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

TL;DR

This paper explores the detailed geometry of a specific ${ m N}=3$ supersymmetric AdS$_4$ solution in massive type IIA supergravity, providing new uplift formulas and analyzing its internal space structure.

Contribution

It constructs a restricted duality hierarchy for the ${ m N}=3$ solution, derives new uplift formulae, and characterizes the internal geometry as a warped fibration, including explicit Killing spinors.

Findings

Derived a new local expression for the ${ m N}=3$ AdS$_4$ solution.

Identified the internal space as a warped fibration of $S^2$ and a hemisphere of $S^4$.

Constructed explicit Killing spinors satisfying massive IIA equations.

Abstract

The geometry of the ${\cal N} = 3$, SO(4)--invariant, AdS$_4$ solution of massive type IIA supergravity that uplifts from the ${\cal N} = 3 $ vacuum of $D=4$ ${\cal N} = 8$ 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 ${\cal 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 ${\cal N}=3$ solution are constructed and shown to obey the massive type IIA Killing spinor equations.