Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}(\hbar^k)$: $k\leqslant 6$ unlike $k\geqslant7$

Abstract

The formula $\star$ mod $\bar{o}(\hbar^k)$ of Kontsevich's star-product with harmonic propagators was known in full at $\hbar^{k\leqslant 6}$ since 2018 for generic Poisson brackets, and since 2022 also at $k=7$ for affine brackets. We discover that the mechanism of associativity for the star-product up to $\bar{o}(\hbar^6)$ is different from the mechanism at order $7$ for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod $\bar{o}(\hbar^6)$, whereas at order $\hbar^7$ and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.