Multi-centered black holes, scaling solutions and pure-Higgs indices from localization

TL;DR

This paper demonstrates that the refined Witten index for Abelian quivers with oriented cycles can be computed via steepest descent, revealing the role of scaling solutions and confirming the Coulomb Branch Formula.

Contribution

It shows the index equals a sum over deformed multi-centered solutions, clarifies the origin of pure-Higgs indices, and confirms the Coulomb Branch Formula for cyclic Abelian quivers.

Findings

Index expressed as sum over deformed multi-centered solutions

Pure-Higgs indices originate from collinear scaling solutions

Part of scaling contributions match stacky invariants

Abstract

The Coulomb Branch Formula conjecturally expresses the refined Witten index for $N=4$ Quiver Quantum Mechanics as a sum over multi-centered collinear black hole solutions, weighted by so-called `single-centered' or `pure-Higgs' indices, and suitably modified when the quiver has oriented cycles. On the other hand, localization expresses the same index as an integral over the complexified Cartan torus and auxiliary fields, which by Stokes' theorem leads to the famous Jeffrey-Kirwan residue formula. Here, by evaluating the same integral using steepest descent methods, we show the index is in fact given by a sum over deformed multi-centered collinear solutions, which encompasses both regular and scaling collinear solutions. As a result, we confirm the Coulomb Branch Formula for Abelian quivers in the presence of oriented cycles, and identify the origin of the pure-Higgs and minimal modification terms as coming from collinear scaling solutions. For cyclic Abelian quivers, we observe that part of the scaling contributions reproduce the stacky invariants for trivial stability, a mathematically well-defined notion whose physics significance had remained obscure.