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
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
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.
 [AB82]
M.
F. Atiyah and R.
Bott, The YangMills equations over Riemann surfaces, Philos.
Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505,
523–615. MR
702806 (85k:14006), http://dx.doi.org/10.1098/rsta.1983.0017
 [AF]
I. M. Anderson and M. E. Fels, Notes on the Palais condition, private communication.
 [AF97]
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
(98k:58208)
 [AF04]
I.
M. Anderson and M.
E. Fels, A cochain 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
(2008f:58002), http://dx.doi.org/10.1142/9789812702142_0001
 [AFT01]
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
(2002h:58068)
 [And]
I. M. Anderson, The Variational Bicomplex, Preprint available online at http://www.math.usu.edu/fgmp/Publications/VB/vb.pdf.
 [And92]
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
(94a:58045), http://dx.doi.org/10.1090/conm/132/1188434
 [Ble81]
David
Bleecker, Gauge theory and variational principles, Global
Analysis Pure and Applied Series A, vol. 1, AddisonWesley Publishing
Co., Reading, Mass., 1981. MR 643361
(83h:53049)
 [CE48]
Claude
Chevalley and Samuel
Eilenberg, Cohomology theory of Lie groups and
Lie algebras, Trans. Amer. Math. Soc. 63 (1948), 85–124.
MR
0024908 (9,567a), http://dx.doi.org/10.1090/S00029947194800249088
 [CJ83]
R.
Coquereaux and A.
Jadczyk, Geometry of multidimensional universes, Comm. Math.
Phys. 90 (1983), no. 1, 79–100. MR 714613
(85e:83053)
 [CJ85]
R.
Coquereaux and A.
Jadczyk, Symmetries of EinsteinYangMills fields and dimensional
reduction, Comm. Math. Phys. 98 (1985), no. 1,
79–104. MR
785262 (86h:53071)
 [CJ86]
R.
Coquereaux and A.
Jadczyk, Consistency of the 𝐺invariant KaluzaKlein
scheme, Nuclear Phys. B 276 (1986), no. 34,
617–628. MR
856538 (87i:83088), http://dx.doi.org/10.1016/05503213(86)900684
 [CJ88]
Robert
Coquereaux and Arkadiusz
Jadcyzk, Riemannian geometry, fiber bundles, KaluzaKlein 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 EinsteinYangMills equations, J.
Math. Phys. 38 (1997), no. 9, 4696–4713. MR 1468661
(98h:53042), http://dx.doi.org/10.1063/1.532116
 [FR06]
M.
E. Fels and A.
G. Renner, Nonreductive homogeneous pseudoRiemannian manifolds of
dimension four, Canad. J. Math. 58 (2006),
no. 2, 282–311. MR 2209280
(2006m:53077), http://dx.doi.org/10.4153/CJM20060121
 [FT02]
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
(2002m:58026), http://dx.doi.org/10.1088/02649381/19/4/303
 [Gar72]
Pedro
L. García, Connections and 1jet fiber bundles, Rend.
Sem. Mat. Univ. Padova 47 (1972), 227–242. MR 0315624
(47 #4173)
 [Got89]
Mark
J. Gotay, Reduction of homogeneous YangMills fields, J. Geom.
Phys. 6 (1989), no. 3, 349–365. MR 1049710
(91c:58039), http://dx.doi.org/10.1016/03930440(89)900090
 [HS53]
G.
Hochschild and J.P.
Serre, Cohomology of Lie algebras, Ann. of Math. (2)
57 (1953), 591–603. MR 0054581
(14,943c)
 [HSV80]
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
(82c:53028), http://dx.doi.org/10.1063/1.524389
 [HTS80]
J.
Harnad, J.
Tafel, and S.
Shnider, Canonical connections on Riemannian symmetric spaces and
solutions to the EinsteinYangMills equations, J. Math. Phys.
21 (1980), no. 8, 2236–2240. MR 579220
(82j:53055), http://dx.doi.org/10.1063/1.524658
 [Jad84]
A.
Jadczyk, Symmetry of EinsteinYangMills systems and dimensional
reduction, J. Geom. Phys. 1 (1984), no. 2,
97–126. MR
794982 (86k:53089), http://dx.doi.org/10.1016/03930440(84)900068
 [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), http://dx.doi.org/10.1007/BF00406391
 [KN63]
S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol.I, John Wiley & Sons, Inc., 1963.
 [Koi90]
Norihito
Koiso, YangMills connections of homogeneous bundles, Osaka J.
Math. 27 (1990), no. 1, 163–174. MR 1049829
(91b:58040)
 [Laq84]
H.
Turner Laquer, Stability properties of the YangMills functional
near the canonical connection, Michigan Math. J. 31
(1984), no. 2, 139–159. MR 752251
(88a:58047), http://dx.doi.org/10.1307/mmj/1029003019
 [Pal79]
Richard
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.
Mathematical Phys. 17 (1976), no. 6, 986–994.
MR
0404362 (53 #8164)
 [Wan58]
Hsienchung
Wang, On invariant connections over a principal fibre bundle,
Nagoya Math. J. 13 (1958), 1–19. MR 0107276
(21 #6001)
 [AB82]
 M. F. Atiyah and R. Bott, The YangMills equations over Riemann surfaces, Phil. Trans. R. Soc. London A 308 (1983), 523615. 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), 609670. MR 1448217 (98k:58208)
 [AF04]
 , A cochain map for the Ginvariant 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. 112. 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. 95108. MR 1870907 (2002h:58068)
 [And]
 I. M. Anderson, The Variational Bicomplex, Preprint available online at http://www.math.usu.edu/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. 5173. MR 1188434 (94a:58045)
 [Ble81]
 D. Bleecker, Gauge Theory and Variational Principles, Global Analysis, Pure and Applied, series A, no. 1, AddisonWesley, 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, 85124. MR 0024908 (9:567a)
 [CJ83]
 R. Coquereaux and A. Jadczyk, Geometry of multidimensional universes, Comm. Math. Phys. 90 (1983), no. 1, 79100. MR 714613 (85e:83053)
 [CJ85]
 , Symmetries of EinsteinYangMills fields and dimensional reduction, Comm. Math. Phys. 98 (1985), no. 1, 79104. MR 785262 (86h:53071)
 [CJ86]
 , Consistency of the Ginvariant KaluzaKlein scheme, Nuclear Phys. B (1986), 617628. MR 856538 (87i:83088)
 [CJ88]
 , Riemannian Geometry, Fiber Bundles, KaluzaKlein 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 EinsteinYangMills equations, J. Math. Phys. 38 (1997), no. 9, 46964713. MR 1468661 (98h:53042)
 [FR06]
 M. E. Fels and A. G. Renner, Nonreductive homogeneous pseudoRiemannian manifolds of dimension four, Canad. J. Math. 58 (2006), no. 2, 282311. 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), 641675. MR 1891490 (2002m:58026)
 [Gar72]
 P. L. Garcia, Connections and jet bundles, Rend. Sem. Mat. Univ. Padova 47 (1972), 227242. MR 0315624 (47:4173)
 [Got89]
 M. J. Gotay, Reduction of homogeneous YangMills fields, J. Geom. Phys. 6 (1989), no. 3, 349365. MR 1049710 (91c:58039)
 [HS53]
 G. Hochschild and J. P. Serre, Cohomology of Lie algebras, Ann. of Math. (2) 57 (1953), no. 3, 591603. 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, 27192724. MR 597586 (82c:53028)
 [HTS80]
 J. Harnad, J. Tafel, and S. Shnider, Canonical connections on Riemannian symmetric spaces and solutions to the EinsteinYangMills equations, J. Math. Phys. 21 (1980), no. 8, 22362240. MR 579220 (82j:53055)
 [Jad84]
 A. Jadczyk, Symmetry of EinsteinYangMills systems and dimensional reduction, J. Geom. Phys. 1 (1984), no. 2, 97126. 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, 97104. MR 740810 (86b:53071)
 [KN63]
 S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol.I, John Wiley & Sons, Inc., 1963.
 [Koi90]
 N. Koiso, YangMills connections of homogeneous bundles, Osaka J. Math. 27 (1990), no. 1, 163174. MR 1049829 (91b:58040)
 [Laq84]
 H. T. Laquer, Stability properties of the YangMills functional near the canonical connection, Michigan Math. J. 31 (1984), no. 2, 139159. MR 752251 (88a:58047)
 [Pal79]
 R. S. Palais, The principle of symmetric criticality, Comm. Math. Phys. 69 (1979), no. 1, 1930. 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, 986994. MR 0404362 (53:8164)
 [Wan58]
 H. C. Wang, On invariant connections over a principal fibre bundle, Nagoya Math. J. 13 (1958), 119. 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 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.
