English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools annual

Global: Zendy Free Plan

Global: Zendy Tools monthly

Global: Zendy Plus monthly

The formula $\star$ mod $\bar{o}(\hbar^k)$ of Kontsevich's star-product withharmonic propagators was known in full at $\hbar^{k\leqslant 6}$ since 2018 forgeneric Poisson brackets, and since 2022 also at $k=7$ for affine brackets. Wediscover 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 thefull star-product and the affine star-product. Namely, at lower orders theneeded consequences of the Jacobi identity are immediately obtained from theassociator mod $\bar{o}(\hbar^6)$, whereas at order $\hbar^7$ and higher, someof the necessary differential consequences are reached from the Kontsevichgraphs in the associator in strictly more than one step.

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