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We propose a formula for computing the (moduli-dependent) contribution ofmulti-centered solutions to the total BPS index in terms of the(moduli-independent) indices associated to single-centered solutions. The maintool in our analysis is the computation of the refined index Tr(-y)^{2J_3} ofconfigurational degrees of freedom of multi-centered BPS black hole solutionsin N=2 supergravity by localization methods. When the charges carried by thecenters do not allow for scaling solutions (i.e. solutions where a subset ofthe centers can come arbitrarily close to each other), the phase space ofclassical BPS solutions is compact and the refined index localizes to a finiteset of isolated fixed points under rotations, corresponding to collinearsolutions. When the charges allow for scaling solutions, the phase space isnon-compact but appears to admit a compactification with finite volume andadditional non-isolated fixed points. We give a prescription for determiningthe contributions of these fixed submanifolds by means of a `minimalmodification hypothesis', which we prove in the special case of dipole haloconfigurations.

A fixed point formula for the index of multi-centered N=2 black holes