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The formal moment map geometry of the space of symplectic connections La Fuente-Gravy, Laurent E-print/Working paper (2021) We deform the moment map picture on the space of symplectic connections on a symplectic manifold. To do that, we study a vector bundle of Fedosov star product algebras on the space of symplectic ... [more ▼] We deform the moment map picture on the space of symplectic connections on a symplectic manifold. To do that, we study a vector bundle of Fedosov star product algebras on the space of symplectic connections. We describe a natural formal connection on this bundle adapted to the star product algebras on the fibers. We study its curvature and show the star product trace of the curvature is a formal symplectic form on the space of symplectic connections. The action of Hamiltonian diffeomorphisms on symplectic connections preserves the formal symplectic structure and we show the star product trace can be interpreted as a formal moment map for this action. Finally, we apply this picture to study automorphisms of star products and Hamiltonian diffeomorphisms. [less ▲] Detailed reference viewed: 64 (0 UL)Quantum moment map and obstructions to the existence of closed Fedosov star products ; La Fuente-Gravy, Laurent in Journal of Geometry and Physics (2021), 163 It is shown that the normalized trace of Fedosov star product for quantum moment map depends only on the path component in the cohomology class of the symplectic form and the cohomology class of the ... [more ▼] It is shown that the normalized trace of Fedosov star product for quantum moment map depends only on the path component in the cohomology class of the symplectic form and the cohomology class of the closed formal 2-form required to define Fedosov connections (Theorem 1.3). As an application we obtain a family of obstructions to the existence of closed Fedosov star products naturally attached to symplectic manifolds (Theorem 1.5) and Kähler manifolds (Theorem 1.6). These obstructions are integral invariants depending only on the path component of the cohomology class of the symplectic form. Restricted to compact Kähler manifolds we re-discover an obstruction found earlier in La Fuente-Gravy (2019). [less ▲] Detailed reference viewed: 46 (1 UL)Moment map and closed Fedosov star products La Fuente-Gravy, Laurent Presentation (2019, September 18) Detailed reference viewed: 86 (0 UL)Moment maps and Fedosov star products La Fuente-Gravy, Laurent Scientific Conference (2019, September 10) Detailed reference viewed: 81 (0 UL)Futaki invariant for Fedosov star products La Fuente-Gravy, Laurent in Journal of Symplectic Geometry (2019), 17(5), 1317-1330 We study obstructions to the existence of closed Fedosov star products on a given Kähler manifold (M, omega, J). In our previous paper [11], we proved that if the Levi-Civita connection of a Kähler ... [more ▼] We study obstructions to the existence of closed Fedosov star products on a given Kähler manifold (M, omega, J). In our previous paper [11], we proved that if the Levi-Civita connection of a Kähler manifold will produce a closed (in the sense of Connes-Flato-Sternheimer [4]) Fedosov’s star product then it is a zero of a moment map μ on the space of symplectic connections. By analogy with the Futaki invariant obstructing the existence of cscK metrics, we build an obstruction for the existence of zero of μ and hence for the existence of closed Fedosov’s star product on a Kähler manifold. [less ▲] Detailed reference viewed: 66 (0 UL)Deformation quantization of Kähler manifolds La Fuente-Gravy, Laurent Presentation (2019, March 11) Detailed reference viewed: 87 (1 UL)Deformation quantization and Kähler geometry with moment maps ; La Fuente-Gravy, Laurent in ICCM 2018 Proceedings, Taiwan 27-29 December 2018 (2019) Detailed reference viewed: 27 (0 UL)Symplectic Dirac operators: Construction and kernels La Fuente-Gravy, Laurent Presentation (2018, July 04) Detailed reference viewed: 46 (1 UL)Moment map and closed Fedosov star products La Fuente-Gravy, Laurent Presentation (2018, June 04) Detailed reference viewed: 37 (1 UL)Moment maps and closed Fedosov's star products La Fuente-Gravy, Laurent Scientific Conference (2017, September 13) Detailed reference viewed: 33 (1 UL)Moment maps and closed Fedosov's star products La Fuente-Gravy, Laurent Scientific Conference (2016, June 06) Detailed reference viewed: 81 (0 UL)Moment maps and Fedosov star products La Fuente-Gravy, Laurent Presentation (2016) Detailed reference viewed: 118 (0 UL)The group of Hamiltonian automorphisms of a star product La Fuente-Gravy, Laurent in Mathematical Physics, Analysis and Geometry (2016), 19(3), Detailed reference viewed: 76 (0 UL)Infinite dimensional moment map geometry and closed Fedosov star products La Fuente-Gravy, Laurent 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 ▲] Detailed reference viewed: 50 (0 UL)Moment map and closed Fedosov star products La Fuente-Gravy, Laurent Presentation (2015) Detailed reference viewed: 147 (0 UL)Moment maps and closed Fedosov star products La Fuente-Gravy, Laurent Presentation (2015) Detailed reference viewed: 80 (0 UL)On Mpc-structures and symplectic Dirac operators ; ; La Fuente-Gravy, Laurent et al in Journal of Geometry and Physics (2014), 86 We prove that the kernels of the restrictions of the symplectic Dirac operator and one of the two symplectic Dirac–Dolbeault operators on natural sub-bundles of polynomial valued spinor fields are finite ... [more ▼] We prove that the kernels of the restrictions of the symplectic Dirac operator and one of the two symplectic Dirac–Dolbeault operators on natural sub-bundles of polynomial valued spinor fields are finite dimensional on a compact symplectic manifold. We compute these kernels explicitly for complex projective spaces and show that the remaining Dirac–Dolbeault operator has infinite dimensional kernels on these finite rank sub-bundles. We construct injections of subgroups of the symplectic group (the pseudo-unitary group and the stabiliser of a Lagrangian subspace) in the Mpc group and classify G-invariant Mpc-structures on symplectic manifolds with a G-action. We prove a variant of Parthasarathy’s formula for the commutator of two symplectic Dirac-type operators on general symmetric symplectic spaces. [less ▲] Detailed reference viewed: 40 (0 UL)The group of Hamiltonian automorphisms of a star product La Fuente-Gravy, Laurent Scientific Conference (2013) Detailed reference viewed: 141 (0 UL) |
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