The gauge algebra of double field theory and Courant brackets

We investigate the symmetry algebra of the recently proposed field theory on a doubled torus that describes closed string modes on a torus with both momentum and winding. The gauge parameters are constrained fields on the doubled space and transform as vectors under T-duality. The gauge algebra defines a T-duality covariant bracket. For the case in which the parameters and fields are T-dual to ones that have momentum but no winding, we find the gauge transformations to all orders and show that the gauge algebra reduces to one obtained by Siegel. We show that the bracket for such restricted parameters is the Courant bracket. We explain how these algebras are realised as symmetries despite the failure of the Jacobi identity.Comment: 25 pages, LaTe

Frame-like Geometry of Double Field Theory

We relate two formulations of the recently constructed double field theory to a frame-like geometrical formalism developed by Siegel. A self-contained presentation of this formalism is given, including a discussion of the constraints and its solutions, and of the resulting Riemann tensor, Ricci tensor and curvature scalar. This curvature scalar can be used to define an action, and it is shown that this action is equivalent to that of double field theory.Comment: 35 pages, v2: minor corrections, to appear in J. Phys.

Nongeometry, Duality Twists, and the Worldsheet

In this paper, we use orbifold methods to construct nongeometric backgrounds, and argue that they correspond to the spacetimes discussed in \cite{dh,wwf}. More precisely, we make explicit through several examples the connection between interpolating orbifolds and spacetime duality twists. We argue that generic nongeometric backgrounds arising from duality twists will not have simple orbifold constructions and then proceed to construct several examples which do have a consistent worldsheet description.Comment: v2-references added; v3-minor correction (eqn. 4.17

Timelike Hopf Duality and Type IIA^* String Solutions

The usual T-duality that relates the type IIA and IIB theories compactified on circles of inversely-related radii does not operate if the dimensional reduction is performed on the time direction rather than a spatial one. This observation led to the recent proposal that there might exist two further ten-dimensional theories, namely type IIA^* and type IIB^*, related to type IIB and type IIA respectively by a timelike dimensional reduction. In this paper we explore such dimensional reductions in cases where time is the coordinate of a non-trivial U(1) fibre bundle. We focus in particular on situations where there is an odd-dimensional anti-de Sitter spacetime AdS_{2n+1}, which can be described as a U(1) bundle over \widetilde{CP}^n, a non-compact version of CP^n corresponding to the coset manifold SU(n,1)/U(n). In particular, we study the AdS_5\times S^5 and AdS_7\times S^4 solutions of type IIB supergravity and eleven-dimensional supergravity. Applying a timelike Hopf T-duality transformation to the former provides a new solution of the type IIA^* theory, of the form \widetilde{CP}^2\times S^1\times S^5. We show how the Hopf-reduced solutions provide further examples of ``supersymmetry without supersymmetry.'' We also present a detailed discussion of the geometrical structure of the Hopf-fibred metric on AdS_{2n+1}, and its relation to the horospherical metric that arises in the AdS/CFT correspondence.Comment: Latex, 26 page

Generalised Geometry for M-Theory

Generalised geometry studies structures on a d-dimensional manifold with a metric and 2-form gauge field on which there is a natural action of the group SO(d,d). This is generalised to d-dimensional manifolds with a metric and 3-form gauge field on which there is a natural action of the group $E_{d}$. This provides a framework for the discussion of M-theory solutions with flux. A different generalisation is to d-dimensional manifolds with a metric, 2-form gauge field and a set of p-forms for $p$ either odd or even on which there is a natural action of the group $E_{d+1}$. This is useful for type IIA or IIB string solutions with flux. Further generalisations give extended tangent bundles and extended spin bundles relevant for non-geometric backgrounds. Special structures that arise for supersymmetric backgrounds are discussed.Comment: 31 page

Flux Compactifications of M-Theory on Twisted Tori

We find the bosonic sector of the gauged supergravities that are obtained from 11-dimensional supergravity by Scherk-Schwarz dimensional reduction with flux to any dimension D. We show that, if certain obstructions are absent, the Scherk-Schwarz ansatz for a finite set of D-dimensional fields can be extended to a full compactification of M-theory, including an infinite tower of Kaluza-Klein fields. The internal space is obtained from a group manifold (which may be non-compact) by a discrete identification. We discuss the symmetry algebra and the symmetry breaking patterns and illustrate these with particular examples. We discuss the action of U-duality on these theories in terms of symmetries of the D-dimensional supergravity, and argue that in general it will take geometric flux compactifications to M-theory on non-geometric backgrounds, such as U-folds with U-duality transition functions.Comment: Latex, 47 page

Duality Twists on a Group Manifold

We study duality-twisted dimensional reductions on a group manifold G, where the twist is in a group \tilde{G} and examine the conditions for consistency. We find that if the duality twist is introduced through a group element \tilde{g} in \tilde{G}, then the flat \tilde{G}-connection A =\tilde{g}^{-1} d\tilde{g} 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,R) twist.Comment: 22 page

A Geometry for Non-Geometric String Backgrounds

A geometric string solution has background fields in overlapping coordinate patches related by diffeomorphisms and gauge transformations, while for a non-geometric background this is generalised to allow transition functions involving duality transformations. Non-geometric string backgrounds arise from T-duals and mirrors of flux compactifications, from reductions with duality twists and from asymmetric orbifolds. Strings in ` T-fold' backgrounds with a local $n$-torus fibration and T-duality transition functions in $O(n,n;\Z)$ are formulated in an enlarged space with a $T^{2n}$ fibration which is geometric, with spacetime emerging locally from a choice of a $T^n$ submanifold of each $T^{2n}$ fibre, so that it is a subspace or brane embedded in the enlarged space. T-duality acts by changing to a different $T^n$ subspace of $T^{2n}$. For a geometric background, the local choices of $T^n$ fit together to give a spacetime which is a $T^n$ bundle, while for non-geometric string backgrounds they do not fit together to form a manifold. In such cases spacetime geometry only makes sense locally, and the global structure involves the doubled geometry. For open strings, generalised D-branes wrap a $T^n$ subspace of each $T^{2n}$ fibre and the physical D-brane is the part of the part of the physical space lying in the generalised D-brane subspace.Comment: 28 Pages. Minor change

Einstein Supergravity and New Twistor String Theories

A family of new twistor string theories is constructed and shown to be free from world-sheet anomalies. The spectra in space-time are calculated and shown to give Einstein supergravities with second order field equations instead of the higher derivative conformal supergravities that arose from earlier twistor strings. The theories include one with the spectrum of N=8 supergravity, another with the spectrum of N=4 supergravity coupled to N=4 super-Yang-Mills, and a family with $N\ge 0$ supersymmetries with the spectra of self-dual supergravity coupled to self-dual super-Yang-Mills. The non-supersymmetric string with N=0 gives self-dual gravity coupled to self-dual Yang-Mills and a scalar. A three-graviton amplitude is calculated for the N=8 and N=4 theories and shown to give a result consistent with the cubic interaction of Einstein supergravity.Comment: LaTeX, 69 pages, no figures; v2: minor corrections made, footnotes and references adde

NS-NS fluxes in Hitchin's generalized geometry

The standard notion of NS-NS 3-form flux is lifted to Hitchin's generalized geometry. This generalized flux is given in terms of an integral of a modified Nijenhuis operator over a generalized 3-cycle. Explicitly evaluating the generalized flux in a number of familiar examples, we show that it can compute three-form flux, geometric flux and non-geometric Q-flux. Finally, a generalized connection that acts on generalized vectors is described and we show how the flux arises from it.Comment: 21 pages, 1 figure; v3: minor change