Kontsevich graphs act on Nambu--Poisson brackets, II. The tetrahedral flow is a coboundary in 4D

TL;DR

This paper proves that the deformation of Nambu--Poisson brackets induced by the tetrahedral graph cocycle is trivial in four dimensions, extending previous results for dimensions two and three.

Contribution

It demonstrates that the tetrahedral graph cocycle induces a trivial deformation of Nambu--Poisson brackets specifically in four-dimensional space.

Findings

Deformation is trivial for d=2 (known since 1996)

Deformation is trivial for d=3 (found in 2020)

Deformation is trivial for d=4 (new result)

Abstract

Kontsevich constructed a map from suitable cocycles in the graph complex to infinitesimal deformations of Poisson bi-vector fields. Under the deformations, the bi-vector fields remain Poisson. We ask, are these deformations trivial, meaning, do they amount to a change of coordinates along a vector field? We examine this question for the tetrahedron, the smallest nontrivial suitable graph cocycle in the Kontsevich graph complex, and for the class of Nambu--Poisson brackets on $\mathbb{R}^d$. Within Kontsevich's graph calculus, we use dimension-specific micro-graphs, in which each vertex represents an ingredient of the Nambu--Poisson bracket. For the tetrahedron, Kontsevich knew that the deformation is trivial for $d = 2$ (1996). In 2020, Buring and the third author found that the deformation is trivial for $d = 3$. Building on these discoveries, we now establish that the deformation is trivial for $d = 4$.