Witten Index and Wall Crossing

TL;DR

Hori, Kim, and Yi develop a first-principles localization framework to compute the Witten index of one-dimensional ${\mathcal{N}}\ge 2$ gauged linear sigma models, including abelian and non-abelian gauge groups, with and without flavor twists. The core idea is to push the gauge coupling to zero and evaluate the index via boundary (JK-residue) contributions from singular hyperplanes, together with controlled infinity contributions on the Coulomb branch; this yields a concrete wall-crossing formula that matches a Coulomb-branch/Mixed-branch analysis across simple phase boundaries. They demonstrate a systematic procedure for higher-rank theories, extend the analysis to ${\mathcal{N}}=4$ quivers and their quiver invariants, and illustrate rich phase structures through detailed examples (Grassmannians, hypersurfaces, Distler-Kachru models, and quiver quantum mechanics). The results connect wall-crossing in 1d to BPS state counting in higher dimensions and provide exact, non-perturbative control over spectrum across phases, including non-compact Higgs branches where twists are essential. The work thus supplies a comprehensive, exact framework for index computations and wall-crossing in 1d SUSY gauge theories with broad implications for BPS state counting and quiver invariants.

Abstract

We compute the Witten index of one-dimensional gauged linear sigma models with at least ${\mathcal N}=2$ supersymmetry. In the phase where the gauge group is broken to a finite group, the index is expressed as a certain residue integral. It is subject to a change as the Fayet-Iliopoulos parameter is varied through the phase boundaries. The wall crossing formula is expressed as an integral at infinity of the Coulomb branch. The result is applied to many examples, including quiver quantum mechanics that is relevant for BPS states in $d=4$ ${\mathcal N}=2$ theories.

Figures (12)

• Figure 1: The moduli space $\mathfrak{M}$ for a $U(1)$ gauge theory
• Figure 2: triangle quiver
• Figure 3: The phases of the model (1,1,1)
• Figure 4: (Possible) phases of the model (k,1,1)
• Figure 5: (Possible) phases of the models with $\vec{k}=(5,2,1)$ (Left) and $\vec{k}=(5,3,2)$ (Right)