Formal geometric quantization
Annales de l'Institut Fourier, Volume 59 (2009) no. 1, p. 199-238

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

𝒬K-(M):=μK^𝒬(Mμ)VμK.

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

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

𝒬K-(M):=μK^𝒬(Mμ)VμK.

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

DOI : https://doi.org/10.5802/aif.2429
Classification:  58F06,  57S15,  19L47,  19L10
Keywords: Geometric quantization, moment map, symplectic reduction, index, transversally elliptic
@article{AIF_2009__59_1_199_0,
     author = {Paradan, Paul-\'Emile},
     title = {Formal geometric quantization},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {59},
     number = {1},
     year = {2009},
     pages = {199-238},
     doi = {10.5802/aif.2429},
     mrnumber = {2514864},
     zbl = {1163.53056},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2009__59_1_199_0}
}
Paradan, Paul-Émile. Formal geometric quantization. Annales de l'Institut Fourier, Volume 59 (2009) no. 1, pp. 199-238. doi : 10.5802/aif.2429. http://www.numdam.org/item/AIF_2009__59_1_199_0/

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