Abelian Zero Modes in Odd Dimensions

TL;DR

This paper presents a new interpretation of zero modes in odd-dimensional abelian Dirac operators using stereographic projection from 4D, generalizing to any odd dimension and relating zero modes to Chern-Simons numbers.

Contribution

It introduces a simple stereographic projection approach to understand zero modes, offering an alternative to Hopf map descriptions and extending the concept to all odd dimensions.

Findings

Zero modes relate to Chern-Simons number nonlinearly.

Stereographic projection provides a straightforward interpretation.

Generalization to any odd dimension is achieved.

Abstract

We show that the Loss-Yau zero modes of the 3d abelian Dirac operator may be interpreted in a simple manner in terms of a stereographic projection from a 4d Dirac operator with a constant field strength of definite helicity. This is an alternative to the conventional viewpoint involving Hopf maps from S^3 to S^2. Furthermore, our construction generalizes in a straightforward way to any odd dimension. The number of zero modes is related to the Chern-Simons number in a nonlinear manner.