On the Coulomb and Higgs branch formulae for multi-centered black holes and quiver invariants

TL;DR

This paper introduces a systematic algorithm for computing the index of multi-centered black holes and their quiver invariants, connecting Coulomb and Higgs branch formulas, and provides a computational tool for these calculations.

Contribution

It presents a general algorithm for enumerating collinear black hole configurations and computing their contributions, linking Coulomb and Higgs branch formulas for quiver moduli spaces.

Findings

Coulomb branch formula agrees with Higgs branch formula for acyclic quivers.

For cyclic quivers, the formula relates to single-centered BPS invariants.

Provides a Mathematica package for practical computation of these invariants.

Abstract

In previous work we have shown that the equivariant index of multi-centered N=2 black holes localizes on collinear configurations along a fixed axis. Here we provide a general algorithm for enumerating such collinear configurations and computing their contribution to the index. We apply this machinery to the case of black holes described by quiver quantum mechanics, and give a systematic prescription -- the Coulomb branch formula -- for computing the cohomology of the moduli space of quiver representations. For quivers without oriented loops, the Coulomb branch formula is shown to agree with the Higgs branch formula based on Reineke's result for stack invariants, even when the dimension vector is not primitive. For quivers with oriented loops, the Coulomb branch formula parametrizes the Poincar\'e polynomial of the quiver moduli space in terms of single-centered (or pure-Higgs) BPS invariants, which are conjecturally independent of the stability condition (i.e. the choice of Fayet-Iliopoulos parameters) and angular-momentum free. To facilitate further investigation we provide a Mathematica package "CoulombHiggs.m" implementing the Coulomb and Higgs branch formulae.