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.

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 background-independent 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 ${\mathrm{E}}_{6(6)}$-covariant extension of 11-dimensional supergravity. This theory requires a $5+27$-dimensional spacetime in which the ``internal'' coordinates transform in the $\overline{\mathbf{27}}$ of ${\mathrm{E}}_{6(6)}$. All fields are ${\mathrm{E}}_{6(6)}$ tensors and transform under (gauged) internal generalized diffeomorphisms. The ``Kaluza-Klein'' vector field acts as a gauge field for the ${\mathrm{E}}_{6(6)}$-covariant ``E-bracket'' rather than a Lie bracket, requiring the presence of 2-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 GL(6) and embeds the action of the complete (i.e. untruncated) 11-dimensional supergravity; the second preserves $\text{GL}(5)\ifmmode\times\else\texttimes\fi{}\text{SL}(2)$ and embeds complete type IIB supergravity. As a byproduct, we thus obtain an off-shell action for type IIB supergravity.

We use double field theory to give a unified description of the low energy limits of type IIA and type IIB superstrings. The Ramond-Ramond potentials fit into spinor representations of the duality group O(D, D) and field-strengths are obtained by acting with the Dirac operator on the potentials. The action, supplemented by a Spin⁺ (D, D)-covariant self-duality condition on field strengths, reduces to the IIA and IIB theories in different frames. As usual, the NS-NS gravitational variables are described through the generalized metric. Our work suggests that the fundamental gravitational variable is a hermitian element of the group Spin(D, D) whose natural projection to O(D, D) gives the generalized metric.

We present the generalized Scherk-Schwarz reduction ansatz for the full supersymmetric exceptional field theory in terms of group valued twist matrices subject to consistency equations. With this ansatz the field equations precisely reduce to those of lower-dimensional gauged supergravity parametrized by an embedding tensor. We explicitly construct a family of twist matrices as solutions of the consistency equations. They induce gauged supergravities with gauge groups SO(p, q) and CSO(p, q, r). Geometrically, they describe compactifications on internal spaces given by spheres and (warped) hyperboloides H ^p,q , thus extending the applicability of generalized Scherk-Schwarz reductions beyond homogeneous spaces. Together with the dictionary that relates exceptional field theory to D = 11 and IIB supergravity, respectively, the construction defines an entire new family of consistent truncations of the original theories. These include not only compactifications on spheres of different dimensions (such as AdS₅ × S ⁵), but also various hyperboloid compactifications giving rise to a higher-dimensional embedding of supergravities with non-compact and non-semisimple gauge groups.

We introduce the exceptional field theory for the group ${\mathrm{E}}_{7(7)}$, based on a $(4+56)$-dimensional spacetime subject to a covariant section condition. The “internal” generalized diffeomorphisms of the coordinates in the fundamental representation of ${\mathrm{E}}_{7(7)}$ 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 $E_{6(6)}$ 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.