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

TL;DR

This paper introduces a micro-graph calculus approach to analyze Kontsevich deformations of Nambu-Poisson structures, demonstrating that certain flows are Poisson coboundaries in specific dimensions, with explicit trivializing vector fields constructed.

Contribution

It develops a new micro-graph calculus method to resolve vertices in Kontsevich's graph calculus, enabling explicit trivializations of the tetrahedral flow for Nambu-determinant Poisson brackets in dimension three.

Findings

The tetrahedral $oldsymbol{ extgamma_3}$-flow is a Poisson coboundary in $oldsymbol{ extbf{R}^3}$.

Explicit trivializing vector fields are constructed using micro-graphs.

The method extends known results from $oldsymbol{ extbf{R}^2}$ to $oldsymbol{ extbf{R}^3}$.

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 $\smash{\mathbb{R}^2}$.