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

Shogo Tanimura
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Abstract

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 $ ϕ: 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 $ α$ $ ( 0 \le α< 1 ) $. When the central extension exists, the winding operator and the zero-mode momentum obey a nontrivial commutator.

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Shogo Tanimura. 1994-08-17. Topology and quantization of abelian sigma model in (1+1) dimensions. https://doi.org/10.1016/0370-2693(94)91297-1

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