In this work the exceptional field theory formulation of supergravity with SL (5) gauge group is considered. This group appears as a U-duality group of D = 7 maximal supergravity. In the formalism presented the hidden global duality group is promoted into a gauge group of a theory in dimensions 7+number of extended directions. This work is a continuation of the series of works for E _8,7,6 , SO (5, 5) and SL (3) × SL (2) duality groups.

We construct the supersymmetric completion of E₆₍₆₎-covariant exceptional field theory. The theory is based on a (5 + 27)-dimensional generalized space-time subject to a covariant section constraint. The fermions are tensors under the local Lorentz group SO(1, 4) × USp(8) and transform as weighted scalars under the E₆₍₆₎ (internal) generalized diffeomorphisms. We present the complete Lagrangian and prove its invariance under supersymmetry. Upon explicit solution of the section constraint the theory embeds full D = 11 supergravity and IIB supergravity, respectively.

We construct Exceptional Field Theory for the group SO(5, 5) based on the extended (6+16)-dimensional spacetime, which after reduction gives the maximal D = 6 supergravity. We present both a true action and a duality-invariant pseudo-action formulations. All the fields of the theory depend on the complete extended spacetime. The U-duality group SO(5, 5) is made a geometric symmetry of the theory by virtue of introducing the generalised Lie derivative that incorporates a duality transformation. Tensor hierarchy appears as a natural consequence of the algebra of generalised Lie derivatives that are viewed as gauge transformations. Upon truncating different subsets of the extra coordinates, maximal supergravities in D = 11 and D = 10 (type IIB) can be recovered from this theory.