The zero loci of Z/2 harmonic spinors in dimension 2, 3 and 4

TL;DR

This paper investigates the structure of the zero set of Z/2 harmonic spinors on 2, 3, and 4-dimensional Riemannian manifolds, revealing geometric properties of these loci in relation to the spinor fields.

Contribution

It provides a detailed analysis of the zero loci of Z/2 harmonic spinors across dimensions two to four, advancing understanding of their geometric and topological features.

Findings

Characterization of zero loci in different dimensions

Relationship between zero set and spinor bundle structure

Insights into the regularity and structure of harmonic spinors

Abstract

Supposing that X is a Riemannian manifold, a Z/2 spinor on X is defined by a data set consisting of a closed set in X to be denoted by Z, a real line bundle over X-Z, and a nowhere zero section on X-Z of the tensor product of the real line bundle and a spinor bundle. The set Z and the spinor are jointly constrained by the following requirement: The norm of the spinor must extend across Z as a continuous function vanishing on Z. In particular, the vanishing locus of the norm of the spinor is the complement of the set where the real line bundle is defined, and hence where the spinor is defined. The Z/2 spinor is said to be harmonic when it obeys a first order Dirac equation on X-Z. This monograph analyzes the structure of the set Z for a Z/2 harmonic spinor on a manifold of dimension either two, three or four.