We construct the non-linear realisation of the semi-direct product of E 11 and its first fundamental representation at lowest order and appropriate to spacetime dimensions four to seven. This leads to a non-linear realisation of the duality groups and introduces fields that depend on a generalised space which possess a generalised vielbein. We focus on the part of the generalised space on which the duality groups alone act and construct an invariant action.

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 the extended geometry of M-theory. A crucial by-product of this construction is the derivation of the physical section condition for M-theory formulated in an extended space.

Extending previous work on generalised geometry, we explicitly construct an E 7(7) -valued vielbein in eleven dimensions that encompasses the would be scalar bosonic degrees of freedom after reduction to four dimensions of D = 11 supergravity, by identifying new "generalised vielbeine" in eleven dimensions associated with the dual 6-form potential and the dual graviton. By maintaining full on-shell equivalence with the original theory at every step our construction furnishes the non-linear ansatz for the dual (magnetic) 7-form flux for any non-trivial compactification of D = 11 supergravity, complementing the known non-linear ansätze for the metric and the 4-form flux. A preliminary analysis of the generalised vielbein postulate for the new vielbein components reveals tantalising hints of new structures beyond D = 11 supergravity and ordinary space-time covariancxe, and also points to the possible D = 11 origins of the embedding tensor. We discuss the extension of these results to E 8(8) .

Abstract:We give the supersymmetric extension of exceptional field theory for E 7(7) , which is based on a (4 + 56)-dimensional generalized spacetime subject to a covariant constraint. The fermions are tensors under the local Lorentz group SO(1, 3) × SU(8) and transform as scalar densities under the E 7(7) (internal) generalized diffeomorphisms. The supersymmetry transformations are manifestly covariant under these symmetries and close, in particular, into the generalized diffeomorphisms of the 56-dimensional space. We give the fermionic field equations and prove supersymmetric invariance. We establish the consistency of these results with the recently constructed generalized geometric formulation of D = 11 supergravity.

We present nonlinear uplift Ansätze for all the bosonic degrees of freedom and dual fields in the S 7 reduction of D ¼ 11 supergravity to maximal SO(8) gauged supergravity and test them for the SOð7Þ AE invariant solutions. In particular, we complete the known Ansätze for the internal components of the metric and four-form flux by constructing a nonlinear Ansatz for the internal components of the dual seven-form flux. Furthermore, we provide Ansätze for the complete set of 56 vector fields, which are given by more general structures than those available from standard Kaluza-Klein theory. The novel features encountered here have no conventional geometric interpretation and provide a new perspective on KaluzaKlein theory. We study the recently found set of generalized vielbein postulates and, for the S 7 compactification, we show that they reduce to the E 7ð7Þ Cartan equation of maximal SO(8) gauged supergravity in four dimensions. The significance of this framework for a higher-dimensional understanding of the embedding tensor and other gauged maximal supergravities is briefly discussed.