  In quiver quantum mechanics with 4 supercharges, supersymmetric ground states
are known to be in one-to-one correspondence with Dolbeault cohomology classes
on the moduli space of stable quiver representations. Using supersymmetric
localization, the refined Witten index can be expressed as a residue integral
with a specific contour prescription, originally due to Jeffrey and Kirwan,
depending on the stability parameters. On the other hand, the physical picture
of quiver quantum mechanics describing interactions of BPS black holes predicts
that the refined Witten index of a non-Abelian quiver can be expressed as a sum
of indices for Abelian quivers, weighted by `single-centered invariants'. In
the case of quivers without oriented loops, we show that this decomposition
naturally arises from the residue formula, as a consequence of applying the
Cauchy-Bose identity to the vector multiplet contributions. For quivers with
loops, the same procedure produces a natural decomposition of the
single-centered invariants, which remains to be elucidated. In the process, we
clarify some under-appreciated aspects of the localization formula. Part of the
results reported herein have been obtained by implementing the Jeffrey-Kirwan
residue formula in a public Mathematica code.
