Kontsevich graphs act on Nambu-Poisson brackets, VI. Open problems

TL;DR

This paper investigates the action of Kontsevich's graphs on Nambu-Poisson brackets, revealing potential infinite identities and open problems in the deformation quantisation framework for multi-vectors.

Contribution

It introduces a dimension-specific calculus for Kontsevich graphs on Nambu-Poisson brackets and explores the (non)triviality of tetrahedral graph cocycles, proposing new identities.

Findings

Detection of a conjecturally infinite set of identities for Jacobian determinants.

Analysis of the (non)triviality of Kontsevich's tetrahedral graph cocycle.

Dimension-specific calculus for Kontsevich graphs on Nambu-Poisson brackets.

Abstract

Kontsevich's graphs from deformation quantisation allow encoding multi-vectors whose coefficients are differential-polynomial in components of Poisson brackets on finite-dimensional affine manifolds. The calculus of Kontsevich graphs can be made dimension-specific for the class of Nambu--Poisson brackets given by Jacobian determinants. Using the Kontsevich--Nambu micro-graphs in dimensions $d\geqslant 2$, we explore the open problem of (non)triviality for Kontsevich's tetrahedral graph cocycle action on the space of Nambu--Poisson brackets. We detect a conjecturally infinite new set of differential-polynomial identities for Jacobian determinants of arbitrary sizes $d\times d$.