Formal geometric quantization

Paradan, Paul-Émile

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 199-238.

Considérons l’action hamiltonienne d’un groupe de Lie compact $K$ sur une variété symplectique $(M,\Omega )$ préquantifiée par un fibré en droites de Kostant-Souriau. On suppose que l’application moment $\Phi$ est propre, ainsi les réductions symplectiques $M_{\mu }:={\Phi }^{-1}(K·\mu )/K$ sont compactes pour tout $\mu$. On peut alors définir la quantification formelle de $M$ comme

$${𝒬}_K^{-\infty }\left(M\right):=\sum _{\mu \in \widehat{K}}𝒬\left(M_{\mu }\right)V_{\mu }^K.$$

Le but de ce travail est l’étude de certaines propriétés fonctorielles de l’application $(M,K)\to {𝒬}_K^{-\infty }\left(M\right)$.

Let $K$ be a compact Lie group acting in a Hamiltonian way on a symplectic manifold $(M,\Omega )$ which is pre-quantized by a Kostant-Souriau line bundle. We suppose here that the moment map $\Phi$ is proper so that the reduced space $M_{\mu }:={\Phi }^{-1}(K·\mu )/K$ is compact for all $\mu$. Then, we can define the “formal geometric quantization” of $M$ as

$${𝒬}_K^{-\infty }\left(M\right):=\sum _{\mu \in \widehat{K}}𝒬\left(M_{\mu }\right)V_{\mu }^K.$$

The aim of this article is to study the functorial properties of the assignment $(M,K)\to {𝒬}_K^{-\infty }\left(M\right)$.

Classification : 58F06, 57S15, 19L47, 19L10
Keywords: Geometric quantization, moment map, symplectic reduction, index, transversally elliptic
Mot clés : quantification géométrique, application moment, réduction symplectique, indice, transversalement elliptique

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