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The tower of Kontsevich deformations for Nambu-Poisson structures on $\mathbb{R}^{d}$: dimension-specific micro-graph calculus
by Ricardo Buring, Arthemy V. Kiselev
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Submission summary
| Authors (as registered SciPost users): | Ricardo Buring |
| Submission information | |
|---|---|
| Preprint Link: | https://arxiv.org/abs/2212.08063v2 (pdf) |
| Date submitted: | 2023-07-14 08:48 |
| Submitted by: | Buring, Ricardo |
| Submitted to: | SciPost Physics Proceedings |
| Proceedings issue: | 34th International Colloquium on Group Theoretical Methods in Physics (GROUP2022) |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by multi-vectors, e.g., Poisson bi-vectors; the Nambu-determinant Poisson brackets are differential-polynomial in the Casimir(s) and density $\varrho$ times Civita symbol. We resolve the old vertices into subgraphs such that every new internal vertex contains one Casimir or one Civita symbol${}\times\varrho$. Using this micro-graph calculus, we show that Kontsevich's tetrahedral $\gamma_3$-flow on the space of Nambu-determinant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we realize the trivializing vector field $\smash{\vec{X}}$ over $\mathbb{R}^3$ using micro-graphs. This $\smash{\vec{X}}$ projects to the known trivializing vector field for the $\gamma_3$-flow over $\mathbb{R}^2$.
Author comments upon resubmission
the submission. The authors are grateful to the Editor-in-Charge and
to the referee; comments of the referee were taken into account when
improving the text (now 10 pages as required), see the List of changes.
Applications of the apparent Poisson-triviality of the
graph-cocycle flows under study will be discussed in subsequent
publication(s), concerning in particular the deformation quantization
of Nambu--Poisson structures. The authors agree with referee's opinion
that it would be interesting to focus not only on the wheel cocycles:
e.g., the referee points out the Kontsevich--Shoikhet cocycle in this
context.
List of changes
In particular, section 1 is extended with Proposition 1 and its
explicit proof. As suggested by the referee, in section 2 we recall
the formula of trivializing vector field (now Theorem 2) to make this
article self-contained. In the encoding of this vector field by using
micro-graphs, sinks are properly indicated. The calculus of
micro-graphs is now introduced with greater care and in more detail.
The list of literature references is updated; technical Proposition
8 is moved to the last page. In new Proposition 9, the object of this
research is furthered to other wheel cocycles in the Kontsevich graph
complex.
Current status:
Reports on this Submission
Report 1 by Kevin Morand on 2023-7-25 (Invited Report)
- Cite as: Kevin Morand, Report on arXiv:2212.08063v2, delivered 2023-07-25, doi: 10.21468/SciPost.Report.7563
Report
See attached file