The Moduli Space of S1-Type Zero Loci for 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 {(Σ,ψ)} where Σ is a C1-embedding S1 curve in M, ψ is a Z/2-harmonic spinor vanishing only on Σ and kψkL21 = 1. We will prove that this moduli space can be parametrized by the space X = { all Riemannian metrics on M } locally as the kernel of a Fredholm operator.