Formality morphism as the mechanism of $\star$-product associativity: how it works

Abstract

The formality morphism $\boldsymbol{\mathcal{F}}=\{{\mathcal{F}}_n$, $n\geqslant1\}$ in Kontsevich's deformation quantization is a collection of maps from tensor powers of the differential graded Lie algebra (dgLa) of multivector fields to the dgLa of polydifferential operators on finite-dimensional affine manifolds. Not a Lie algebra morphism by its term ${\mathcal{F}}_1$ alone, the entire set $\boldsymbol{{\mathcal{F}}}$ is an $L_\infty$-morphism instead. It induces a map of the Maurer-Cartan elements,  taking Poisson bivectors to deformations $\mu_A\mapsto\star_{A[[\hbar]]}$ of the usual multiplication
of functions into associative noncommutative $\star$-products of power series in $\hbar$. The associativity of $\star$-products is then realized, in terms of the Kontsevich graphs which encode polydifferential operators, by differential consequences of the Jacobi identity. The aim of this paper is to illustrate the work of this algebraic mechanism for the Kontsevich $\star$-products (in particular, with harmonic propagators).