Wrapping M5-branes on a non-spin manifold: Spin${}^h$ structures and AdS${}_7$ black holes

We construct a solution of seven-dimensional maximal gauged supergravity of the form AdS${}_2\times M_5$, where $M_5$ is the five-dimensional Wu manifold $SU(3)\mathrm{/}SO(3)$. The solution is supersymmetric, despite the fact that the Wu manifold admits neither a spin structure nor a spin${}^c$ structure. Rather, it admits a spin${}^h$ structure, wherein the ill-definedness of ordinary spinors due to a minus sign in their holonomy is compensated by a further minus sign arising through the spinors carrying $SO(3)$ Yang-Mills charge, in a half-integer isospin representation of the $SU(2)$ double cover of the $SO(3)$. We show how the configuration can be lifted to a solution of $D=11$ supergravity of the form AdS${}_2\times M_9$, where $M_9$ is the unit tangent bundle of the Wu manifold. We discuss the geometry and topology of $M_9$, and study the supersymmetry from the eleven-dimensional standpoint. We extend the seven-dimensional solution to a black hole, and discuss some aspects of its holographic interpretation and the UV to IR flow. We revisit a previously-known AdS${}_2\times H^3\times H^2$ solution, and its UV to IR flow, showing that the flow equations in fact admit of an exact solution. Additionally, we give a detailed discussion of the lift of the AdS${}_2\times S^5$ solution to $D=11$, exhibiting similarities and differences between the properties of its internal space $M_9$ (the unit tangent bundle of $S^5$) and the Wu manifold example of our main discussion. We also construct the non-compact dual of the Wu manifold solution, where $M_5$ is taken to be $SL(3,\mathbb{R})\mathrm{/}SO(3)$.