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 Kahler manifold \((M,\omega ,J)\), we define a Calabi-type functional \(\mathscr {F}\) on the space \(\mathcal {M}_{\Theta }\) of Kahler 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 Kahler manifolds, a geometric characterization of a space of Fedosov’s star products that are closed up to order 3 in \(\nu \).