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

TL;DR

This work develops a complete, algorithmic framework for computing the refined Coulomb index $g_{\rm Coulomb}$ of multi-centered black holes, enabling reconstruction of the total index $\Omega(\gamma;y)$ from single-centered invariants $\Omega_{\rm S}(\alpha)$. It provides explicit Abelian no-loop formulas, introduces inductive constructions $F$ and $G$ to capture collinear and scaling solutions, and extends to generic Abelian and non-Abelian quivers, including non-primitive dimension vectors. The authors prove the Coulomb/Higgs equivalence for quivers without oriented loops by connecting the Coulomb branch formula to Reineke’s Higgs-branch results and by employing Abelianization, with a Mathematica package CoulombHiggs.m to implement the calculations. The framework unifies the quiver moduli space cohomology computations across Higgs and Coulomb perspectives, enabling systematic access to Poincaré–Laurent and Dolbeault polynomials via single-centered BPS 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 Poincaré 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.

Figures (1)

• Figure 1: Left: The physical domain for 3-center collinear solutions has 5 boundary components at which the superpotential $W$ diverges. The sign of $W$ on each component is that of the quantity indicated on the corresponding edge. Right: the physical domain for 4-center collinear scaling solutions also has 5 boundary components, at which the superpotential $\hat{W}$ diverges. The sign of $W$ on each component is that of the linear combination of $\hat{\alpha}_{ij}$ indicated on the corresponding edge or vertex. In both cases, by considering the topology of the gradient flow (indicated by the arrows for some suitable choice of signs on the boundary components), it is easy to convince oneself that a critical point exists in the physical domain if and only if the signs on the 5 boundary components are identical, or flip 4 times around the boundary.