1Laboratoire de Mathématiques, Université de Haute-Alsace Mulhouse, 4, Rue des Frères Lumière, 68093 Mulhouse, France. e-mail: Martin.Bordemann@uha.fr
2Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, Hermann Herder Straße 3, 79104 Freiburg, Germany. e-mail: Nikolai.Neumaier@physik.uni-freiburg.de
3Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, Hermann Herder Straße 3, 79104 Freiburg, Germany. e-mail: Stefan.Waldmann@physik.uni-freiburg.de
4Fakultät für Mathematik und Physik, Albert-Ludwigs-Universität Freiburg, Physikalisches Institut, Hermann Herder Straße 3, 79104 Freiburg, Germany. e-mail: Stefan.Weiss@physik.uni-freiburg.de
Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal). Volume 2010, Issue 639, Pages 1–38, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102, DOI: 10.1515/crelle.2010.009, January 2010
Abstract
In this paper we establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of a principal fibre bundle: the functions on the total space are deformed into a right module for the star product algebra of the functions on the base manifold. In the case of a principal fibre bundle we additionally require invariance under the principal action. We prove existence and uniqueness of such deformations. The commutant within all di¤erential operators on the total space is computed and gives a deformation of the algebra of vertical di¤erential operators. Applications to noncommutative gauge field theories and phase space reduction of star products are discussed.
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