The formal moment map geometry of the space of symplectic connections

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 ▲]

Moment map and closed Fedosov star products

Presentation (2019, September 18)

Futaki invariant for Fedosov star products

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 ▲]

Deformation quantization and Kähler geometry with moment maps

in ICCM 2018 Proceedings, Taiwan 27-29 December 2018 (2019)

Moment map and closed Fedosov star products

Presentation (2018, June 04)

Moment maps and closed Fedosov's star products

Scientific Conference (2016, June 06)

The group of Hamiltonian automorphisms of a star product

in Mathematical Physics, Analysis and Geometry (2016), 19(3),

Moment map and closed Fedosov star products

Presentation (2015)

On Mpc-structures and symplectic Dirac operators

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 ▲]