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

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

Ramond-Ramond Cohomology and O(D,D) T-duality

In the name of supersymmetric double field theory, superstring effective actions can be reformulated into simple forms. They feature a pair of vielbeins corresponding to the same spacetime metric, and hence enjoy double local Lorentz symmetries. In a manifestly covariant manner --with regard to O(D,D) T-duality, diffeomorphism, B-field gauge symmetry and the pair of local Lorentz symmetries-- we incorporate R-R potentials into double field theory. We take them as a single object which is in a bi-fundamental spinorial representation of the double Lorentz groups. We identify cohomological structure relevant to the field strength. A priori, the R-R sector as well as all the fermions are O(D,D) singlet. Yet, gauge fixing the two vielbeins equal to each other modifies the O(D,D) transformation rule to call for a compensating local Lorentz rotation, such that the R-R potential may turn into an O(D,D) spinor and T-duality can flip the chirality exchanging type IIA and IIB supergravities.Comment: 1+37 pages, no figure; Structure reorganized, References added, To appear in JHEP. cf. Gong Show of Strings 2012 (http://wwwth.mpp.mpg.de/members/strings/strings2012/strings_files/program/Talks/Thursday/Gongshow/Lee.pdf

The local symmetries of M-theory and their formulation in generalised geometry

In the doubled field theory approach to string theory, the T-duality group is promoted to a manifest symmetry at the expense of replacing ordinary Riemannian geometry with generalised geometry on a doubled space. The local symmetries are then given by a generalised Lie derivative and its associated algebra. This paper constructs an analogous structure for M-theory. A crucial by-product of this is the derivation of the physical section condition for M-theory formulated in an extended space.Comment: 20 pages, v2: Author Name corrected, v3: typos correcte

Geometry and dynamics of higher-spin frame fields

We give a systematic account of unconstrained free bosonic higher-spin fields on D-dimensional Minkowski and (Anti-)de Sitter spaces in the frame formalism. The generalized spin connections are determined by solving a chain of torsion-like constraints. Via a generalization of the vielbein postulate these allow to determine higher-spin Christoffel symbols, whose relation to the de Wit--Freedman connections is discussed. We prove that the generalized Einstein equations, despite being of higher-derivative order, give rise to the AdS Fronsdal equations in the compensator formulation. To this end we derive Damour-Deser identities for arbitrary spin on AdS. Finally we discuss the possibility of a geometrical and local action principle, which is manifestly invariant under unconstrained higher-spin symmetries.Comment: 30 pages, uses youngtab.sty, v2: minor changes, references adde

Duality Invariant M-theory: Gauged supergravities and Scherk-Schwarz reductions

We consider the reduction of the duality invariant approach to M-theory by a U-duality group valued Scherk-Schwarz twist. The result is to produce potentials for gauged supergravities that are normally associated with non-geometric compactifications. The local symmetry reduces to gauge transformations with the gaugings exactly matching those of the embedding tensor approach to gauged supergravity. Importantly, this approach now includes a nontrivial dependence of the fields on the extra coordinates of the extended space.Comment: 22 pages Latex; v2: typos corrected and references adde

A Double Sigma Model for Double Field Theory

We define a sigma model with doubled target space and calculate its background field equations. These coincide with generalised metric equation of motion of double field theory, thus the double field theory is the effective field theory for the sigma model.Comment: 26 pages, v1: 37 pages, v2: references added, v3: updated to match published version - background and detail of calculations substantially condensed, motivation expanded, refs added, results unchange

Classification of non-Riemannian doubled-yet-gauged spacetime

Assuming $\mathbf{O}(D,D)$ covariant fields as the `fundamental' variables, Double Field Theory can accommodate novel geometries where a Riemannian metric cannot be defined, even locally. Here we present a complete classification of such non-Riemannian spacetimes in terms of two non-negative integers, $(n,\bar{n})$, $0\leq n+\bar{n}\leq D$. Upon these backgrounds, strings become chiral and anti-chiral over $n$ and $\bar{n}$ directions respectively, while particles and strings are frozen over the $n+\bar{n}$ directions. In particular, we identify $(0,0)$ as Riemannian manifolds, $(1,0)$ as non-relativistic spacetime, $(1,1)$ as Gomis-Ooguri non-relativistic string, $(D{-1},0)$ as ultra-relativistic Carroll geometry, and $(D,0)$ as Siegel's chiral string. Combined with a covariant Kaluza-Klein ansatz which we further spell, $(0,1)$ leads to Newton-Cartan gravity. Alternative to the conventional string compactifications on small manifolds, non-Riemannian spacetime such as $D=10$, $(3,3)$ may open a new scheme of the dimensional reduction from ten to four.Comment: 1+41 pages; v2) Refs added; v3) Published version; v4) Sign error in (2.51) correcte

On massive gravitons in 2+1 dimensions

The Fierz-Pauli (FP) free field theory for massive spin 2 particles can be extended, in a spacetime of (1+2) dimensions (3D), to a generally covariant parity-preserving interacting field theory, in at least two ways. One is "new massive gravity" (NMG), with an action that involves curvature-squared terms. Another is 3D "bigravity", which involves non-linear couplings of the FP tensor field to 3D Einstein-Hilbert gravity. We review the proof of the linearized equivalence of both "massive 3D gravity" theories to FP theory, and we comment on their similarities and differences.Comment: 6 pages, to appear in the proceedings of the Spanish Relativity Meeting ERE2009, Bilbao; minor changes, reference adde

Differential geometry with a projection: Application to double field theory

In recent development of double field theory, as for the description of the massless sector of closed strings, the spacetime dimension is formally doubled, i.e. from D to D+D, and the T-duality is realized manifestly as a global O(D,D) rotation. In this paper, we conceive a differential geometry characterized by a O(D,D) symmetric projection, as the underlying mathematical structure of double field theory. We introduce a differential operator compatible with the projection, which, contracted with the projection, can be covariantized and may replace the ordinary derivatives in the generalized Lie derivative that generates the gauge symmetry of double field theory. We construct various gauge covariant tensors which include a scalar and a tensor carrying two O(D,D) vector indices.Comment: 1+22 pages, No figure; a previous result on 4-index tensor removed, presentation improve