The differential geometry of Fedosov's quantization

TL;DR

This paper explores the geometric foundations of Fedosov's deformation quantization, analyzing classical analogs of Fedosov's connection operations and their implications for symplectic geometry and index theory.

Contribution

It provides a classical geometric interpretation of Fedosov's quantization process and extends the analysis to symplectic versions, linking to index theorems.

Findings

Classical analogs of Fedosov's operations produce the exponential map of a linear connection.

Symplectic versions of these operations are also characterized.

Implications for deformation quantization and Fedosov's index theorem are discussed.

Abstract

B. Fedosov has given a simple and very natural construction of a deformation quantization for any symplectic manifold, using a flat connection on the bundle of formal Weyl algebras associated to the tangent bundle of a symplectic manifold. The connection is obtained by affinizing, nonlinearizing, and iteratively flattening a given torsion free symplectic connection. In this paper, a classical analog of Fedosov's operations on connections is analyzed and shown to produce the usual exponential mapping of a linear connection on an ordinary manifold. A symplectic version is also analyzed. Finally, some remarks are made on the implications for deformation quantization of Fedosov's index theorem on general symplectic manifolds.