All solutions of the localization equations for N=2 quantum black hole entropy

We find the most general bosonic solution to the localization equations describing the contributions to the quantum entropy of supersymmetric black holes in four-dimensional N=2 supergravity coupled to n_v vector multiplets. This requires the analysis of the BPS equations of the corresponding off-shell supergravity (including fluctuations of the auxiliary fields) with AdS2 \times S2 attractor boundary conditions. Our work completes and extends the results of arXiv:1012.0265 that were obtained for the vector multiplet sector, to include the fluctuations of all the fields of the off-shell supergravity. We find that, when the auxiliary SU(2) gauge field strength vanishes, the most general supersymmetric configuration preserving four supercharges is labelled by n_v+1 real parameters corresponding to the excitations of the conformal mode of the graviton and the scalars of the n_v vector multiplets. In the general case, the localization manifold is labelled by an additional SU(2) triplet of one-forms and a scalar function.