# Formal geometric quantization

• [1] Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier (I3M) Place Eugène Bataillon 34095 MONTPELLIER (France)
• Volume: 59, Issue: 1, page 199-238
• ISSN: 0373-0956

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## Abstract

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Let $K$ be a compact Lie group acting in a Hamiltonian way on a symplectic manifold $\left(M,\Omega \right)$ 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 ${M}_{\mu }:={\Phi }^{-1}\left(K·\mu \right)/K$ is compact for all $\mu$. Then, we can define the “formal geometric quantization” of $M$ as${𝒬}_{K}^{-\infty }\left(M\right):=\sum _{\mu \in \stackrel{^}{K}}𝒬\left({M}_{\mu }\right){V}_{\mu }^{K}.$The aim of this article is to study the functorial properties of the assignment $\left(M,K\right)\to {𝒬}_{K}^{-\infty }\left(M\right)$.

## How to cite

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Paradan, Paul-Émile. "Formal geometric quantization." Annales de l’institut Fourier 59.1 (2009): 199-238. <http://eudml.org/doc/10390>.

abstract = {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 $M_\{\mu \}:=\Phi ^\{-1\}(K\cdot \mu )/K$ is compact for all $\mu$. Then, we can define the “formal geometric quantization” of $M$ as$\mathcal\{Q\}\_K^\{-\infty \}(M):=\sum \_\{\mu \in \widehat\{K\}\} \mathcal\{Q\}(M\_\{\mu \}) V\_\mu ^K.$The aim of this article is to study the functorial properties of the assignment $(M,K)\rightarrow \mathcal\{Q\}_K^\{-\infty \}(M)$.},
affiliation = {Université Montpellier II Institut de Mathématiques et de Modélisation de Montpellier (I3M) Place Eugène Bataillon 34095 MONTPELLIER (France)},
journal = {Annales de l’institut Fourier},
keywords = {Geometric quantization; moment map; symplectic reduction; index; transversally elliptic; geometric quantization},
language = {eng},
number = {1},
pages = {199-238},
publisher = {Association des Annales de l’institut Fourier},
title = {Formal geometric quantization},
url = {http://eudml.org/doc/10390},
volume = {59},
year = {2009},
}

TY - JOUR
TI - Formal geometric quantization
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 1
SP - 199
EP - 238
AB - 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 $M_{\mu }:=\Phi ^{-1}(K\cdot \mu )/K$ is compact for all $\mu$. Then, we can define the “formal geometric quantization” of $M$ as$\mathcal{Q}_K^{-\infty }(M):=\sum _{\mu \in \widehat{K}} \mathcal{Q}(M_{\mu }) V_\mu ^K.$The aim of this article is to study the functorial properties of the assignment $(M,K)\rightarrow \mathcal{Q}_K^{-\infty }(M)$.
LA - eng
KW - Geometric quantization; moment map; symplectic reduction; index; transversally elliptic; geometric quantization
UR - http://eudml.org/doc/10390
ER -

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