Deformations of $\mathbb Z_2$-Harmonic Spinors on 3-Manifolds

TL;DR

This paper studies the local structure of the moduli space of $Z_2$-harmonic spinors on 3-manifolds, revealing that near smooth singular sets, the space projects to a codimension 1 submanifold, using advanced analysis techniques.

Contribution

It provides a detailed analysis of the universal moduli space of $Z_2$-harmonic spinors, incorporating the effects of singular sets and employing the Nash-Moser theorem for regularity issues.

Findings

Moduli space projects to a codimension 1 submanifold near smooth singular sets.

Analysis accounts for infinite-dimensional obstruction bundle complexities.

Uses Nash-Moser theorem to handle loss of regularity in deformations.

Abstract

A $\mathbb Z_2$-harmonic spinor on a 3-manifold $Y$ is a solution of the Dirac equation on a bundle that is twisted around a submanifold $\mathcal Z$ of codimension 2 called the singular set. This article investigates the local structure of the universal moduli space of $\mathbb Z_2$-harmonic spinors over the space of parameters $(g,B)$ consisting of a metric and perturbation to the spin connection. The main result states that near a $\mathbb Z_2$-harmonic spinor with $\mathcal Z$ smooth, the universal moduli space projects to a codimension 1 submanifold in the space of parameters. The analysis is complicated by the presence of an infinite-dimensional obstruction bundle and a loss of regularity in the first variation of the Dirac operator with respect to deformations of the singular set $\mathcal Z$, necessitating the use of the Nash-Moser Implicit Function Theorem.