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Fermion zero modes on parallelized and squashed group manifolds

It is shown that the Cartan-Schouten parallelism should be replaced by a metric-consistent parallelism on a parallelized and squashed group manifold. If the isometry is G x G' (G'is contained inG), then spin-half fermion zero modes only have G' as the gauge symmetry. Their quantum numbers are determined by a reduction of the representation ..gamma.. under G', where ..gamma.. is an irreducible representation of G with the highest weight (1,1,...,1).