On the Kontsevich $\star$-product associativity mechanism

TL;DR

This paper explores how Kontsevich's deformation quantization constructs an associative star-product on Poisson manifolds, illustrating how the Jacobi identity ensures associativity through graph-based mechanisms up to third order in deformation parameter.

Contribution

It provides an explicit expansion of the star-product up to third order, demonstrating the conversion of the Jacobi identity into associativity via graph-based mechanisms.

Findings

Explicit expansion of star-product up to ^3

Illustration of associativity mechanism via graphs with loops

Connection between Jacobi identity and associativity

Abstract

The deformation quantization by Kontsevich [arXiv:q-alg/9709040] is a way to construct an associative noncommutative star-product $\star=\times+\hbar \{\ ,\ \}_{P}+\bar{o}(\hbar)$ in the algebra of formal power series in $\hbar$ on a given finite-dimensional affine Poisson manifold: here $\times$ is the usual multiplication, $\{\ ,\ \}_{P}\neq0$ is the Poisson bracket, and $\hbar$ is the deformation parameter. The product $\star$ is assembled at all powers $\hbar^{k\geq0}$ 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 $\star$ using $k$ copies of $\{\ ,\ \}_{P}$. Cattaneo and Felder [ arXiv:math/9902090 [math.QA] ] interpreted these topological portraits as the genuine Feynman diagrams in the Ikeda-Izawa model [arXiv:hep-th/9312059] for quantum gravity. By expanding the star-product up to $\bar{o}(\hbar^3)$, i.e., with respect to graphs with at most five vertices but possibly containing loops, we illustrate the mechanism Assoc = Operator(Poisson) that converts the Jacobi identity for the bracket $\{\ ,\ \}_{P}$ into the associativity of $\star$. Key words: Deformation quantization, associative algebra, Poisson bracket, graph complex, star-product PACS: 02.40.Sf, 02.10.Ox, 02.40.Gh, also 04.60.-m