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 ${\rm SO}(1,3)\times {\rm SU}(8)$ 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.
Highlights
‘exceptional field theory’ (EFT) [14, 15]
The supersymmetry transformations are manifestly covariant under these symmetries and close, in particular, into the generalized diffeomorphisms of the 56-dimensional space
Those parts of the bosonic sector which lead to scalar and vector fields in the dimensionally reduced maximal supergravity can be assembled into E7(7) objects, namely a 56-bein encoding the internal field components and a 56-plet of vectors combining the 28 electric and 28 magnetic vectors of N = 8 supergravity; their supersymmetry transformations can be shown to take the precise form of the four-dimensional maximal gauged supergravity
Summary
Expressed via the standard decomposition of the Cartan form V−1∂M V along the compact and non-compact parts of the E7(7) Lie algebra. The full set of equations of motion is invariant under generalized diffeomorphisms in the external coordinates acting as δξeμα = ξν Dν eμα + Dμξν eν α , δξMMN = ξμ DμMMN , δξ Aμ M ξν. Variation with respect to the two-forms Bμν α and Bμν M yields projections of the first-order vector field equations (2.15). The variation of the action with respect to the vector fields leads to second order field equations. 2i e−1 ∂N e Pμ ABCDVN ABVM CD − c.c. Equation (2.35) may be compared to the second order field equations obtained from combining the derivative of (2.15) with the Bianchi identities. The first-order equations (2.39) show that the two-form fields do not bring in additional degrees of freedom to the theory
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