor field that transforms under Spin(d, d). If the...

U -duality extension of Drinfel’d double

“…A new formulation of DFT on group manifolds (called DFT WZW ) has been developed in [20][21][22], and in the recent works [23,24], the PL T -duality has been studied in the framework of DFT WZW . In more recent papers [7,25], the non-Abelian T -duality and the PL T -duality have been discussed by using another approach, called the gauged DFT [26][27][28][29][30][31] (see also [32,33] for recent discussion on the Drinfel'd double and related aspects in DFT). Thus, DFT is now not restricted to Abelian T -duality but can be applied also to the non-Abelian extensions.…”

Yang–Baxter deformation as an $\boldsymbol {O(d,d)}$ transformation

“…Recently in [25], we showed that the NATD transformation rules obtained in [24] for a generic GS string with isometry group G can be described through the action of a coordinate dependent 1 See also [23], where they arrive at the same result by utilizing the O(d, d) structure of NATD.…”

“…Recently, Borsato and Wulff extended their work to homogeneous YB deformations of more general sigma models than PCMs. In their paper [24], they derived the rules for NATD for a generic Green-Schwarz (GS) string with isometry group G, where NATD is applied with respect to a subgroup of G. Then, by using the connection between NATD and homogeneous YB models that they had established in [21], [22], they proposed the form of homogeneous YB deformation for a generic GS sigma model and showed that this gave a natural generalization of the YB deformation of PCMs and supercosets.Recently in [25], we showed that the NATD transformation rules obtained in [24] for a generic GS string with isometry group G can be described through the action of a coordinate dependent 1 See also [23], where they arrive at the same result by utilizing the O(d, d) structure of NATD.…”

“…Given the relation between NATD and homogeneous YB deformations discussed briefly above, the results of [25] implies that it should be possible to describe YB deformations also as O(d, d) transformations and hence to embed it in DFT. In fact, this approach has already been adopted by several groups.…”

“…The YB matrix we study in this paper is related to the NATD matrix we studied in [25] by a constant O(d, d) transformation. That the two matrices are related is of course not surprising at all, given the fact that [24] constructed the YB deformed models by utilizing the connection with NATD.…”