The Algebra of Chern—Simons Classes, the Poisson Bracket on it, and the Action of the Gauge Group
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.
Unable to display preview. Download preview PDF.
- [Kon]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.Google Scholar
- [LZ1]B.H. Lian and G.J. Zuckerman, New Perspectives on the BRST-Algebraic Structure of String Theory, hep-th/9211072.Google Scholar
- [LZ2]B.H. Lian and G.J. Zuckerman, Some Classical and Quantum Algebras, Lie Theory and Geometry, Birkhäuser, 1994, this volume pp. 509–529Google Scholar
- [Ml]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.Google Scholar
- [M2]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.Google Scholar
- [O]Olver, P.J., Applications of Lie groups to the Partial Differential Equations, Springer, Berlin 1985.Google Scholar
- [You]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.Google Scholar