Exceptional Field Theory I: $E_{6(6)}$ covariant Form of M-Theory and Type IIB

We present the details of the recently constructed $E_{6(6)}$ covariant extension of 11-dimensional supergravity. This theory requires a 5+27 dimensional spacetime in which the internal' coordinates transform in the $\bar{\bf 27}$ of $E_{6(6)}$. All fields are $E_{6(6)}$ tensors and transform under (gauged) internal generalized diffeomorphisms. The Kaluza-Klein’ vector field acts as a gauge field for the $E_{6(6)}$ covariant `E-bracket’ rather than a Lie bracket, requiring the presence of two-forms akin to the tensor hierarchy of gauged supergravity. We construct the complete and unique action that is gauge invariant under generalized diffeomorphisms in the internal and external coordinates. The theory is subject to covariant section constraints on the derivatives, implying that only a subset of the extra 27 coordinates is physical. We give two solutions of the section constraints: the first preserves GL(6) and embeds the action of the complete (i.e. untruncated) 11-dimensional supergravity; the second preserves GL(5) x SL(2) and embeds complete type IIB supergravity. As a by-product, we thus obtain an off-shell action for type IIB supergravity.