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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
Submission information
Preprint Link:  (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
Academic field: Physics
  • Mathematical Physics
Approaches: Theoretical, Computational


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)

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