Invariant YangMills connections over nonreductive pseudoRiemannian homogeneous spaces
Author:
Dennis The
Journal:
Trans. Amer. Math. Soc. 361 (2009), 38793914
MSC (2000):
Primary 70S15; Secondary 34A26, 53C30
Published electronically:
February 10, 2009
MathSciNet review:
2491904
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Abstract: We study invariant gauge fields over the 4dimensional nonreductive pseudoRiemannian homogeneous spaces recently classified by Fels and Renner (2006). Given compact semisimple, classification results are obtained for principal bundles over admitting: (1) a action (by bundle automorphisms) projecting to left multiplication on the base, and (2) at least one invariant connection. There are two cases which admit nontrivial examples of such bundles, and all invariant connections on these bundles are YangMills. The validity of the principle of symmetric criticality (PSC) is investigated in the context of the bundle of connections and is shown to fail for all but one of the FelsRenner cases. This failure arises from degeneracy of the scalar product on pseudotensorial forms restricted to the space of symmetric variations of an invariant connection. In the exceptional case where PSC is valid, there is a unique invariant connection which is moreover universal; i.e., it is the solution of the EulerLagrange equations associated to any invariant Lagrangian on the bundle of connections. This solution is a canonical connection associated with a weaker notion of reductivity which we introduce.
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Additional Information
Dennis The
Affiliation:
Department of Mathematics and Statistics, McGill University, 805 Sherbrooke Street West, Montreal, Quebec, Canada H3A 2K6
Address at time of publication:
Department of Mathematics, Mailstop 3368, Texas A&M University, College Station, Texas 778433368
Email:
dthe@math.mcgill.ca, dthe@math.tamu.edu
DOI:
http://dx.doi.org/10.1090/S0002994709047977
PII:
S 00029947(09)047977
Keywords:
YangMills,
invariant connection,
Lie groups,
nonreductive,
pseudoRiemannian,
homogeneous space
Received by editor(s):
October 9, 2007
Published electronically:
February 10, 2009
Additional Notes:
The author was supported in part by an NSERC CGSD and a Quebec FQRNT Fellowship.
Article copyright:
© Copyright 2009
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
