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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:
  • Mathematical Physics
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 $\mathbb{R}^2$.

Current status:
Has been resubmitted


Author comments upon resubmission

By following the Editorial Recommendation, the authors have revised
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.

Submission & Refereeing History

Resubmission 2212.08063v3 on 16 August 2023

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Resubmission 2212.08063v2 on 14 July 2023

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

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  • validity: good
  • significance: low
  • originality: ok
  • clarity: good
  • formatting: excellent
  • grammar: excellent

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