Fermion Zero Modes in Odd Dimensions

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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 S²ⁿ⁻¹ has symmetries of SU(n)(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.