Infinite dimensional moment map geometry and closed Fedosov star products

We study the Cahen–-Gutt moment map on the space of symplectic connections of a
symplectic manifold. Given a Kähler manifold (M, ω, J ), we define a Calabi-type functional
F on the space M of Kähler metrics in the class [ω]. We study the space of zeroes
of F. When (M, ω, J ) has non-negative Ricci tensor and ω is a zero of F, we show the
space of zeroes of F near ω has the structure of a smooth finite dimensional submanifold.
We give a new motivation, coming from deformation quantization, for the study of moment
maps on infinite dimensional spaces. More precisely, we establish a strong link between
trace densities for star products (obtained from Fedosov-type methods) and moment map
geometry on infinite dimensional spaces. As a byproduct, we provide, on certain Kähler
manifolds, a geometric characterization of a space of Fedosov star products that are closed
up to order 3.