Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}(\hbar^k)$: $k\leqslant 6$ unlike $k\geqslant7$
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
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2023
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| _version_ | 1866910297604751360 |
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| author | Buring, Ricardo Kiselev, Arthemy V. |
| author_facet | Buring, Ricardo Kiselev, Arthemy V. |
| contents | The formula $\star$ mod $\bar{o}(\hbar^k)$ of Kontsevich's star-product with harmonic propagators was known in full at $\hbar^{k\leqslant 6}$ since 2018 for generic Poisson brackets, and since 2022 also at $k=7$ for affine brackets. We discover that the mechanism of associativity for the star-product up to $\bar{o}(\hbar^6)$ is different from the mechanism at order $7$ for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod $\bar{o}(\hbar^6)$, whereas at order $\hbar^7$ and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_16664 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}(\hbar^k)$: $k\leqslant 6$ unlike $k\geqslant7$ Buring, Ricardo Kiselev, Arthemy V. Quantum Algebra Mathematical Physics Combinatorics Symplectic Geometry 05C22, 53D55, 68R10, 16Z05, 53D17, 81R60, 81Q30 The formula $\star$ mod $\bar{o}(\hbar^k)$ of Kontsevich's star-product with harmonic propagators was known in full at $\hbar^{k\leqslant 6}$ since 2018 for generic Poisson brackets, and since 2022 also at $k=7$ for affine brackets. We discover that the mechanism of associativity for the star-product up to $\bar{o}(\hbar^6)$ is different from the mechanism at order $7$ for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod $\bar{o}(\hbar^6)$, whereas at order $\hbar^7$ and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step. |
| title | Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}(\hbar^k)$: $k\leqslant 6$ unlike $k\geqslant7$ |
| topic | Quantum Algebra Mathematical Physics Combinatorics Symplectic Geometry 05C22, 53D55, 68R10, 16Z05, 53D17, 81R60, 81Q30 |
| url | https://arxiv.org/abs/2309.16664 |