Attractor flow trees, BPS indices and quivers

Inspired by the split attractor flow conjecture for multi-centered black hole solutions in N=2 supergravity, we propose a formula expressing the BPS index $\Omega(\gamma,z)$ in terms of `attractor indices' $\Omega_*(\gamma_i)$. The latter count BPS states in their respective attractor chamber. This formula expresses the index as a sum over stable flow trees weighted by products of attractor indices. We show how to compute the contribution of each tree directly in terms of asymptotic data, without having to integrate the attractor flow explicitly. Furthermore, we derive new representations for the index which make it manifest that discontinuities associated to distinct trees cancel in the sum, leaving only the discontinuities consistent with wall-crossing. We apply these results in the context of quiver quantum mechanics, providing a new way of computing the Betti numbers of quiver moduli spaces, and compare them with the Coulomb branch formula, clarifying the relation between attractor and single-centered indices.