The trilogy of the momentum map
type: Item
instance of: scholarly article
title: The trilogy of the momentum map
zbMATH Open document ID: 0836.58001
publication date: 1995-07-12T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/75140
review text: Let \(\omega\) be a closed 2-form on a convex differentiable manifold \(X\) and denote by \(P_\omega\) the group of periods of the form \(\omega\). The factor space \(T_\omega= \mathbb{R}/P_\omega\) is called the torus of periods of the form \(\omega\). Every principal bundle of base \(X\) with the structural group \(T_\omega\) that admits a connection of curvature \(\omega\) is called an integration bundle of the form \(\omega\). In the first section of the paper, the author proves that the set of integration bundles of the form \(\omega\) is determined, up to an equivalence of principal bundles, by the first group of extensions of \(H_1(X, \mathbb{Z})\) with coefficients in \(P_\omega\), i.e. \(\text{Ext}(H_1(X, \mathbb{Z}), P_\omega)\) (Theorem 1.5). In the following three sections, the author generalizes the known construction from the case where the form \(\omega\) is integer (i.e. the group of periods \(P_\omega\) is either null or isomorphic to \(\mathbb{Z}\)) of the trilogy of the Moment Map to the case where \(\omega\) is an arbitrary closed 2-form. In section 5 the obtained construction is illustrated by some examples. Finally, the author gives an appendix which presents the category of differentiable spaces used in this paper. This category of differentiable spaces is necessary because the torus of periods \(T_\omega\) is not a Lie group if the form \(\omega\) is not integer.
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 37J99
Mathematics Subject Classification ID: 58A05
Mathematics Subject Classification ID: 53D50
zbMATH DE Number: 773330
zbMATH Keywords: symplectic geometry
zbMATH Keywords: momentum map
zbMATH Keywords: central extensions
zbMATH Keywords: geometric quantization
zbMATH Keywords: torus of periods
zbMATH Keywords: integration bundle
author: Patrick Iglesias-Zemmour
reviewed by: Neculai Papaghiuc
MaRDI profile type: Publication
OpenAlex ID: W2328372078
cites work: Q4176171
cites work: Loop spaces, characteristic classes and geometric quantization
cites work: Iterated path integrals
cites work: Q3963261
cites work: Q5568607
cites work: Q5580943
cites work: Q3327362
DOI: 10.5802/AIF.1476
published in: Annales de l’institut Fourier
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description: scientific article; zbMATH DE number 773330