Abstract
Developing ideas of formal geometry [Gl], [GKF], and ideas based on combinatorial formulas for characteristic classes, we introduce the algebraic structure that models the case of N connections given on the vector bundle over an oriented manifold. First we construct a graded free associative algebra A with a differential d. Then we go to the space V of cyclic words of A. Certain elements of V correspond to the secondary characteristic classes associated to k connections. That construction allows us to give easily the explicit formulas for some known secondary classes and to construct the new ones. The space V has new operations: it is a graded Lie algebra with respect to the Poisson bracket. There is an analogy between our algebra and the Kontsevich version of the noncommutative symplectic geometry. We consider then an algebraic model of the action of the gauge group. We describe how elements of our algebra corresponding to the secondary characteristic classes change under this action.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
R. Bott, On the Chern-Weil homomorphism and the continuous cohomology of Lie groups, Advances in Math. 11 (1973), 289–303.
F. Beresin and V. Retakh, A method of computing characteristic classes of vector bundles, Reports on Mathematical Physics 18 (1980), 363–378.
R. Bott, H. Shulman, J. Stasheff, On the de Rham theory of certain classifying spaces, Advances in Math. 20 (1976), 43–56.
J. Cheeger, Spectral geometry of singular Riemannian spaces, J. Differential Geom. 18 (1983), 575–657.
S.S. Chern, J. Simons, Characteristic forms and geometric invariants, Ann. Math. 99 (1974), 48–69.
J. Dupont, The Dilogarithm as a characteristic class of a flat bundle, J. Pure and Appl. Algebra 44 (1987), 137–164.
I.B. Frenkel and Y. Zhu, Vertex Operator Algebras Associated to Affine Lie Algebras, Duke Math. J. 66 (1992), 123
I.M. Gelfand, D.B. Fuks, and D.A. Kazhdan, The actions of infinite dimensional Lie algebras, Funct. Anal. Appl. 6 (1972), 9–13.
A. Gabrielov, I.M. Gelfand, and M. Losik, Combinatorial calculation of characteristic class, Funktsional Anal, i Prilozhen. (Fund. Anal. Appl.) 9 no. 2, 12–28 no. 3, 5–26, (1975), 54–55.
I.M. Gelfand and R. MacPherson, A combinatorial formula for the Pontrjagin classes, Bull AMS 26 no.2, 1992, 304–309.
I.M. Gelfand and B.L. Tsygan, On the localization of topological invariants, Comm. Math. Phys. 146 (1992), 73–90.
F. Hirzebruch, Topological Methods in Algebraic Geometry, Springer-Verlag, Berlin, 1966.
I.S. Krasil’shchik, V.V. Lychagin, and A.M. Vinogradov, Geometry of jet spaces and nonlinear partial differential equations, Gordon andBreach, NY 1986.
M. Kontsevich, Formal (Non)-Commutative Symplectic Geometry, The Gelfand Mathematical Seminars 1990–92, L.Corwin, I.Gelfand, and J. Lepowsky, eds., 173–189, Birkhäuser, Boston 1993.
J.L. Loday, Cyclic homology, Springer, Berlin, 1992.
J.L. Loday and D. Quillen, Cyclic homology and the Lie algebra of matrices, Topology 24 (1985), 89–95.
B.H. Lian and G.J. Zuckerman, New Perspectives on the BRST-Algebraic Structure of String Theory, hep-th/9211072.
B.H. Lian and G.J. Zuckerman, Some Classical and Quantum Algebras, Lie Theory and Geometry, Birkhäuser, 1994, this volume pp. 509–529
R. MacPherson, The combinatorial formula of Garielov, Gelfand, and Losik for the first Pontrjagin class, Séminaire Bourbaki No. 497, Lecture Notes in Math. 667, Springer, Heidelberg 1977.
R. MacPherson, Combinatorial differential manifolds, in: Topological Methods in Modern Mathematics: A Symposium in Honor of John Milnor’s Sixtieth Birthday, Lisa R. Goldberg and Anthony V. Phillips, eds., Publish or Perish, Inc., Houston 1993.
V. Mathai and D Quillen, Superconnections, Thom classes, and equivariant differential forms, Topology 25 (1986), 85–110.
Olver, P.J., Applications of Lie groups to the Partial Differential Equations, Springer, Berlin 1985.
L. Positselskii, Quadratic duality and curvature, Funct. Anal. Appl. 27 (1993), 197–204.
D. Quillen, Superconnections and the Chern character, Topology 24 no.l, 1985, 89–95.
Tsujishita, T., On variation bicomplex associated to differential equations, Osaka J. Math. 19 (1982), 311–363.
B.V.Youssin, Sur les formes S p,q apparaissant dans le calcul combinatoire de la deuxième classe de Pontriaguine par la méthode de Gabrielov, Gel’fand et Losik, C.R. Acad. Sci. Paris, t.292, 641–644.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1994 Springer Science+Business Media New York
About this chapter
Cite this chapter
Gelfand, I.M., Smirnov, M.M. (1994). The Algebra of Chern—Simons Classes, the Poisson Bracket on it, and the Action of the Gauge Group. In: Brylinski, JL., Brylinski, R., Guillemin, V., Kac, V. (eds) Lie Theory and Geometry. Progress in Mathematics, vol 123. Birkhäuser, Boston, MA. https://doi.org/10.1007/978-1-4612-0261-5_9
Download citation
DOI: https://doi.org/10.1007/978-1-4612-0261-5_9
Publisher Name: Birkhäuser, Boston, MA
Print ISBN: 978-1-4612-6685-3
Online ISBN: 978-1-4612-0261-5
eBook Packages: Springer Book Archive