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

We study the Cahen-Gutt moment map on the space of symplectic connections of a symplectic manifold. On a Kähler manifold, we define a Calabi-type functional $\mathscr{F}$ on the space of Kähler metrics in a given Kähler class. We study the zeroes of $\mathscr{F}$. Given a zero of $\mathscr{F}$ with non-negative Ricci tensor, we show the space of zeroes around the given one has the structure of a finite dimensional embedded submanifold. We give a new motivation, coming from deformation quantisation, 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 consequence, 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 $ν$.

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