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.
KeywordsGauge Group Vector Bundle Gauge Transformation Poisson Bracket Vertex Operator Algebra
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