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See detailInfinite dimensional moment map geometry and closed Fedosov star products
La Fuente-Gravy, Laurent UL

in Annals of Global Analysis and Geometry (2016), 49(1), 1-22

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 ... [more ▼]

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. [less ▲]

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