Multiple zero modes of the Dirac operator in three dimensions

TL;DR

This paper explores the existence of multiple zero modes in three-dimensional Dirac operators, generalizing recent constructions and linking zero modes to topological Hopf maps, revealing new mathematical relationships.

Contribution

It extends the class of Dirac operators known to have zero modes and establishes a connection between zero modes and Hopf maps, broadening understanding of their topological properties.

Findings

Construction of multiple zero modes for a wider class of Dirac operators

Relation between zero modes and Hopf index

Generalization of recent results on zero mode degeneracy

Abstract

One of the key properties of Dirac operators is the possibility of a degeneracy of zero modes. For the Abelian Dirac operator in three dimensions the construction of multiple zero modes has been sucessfully carried out only very recently. Here we generalise these results by discussing a much wider class of Dirac operators together with their zero modes. Further we show that those Dirac operators that do admit zero modes may be related to Hopf maps, where the Hopf index is related to the number of zero modes in a simple way.