The formal moment map geometry of the space of symplectic connections

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 algebra on the space of symplectic connections E. 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 * -product trace of the curvature is a formal symplectic form on E. The action of Hamiltonian diffeomorphisms on E preserves the formal symplectic structure and we show the * -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.