The abelian sigma model in (1 + 1) dimensions has a manifold-valued fi
eld phi: S-1 --> S-1. An algebra of the quantum field is defined respe
cting the topological aspect of the model. It is shown that when a cen
tral extension is introduced into the algebra, the winding operator an
d the momenta operators satisfy anomalous commutators. We obtain an in
finite number of inequivalent Hilbert spaces, which are characterized
by a central extension and a continuous parameter alpha (0 less than o
r equal to alpha less than or equal to 1).