T-duality and non-geometric solutions from double geometry

Although the introduction of generalised and extended geometry has been motivated mainly by the appearance of dualities upon reductions on tori, it has until now been unclear how (all) the duality transformations arise from first principles in extended geometry. A proposal for solving this problem is given in the framework of double field theory. It is based on a clearly defined extension of the definition of gauge symmetry by isometries of an underlying pseudo-Riemannian manifold. The ensuing relation between transformations of coordinates and fields, which is now derived from first principles, differs from earlier proposals.Comment: 13 pp., plain te

Leibniz Gauge Theories and Infinity Structures

We formulate gauge theories based on Leibniz(-Loday) algebras and uncover their underlying mathematical structure. Various special cases have been developed in the context of gauged supergravity and exceptional field theory. These are based on `tensor hierarchies', which describe towers of $p$-form gauge fields transforming under non-abelian gauge symmetries and which have been constructed up to low levels. Here we define `infinity-enhanced Leibniz algebras' that guarantee the existence of consistent tensor hierarchies to arbitrary level. We contrast these algebras with strongly homotopy Lie algebras ($L_{\infty}$ algebras), which can be used to define topological field theories for which all curvatures vanish. Any infinity-enhanced Leibniz algebra carries an associated $L_{\infty}$ algebra, which we discuss.Comment: 50 pages, v2: refs added, new subsection 3.2, version to appear in Comm. Math. Phy

Perturbative Double Field Theory on General Backgrounds

We develop the perturbation theory of double field theory around arbitrary solutions of its field equations. The exact gauge transformations are written in a manifestly background covariant way and contain at most quadratic terms in the field fluctuations. We expand the generalized curvature scalar to cubic order in fluctuations and thereby determine the cubic action in a manifestly background covariant form. As a first application we specialize this theory to group manifold backgrounds, such as $SU(2) \simeq S^3$ with $H$-flux. In the full string theory this corresponds to a WZW background CFT. Starting from closed string field theory, the cubic action around such backgrounds has been computed before by Blumenhagen, Hassler and L\"ust. We establish precise agreement with the cubic action derived from double field theory. This result confirms that double field theory is applicable to arbitrary curved background solutions, disproving assertions in the literature to the contrary.Comment: 36 pages, v2: minor clarification

Massive Type II in Double Field Theory

We provide an extension of the recently constructed double field theory formulation of the low-energy limits of type II strings, in which the RR fields can depend simultaneously on the 10-dimensional space-time coordinates and linearly on the dual winding coordinates. For the special case that only the RR one-form of type IIA carries such a dependence, we obtain the massive deformation of type IIA supergravity due to Romans. For T-dual configurations we obtain a `massive' but non-covariant formulation of type IIB, in which the 10-dimensional diffeomorphism symmetry is deformed by the mass parameter.Comment: 21 page

U-duality covariant gravity

We extend the techniques of double field theory to more general gravity theories and U-duality symmetries, having in mind applications to the complete D=11 supergravity. In this paper we work out a (3+3)-dimensional `U-duality covariantization' of D=4 Einstein gravity, in which the Ehlers group SL(2,R) is realized geometrically, acting in the 3 representation on half of the coordinates. We include the full (2+1)-dimensional metric, while the `internal vielbein' is a coset representative of SL(2,R)/SO(2) and transforms under gauge transformations via generalized Lie derivatives. In addition, we introduce a gauge connection of the `C-bracket', and a gauge connection of SL(2,R), albeit subject to constraints. The action takes the form of (2+1)-dimensional gravity coupled to a Chern-Simons-matter theory but encodes the complete D=4 Einstein gravity. We comment on generalizations, such as an `$E_{8(8)}$ covariantization' of M-theory.Comment: 36 pages, v2: refs. added, to appear in JHE

On the Riemann Tensor in Double Field Theory

Double field theory provides T-duality covariant generalized tensors that are natural extensions of the scalar and Ricci curvatures of Riemannian geometry. We search for a similar extension of the Riemann curvature tensor by developing a geometry based on the generalized metric and the dilaton. We find a duality covariant Riemann tensor whose contractions give the Ricci and scalar curvatures, but that is not fully determined in terms of the physical fields. This suggests that \alpha' corrections to the effective action require \alpha' corrections to T-duality transformations and/or generalized diffeomorphisms. Further evidence to this effect is found by an additional computation that shows that there is no T-duality invariant four-derivative object built from the generalized metric and the dilaton that reduces to the square of the Riemann tensor.Comment: 36 pages, v2: minor changes, ref. added, v3: appendix on frame formalism added, version to appear in JHE