On a connection used in deformation quantization

TL;DR

This paper provides a coordinate-free proof that the connection used in deformation quantization on cotangent bundles is derived from the symplectification of the Levi-Civita connection's complete lift, clarifying its geometric origin.

Contribution

It offers a coordinate-free geometric proof linking the Fedosov connection to the symplectification of the Levi-Civita connection's lift, extending prior coordinate-based results.

Findings

The Fedosov connection is obtained via symplectification of the complete lift of the Levi-Civita connection.

The proof is coordinate-free, providing a more geometric understanding.

Connects deformation quantization constructions with classical differential geometry.

Abstract

Using natural lifting operations, we give a coordinate-free proof of the fact that the connection used by Bordemann, Neumaier and Waldmann to construct the Fedosov standard ordered star product on the cotangent bundle of a Riemannian manifold is obtained by symplectification of the complete lift of the corresponding Levi-Civita connection, in the sense of Yano and Patterson. In terms of local coordinates, this has already been shown by Plebanski, Przanowski and Turrubiates.