Examples of singularity models for $\mathbb{Z}/2$ harmonic 1-forms and spinors in dimension 3

TL;DR

This paper constructs local models for $ obreak ext{Z}/2$ harmonic 1-forms and spinors in three dimensions near singular points, using symmetries of Platonic solids to describe their zero loci as rays from the origin.

Contribution

It introduces explicit local models for $ obreak ext{Z}/2$ harmonic forms and spinors in 3D based on Platonic symmetries, advancing understanding of their singularities.

Findings

Models are homogeneous under rescaling.

Zero loci consist of rays pointing to Platonic solid vertices.

Symmetry-based construction of singularity models.

Abstract

We use the symmetries of the tetrahedron, octahedron and icosahedron to construct local models for a $\mathbb{Z}/2$ harmonic 1-form or spinor in 3-dimensions near a singular point in its zero loci. The local models are $\mathbb{Z}/2$ harmonic 1-forms or spinors on $\mathbb{R}^3$ that are homogeneous with respect to rescaling of $\mathbb{R}^3$ with their zero locus consisting of four or more rays from the origin. The rays point from the origin to the vertices of a centered tetrahedron in one example; and they point from those of a centered octahedron and a centered icosahedron in two others.