Attractor flow trees, BPS indices and quivers

TL;DR

This paper introduces a formula connecting BPS indices to attractor indices via flow trees, enabling direct computation and clarifying wall-crossing behavior, with applications to quiver quantum mechanics.

Contribution

It proposes a new formula for BPS indices based on attractor flow trees, simplifying calculations and elucidating wall-crossing phenomena in supergravity and quiver theories.

Findings

Derived a sum-over-trees formula for BPS indices.

Provided a method to compute contributions from asymptotic data.

Compared attractor and single-centered indices, clarifying their relation.

Abstract

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.