Consistent ${\cal N}=8$ truncation of massive IIA on $S^6$

Massive type IIA supergravity is shown to admit a consistent truncation on the six-sphere to maximal supergravity in four dimensions with a dyonic ISO(7) gauging. We obtain the complete, non-linear embedding of all the $D=4$ fields into the IIA metric and form potentials, and show its consistency. We first rewrite the IIA theory in an $\textrm{SO}(1,3) \times \textrm{SL}(7)$--covariant way. Then, we employ an ${\cal N}=8$ SL(7)--covariant restriction of the $D=4$ tensor hierarchy in order to find the full embedding. The redundant $D=4$ degrees of freedom introduced by the tensor hierarchy can be eliminated by writing the embedding in terms of the field strengths and exploiting the restricted duality hierarchy. In particular, closed expressions for the Freund-Rubin term are found using this technique which reveal a pattern valid for other truncations. Finally, we show that the present ${\cal N}=8$ truncation of massive IIA on $S^6$ and the ${\cal N}=2$ truncation obtained when $S^6$ is equipped with its nearly-K\"ahler structure, overlap in the ${\cal N}=1$, G$_2$--invariant sector of the former.