Abstract
We prove the existence of singular harmonic $\mathbf{Z}_2$ spinors on $3$‑manifolds with $b_1 \gt 1$. The proof relies on a wall-crossing formula for solutions to the Seiberg–Witten equation with two spinors. The existence of singular harmonic $\mathbf{Z}_2$ 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 $G_2$-instantons [DS11].
Citation
Aleksander Doan. Thomas Walpuski. "On the existence of harmonic $Z_2$ spinors." J. Differential Geom. 117 (3) 395 - 449, March 2021. https://doi.org/10.4310/jdg/1615487003
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