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

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 $\vec{X}$ over ${\mathbb{R}}^3$ using micro-graphs. This $\vec{X}$ projects to the known trivializing vector field for the ${\gamma }_3$-flow over ${\mathbb{R}}^2$.