Préquantification de Certaines Variétés de Poisson

Abstract

A surjective submersion $π: M \to B$ carrying a field of simplectic structures on the fibres is symplectic if this Poisson structure is minimal. A symplectic submersion may be interpreted as a family of mechanical systems depending on a parameter in $B$. We give some conditions to find a closed form which represent the foliated form $σ$ gluing the symplectic forms on the fibres. This is the first step to prequantize all these systems at once. We will indeed exhibit an integrality condition which does not depend on the closed form representing $σ$: if the fibres are 1-connected and $H^3(B;Z)=0$, then there exists a $S^1$-principal fibre bundle with a connection whose curvature represents $σ$ iff the group of spherical periods of $σ$ is a discrete subgroup of R. The symplectic integration of a Poisson manifold $(M,Λ)$ is a symplectic groupoid $(Γ,η)$ with 1-connected fibres such that the space of units with the induced Poisson structure is isomorphic to $(M,Λ)$. This notion was introduced by A. Weinstein in order to quantize Poisson manifolds by quantizing their symplectic integration. We show that if the symplectic integration is prequantizable, then there exists a unique prequantization which is trivial over $M$. We show that the symplectic integration of a minimal Poisson manifold is prequantizable iff the group of spherical periods is discrete. Moreover we prove that a {\em totally aspherical} Poisson manifold (any vanishing cycle is trivial and the $π_2$ of the leaves is zero) is prequantized in the sense of Weinstein by a trivial fibre bundle.