The moduli space of $S^1$-type zero loci for $\mathbb{Z}/2$-harmonic spinors in dimension $3$

Let $M$ be a compact oriented $3$-dimensional smooth manifold. In this paper, we construct a moduli space consisting of pairs $(\Sigma,\psi)$ where $\Sigma$ is a $C^1$-embedding simple closed curve in $M$, $\psi$ is a $\mathbb{Z}/2$-harmonic spinor vanishing only on $\Sigma$, and ${\lVert \psi \rVert}_{L^2_1} \neq 0$. We prove that when $\Sigma$ is $C^2$, a neighborhood of $(\Sigma,\psi)$ in the moduli space can be parametrized by the space of Riemannian metrics on $M$ locally as the kernel of a Fredholm operator.

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Received 4 February 2018

Accepted 16 August 2020

Published 21 September 2023