  The abelian sigma model in (1+1) dimensions is a field theoretical model
which has a field $ \phi : S^1 \to S^1 $. An algebra of the quantum field is
defined respecting the topological aspect of the model. It is shown that the
zero-mode has an infinite number of inequivalent quantizations. It is also
shown that when a central extension is introduced into the algebra, the winding
operator and the momenta operators satisfy anomalous commutators.
