Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values?

TL;DR

This paper computes the affine Kontsevich star-product up to order 7, revealing that the zeta value b3(3)^2/c6^6 cancels out, leading to a simplified formula with only rational coefficients.

Contribution

It provides explicit formulas for the affine Kontsevich star-product up to order 7 and shows the disappearance of certain zeta values, simplifying the expression.

Findings

Computed star-product up to order 7 with harmonic propagators.

Verified weights satisfy cyclic weight relations and match software computations.

Discovered the vanishing of b3(3)^2/c6^6 in the formula, leading to a rational coefficient version.

Abstract

The Kontsevich star-product admits a well-defined restriction to the class of affine -- in particular, linear -- Poisson brackets; its graph expansion consists only of Kontsevich's graphs with in-degree $\leqslant 1$ for aerial vertices. We obtain the formula $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar^7)$ with harmonic propagators for the graph weights (over $n\leqslant 7$ aerial vertices); we verify that all these weights satisfy the cyclic weight relations by Shoikhet--Felder--Willwacher, that they match the computations using the $\textsf{kontsevint}$ software by Panzer, and the resulting affine star-product is associative modulo $\bar{o}(\hbar^7)$. We discover that the Riemann zeta value $\zeta(3)^2/\pi^6$, which enters the harmonic graph weights (up to rationals), actually disappears from the analytic formula of $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar^7)$ \textit{because} all the $\mathbb{Q}$-linear combinations of Kontsevich graphs near $\zeta(3)^2/\pi^6$ represent differential consequences of the Jacobi identity for the affine Poisson bracket, hence their contribution vanishes. We thus derive a ready-to-use shorter formula $\star_{\text{aff}}^{\text{red}}$ mod~$\bar{o}(\hbar^7)$ with only rational coefficients.