Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values?
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2022
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| author | Buring, Ricardo Kiselev, Arthemy V. |
| author_facet | Buring, Ricardo Kiselev, Arthemy V. |
| contents | 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 $ζ(3)^2/π^6$, 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 $ζ(3)^2/π^6$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_14438 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values? Buring, Ricardo Kiselev, Arthemy V. Quantum Algebra Mathematical Physics Combinatorics Symplectic Geometry 05C22, 53D55, 68R10, also 11M32, 16Z05, 53D17. 81R60, 81Q30 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 $ζ(3)^2/π^6$, 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 $ζ(3)^2/π^6$ 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. |
| title | Kontsevich's star-product up to order 7 for affine Poisson brackets: where are the Riemann zeta values? |
| topic | Quantum Algebra Mathematical Physics Combinatorics Symplectic Geometry 05C22, 53D55, 68R10, also 11M32, 16Z05, 53D17. 81R60, 81Q30 |
| url | https://arxiv.org/abs/2209.14438 |