G-structures for black hole near-horizon geometries

Abstract

We derive necessary and sufficient conditions for warped AdS2 solutions of Type II supergravity to preserve $\mathcal{N}=1$ supersymmetry, in terms of bispinors. Such solutions generically support an $SU(3)$-structure on their internal manifold M8, which can experience an enhancement to a $\textrm{G}_2$-structure. We perform an $SU(3)$-structure torsion classes analysis and express the fluxes and other physical fields in terms of these, in general. We use our results to derive two new classes of AdS2 solutions. In (massive) Type IIA supergravity we derive an $\mathcal{N=1}$ supersymmetric class for which M8 decomposes as a weak $\textrm{G}_2$ manifold foliated over an interval and which is locally defined in terms of a degree three polynomial. In Type IIB supergravity we find a class of $\textrm{AdS}_2×\textrm{CY}_2×\Sigma_2$ solutions governed by a harmonic function on $\Sigma_2$ and partial differential equations reminiscent of D3-D7-brane configurations.