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The Coulomb Branch Formula conjecturally expresses the refined Witten indexfor $N=4$ Quiver Quantum Mechanics as a sum over multi-centered collinear blackhole solutions, weighted by so-called `single-centered' or `pure-Higgs'indices, and suitably modified when the quiver has oriented cycles. On theother hand, localization expresses the same index as an integral over thecomplexified Cartan torus and auxiliary fields, which by Stokes' theorem leadsto the famous Jeffrey-Kirwan residue formula. Here, by evaluating the sameintegral using steepest descent methods, we show the index is in fact given bya sum over deformed multi-centered collinear solutions, which encompasses bothregular and scaling collinear solutions. As a result, we confirm the CoulombBranch Formula for Abelian quivers in the presence of oriented cycles, andidentify the origin of the pure-Higgs and minimal modification terms as comingfrom collinear scaling solutions. For cyclic Abelian quivers, we observe thatpart of the scaling contributions reproduce the stacky invariants for trivialstability, a mathematically well-defined notion whose physics significance hadremained obscure.

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