Formal geometric quantization

Paul-Émile Paradan[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 (2009)

  • Volume: 59, Issue: 1, page 199-238
  • ISSN: 0373-0956

Abstract

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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 ) .

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>.

@article{Paradan2009,
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)},
author = {Paradan, Paul-Émile},
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
AU - Paradan, Paul-Émile
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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