The 3d Twisted Index and Wall-Crossing

TL;DR

This paper investigates how the twisted index of 3d supersymmetric gauge theories on $S^1 imes ext{Riemann surface}$ changes with a real FI parameter, revealing wall-crossing phenomena and deriving a general wall-crossing formula.

Contribution

It introduces a modified contour prescription for the twisted index in the presence of a 1d FI parameter and derives a general wall-crossing formula for abelian theories.

Findings

Wall-crossing phenomena occur due to the 1d FI parameter.

A general wall-crossing formula for abelian theories is derived.

Illustrations include $ ext{N}=4$ abelian theories and non-abelian examples.

Abstract

We study the twisted index of 3d $\mathcal{N}=2$ supersymmetric gauge theories on $S^1 \times \Sigma$ in the presence of a real FI parameter deformation. This parameter induces a 1d FI parameter for the effective supersymmetric quantum mechanics on $S^1$. Using supersymmetric localisation, the twisted index can be expressed as a contour integral. We show that the contour prescription is modified in the presence of the 1d FI parameter, leading to wall-crossing phenomena for the twisted index. In particular, we derive a general wall-crossing formula for abelian gauge theories. We also examine the origin of wall-crossing as change of stability condition in the algebro-geometric interpretation of the twisted index. These ideas are illustrated for abelian theories with $\mathcal{N}=4$ supersymmetry and in a non-abelian example that reproduces wall-crossing phenomena associated to moduli spaces of stable pairs.