Quantum entropy of supersymmetric spinning black holes in 5 dimensions

In this thesis, we propose a formula for the quantum entropy function of the BMPV black hole using localisation methods. We use the framework of off-shell N = 2, D = 5 supergravity to study the quantum entropy function of a black hole proposed by Sen. We analyse the localization equations relevant to the quantum entropy of the BMPV black holes. The precise problem is to classify all solutions to the off-shell supersymmetry equations in N = 2 supergravity coupled to nv + 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 lo-calisation 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 localisation solution for the five-dimensional black hole problem is an (nv + 1)-dimensional manifold, which is precisely the lift of the localisation manifold for supersymmetric black holes in four-dimensional asymptotically flat space. We then evaluate the action on the black hole, including the vector multiplet fluctuations. We lift the localisation solutions from 4d for the Weyl multiplet and evaluate the second derivative action. We then propose a formula for the full action, including higher derivatives, over the full localisation manifold. By comparing with the low energy logarithmic corrections found by Sen, we propose an ansatz for the one loop determinant. Lastly, we evaluate the quantum entropy function at second derivative level.