An Index Formula for Supersymmetric Quantum Mechanics

TL;DR

The paper addresses counting protected ground states in gauged quantum mechanics with four supercharges, developing a localization-based residue formula for the refined index. The index is computed as a Jeffrey-Kirwan residue on the complexified Cartan, with the contour dictated by Fayet-Iliopoulos parameters and superpotential data, enabling precise tracking of wall-crossing. It connects the refined index to Higgs-branch cohomology, BPS spectra in 4d N=2 systems, and quiver moduli spaces, and validates the method through a range of Abelian and non-Abelian examples including Dyon Chains, Electron Halos, and superpotential deformations such as XYZ-type models and SU(3) YM. The results provide an efficient, exact tool for computing ground-state spectra and elucidate the geometric and physical structure behind wall-crossing in supersymmetric theories.

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.

Figures (12)

• Figure 1: The general abelian linear quiver which governs the bounds states of the specified dyons. The integers at the nodes denote the ranks of the associated gauge groups, while $k_{i}$ are the number of bifundamentals (arrows).
• Figure 2: The quiver relevant for studying the bound states of a monopole and a cloud of electrons. The integers at the nodes denote the ranks of the associated gauge groups, while $k$ is the number of bifundamentals (arrows).
• Figure 3: The two-node linear quiver. The integers at the nodes denote the ranks of the associated gauge groups, while $k$ is the number of bifundamentals (arrows). In (b) and (c), the two ways of decoupling a $U(1).$
• Figure 4: The three-node linear quiver. The integers at the nodes denote the ranks of the associated gauge groups, while $k_{i}$ are the number of bifundamentals (arrows). In (b), (c), and (d), the three ways of decoupling a $U(1).$ In (b), the quiver has become disconnected and the model factorizes.
• Figure 5: Three-node quiver with the first node decoupled. The figure shows the charge covectors $Q_i$ in $\frak{h}^*= (\frak{u}(1)^2)^*\cong \mathbb{R}^2$.