Formal geometric quantization

Paradan, Paul-Émile

doi:10.5802/aif.2429

Formal geometric quantization
[Quantification géométrique formelle]

Paradan, Paul-Émile ¹

¹ Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier (I3M) Place Eugène Bataillon 34095 MONTPELLIER (France)

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

Résumé
Abstract

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 $M_{μ} : = Φ^{- 1} (K \cdot μ) / K$ sont compactes pour tout $μ$ . On peut alors définir la quantification formelle de $M$ comme

𝒬_{K}^{- \infty} (M) : = \sum_{μ \in \hat{K}} 𝒬 (M_{μ}) V_{μ}^{K} .

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

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 $M_{μ} : = Φ^{- 1} (K \cdot μ) / K$ is compact for all $μ$ . Then, we can define the “formal geometric quantization” of $M$ as

𝒬_{K}^{- \infty} (M) : = \sum_{μ \in \hat{K}} 𝒬 (M_{μ}) V_{μ}^{K} .

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

MR Zbl

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

Affiliations des auteurs :

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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     author = {Paradan, Paul-\'Emile},
     title = {Formal geometric quantization},
     journal = {Annales de l'Institut Fourier},
     pages = {199--238},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     doi = {10.5802/aif.2429},
     zbl = {1163.53056},
     mrnumber = {2514864},
     language = {en},
     url = {https://www.numdam.org/articles/10.5802/aif.2429/}
}

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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. https://www.numdam.org/articles/10.5802/aif.2429/

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