Remote Access Transactions of the American Mathematical Society
Green Open Access

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
DOI: https://doi.org/10.1090/S0002-9947-09-04797-7
Published electronically: February 10, 2009
MathSciNet review: 2491904
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.


References [Enhancements On Off] (What's this?)

  • [AB82] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. London A 308 (1983), 523-615. MR 702806 (85k:14006)
  • [AF] I. M. Anderson and M. E. Fels, Notes on the Palais condition, private communication.
  • [AF97] -, Symmetry reduction of variational bicomplexes and the principle of symmetric criticality, Amer. J. Math. 119 (1997), 609-670. MR 1448217 (98k:58208)
  • [AF04] -, A cochain map for the G-invariant de Rham complex, Symmetry and Perturbation Theory: Proceedings of the International Conference SPT 2004 (Cala Gonone, Italy, 30 May - 6 June 2004), World Scientific, 2004, pp. 1-12. MR 2331199 (2008f:58002)
  • [AFT01] I. M. Anderson, M. E. Fels, and C. G. Torre, Group invariant solutions without transversality and the principle of symmetric criticality, Backlund and Darboux transformations. The geometry of solitons (Halifax, NS, 1999), vol. 29, Amer. Math. Soc., 2001, pp. 95-108. MR 1870907 (2002h:58068)
  • [And] I. M. Anderson, The Variational Bicomplex, Preprint available online at http://www.math.usu.edu/$ \sim$fgmp/Publications/VB/vb.pdf.
  • [And92] -, Introduction to the variational bicomplex, Mathematical Aspects of Classical Field Theory (M. Gotay, J. Marsden, and V. Moncrief, eds.), Contemp. Math., vol. 132, Amer. Math. Soc., Providence, RI., 1992, pp. 51-73. MR 1188434 (94a:58045)
  • [Ble81] D. Bleecker, Gauge Theory and Variational Principles, Global Analysis, Pure and Applied, series A, no. 1, Addison-Wesley, London, 1981. MR 643361 (83h:53049)
  • [CE48] C. Chevalley and S. Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), no. 1, 85-124. MR 0024908 (9:567a)
  • [CJ83] R. Coquereaux and A. Jadczyk, Geometry of multidimensional universes, Comm. Math. Phys. 90 (1983), no. 1, 79-100. MR 714613 (85e:83053)
  • [CJ85] -, Symmetries of Einstein-Yang-Mills fields and dimensional reduction, Comm. Math. Phys. 98 (1985), no. 1, 79-104. MR 785262 (86h:53071)
  • [CJ86] -, Consistency of the G-invariant Kaluza-Klein scheme, Nuclear Phys. B (1986), 617-628. MR 856538 (87i:83088)
  • [CJ88] -, Riemannian Geometry, Fiber Bundles, Kaluza-Klein theories, and all that..., World Scientific Lecture Notes in Physics, vol. 16, World Scientific Publishing Co., Singapore, 1988. MR 940468 (89e:53108)
  • [Dar97] B. K. Darian, Gauge fields in homogeneous and inhomogeneous cosmologies, Ph.D. thesis, University of Alberta, 1997.
  • [DK97] B. K. Darian and H. P. Künzle, Cosmological Einstein-Yang-Mills equations, J. Math. Phys. 38 (1997), no. 9, 4696-4713. MR 1468661 (98h:53042)
  • [FR06] M. E. Fels and A. G. Renner, Non-reductive homogeneous pseudo-Riemannian manifolds of dimension four, Canad. J. Math. 58 (2006), no. 2, 282-311. MR 2209280 (2006m:53077)
  • [FT02] M. E. Fels and C. G. Torre, The principle of symmetric criticality in general relativity, Class. Quantum Gravity 19 (2002), 641-675. MR 1891490 (2002m:58026)
  • [Gar72] P. L. Garcia, Connections and $ 1$-jet bundles, Rend. Sem. Mat. Univ. Padova 47 (1972), 227-242. MR 0315624 (47:4173)
  • [Got89] M. J. Gotay, Reduction of homogeneous Yang-Mills fields, J. Geom. Phys. 6 (1989), no. 3, 349-365. MR 1049710 (91c:58039)
  • [HS53] G. Hochschild and J. P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), no. 3, 591-603. MR 0054581 (14:943c)
  • [HSV80] J. Harnad, S. Shnider, and L. Vinet, Group actions on principal bundles and invariance conditions for gauge fields, J. Math. Phys. 21 (1980), no. 12, 2719-2724. MR 597586 (82c:53028)
  • [HTS80] J. Harnad, J. Tafel, and S. Shnider, Canonical connections on Riemannian symmetric spaces and solutions to the Einstein-Yang-Mills equations, J. Math. Phys. 21 (1980), no. 8, 2236-2240. MR 579220 (82j:53055)
  • [Jad84] A. Jadczyk, Symmetry of Einstein-Yang-Mills systems and dimensional reduction, J. Geom. Phys. 1 (1984), no. 2, 97-126. MR 794982 (86k:53089)
  • [JP84] A. Jadczyk and K. Pilch, Geometry of gauge fields in a multidimensional universe, Lett. Math. Phys. 8 (1984), no. 2, 97-104. MR 740810 (86b:53071)
  • [KN63] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol.I, John Wiley & Sons, Inc., 1963.
  • [Koi90] N. Koiso, Yang-Mills connections of homogeneous bundles, Osaka J. Math. 27 (1990), no. 1, 163-174. MR 1049829 (91b:58040)
  • [Laq84] H. T. Laquer, Stability properties of the Yang-Mills functional near the canonical connection, Michigan Math. J. 31 (1984), no. 2, 139-159. MR 752251 (88a:58047)
  • [Pal79] R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19-30. MR 547524 (81c:58026)
  • [PSWZ76] J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, Invariants of real low dimension Lie algebras, J. Math. Phys. 17 (1976), no. 6, 986-994. MR 0404362 (53:8164)
  • [Wan58] H. C. Wang, On invariant connections over a principal fibre bundle, Nagoya Math. J. 13 (1958), 1-19. MR 0107276 (21:6001)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 70S15, 34A26, 53C30

Retrieve articles in all journals with MSC (2000): 70S15, 34A26, 53C30


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: https://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.

American Mathematical Society