Deformation Quantization of Principal Fibre Bundles and Classical Gauge Theories

TL;DR

This paper develops a geometric framework for deformation quantization of principal fibre bundles and gauge theories on noncommutative space-times, establishing existence, uniqueness, and explicit structures of these quantizations.

Contribution

It introduces a comprehensive theory of deformation quantization for principal fibre bundles and surjective submersions, including explicit calculations of their commutants and implications for gauge theories.

Findings

Deformation quantizations of surjective submersions and principal fibre bundles always exist and are unique up to equivalence.

Explicit computation of commutants of deformed module structures within differential operators.

Deformation quantization of principal fibre bundles induces quantizations of associated vector bundles.

Abstract

In this dissertation the notion of deformation quantization of principal fibre bundles is established and investigated in order to find a geometric formulation of classical gauge theories on noncommutative space-times. As a generalization, the notion of deformation quantization of surjective submersions is also discussed. It is shown that deformation quantizations of surjective submersions and principal fibre bundles always exist and are unique up to equivalence. These statements concerning complex-valued functions are moreover formulated and proved for sections of arbitrary vector bundles over the total space, in particular equivariant vector bundles. The commutants of the deformed right module structures within the differential operators, playing an inportant role with regard to the infinitesimal gauge transformations, are computed explicitly in each case. Depending on the choice of specific covariant derivatives and connections the commutants are isomorphic to the formal power series of the respective vertical differential operators which thus inherit a deformation of the algebra structure. The resulting deformed bimodules are again unique up to equivalence. With respect to further applications it is finally shown that every deformation quantization of a principal fibre bundle induces a deformation quantization of any associated vector bundle.