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E$_{8(8)}$ Exceptional Field Theory: Geometry, Fermions and Supersymmetry

We present the supersymmetric extension of the recently constructed E$_{8(8)}$ exceptional field theory -- the manifestly U-duality covariant formulation of the untruncated ten- and eleven-dimensional supergravities. This theory is formulated on a (3+248) dimensional spacetime (modulo section constraint) in which the extended coordinates transform in the adjoint representation of E$_{8(8)}$. All bosonic fields are E$_{8(8)}$ tensors and transform under internal generalized diffeomorphisms. The fermions are tensors under the generalized Lorentz group SO(1,2)$\times$SO(16), where SO(16) is the maximal compact subgroup of E$_{8(8)}$. Vanishing generalized torsion determines the corresponding spin connections to the extent they are required to formulate the field equations and supersymmetry transformation laws. We determine the supersymmetry transformations for all bosonic and fermionic fields such that they consistently close into generalized diffeomorphisms. In particular, the covariantly constrained gauge vectors of E$_{8(8)}$ exceptional field theory combine with the standard supergravity fields into a single supermultiplet. We give the complete extended Lagrangian and show its invariance under supersymmetry. Upon solution of the section constraint the theory reduces to full D=11 or type IIB supergravity.