A particular higher-derivative extension of the Einstein-Hilbert action in three spacetime dimensions is shown to be equivalent at the linearized level to the (unitary) Pauli-Fierz action for a massive spin-2 field. A more general model, which also includes "topologically-massive" gravity as a special case, propagates the two spin-2 helicity states with different masses. We discuss the extension to massive N-extended supergravity, and we present a "cosmological" extension that admits an anti-de Sitter vacuum.

The generalized metric is a T-duality covariant symmetric matrix constructed from the metric and two-form gauge field and arises in generalized geometry. We view it here as a metric on the doubled spacetime and use it to give a simple formulation with manifest T-duality of the double field theory that describes the massless sector of closed strings. The gauge transformations are written in terms of a generalized Lie derivative whose commutator algebra is defined by a double field theory extension of the Courant bracket.Comment: 36 pages, v2: minor correction

Double field theory describes a massless subsector of closed string theory with both momentum and winding excitations. The gauge algebra is governed by the Courant bracket in certain subsectors of this double field theory. We construct the associated nonlinear backgroundindependent action that is T-duality invariant and realizes the Courant gauge algebra. The action is the sum of a standard action for gravity, antisymmetric tensor, and dilaton fields written with ordinary derivatives, a similar action for dual fields with dual derivatives, and a mixed term that is needed for gauge invariance.

We present the details of the recently constructed E 6p6q covariant extension of 11dimensional supergravity. This theory requires a 5`27 dimensional spacetime in which the 'internal' coordinates transform in the27 of E 6p6q . All fields are E 6p6q tensors and transform under (gauged) internal generalized diffeomorphisms. The 'Kaluza-Klein' vector field acts as a gauge field for the E 6p6q covariant 'E-bracket' rather than a Lie bracket, requiring the presence of two-forms akin to the tensor hierarchy of gauged supergravity. We construct the complete and unique action that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. The theory is subject to covariant section constraints on the derivatives, implying that only a subset of the extra 27 coordinates is physical. We give two solutions of the section constraints: the first preserves GLp6q and embeds the action of the complete (i.e. untruncated) 11-dimensional supergravity; the second preserves GLp5qˆSLp2q and embeds complete type IIB supergravity. As a by-product, we thus obtain an off-shell action for type IIB supergravity.

We introduce the exceptional field theory for the group E 7ð7Þ , based on a ð4 þ 56Þ-dimensional spacetime subject to a covariant section condition. The "internal" generalized diffeomorphisms of the coordinates in the fundamental representation of E 7ð7Þ are governed by a covariant "E-bracket," which is gauged by 56 vector fields. We construct the complete and unique set of field equations that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. Among them are featured the non-Abelian twisted self-duality equations for the 56 gauge vectors. We discuss the explicit solutions of the section condition describing the embedding of the full, untruncated 11-dimensional and type IIB supergravity, respectively. As a new feature compared to the previously constructed E 6ð6Þ formulation, some components among the 56 gauge vectors descend from the 11-dimensional dual graviton but nevertheless allow for a consistent coupling by virtue of a covariantly constrained compensating 2-form gauge field.