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...

We consider generalised Scherk Schwarz reductions of supergravity and superstring theories with twists by electromagnetic dualities that are symmetries of the equations of motion but not of the action, such as the S-duality of D = 4, N = 4 super-Yang-Mills coupled to supergravity. The reduction cannot be done on the action itself, but must be done either on the field equations or on a duality invariant form of the action, such as one in the doubled formalism in which potentials are introduced for both electric and magnetic fields. The resulting theory in odd-dimensions has massive form fields satisfying a selfduality condition dA ∼ m * A. We construct such theories in D = 3, 5, 7.

We examine the solution generating symmetries by which Lunin and Maldacena have generated the gravity duals of β-deformations of certain field theories. We identify the O(2, 2, IR) matrix, which acts on the background matrix E = g + B, where g and B are the metric and the B-field of the undeformed background, respectively. This simplifies the calculations and makes some features of the deformed backgrounds more transparent. We also find a new three-parameter deformation of the Sasaki-Einstein manifolds T 1,1 and Y p,q . Following the recent literature on the three-parameter deformation of AdS 5 × S 5 , one would expect that our new solutions should correspond to non-supersymmetric marginal deformations of the relevant dual field theories.

We construct new M-theory solutions starting from those that contain 5 U (1) isometries. We do this by reducing along one of the 5-torus directions, then T-dualizing via the action of an O(4, 4) matrix and lifting back to 11-dimensions. The particular T-duality transformation is a sequence of O(2, 2) transformations embedded in O(4, 4), where the action of each O(2, 2) gives a Lunin-Maldacena deformation in 10-dimensions. We find general formulas for the metric and 4-form field of single and multiparameter deformed solutions, when the 4-form of the initial 11-dimensional background has at most one leg along the 5-torus. All the deformation terms in the new solutions are given in terms of subdeterminants of a 5 × 5 matrix, which represents the metric on the 5-torus. We apply these results to several M-theory backgrounds of the type AdS r × X 11−r . By appropriate choices of the T-duality and reduction directions we obtain analogues of beta, dipole and noncommutative deformations. We also provide formulas for backgrounds with only 3 or 4 U (1) isometries and study a case, for which our assumption for the 4-form field is violated.

We study duality-twisted dimensional reductions on a group manifold G, where the twist is in a groupG and examine the conditions for consistency. We find that if the duality twist is introduced through a group elementg inG, then the flatG-connection A =g −1 dg must have constant components M n with respect to the basis 1-forms on G, so that the dependence on the internal coordinates cancels out in the lower dimensional theory. This condition can be satisfied if and only if M n forms a representation of the Lie algebra of G, which then ensures that the lower dimensional gauge algebra closes. We find the form of this gauge algebra and compare it to that arising from flux compactifications on twisted tori. As an example of our construction, we find a new five dimensional gauged, massive supergravity theory by dimensionally reducing the eight dimensional Type II supergravity on a three dimensional unimodular, non-semi-simple, non-abelian group manifold with an SL(3, IR) twist.