Global Anomalies and Anyons in 1+1 Dimensions

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We consider the analog in one spatial dimension of the Bose-Fermi transmutation for planar systems. A quantum mechanical system of a spin 1/2 particle coupled to an abelian gauge field, which is classically invariant under gauge transformations and charge conjugation is studied. It is found that unless the flux enclosed by the particle orbits is quantized, and the spin takes a value n+ 1/2, at least one of the two symmetries would be anomalous. Thus, charge conjugation invariance and the existence of abelian instantons simultaneously force the particles to be either bosons or fermions, but not anyons.