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

TL;DR

This paper investigates the associativity mechanisms of Kontsevich's star-product mod $ar{o}( abla^k)$ for $k \

Contribution

It reveals that the associativity mechanism changes at order 7, requiring more complex graph-based reasoning compared to lower orders.

Findings

Associativity at orders ≤6 is directly derived from the Jacobi identity.

At order 7 and above, associativity involves multi-step graph-based differential consequences.

Different mechanisms govern the star-product's associativity below and above order 6.

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.