Fermion Zero Modes in Odd Dimensions

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.