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arXiv · 2409.15932

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

Floor Schipper·Mollie S Jagoe Brown·Arthemy V Kiselev
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Abstract

Kontsevich constructed a map between `good' graph cocycles $γ$ and infinitesimal deformations of Poisson bivectors on affine manifolds, that is, Poisson cocycles in the second Lichnerowicz--Poisson cohomology. For the tetrahedral graph cocycle $γ_3$ 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, $\dot{P}=[[ P, \vec{X}^{γ_3}_{\text{dim}}]]$, by using dimension-dependent vector fields $\vec{X}^{γ_3}_{\text{dim}}$ expressed by Kontsevich (micro-) graphs. We establish that these trivializing vector fields $\vec{X}^{γ_3}_{\text{dim}}$ 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.

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Floor Schipper, Mollie S Jagoe Brown, Arthemy V Kiselev. 2024-11-23. Kontsevich graphs act on Nambu-Poisson brackets, III. Uniqueness aspects. https://doi.org/10.1088/1742-6596%2F2912%2F1%2F012035

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