Fermion Zero Modes in Odd Dimensions

TL;DR

This paper investigates fermion zero modes of the Abelian Dirac operator in odd dimensions, revealing symmetry properties and explicitly constructing zero modes using group theory and stereographic projection.

Contribution

It introduces a method to determine zero modes in odd dimensions using stereographic projection and symmetry analysis, providing explicit forms and counts of zero modes.

Findings

Dirac operator zero modes are characterized in odd dimensions.

Symmetries of the gauge field are identified as SU(n)×U(1).

Explicit zero mode solutions are constructed using group theory.

Abstract

We study the zero modes of the Abelian Dirac operator in any odd dimension. We use the stereographic projection between a $(2n-1)$ dimensional space and a $(2n-1)$ sphere embedded in a $2n$ dimensional space. It is shown that the Dirac operator with a gauge field of uniform field strengths in $S^{2n-1}$ has symmetries of SU($n$)$\times$U(1) which is a subgroup of SO($2n$). Using group representation theory, we obtain the number of fermion zero modes, as well as their explicit forms, in a simple way.