The geometry of $\mathcal{N}=3$ AdS${}_4$ in massive IIA

The geometry of the $\mathcal{N}=3$, SO(4)–invariant, AdS${}_4$ solution of massive type IIA supergravity that uplifts from the $\mathcal{N}=3$ vacuum of $D=4$ $\mathcal{N}=8$ dyonic ISO(7) supergravity is investigated. Firstly, a $D=4$, SO(4)–invariant restricted duality hierarchy is constructed and used to uplift the entire, dynamical SO(4)–invariant sector to massive type IIA. The resulting consistent uplift formulae are used to obtain a new local expression for the $\mathcal{N}=3$ AdS${}_4$ solution in massive IIA and analyse its geometry. Locally, the internal $S^6$ geometry corresponds to a warped fibration of $S^2$ and a hemisphere of $S^4$. This can be regarded as a warped generalisation of the usual twistor fibration geometry. Finally, the triplet of Killing spinors corresponding to the $\mathcal{N}=3$ solution are constructed and shown to obey the massive type IIA Killing spinor equations.