The orbit space of the action of gauge transformation group on connections
type: Item
instance of: scholarly article
title: The orbit space of the action of gauge transformation group on connections
zbMATH Open document ID: 0718.58013
DOI: 10.1016/0393-0440(89)90025-9
author: Maria Cristina Abbati
author: Alessandro Manià
author: Renzo Cirelli
published in: Journal of Geometry and Physics
publication date: 1989-01-01T00:00:00Z
review text: For a principal bundle P with compact base manifold M and structure group a Lie group G which is the direct product of a compact group and a vector space, the action of the group \({\mathcal G}\) of all smooth gauge transformations on the spaces \({\mathcal C}\) of all smooth principal connections is investigated. It is shown that each orbit is a closed splitting tame submanifold and admits smooth tame slices, so that the orbit space \({\mathcal C}/{\mathcal G}\) is stratified in tame submanifolds corresponding to conjugacy classes of holonomy groups of connections. The method follows \textit{W. Kondracki} and \textit{J. Rogulski}, Prepr., Inst. Math. Pol. Acad. Sci., Warsz. 281, 1-110 (1983; Zbl 0522.58025) who showed this result for suitable Sobolev completions (for compact G), but replaces Sobolev space techniques mainly by the Nash-Moser inverse function theorem given by \textit{R. S. Hamilton} in Bull. Am. Math. Soc., New Ser. 7, 65-222 (1982; Zbl 0499.58003).
Mathematics Subject Classification ID: 58D15
Mathematics Subject Classification ID: 58D27
Mathematics Subject Classification ID: 53C07
Mathematics Subject Classification ID: 57N80
Mathematics Subject Classification ID: 81T13
zbMATH DE Number: 4184452
zbMATH Keywords: stratification
zbMATH Keywords: smooth gauge transformations
zbMATH Keywords: connections
reviewed by: Peter W. Michor
MaRDI profile type: Publication
full work available at URL: 0393-0440(89)90025-9
OpenAlex ID: W1976456557
cites work: The group of gauge transformations as a Schwartz–Lie group
cites work: Smoothness of the action of the gauge transformation group on connections
cites work: The Riemannian geometry of the configuration space of gauge theories
cites work: Some remarks on the Gribov ambiguity
cites work: The Geometry of the Orbit Space for Non-Abelian Gauge Theories
cites work: Q5605734
cites work: Q4056764
cites work: The inverse function theorem of Nash and Moser
cites work: Q3672764
cites work: Weak Riemannian structures on gauge-group orbits
cites work: The Yang-Mills equations over Riemann surfaces
cites work: Symmetry and bifurcations of momentum mappings
cites work: Q3871594
cites work: Q4084975
cites work: Elliptic operators and the decomposition of tensor fields
cites work: The classification of 𝐺-spaces
cites work: Q5649693
cites work: Q3037685
cites work: Q3805428
cites work: Q5599559
cites work: Q3966802
description: scientific article