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

TL;DR

This work develops a fixed-point localization framework to compute the moduli-dependent BPS index for multi-centered N=2 black holes by expressing it through single-centered indices and a configurational refined index. It distinguishes between compact phase spaces (no scaling) and non-compact ones with scaling regions, and introduces a minimal modification hypothesis to consistently include scaling contributions, validated in dipole halo configurations and through explicit one-modulus model examples. The main result is a practical, inductive scheme that yields SU(2) characters for the total index by combining fixed-point data with controlled corrections from scaling regions, preserving wall-crossing and split attractor flow structures. This provides a concrete, computable bridge between microscopic single-centered data and the spectrum of multi-centered BPS states across moduli space, with potential applications to quiver quantum mechanics and black hole microstate counting. The approach advances our understanding of how non-compact solution spaces contribute to BPS spectra and offers a robust tool for studying moduli-dependent black hole indices in string theory.

Abstract

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