The formal moment map geometry of the space of symplectic connections

TL;DR

This paper develops a formal geometric framework for the space of symplectic connections using Fedosov star products, revealing a formal moment map structure and its implications for automorphisms and Hamiltonian diffeomorphisms.

Contribution

It introduces a formal moment map picture on symplectic connections via Fedosov star products, connecting deformation quantization with symplectic geometry.

Findings

Star product trace defines a formal symplectic form on the space of symplectic connections.

The star product trace acts as a formal moment map for Hamiltonian diffeomorphisms.

The formal connection's curvature relates to the formal symplectic structure and automorphisms.

Abstract

We deform the moment map picture on the space of symplectic connections on a symplectic manifold. To do that, we study a vector bundle of Fedosov star product algebras on the space of symplectic connections. We describe a natural formal connection on this bundle adapted to the star product algebras on the fibers. We study its curvature and show the star product trace of the curvature is a formal symplectic form on the space of symplectic connections. The action of Hamiltonian diffeomorphisms on symplectic connections preserves the formal symplectic structure and we show the star product trace can be interpreted as a formal moment map for this action. Finally, we apply this picture to study automorphisms of star products and Hamiltonian diffeomorphisms.