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Deformation quantization of surjective submersions and principal fibre bundles. (English) Zbl 1188.53104
Principle fibre bundles and surjective submersions are omnipresent in differential geometry. On the other hand, deformation quantization has great popularity in various applications in mathematical physics and involving many applications: in quantum theory of classical mechanical system, noncommutative space-times, etc. Having these applications in mind, in this paper, the authors establish a notion of deformation quantization of a surjective submersion which is specialized further to the case of principle bundles: 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 principle bundles, the invariance under the principal action is additionally required. The existence and uniqueness of such deformations are proved. The commutant within all differential operators on the total space is computed and gives a deformation of the algebra of vertical differential operators. The paper concludes by discussing some applications to noncommutative gauge field theories and phase space reduction of star products.

MSC:
53D55 Deformation quantization, star products
46L65 Quantizations, deformations for selfadjoint operator algebras
53D50 Geometric quantization
55R10 Fiber bundles in algebraic topology
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