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 AdS(2) x S-2 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.