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 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 $\Vert \psi {\Vert }_{L_1^2}\ne 0$. 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.

arXiv comment17/8/2020 Important correction: In this version we need further regularity for $\Sigma$ now ($\Sigma$ is in C^2). The result is not as strong as the previous version. The current version will appear on Communication in Analysis and Geometry(CAG)