Kontsevich graphs act on Nambu--Poisson brackets, II. The tetrahedral flow is a coboundary in 4D
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| Format: | Preprint |
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2024
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| author | Brown, Mollie S. Jagoe Schipper, Floor Kiselev, Arthemy V. |
| author_facet | Brown, Mollie S. Jagoe Schipper, Floor Kiselev, Arthemy V. |
| contents | Kontsevich constructed a map from suitable cocycles in the graph complex to infinitesimal deformations of Poisson bi-vector fields. Under the deformations, the bi-vector fields remain Poisson. We ask, are these deformations trivial, meaning, do they amount to a change of coordinates along a vector field? We examine this question for the tetrahedron, the smallest nontrivial suitable graph cocycle in the Kontsevich graph complex, and for the class of Nambu--Poisson brackets on $\mathbb{R}^d$.
Within Kontsevich's graph calculus, we use dimension-specific micro-graphs, in which each vertex represents an ingredient of the Nambu--Poisson bracket. For the tetrahedron, Kontsevich knew that the deformation is trivial for $d = 2$ (1996). In 2020, Buring and the third author found that the deformation is trivial for $d = 3$. Building on these discoveries, we now establish that the deformation is trivial for $d = 4$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_12555 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Kontsevich graphs act on Nambu--Poisson brackets, II. The tetrahedral flow is a coboundary in 4D Brown, Mollie S. Jagoe Schipper, Floor Kiselev, Arthemy V. Quantum Algebra 53D55, 17B63, 05C22, 68R10, 53D17 Kontsevich constructed a map from suitable cocycles in the graph complex to infinitesimal deformations of Poisson bi-vector fields. Under the deformations, the bi-vector fields remain Poisson. We ask, are these deformations trivial, meaning, do they amount to a change of coordinates along a vector field? We examine this question for the tetrahedron, the smallest nontrivial suitable graph cocycle in the Kontsevich graph complex, and for the class of Nambu--Poisson brackets on $\mathbb{R}^d$. Within Kontsevich's graph calculus, we use dimension-specific micro-graphs, in which each vertex represents an ingredient of the Nambu--Poisson bracket. For the tetrahedron, Kontsevich knew that the deformation is trivial for $d = 2$ (1996). In 2020, Buring and the third author found that the deformation is trivial for $d = 3$. Building on these discoveries, we now establish that the deformation is trivial for $d = 4$. |
| title | Kontsevich graphs act on Nambu--Poisson brackets, II. The tetrahedral flow is a coboundary in 4D |
| topic | Quantum Algebra 53D55, 17B63, 05C22, 68R10, 53D17 |
| url | https://arxiv.org/abs/2409.12555 |