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

TL;DR

This paper demonstrates a consistent truncation of massive IIA supergravity on a six-sphere to a four-dimensional maximal supergravity with a dyonic ISO(7) gauging, providing explicit embeddings and revealing structural patterns.

Contribution

It provides the first complete, non-linear embedding of all four-dimensional fields into massive IIA supergravity on S^6, including the derivation of explicit expressions and the analysis of truncation overlaps.

Findings

Complete embedding of 4D fields into IIA supergravity achieved.

Explicit expressions for the Freund-Rubin term derived.

Overlap between N=8 and N=2 truncations identified.

Abstract

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.