Formal geometric quantization
[Quantification géométrique formelle]
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,Ω) préquantifiée par un fibré en droites de Kostant-Souriau. On suppose que l’application moment Φ est propre, ainsi les réductions symplectiques M μ :=Φ -1 (K·μ)/K sont compactes pour tout μ. On peut alors définir la quantification formelle de M comme

𝒬 K - ( M ) : = μ K ^ 𝒬 ( M μ ) V μ K .

Le but de ce travail est l’étude de certaines propriétés fonctorielles de l’application (M,K)𝒬 K - (M).

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

𝒬 K - ( M ) : = μ K ^ 𝒬 ( M μ ) V μ K .

The aim of this article is to study the functorial properties of the assignment (M,K)𝒬 K - (M).

DOI : 10.5802/aif.2429
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
Paradan, Paul-Émile 1

1 Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier (I3M) Place Eugène Bataillon 34095 MONTPELLIER (France)
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Paradan, Paul-Émile. Formal geometric quantization. Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 199-238. doi : 10.5802/aif.2429. http://www.numdam.org/articles/10.5802/aif.2429/

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