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Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values? (2209.14438v3)

Published 28 Sep 2022 in math.QA, math-ph, math.CO, math.MP, and math.SG

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 $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar7)$ 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 $\bar{o}(\hbar7)$. We discover that the Riemann zeta value $\zeta(3)2/\pi6$, which enters the harmonic graph weights (up to rationals), actually disappears from the analytic formula of $\star_{\text{aff}}\text{ mod }\bar{o}(\hbar7)$ \textit{because} all the $\mathbb{Q}$-linear combinations of Kontsevich graphs near $\zeta(3)2/\pi6$ 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~$\bar{o}(\hbar7)$ with only rational coefficients.

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