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

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 \varrho𝜚 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×\varrho𝜚. Using this micro - graph calculus, we show that Kontsevich’s tetrahedral \gamma_3γ3-flow on the space of Nambu - determinant Poisson brackets over \mathbb{R}^3ℝ3 is a Poisson coboundary: we realize the trivializing vector field X over \mathbb{R}^3ℝ3 using micro - graphs. This X projects to the known trivializing vector field for the \gamma_3γ3-flow over \mathbb{R}^2ℝ2.