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arXiv · 2409.12555

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

Mollie S. Jagoe Brown·Floor Schipper·Arthemy V. Kiselev
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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 $d = 4$.

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Mollie S. Jagoe Brown, Floor Schipper, Arthemy V. Kiselev. 2024-10-09. Kontsevich graphs act on Nambu--Poisson brackets, II. The tetrahedral flow is a coboundary in 4D. https://doi.org/10.1088/1742-6596%2F2912%2F1%2F012042

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