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The tower of Kontsevich deformations for Nambu-Poisson structures on $\mathbb{R}^d$: Dimension-specific micro-graph calculus

Ricardo Buring, Arthemy V. Kiselev

SciPost Phys. Proc. 14, 020 (2023) · published 23 November 2023

Proceedings event

34th International Colloquium on Group Theoretical Methods in Physics


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×$\varrho$. 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 $\vec{X}$ over $\mathbb{R}^3$ using micro - graphs. This $\vec{X}$ projects to the known trivializing vector field for the $\gamma_3$-flow over $\mathbb{R}^2$.

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