The Algebra of Chern-Simons Classes and the Poisson Bracket on it

Abstract

Developing ideas based on combinatorial formulas for characteristic classes we introduce the algebra modeling secondary characteristic classes associated to $N$ connections. Certain elements of the algebra correspond to the ordinary and secondary characteristic classes.That construction allows us to give easily the explicit formulas for some known secondary classes and to construct the new ones. We write how $i$-th differential and $i$-th homotopy operator in the algebra are connected with the Poisson bracket defined in this algebra. There is an analogy between this algebra and the noncommutative symplectic geometry of Kontsevich. We consider then an algebraic model of the action of the gauge group. We describe how elements in the algebra corresponding to the secondary characteristic classes change under this action.