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 $\|\psi\|_{L^2_1}\neq 0$. 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.