We use exceptional field theory as a tool to work out the full non-linear reduction ansaetze for the AdS$_5times S^5$ compactification of IIB supergravity and its non-compact counterparts in which the sphere $S^5$ is replaced by the inhomogeneous hyp
erboloidal space $H^{p,q}$. The resulting theories are the maximal 5D supergravities with gauge groups SO(p,q). They are consistent truncations in the sense that every solution of 5D supergravity lifts to a solution of IIB supergravity. In particular, every stationary point and every holographic RG flow of the scalar potentials for the compact and non-compact 5D gaugings directly lift to solutions of IIB supergravity.
We review E$_{6(6)}$ exceptional field theory with a particular emphasis on the embedding of type IIB supergravity, which is obtained by picking the GL$(5)times {rm SL}(2)$ invariant solution of the section constraint. We work out the precise decompo
sition of the E$_{6(6)}$ covariant fields on the one hand and the Kaluza-Klein-like decomposition of type IIB supergravity on the other. Matching the symmetries, this allows us to establish the precise dictionary between both sets of fields. Finally, we establish on-shell equivalence. In particular, we show how the self-duality constraint for the four-form potential in type IIB is reconstructed from the duality relations in the off-shell formulation of the E$_{6(6)}$ exceptional field theory.
We give a reformulation of non-linear Einstein gravity, which contains the dual graviton together with the ordinary metric and a shift gauge field. The metric does not enter through a `kinetic Einstein-Hilbert term, but via topological couplings, and
so the theory does not lead to a doubling of degrees of freedom. The field equations take the form of first-order duality relations. We analyze the gauge symmetries and comment on their meaning with regard to the E11 proposal.
