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 | References | Similar Articles | Additional Information

Abstract: We study invariant gauge fields over the 4-dimensional non-reductive pseudo-Riemannian homogeneous spaces recently classified by Fels and Renner (2006). Given compact semi-simple, 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 non-trivial examples of such bundles, and all -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 -invariant connection which is moreover universal; i.e., it is the solution of the Euler-Lagrange 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 77843-3368

Email:
dthe@math.mcgill.ca, dthe@math.tamu.edu

DOI:
http://dx.doi.org/10.1090/S0002-9947-09-04797-7

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.