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

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 ⋆_aff mod ō(ℏ⁷) 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 kontsevint software by Panzer, and the resulting affine star-product is associative modulo ō(ℏ⁷). We discover that the Riemann zeta value ζ(3)²/π⁶, which enters the harmonic graph weights (up to rationals), actually disappears from the analytic formula of ⋆_aff mod ō(ℏ⁷)because alltheQ-linearcombinationsofKontsevichgraphsnearζ(3)²/π⁶ 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 ⋆^red_aff mod ō(ℏ⁷) with only rational coefficients.