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Topology and Inequivalent Quantizations of Abelian Sigma Model

The abelian sigma model in (1+1) dimensions is a field theoretical model which has a field $ ϕ: 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.