The Twisted Index and Topological Saddles

TL;DR

This paper explores the contributions of topological saddle points to the twisted index of 3d $ $=2 gauge theories, revealing their role in the Jeffrey-Kirwan residue and wall-crossing phenomena.

Contribution

It introduces the analysis of topological saddle points in supersymmetric localization, connecting them to the Jeffrey-Kirwan residue and wall-crossing in the twisted index.

Findings

Topological saddles contribute to the twisted index in abelian theories.

Combined saddle contributions reproduce the Jeffrey-Kirwan residue prescription.

The work elucidates wall-crossing behavior in the twisted index.

Abstract

The twisted index of 3d $\mathcal{N}=2$ gauge theories on $S^1 \times \Sigma$ has an algebro-geometric interpretation as the Witten index of an effective supersymmetric quantum mechanics. In this paper, we consider the contributions to the supersymmetric quantum mechanics from topological saddle points in supersymmetric localisation of abelian gauge theories. Topological saddles are configurations where the matter fields vanish and the gauge symmetry is unbroken, which exist for non-vanishing effective Chern-Simons levels. We compute the contributions to the twisted index from both topological and vortex-like saddles points and show that their combination recovers the Jeffrey-Kirwan residue prescription for the twisted index and its wall-crossing.