Deformation Quantization of Symplectic Fibrations

TL;DR

This paper explores how deformation quantization applies to symplectic fibrations, relating the quantization of base, fibre, and total space, and provides methods to compute the Fedosov connection explicitly.

Contribution

It establishes the relation between deformation quantization of the base, fibre, and total space of symplectic fibrations and computes the characteristic class of such deformations.

Findings

Relation between deformation quantization of base, fibre, and total space

Explicit calculation of Fedosov connection and star-product

Quantization of classical moment maps with equivariant connections

Abstract

A symplectic fibration is a fibre bundle in the symplectic category. We find the relation between deformation quantization of the base and the fibre, and the total space. We use the weak coupling form of Guillemin, Lerman, Sternberg and find the characteristic class of deformation of symplectic fibration. We also prove that the classical moment map could be quantized if there exists an equivariant connection. Along the way we touch upon the general question of quantization with values in a bundle of algebras. We consider Fedosov's construction of deformation quantization in general. In the Appendix we show how to calculate step by step the Fedosov connection, flat sections of the Weyl algebra bundle corresponding to functions and their star-product.