arXiv subjects
Publications and source records attributed to Laurent La Fuente-Gravy.
In the framework of formal deformation quantization, we apply our formal moment map construction on the space of almost complex structures to recover the Donaldson-Fujiki moment map picture of the Hermitian scalar curvature. In the integrable case, it yields a formal moment map deforming the scalar curvature moment map.
In the framework of deformation quantization, we obtain a deformation of Donaldson moment map on $\textrm{Diff}_0(M)$, the connected component of the group of diffeomorphisms of a symplectic manifold $(M,ω)$ admitting another symplectic structure $χ$.
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.
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 [29].
In the first part of this paper we outline the constructions and properties of Fedosov star product and Berezin-Toeplitz star product. In the second part we outline the basic ideas and recent developments on Yau-Tian-Donaldson conjecture on the existence of Kähler metrics of constant scalar curvature. In the third part of the paper we outline recent results of both authors, and in particular show that the constant scalar curvature Kähler metric problem and the study of deformation quantization meet at the notion of trace (density) for star product. We formulate a cohomology formula for the invariant of K-stability condition on Kähler metrics with constant Cahen-Gutt momentum.
We study obstructions to the existence of closed Fedosov's star products on a given Kähler manifold. In our previous paper, we proved that the Levi-Civita connection of a Kähler manifold will produce a closed (in the sense of Connes-Flato-Sternheimer) Fedosov's star product only if it is a zero of the Cahen-Gutt moment map 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 the moment map and hence for the existence of closed Fedosov's star products on a Kähler manifold.
We deform the group of Hamiltonian diffeomorphisms into the group of Hamiltonian automorphisms of a formal star product on a symplectic manifold. We study the geometry of that group and deform the Flux morphism in the framework of deformation quantization.
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 $ν$.