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The tower of Kontsevich deformations for Nambu-Poisson structures on $\mathbb{R}^{d}$: dimension-specific micro-graph calculus
by Ricardo Buring, Arthemy V. Kiselev
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Submission summary
| Authors (as registered SciPost users): | Ricardo Buring |
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| Preprint Link: | https://arxiv.org/abs/2212.08063v3 (pdf) |
| Date accepted: | 2023-08-29 |
| Date submitted: | 2023-08-16 09:11 |
| Submitted by: | Buring, Ricardo |
| Submitted to: | SciPost Physics Proceedings |
| Proceedings issue: | 34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) |
| Ontological classification | |
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| Academic field: | Physics |
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| Approaches: | Theoretical, Computational |
Abstract
In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by multi-vectors, e.g., Poisson bi-vectors; the Nambu-determinant Poisson brackets are differential-polynomial in the Casimir(s) and density $\varrho$ times Levi-Civita symbol. We resolve the old vertices into subgraphs such that every new internal vertex contains one Casimir or one Levi-Civita symbol${}\times\varrho$. Using this micro-graph calculus, we show that Kontsevich's tetrahedral $\gamma_3$-flow on the space of Nambu-determinant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we realize the trivializing vector field $\smash{\vec{X}}$ over $\smash{\mathbb{R}^3}$ using micro-graphs. This $\smash{\vec{X}}$ projects to the known trivializing vector field for the $\gamma_3$-flow over $\smash{\mathbb{R}^2}$.
List of changes
- All the typos which have been pointed out are fixed, and instances of "Civita symbol" are replaced with "Levi-Civita symbol".
- Proposition 1 is extended with a reference to the second Poisson cohomology and with an encoding and formula of the "sunflower" graph and its 1-vector field.
Published as SciPost Phys. Proc. 14, 020 (2023)