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

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.

arXiv comment45 pages; v2: added subsection 3.4.2 on Abelian quivers with multiple oriented cycles, plus various clarifications and several references. Converted to SciPost style

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Cited by in corpus (8)

• Toroidal and Elliptic Quiver BPS Algebras and Beyond
• Scaling Black Holes and Modularity
• BPS States Meet Generalized Cohomology
• Wall-Crossing Effects on Quiver BPS Algebras
• Supersymmetric black holes and $T\bar{T}$ deformation
• The geometry of gauged (super)conformal mechanics
• Black holes and the loss landscape in machine learning
• On the existence of scaling multi-centered black holes