On the existence of harmonic $\mathbf{Z}_2$ spinors

TL;DR

This paper proves the existence of singular harmonic Z_2 spinors on certain 3-manifolds, using a wall-crossing formula related to Seiberg-Witten equations, providing insights into G_2-instanton counting.

Contribution

It introduces a new existence proof for singular harmonic Z_2 spinors on 3-manifolds with b_1 > 1, utilizing a novel wall-crossing formula for Seiberg-Witten solutions.

Findings

Existence of singular harmonic Z_2 spinors on specific 3-manifolds.

Development of a wall-crossing formula for Seiberg-Witten equations with two spinors.

Implications for Donaldson and Segal's G_2-instanton counting.

Abstract

We prove the existence of singular harmonic ${\bf Z}_2$ spinors on $3$-manifolds with $b_1 > 1$. The proof relies on a wall-crossing formula for solutions to the Seiberg-Witten equation with two spinors. The existence of singular harmonic ${\bf Z}_2$ spinors and the shape of our wall-crossing formula shed new light on recent observations made by Joyce regarding Donaldson and Segal's proposal for counting $G_2$-instantons.