Kontsevich graphs act on Nambu-Poisson brackets, III. Uniqueness aspects

TL;DR

This paper investigates the uniqueness of trivializing vector fields associated with Nambu-Poisson brackets under Kontsevich graph deformations, showing they are unique up to Hamiltonian vector fields but the graph representations are not unique.

Contribution

It proves the trivializing vector fields are unique modulo Hamiltonian vector fields and highlights the non-uniqueness of Kontsevich graph representations for these multivectors.

Findings

Trivializing vector fields are unique modulo Hamiltonian vector fields.

Kontsevich graph representations of multivectors are not unique.

The study extends understanding of deformation quantization in Nambu-Poisson structures.

Abstract

Kontsevich constructed a map between `good' graph cocycles $\gamma$ and infinitesimal deformations of Poisson bivectors on affine manifolds, that is, Poisson cocycles in the second Lichnerowicz--Poisson cohomology. For the tetrahedral graph cocycle $\gamma_3$ and for the class of Nambu-determinant Poisson bivectors $P$ over $\mathbb{R}^2$, $\mathbb{R}^3$ and $\mathbb{R}^4$, we know the fact of trivialization, $\dot{P}=[[ P, \vec{X}^{\gamma_3}_{\text{dim}}]]$, by using dimension-dependent vector fields $\vec{X}^{\gamma_3}_{\text{dim}}$ expressed by Kontsevich (micro-) graphs. We establish that these trivializing vector fields $\vec{X}^{\gamma_3}_{\text{dim}}$ are unique modulo Hamiltonian vector fields $\vec{X}_{H}=d_P(H)= [[ P, H]]$, where $d_P$ is the Lichnerowicz--Poisson differential and where the Hamiltonians $H$ are also represented by Kontsevich (micro-)graphs. However, we find that the choice of Kontsevich (micro-)graphs to represent the aforementioned multivectors is not unique.