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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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Authors (as registered SciPost users): Ricardo Buring
Submission information
Preprint Link: scipost_202212_00033v1  (pdf)
Date submitted: 2022-12-16 14:27
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
  • Mathematical Physics
Approaches: Theoretical, Computational


In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by copies of a given Poisson structure; in turn, the Nambu-determinant Poisson brackets themselves are differential polynomial in the Casimir(s) and a density $\varrho$ times the Civita symbol. We now 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 establish that Kontsevich's tetrahedral $\gamma_3$-flow on the space of Nambu-determinant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we obtain a micro-graph realization $X^\gamma$ of the trivializing vector field $\smash{\vec{X}}$ over $\mathbb{R}^3$. This $\smash{\vec{X}}$ does project to the known trivializing vector field for the $\gamma_3$-flow over $\mathbb{R}^2$. We conjecture that over all $\mathbb{R}^{d\geqslant 3}$, Kontsevich's $\gamma_3$-flows of Nambu-Poisson brackets are coboundaries; the trivializing vector fields then project down under $\mathbb{R}^{d} \to \mathbb{R}^{d-1}$.

Current status:
Has been resubmitted

Submission & Refereeing History

Resubmission 2212.08063v3 on 16 August 2023

Reports on this Submission

Report 1 by Kevin Morand on 2023-1-28 (Invited Report)

  • Cite as: Kevin Morand, Report on arXiv:scipost_202212_00033v1, delivered 2023-01-28, doi: 10.21468/SciPost.Report.6571


See the attached pdf


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

Author:  Ricardo Buring  on 2023-02-06  [id 3312]

(in reply to Report 1 by Kevin Morand on 2023-01-28)

The authors are grateful to the Referee for careful reading of this short paper for conference proceedings of Group34. The Referee may be aware that the 8-page limit was absolute for all submissions, hence if anything be added or expanded at Referee's request, this would only now be possible (on the basis of Referee's advice).

The paper under review is aimed to prove in full our Theorem 8 on p.206 in OCNMP (ref.[8] in the submission = ref.[2] in the report). That is --due to the volume restriction of 8 pages,-- the "main theorem" is in the citation: it was not repeated here until the Referee pointed this out.

The paper under review reflects the talk given at Group34. Not every conference proceedings text should be expected --in Referee's report, twice!-- to be a crucially meaningful addition to the field. To the same extent as not all surgeries must be amputations. Let us introduce sanity into the refereeing process.

The paper under review narrates on the problem posed by M.Kontsevich to the second author in March 2008. The problem is the issue of (non)triviality of graph cocycle flows for (classes of) Poisson brackets: to find NON-trivial case(s). Its relation to deformation quantisation is, strictly speaking, outside the 8-page frames of this conference text. Yet of course, the Referee is right: further problems and developments can be mentioned (on new pages 9,10,etc.).

The paper under review reported on the state-of-the-art by mid-December 2022, that is after Group34 and discussions with Kontsevich at the IHES in November, still before the breakthrough on 25 December 2022: the tetrahedral graph flow on the space of Nambu--Poisson brackets is trivial in all dimensions >1, because the trivializing vector field is the "sunflower" graph (from dimension two) in which every internal vertex is expanded to the Nambu--Poisson bracket's micro-graph and all the in-coming arrows work by their respective Leibniz rules. The same applies to the pentagon- and heptagon-wheel graph cocycles, for which we know the trivializations in dimension two for the Nambu--Poisson brackets. This now being said, the solution is simple and natural, yet not at once obvious. In particular, this solution is absent from the carefully written, long referee report. --- We agree that, as suggested in Report.(3.Opinion).(2), a conjecture about all the wheel generators in all dimensions would now be appropriate. The method and reasoning in the paper under review justify this conjecture.

In this sense, this conference paper almost concludes a long line of our research, whence the many references to our previous work. (We thank the Referee for recalling about Shoikhet's cocycle; this and other literature references can be added --- again, using pages 9,10,etc., beyond the 8-page length limit.)

Likewise, the notion of Leibniz (micro-)graphs, providing the right-hand side in Eq.(2), has become standard in Kontsevich's graph calculus. It was implicitly present in the main result, Formality Theorem by Kontsevich (1997) about the Jacobi identity as the only obstruction to deformation quantisation at all orders. The term itself was coined in 2016 for our arXiv:1608.01710. To clarify a possible misunderstanding, we draw Referee's attention to our Lemma 2 on p.3 in the manuscript under review: the Jacobiator for Nambu--Poisson brackets is a NON-trivial linear combination of micro-graphs. This Lemma 2 is a once-for-ever remedy agains "naive expectation" in Report.(4.Questions).(2) that there could be no need to mod out by the Jacobi identity at the level of (micro-)graphs. This is the known distinction between graphs and analytic expressions (where the Jacobiator vanishes for Poisson brackets).

The Referee is expert on the undirected graph complex from the grt. But in the world of graph flows and star-products, the Kontsevich graph calculus explicitly uses sink vertices for product's or multi-vector's arguments. And this indeed is "calculus" because it simply is the differential calculus of multi-vectors. E.g., consider the Schouten bracket [[P,P]] of (Nambu-)Poisson bi-vectors: one graph acts on the other minus vice versa --- but not in the operadic sense of the grt graph complex. The two formalisms are akin still they are not the same. We draw Referee's attention to this distinction (cf. Report.(4.Questions).(1)). --- Modulo the above explanation, the Referee is sure welcome to propose better variants of the title for the paper under review.

Finally, about tadpoles.
The invariable necessity of tadpoles is a discovery (so, "result") in this submission; we show that, and where explicitly, all the tadpoles in vector fields X do not survive into the graph cocycles [[P,X]] for Nambu--Poisson brackets. Tadpoles were basically absent from Willwacher's work on grt until he brought BV into the graph formalism; in the Cattaneo--Felder path integral interpretation of Kontsevich's star-products, tadpoles were regularized by using brute force QFT techniques. We discover now that "nothing must be done at all" with these tadpoles in this piece of Poisson formalism: all the tadpoles are contained only in the differential consequences of the Jacobi identity. --- This can be emphasized in a revision (at the price of stepping over the 8-page volume limit). The same concerns Referee's Report.(4.Questions).(3) about tadpoles in our Open Problem 1.

We thank the Referee for attentive reading and much effort. We hope that this letter would clarify the scope, content, and rules of the game, and that is would offer better understanding of micro-graphs, Leibniz (micro-)graphs, and of extent to which our conjectures are now established.

(Letter "q" in Definition 1 was a typo: it should be "1" next to it on the keyboard.)

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