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

Abstract

Let $M$ be a compact oriented 3-dimensional smooth manifold. In this paper, we will construct a moduli space consisting of the following date $\{(\Sigma, \psi)\}$ where $\Sigma$ is a $C^1$-embedding $S^1$ curve in $M$, $\psi$ is a $\mathbb{Z}/2$-harmonic spinor vanishing only on $\Sigma$ and $\|\psi\|_{L^2_1}=1$. We will prove that this moduli space can be parametrized by the space $\mathcal{X}=$ all Riemannian metrics on M locally as the kernel of a Fredholm operator.