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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Invariant Yang-Mills connections over non-reductive pseudo-Riemannian homogeneous spaces


Author: Dennis The
Journal: Trans. Amer. Math. Soc. 361 (2009), 3879-3914
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 4-dimensional non-reductive pseudo-Riemannian homogeneous spaces $ G/K$ recently classified by Fels and Renner (2006). Given $ H$ compact semi-simple, classification results are obtained for principal $ H$-bundles over $ G/K$ admitting: (1) a $ G$-action (by bundle automorphisms) projecting to left multiplication on the base, and (2) at least one $ G$-invariant connection. There are two cases which admit non-trivial examples of such bundles, and all $ G$-invariant connections on these bundles are Yang-Mills. 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 Fels-Renner cases. This failure arises from degeneracy of the scalar product on pseudo-tensorial forms restricted to the space of symmetric variations of an invariant connection. In the exceptional case where PSC is valid, there is a unique $ G$-invariant connection which is moreover universal; i.e., it is the solution of the Euler-Lagrange equations associated to any $ G$-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 77843-3368
Email: dthe@math.mcgill.ca, dthe@math.tamu.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04797-7
PII: S 0002-9947(09)04797-7
Keywords: Yang--Mills, invariant connection, Lie groups, non-reductive, pseudo-Riemannian, 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 CGS-D 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.