Publication:
Gauge interpretation of characteristic classes

Loading...
Thumbnail Image
Full text at PDC
Publication Date
2001
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
International Press
Citations
Google Scholar
Research Projects
Organizational Units
Journal Issue
Abstract
Let π:P→M be a principal G-bundle. Then one can consider the following diagram of fibre bundles: \CD J^{1}(P) @>\pi_{10}>> P\\ @VqVV @VV\pi V\\ C(P) @>p>> M\endCD where p is the bundle of connections of π. As is well known, q is also a principal G-bundle, and the canonical contact form θ on J1(P) can be considered as a connection form on q, with curvature form Θ. One defines aut P as the Lie algebra of G-invariant vector fields on P and gau P as the ideal of π-vertical G-invariant vector fields on P. If X∈autP⊂X(P), then one defines the infinitesimal contact transformation associated to X, X1∈X(J1(P)), and its q-projection XC∈X(C(P)). A differential form Ω on C(P) is said to be aut P-invariant [resp. gauge invariant] if LXCΩ=0 for every X∈autP [resp. X∈gauP]. On the other hand, let us denote by g the Lie algebra of G. An element of the symmetric algebra of g∗ will be called a Weil polynomial. The main result of the paper is the following theorem: If G is connected, for every gauge invariant form Ω on C(P) there exist differential forms ω1,…,ωk on M and Weil polynomials f1,…,fk such that Ω=p∗(ω1)∧f1(Θ)+⋯+p∗(ωk)∧fk(Θ). As a consequence, the authors prove that a differential form Ω on C(P) is aut P-invariant iff Ω=f(Θ), where f is a Weil polynomial, and then Ω is closed. Explicit examples are shown and the link between the above theorem and the geometric formulation of Utiyama's theorem is explained.
Description
Keywords
Citation
M. F. Atiyah, Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc. 85 (1957), 181–207. D. Betounes, The geometry of gauge-particle field interaction: a generalization of Utiyama's theorem, J. Geom. Phys. 6 (1989), 107–125. MR1027299 (90j:53095) D. Bleecker, Gauge Theory and Variational Principles, Addison-Wesley Publishing Company, Inc., Reading, MA, 1981. M. Castrillón López and J. Mu~noz Masqué, The geometry of the bundle of connections, Math. Z. 236 (2001), 797–811. D. J. Eck, Gauge-natural bundles and generalized gauge theories, Mem. Amer. Math. Soc. 33 (1981). P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, CRC Press, Boca Raton, FL, 1995. V. Guillemin and S. Sternberg, Symplectic techniques in physics, Cambridge University Press, Cambridge, 1983. M. Keyl, About the geometric structure of symmetry-breaking, J. Math. Phys. 32 (1991), 1065–1071. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry Volume I and Volume II, John Wiley & Sons, Inc. (Interscience Division), New York, 1963, 1969. D. J. Saunders, The Geometry of Jet Bundles, Cambridge University Press, Cambridge, UK, 1989. V. S. Varadarajan, Lie groups, Lie algebras, and their representations, Prentice Hall, Inc., N. J., 1974.
Collections