An Index Formula for Supersymmetric Quantum Mechanics

TL;DR

This paper develops a residue integral formula for the refined index in supersymmetric quantum mechanics, enabling efficient computation of BPS state counts and understanding wall-crossing phenomena in related theories.

Contribution

It introduces a localization residue formula for the refined index in gauged quantum mechanics with four supercharges, connecting it to quiver moduli spaces and BPS state counting.

Findings

Residue integral formula for the refined index derived

Efficient computation method for quiver moduli cohomology established

Wall-crossing phenomena explained via contour variations in the integral

Abstract

We derive a localization formula for the refined index of gauged quantum mechanics with four supercharges. Our answer takes the form of a residue integral on the complexified Cartan subalgebra of the gauge group. The formula captures the dependence of the index on Fayet-Iliopoulos parameters and the presence of a generic superpotential. The residue formula provides an efficient method for computing cohomology of quiver moduli spaces. Our result has broad applications to the counting of BPS states in four-dimensional N=2 systems. In that context, the wall-crossing phenomenon appears as discontinuities in the value of the residue integral as the integration contour is varied. We present several examples illustrating the various aspects of the index formula.