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

TL;DR

This paper constructs a moduli space of pairs consisting of a simple closed curve and a $Z/2$-harmonic spinor vanishing on it in a 3-manifold, showing local parametrization by Riemannian metrics via Fredholm operators.

Contribution

It introduces a new moduli space for $Z/2$-harmonic spinors with zero loci of $S^1$-type and establishes local parametrization by Riemannian metrics.

Findings

The moduli space includes pairs $( ext{curve}, ext{spinor})$ with specific properties.

Local structure of the moduli space is described by the kernel of a Fredholm operator.

Neighborhoods of points in the moduli space can be parametrized by Riemannian metrics.

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