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

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.

Groups actions are corrected and more details are given about them. An appendix has been added proving that if Z has only 2 rays from the origin, then the rays are antipodal

References in corpus (1)

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