Supersymmetric E${}_{7(7)}$ Exceptional Field Theory

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 $\mathrm{S}\mathrm{O}(1,3)\times \mathrm{S}\mathrm{U}(8)$ 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.