Complete $D=11$ embedding of SO(8) supergravity

The truncation formulas of $D=11$ supergravity on $S^7$ to $D=4$ $\mathcal{N}=8$ SO(8)-gauged supergravity are completed to include the full nonlinear dependence of the $D=11$ three-form potential ${\widehat{A}}_{(3)}$ on the $D=4$ fields, and their consistency is shown. The full embedding into ${\widehat{A}}_{(3)}$ is naturally expressed in terms of a restricted version, still $\mathcal{N}=8$ but only SL(8)–covariant, of the $D=4$ tensor hierarchy. The redundancies introduced by this approach are removed at the level of the field strength ${\widehat{F}}_{(4)}$ by exploiting $D=4$ duality relations. Finally, new expressions for the full consistent truncation formulas are given that are explicit in all $D=11$ and $D=4$ fields.