G-structures for black hole near-horizon geometries

We derive necessary and sufficient conditions for warped AdS$_2$ solutions of Type II supergravity to preserve ${\cal N}=1$ supersymmetry, in terms of bispinors. Such solutions generically support an SU$(3)$-structure on their internal manifold M$_8$, which can experience an enhancement to a 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 AdS$_2$ solutions. In (massive) Type IIA supergravity we derive an ${\mathcal N}=1$ supersymmetric class for which M$_8$ decomposes as a weak 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 AdS$_2\times\text{S}^2\times\text{CY}_2\times\Sigma_2$ solutions preserving small ${\cal N}=4$ supersymmetry, governed by a harmonic function on $\Sigma_2$ and partial differential equations reminiscent of D3-D7-brane configurations.