Formal geometric quantization

Paradan, Paul-Émile

Formal geometric quantization
[Quantification géométrique formelle]

Annales de l'Institut Fourier, Tome 59 (2009), p. 199-238 / Harvested from Numdam

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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 $(M, K) \to 𝒬_{K}^{- \infty} (M)$ .

Publié le : 2009-01-01

MR 2514864 | Zbl 1163.53056

DOI : https://doi.org/10.5802/aif.2429
Classification: 58F06, 57S15, 19L47, 19L10
Mots clés: quantification géométrique, application moment, réduction symplectique, indice, transversalement elliptique

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     author = {Paradan, Paul-\'Emile},
     title = {Formal geometric quantization},
     journal = {Annales de l'Institut Fourier},
     volume = {59},
     year = {2009},
     pages = {199-238},
     doi = {10.5802/aif.2429},
     zbl = {1163.53056},
     mrnumber = {2514864},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2009__59_1_199_0}
}

Paradan, Paul-Émile. Formal geometric quantization. Annales de l'Institut Fourier, Tome 59 (2009) pp. 199-238. doi : 10.5802/aif.2429. http://gdmltest.u-ga.fr/item/AIF_2009__59_1_199_0/

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