On the existence of harmonic $Z_2$ spinors

Abstract

We prove the existence of singular harmonic Z2 spinors on 3–manifolds with b1 > 1. The proof relies on a wall-crossing formula for solutions to the Seiberg–Witten equation with two spinors. The existence of singular harmonic Z2 spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce [Joy17] regarding Donaldson and Segal’s proposal for counting G2–instantons [DS11].