Degeneracy of zero modes of the Dirac operator in three dimensions

TL;DR

This paper proves that degeneracy of zero modes exists for the Abelian Dirac operator in three dimensions by explicitly constructing examples, and explores potential links to topological Hopf maps.

Contribution

It provides the first explicit construction demonstrating zero mode degeneracy in 3D Abelian Dirac operators and discusses its topological implications.

Findings

Degeneracy of zero modes is possible in 3D Abelian Dirac operators.

Explicit examples of Dirac operators with multiple zero modes are constructed.

Potential connection between zero mode degeneracy and Hopf map topology is discussed.

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 question whether such multiple zero modes may exist has remained unanswered until now. Here we prove that the feature of zero mode degeneracy indeed occurs for the Abelian Dirac operator in three dimensions, by explicitly constructing a class of Dirac operators together with their multiple zero modes. Further, we discuss some implications of our results, especially a possible relation to the topological feature of Hopf maps.