On the Kontsevich ★-product associativity mechanism

Abstract

The deformation quantization by Kontsevich is a way to construct an associative noncommutative star-product $\star = \times + \hbar {\{ ,\} _{\rm P}} + \overline o \left( \hbar \right)$ in the algebra of formal power series in h on a given finite-dimensional affine Poisson manifold: here × is the usual multiplication, {,} _P ≠ 0 is the Poisson bracket, and h is the deformation parameter. The product ★ is assembled at all powers h ^{k ≥ 0} via summation over a certain set of weighted graphs with k + 2 vertices; for each k > 0, every such graph connects the two co-multiples of ★ using k copies of {,} _P . Cattaneo and Felder interpreted these topological portraits as genuine Feynman diagrams in the Ikeda–Izawa model for quantum gravity. By expanding the star-product up to $\bar o\left( {{\hbar ^3}} \right)$, i.e., with respect to graphs with at most five vertices but possibly containing loops, we illustrate the mechanism Assoc = ≦ (Poisson) that converts the Jacobi identity for the bracket {,} _P into the associativity of ★.