Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}({\hslash }^k)$: $k⩽6$ unlike $k⩾7$

Abstract

The formula $\star$ mod $\bar{o}({\hslash }^k)$ of Kontsevich's star-product with harmonic propagators was known in full at ${\hslash }^{k⩽6}$ 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 $\bar{o}({\hslash }^6)$ 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 $\bar{o}({\hslash }^6)$, whereas at order ${\hslash }^7$ and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.

Expanded extract from arXiv:2209.14438 [q-alg]; Proc. international symposium on Quantum Theory and Symmetries (QTS12) on 24-28 July 2023 in CVUT Prague, Czech Republic; 8 pages + Appendices (7pp.), 2 tables, 1 figure

References in corpus (1)

• The expansion $\star$ mod $\bar{o}({\hslash }^4)$ and computer-assisted proof schemes in the Kontsevich deformation quantization