On the localization manifold of 5d supersymmetric spinning black holes

We analyze the localization equations relevant to the quantum entropy of spinning supersymmetric black holes in five-dimensional asymptotically flat space. The precise problem is to classify all solutions to the off-shell supersymmetry equations in N=2 supergravity coupled to $n_\text{v}+1$ vector multiplets around the near-horizon black hole. We rewrite these equations in terms of the bosonic spinor bilinears that exist in the geometry for an arbitrary background. We then focus on the vector multiplet fluctuations around the near-horizon attractor region of the supersymmetric black hole, and classify all smooth solutions to the localization equations in this background for different choices of analytic continuation. For the choice of analytic continuation consistent with the 4d/5d lift, we find that the most general localization solution for the five-dimensional black hole problem is an~$(n_\text{v}+1)$-dimensional manifold, which is precisely the lift of the localization manifold for supersymmetric black holes in four-dimensional asymptotically flat space.