Topology and Inequivalent Quantizations of Abelian Sigma Model

TL;DR

This paper explores the topological features of the (1+1)-dimensional abelian sigma model, revealing multiple inequivalent quantizations of the zero-mode and anomalous commutators arising from central extensions.

Contribution

It demonstrates the existence of infinitely many inequivalent quantizations of the zero-mode and analyzes the effects of central extensions on operator commutation relations.

Findings

Zero-mode admits infinitely many inequivalent quantizations.

Central extension induces anomalous commutators.

Topological structure influences quantum algebra.

Abstract

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.