# Quiver indices and Abelianization from Jeffrey-Kirwan residues

## TL;DR

This paper explores how Jeffrey-Kirwan residues can be used to compute quiver indices and their abelianization, revealing a natural decomposition related to the structure of the quiver, with implications for understanding BPS black holes.

## Contribution

It demonstrates that the Jeffrey-Kirwan residue formula naturally yields a decomposition of quiver indices into Abelian parts, clarifying aspects of localization in quiver quantum mechanics.

## Key findings

- Residue formula leads to Abelian decomposition for acyclic quivers.
- The same approach suggests a decomposition of single-centered invariants for cyclic quivers.
- Implementation of Jeffrey-Kirwan residues in Mathematica aids in calculations.

## Abstract

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.