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The tower of Kontsevich deformations for NambuPoisson structures on $\mathbb{R}^{d}$: dimensionspecific micrograph 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:  20230714 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: 

Approaches:  Theoretical, Computational 
Abstract
In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by multivectors, e.g., Poisson bivectors; the Nambudeterminant Poisson brackets are differentialpolynomial 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 micrograph calculus, we show that Kontsevich's tetrahedral $\gamma_3$flow on the space of Nambudeterminant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we realize the trivializing vector field $\smash{\vec{X}}$ over $\mathbb{R}^3$ using micrographs. This $\smash{\vec{X}}$ projects to the known trivializing vector field for the $\gamma_3$flow over $\mathbb{R}^2$.
Current status:
Author comments upon resubmission
the submission. The authors are grateful to the EditorinCharge 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 Poissontriviality of the
graphcocycle flows under study will be discussed in subsequent
publication(s), concerning in particular the deformation quantization
of NambuPoisson 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 KontsevichShoikhet 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 selfcontained. In the encoding of this vector field by using
micrographs, sinks are properly indicated. The calculus of
micrographs 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
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Reports on this Submission
Report 1 by Kevin Morand on 2023725 (Invited Report)
 Cite as: Kevin Morand, Report on arXiv:2212.08063v2, delivered 20230725, doi: 10.21468/SciPost.Report.7563
Report
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