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

The, Dennis

doi:10.1090/S0002-9947-09-04797-7

Invariant Yang-Mills connections over non-reductive pseudo-Riemannian homogeneous spaces
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Trans. Amer. Math. Soc. 361 (2009), 3879-3914 Request permission

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

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI 10.1098/rsta.1983.0017
I. M. Anderson and M. E. Fels, Notes on the Palais condition, private communication.
Ian M. Anderson and Mark E. Fels, Symmetry reduction of variational bicomplexes and the principle of symmetric criticality, Amer. J. Math. 119 (1997), no. 3, 609–670. MR 1448217
I. M. Anderson and M. E. Fels, A co-chain map for the $G$ invariant de Rham complex, SPT 2004—Symmetry and perturbation theory, World Sci. Publ., Hackensack, NJ, 2005, pp. 1–12. MR 2331199, DOI 10.1142/9789812702142_{0}001
Ian M. Anderson, Mark E. Fels, and Charles G. Torre, Group invariant solutions without transversality and the principle of symmetric criticality, Bäcklund and Darboux transformations. The geometry of solitons (Halifax, NS, 1999) CRM Proc. Lecture Notes, vol. 29, Amer. Math. Soc., Providence, RI, 2001, pp. 95–108. MR 1870907, DOI 10.1090/crmp/029/06
I. M. Anderson, The Variational Bicomplex, Preprint available online at http://www.math.usu.edu/~fg_mp/Publications/VB/vb.pdf.
Ian M. Anderson, Introduction to the variational bicomplex, Mathematical aspects of classical field theory (Seattle, WA, 1991) Contemp. Math., vol. 132, Amer. Math. Soc., Providence, RI, 1992, pp. 51–73. MR 1188434, DOI 10.1090/conm/132/1188434
David Bleecker, Gauge theory and variational principles, Global Analysis Pure and Applied: Series A, vol. 1, Addison-Wesley Publishing Co., Reading, Mass., 1981. MR 643361
Claude Chevalley and Samuel Eilenberg, Cohomology theory of Lie groups and Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124. MR 24908, DOI 10.1090/S0002-9947-1948-0024908-8
R. Coquereaux and A. Jadczyk, Geometry of multidimensional universes, Comm. Math. Phys. 90 (1983), no. 1, 79–100. MR 714613
R. Coquereaux and A. Jadczyk, Symmetries of Einstein-Yang-Mills fields and dimensional reduction, Comm. Math. Phys. 98 (1985), no. 1, 79–104. MR 785262
R. Coquereaux and A. Jadczyk, Consistency of the $G$-invariant Kaluza-Klein scheme, Nuclear Phys. B 276 (1986), no. 3-4, 617–628. MR 856538, DOI 10.1016/0550-3213(86)90068-4
Robert Coquereaux and Arkadiusz Jadcyzk, Riemannian geometry, fiber bundles, Kaluza-Klein theories and all that$\,\ldots$, World Scientific Lecture Notes in Physics, vol. 16, World Scientific Publishing Co., Singapore, 1988. MR 940468, DOI 10.1142/9789812799289
B. K. Darian, Gauge fields in homogeneous and inhomogeneous cosmologies, Ph.D. thesis, University of Alberta, 1997.
B. K. Darian and H. P. Künzle, Cosmological Einstein-Yang-Mills equations, J. Math. Phys. 38 (1997), no. 9, 4696–4713. MR 1468661, DOI 10.1063/1.532116
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, DOI 10.4153/CJM-2006-012-1
Mark E. Fels and Charles G. Torre, The principle of symmetric criticality in general relativity, Classical Quantum Gravity 19 (2002), no. 4, 641–675. MR 1891490, DOI 10.1088/0264-9381/19/4/303
Pedro L. García, Connections and $1$-jet fiber bundles, Rend. Sem. Mat. Univ. Padova 47 (1972), 227–242. MR 315624
Mark J. Gotay, Reduction of homogeneous Yang-Mills fields, J. Geom. Phys. 6 (1989), no. 3, 349–365. MR 1049710, DOI 10.1016/0393-0440(89)90009-0
G. Hochschild and J.-P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), 591–603. MR 54581, DOI 10.2307/1969740
J. Harnad, S. Shnider, and Luc Vinet, Group actions on principal bundles and invariance conditions for gauge fields, J. Math. Phys. 21 (1980), no. 12, 2719–2724. MR 597586, DOI 10.1063/1.524389
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, DOI 10.1063/1.524658
A. Jadczyk, Symmetry of Einstein-Yang-Mills systems and dimensional reduction, J. Geom. Phys. 1 (1984), no. 2, 97–126. MR 794982, DOI 10.1016/0393-0440(84)90006-8
A. Jadczyk and K. Pilch, Geometry of gauge fields in a multidimensional universe, Lett. Math. Phys. 8 (1984), no. 2, 97–104. MR 740810, DOI 10.1007/BF00406391
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol.I, John Wiley & Sons, Inc., 1963.
Norihito Koiso, Yang-Mills connections of homogeneous bundles, Osaka J. Math. 27 (1990), no. 1, 163–174. MR 1049829
H. Turner Laquer, Stability properties of the Yang-Mills functional near the canonical connection, Michigan Math. J. 31 (1984), no. 2, 139–159. MR 752251, DOI 10.1307/mmj/1029003019
Richard S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 19–30. MR 547524
J. Patera, R. T. Sharp, P. Winternitz, and H. Zassenhaus, Invariants of real low dimension Lie algebras, J. Mathematical Phys. 17 (1976), no. 6, 986–994. MR 404362, DOI 10.1063/1.522992
Hsien-chung Wang, On invariant connections over a principal fibre bundle, Nagoya Math. J. 13 (1958), 1–19. MR 107276