This research paper investigates the deformation of Nambu-Poisson brackets under the action of the Kontsevich tetrahedron graph (γ3) in dimensions 2, 3, and 4. The authors utilize Kontsevich's graph calculus and a series of simplifications to address the computational challenges posed by higher dimensions.
Research Objective:
The study aims to determine whether the deformation of Nambu-Poisson brackets by the γ3 graph is trivial, i.e., if it can be realized through a change of coordinates.
Methodology:
The authors employ Kontsevich's graph calculus, representing mathematical expressions as graphs. They introduce "Nambu micro-graphs," which are graphs built using the Nambu-Poisson bracket as a subgraph. To simplify the problem, they focus on "d-descendants" of lower-dimensional solutions and exploit the skew-symmetry property of the Nambu-Poisson bracket.
Key Findings:
The research demonstrates that the γ3-flow, representing the deformation of the Nambu-Poisson bracket by γ3, is trivial in dimensions 2, 3, and 4. This implies the existence of a vector field along which a change of coordinates can produce the deformation.
Main Conclusions:
The study concludes that the deformation of Nambu-Poisson brackets by the Kontsevich tetrahedron graph is trivial up to dimension 4. This suggests that the Nambu-Poisson system is "isolated" in these dimensions, meaning it remains unchanged under this specific deformation.
Significance:
This research contributes to the understanding of deformation quantization and the behavior of Nambu-Poisson brackets under deformation. It provides insights into the properties of these mathematical structures and their potential applications in areas such as mathematical physics.
Limitations and Future Research:
The study focuses on a specific graph (γ3) and the class of Nambu-Poisson brackets. Further research could explore the triviality of deformations induced by other Kontsevich graphs and different classes of Poisson brackets. Investigating higher dimensions beyond 4 also remains an open problem.
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by Mollie S. Ja... at arxiv.org 10-10-2024
https://arxiv.org/pdf/2409.12555.pdfDeeper Inquiries