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

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 $⩽1$ for aerial vertices. We obtain the formula ${\star }_{\text{aff}}\text{ mod }\bar{o}({\hslash }^7)$ with harmonic propagators for the graph weights (over $n⩽7$ aerial vertices); we verify that all these weights satisfy the cyclic weight relations by Shoikhet--Felder--Willwacher, that they match the computations using the $\text{kontsevint}$ software by Panzer, and the resulting affine star-product is associative modulo $\bar{o}({\hslash }^7)$. 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 ${\star }_{\text{aff}}\text{ mod }\bar{o}({\hslash }^7)$ \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~$\bar{o}({\hslash }^7)$ with only rational coefficients.

This paper is an extract from the PhD thesis of R.B. and is based on a series of talks and posters in Banff and Strasbourg; contains 44 pages, 2 tables, 2 appendices. Old chapter 3 and old appendices 3--6 were extracted from original version 1 and, in expanded form, published separately, see arXiv:2309.16664 [math.QA]