Topology and quantization of abelian sigma model in (1+1) dimensions

It is known that there exist an infinite number of inequivalent quantizations on a topologically nontrivial manifold even if it is a finite-dimensional manifold. In this paper we consider the abelian sigma model in (1+1) dimensions to explore a system having infinite degrees of freedom. The model has a field variable $\phi : S^1 \to S^1$. An algebra of the quantum field is defined respecting the topological aspect of this model. A central extension of the algebra is also introduced. It is shown that there exist an infinite number of unitary inequivalent representations, which are characterized by a central extension and a continuous parameter $\alpha$ $( 0 \le \alpha < 1 )$. When the central extension exists, the winding operator and the zero-mode momentum obey a nontrivial commutator.