The moduli space of $S^1$-type zero loci for $\mathbb{Z}\mathrm{/}2$-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 $(\mathrm{\Sigma },\psi )$ where $\mathrm{\Sigma }$ is a $C^1$-embedding simple closed curve in $M$, $\psi$ is a $\mathbb{Z}\mathrm{/}2$-harmonic spinor vanishing only on $\mathrm{\Sigma }$, and $\mathrm{\parallel }\psi {\mathrm{\parallel }}_{L_1^2}\mathrm{\ne }0$. We prove that when $\mathrm{\Sigma }$ is $C^2$, a neighborhood of $(\mathrm{\Sigma },\psi )$ in the moduli space can be parametrized by the space of Riemannian metrics on $M$ locally as the kernel of a Fredholm operator.