The universal moduli space of non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic spinors

Abstract

This article equips the universal moduli spaces of non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic spinors and eigenspinors with tame Fréchet manifold structures and proves that their natural projections to the corresponding parameter spaces are uniformly Fredholm. A unified construction yields the deformation theories of harmonic spinors and eigenspinors in dimension three and of chiral harmonic spinors in dimension four, recovering earlier work of Donaldson, Parker, and Takahashi. As an application, a deformation theory for non-degenerate $\mathbb{Z}/2\mathbb{Z}$ harmonic self-dual $2$-forms in dimension four is established.