Eleven-dimensional supergravity reveals large exceptional symmetries upon reduction, in accordance with the $U$-duality groups of $M$ theory, but their higher-dimensional geometric origin has remained a mystery. In this Letter, we show that $D=11$ supergravity can be extended to be fully covariant under the exceptional groups ${E}_{n(n)}$, $n=6$, 7, 8. Motivated by a similar formulation of double field theory we introduce an extended ``exceptional spacetime.'' We illustrate the construction by giving the explicit ${E}_{6(6)}$ covariant form: the full $D=11$ supergravity, in a $5+6$ splitting of coordinates but without truncation, embeds into an ${E}_{6(6)}$ covariant $5+27$ dimensional theory. We argue that this covariant form likewise comprises type IIB supergravity.

We present the details of the recently constructed ${\mathrm{E}}_{6(6)}$-covariant extension of 11-dimensional supergravity. This theory requires a $5+27$-dimensional spacetime in which the ``internal'' coordinates transform in the $\overline{\mathbf{27}}$ of ${\mathrm{E}}_{6(6)}$. All fields are ${\mathrm{E}}_{6(6)}$ tensors and transform under (gauged) internal generalized diffeomorphisms. The ``Kaluza-Klein'' vector field acts as a gauge field for the ${\mathrm{E}}_{6(6)}$-covariant ``E-bracket'' rather than a Lie bracket, requiring the presence of 2-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 $\text{GL}(5)\ifmmode\times\else\texttimes\fi{}\text{SL}(2)$ and embeds complete type IIB supergravity. As a byproduct, we thus obtain an off-shell action for type IIB supergravity.

We construct maximally supersymmetric gauged N = 16 supergravity in three dimensions, thereby obtaining an entirely new class of anti--de Sitter supergravities. These models apparently cannot be derived from any known higher-dimensional theory and point to the existence of a new type of supergravity beyond D = 11. One of their noteworthy features is a non-Abelian generalization of the duality between scalar and vector fields in three dimensions. Among the possible gauge groups, SO(8) x SO(8) is distinguished as the maximal compact gauge group, but there are also more exotic possibilities such as F(4(-20)) x G2.