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.
Highlights
The Quiver Quantum Mechanics (QQM) has SU(2)+ × SU(2)− global R-symmetry, with Cartan torus U(1)+ × U(1)− acting with the following charge assignments vt x3 λ− D σ λ+ φ ψ+ ψ−
We conjecture that the condition for existence of scaling bound states is strengthened to κi j ≥ K − 1 and cyclic perm. , i< j generalizing the known condition for cyclic quivers [12]
Assuming that γ is a linear combination of the basis vectors γa with coefficients at most 1, the Coulomb branch formula (2.18) simplifies to n gC ({αi, ζi}, y) ΩS(αi, y), γ=
Summary
When the quiver has oriented cycles, this prescription fails to produce a bona-fide character of the rotation group, rather it produces a rational function of y with a pole at y = 1 This issue can be traced to the existence of fixed points of J3 which do not correspond to any collinear configuration, but rather to ‘scaling solutions’, where the centers become arbitrarily close to each other, with almost vanishing angular momentum [15,16,17]. The Witten index in any chamber can be recovered from those by using attractor flow tree formulae [18,19,20,21,22]
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