Deformation Quantization of Surjective Submersions and Principal Fibre Bundles

TL;DR

This paper develops a framework for deformation quantization of surjective submersions and principal fibre bundles, establishing existence, uniqueness, and applications to noncommutative gauge theories and phase space reduction.

Contribution

It introduces a novel deformation quantization approach for surjective submersions and principal fibre bundles, including invariance conditions and explicit computations of the commutant.

Findings

Existence and uniqueness of the deformation quantization are proven.

The commutant of differential operators yields a deformation of vertical differential operators.

Applications to noncommutative gauge theories and phase space reduction are discussed.

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 case of a principal fibre bundle we require in addition invariance under the principal action. We prove existence and uniqueness of such deformations. The commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operators. Applications to noncommutative gauge field theories and phase space reduction of star products are discussed.