Infinite dimensional moment map geometry and closed Fedosov’s star products

Abstract

We study the Cahen–Gutt moment map on the space of symplectic connections of a symplectic manifold. Given a Kähler manifold $(M,\omega ,J)$, we define a Calabi-type functional $\mathscr {F}$ on the space $\mathcal {M}_{\Theta }$ of Kähler metrics in the class $\Theta :=[\omega ]$. We study the space of zeroes of $\mathscr {F}$. When $(M,\omega ,J)$ has non-negative Ricci tensor and $\omega$ is a zero of $\mathscr {F}$, we show the space of zeroes of $\mathscr {F}$ near $\omega$ 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’s 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’s star products that are closed up to order 3 in $\nu$.