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

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 $d=4$.

References in corpus (4)

• The Kontsevich tetrahedral flow in 2D: a toy model
• Kontsevich graphs act on Nambu-Poisson brackets, III. Uniqueness aspects
• Kontsevich graphs act on Nambu-Poisson brackets, I. New identities for Jacobian determinants
• Kontsevich graphs act on Nambu--Poisson brackets, II. The tetrahedral flow is a coboundary in 4D

Cited by in corpus (2)

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