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Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}(\hbar^k)$: $k\leqslant 6$ unlike $k\geqslant7$

Ricardo Buring, Arthemy V. Kiselev

Abstract

The formula $\star$ mod $\bar{o}(\hbar^k)$ of Kontsevich's star-product with harmonic propagators was known in full at $\hbar^{k\leqslant 6}$ since 2018 for generic Poisson brackets, and since 2022 also at $k=7$ for affine brackets. We discover that the mechanism of associativity for the star-product up to $\bar{o}(\hbar^6)$ is different from the mechanism at order $7$ for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod $\bar{o}(\hbar^6)$, whereas at order $\hbar^7$ and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.

Associativity certificates for Kontsevich's star-product $\star$ mod $\bar{o}(\hbar^k)$: $k\leqslant 6$ unlike $k\geqslant7$

Abstract

The formula mod of Kontsevich's star-product with harmonic propagators was known in full at since 2018 for generic Poisson brackets, and since 2022 also at for affine brackets. We discover that the mechanism of associativity for the star-product up to is different from the mechanism at order for both the full star-product and the affine star-product. Namely, at lower orders the needed consequences of the Jacobi identity are immediately obtained from the associator mod , whereas at order and higher, some of the necessary differential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.
Paper Structure (4 sections, 6 theorems, 4 equations, 2 tables)

This paper contains 4 sections, 6 theorems, 4 equations, 2 tables.

Table of Contents

  1. Corrigendum: The encoding and use of Leibniz graphs
  2. Factorization of associator $\mathop{\mathrm{Assoc}}\nolimits(\star)\text{ mod }\bar{o}(\hbar^6)$ via Leibniz graphs
  3. Beyond the 0th layer of Leibniz graphs in factorization of $\mathop{\mathrm{Assoc}}\nolimits(\star_{\text{aff}})\text{ mod }\bar{o}(\hbar^7)$
  4. The 1st layer of Leibniz graphs much used in any factorization of the associator $\mathop{\mathrm{Assoc}}\nolimits(\star_{\text{aff}}^{\text{red}})\text{ mod }\bar{o}(\hbar^7)$ for the reduced affine star-product

Key Result

Theorem 1

For every Poisson bi-vector $P$ on a finite-dimensional affine real manifold $M$ and an infinitesimal deformation $\times \mapsto \times + \hbar\,\{{\cdot},{\cdot}\}_P + \bar{o}(\hbar)$ towards the respective Poisson bracket, there exists a system of weights $w(\Gamma)$, uniformly given by an integr is associative; here $\hat{G}^n_m \subset G^n_m$ is the subset of Kontsevich graphs built of wedges

Theorems & Definitions (13)

  • Theorem 1: MK97
  • Definition 1
  • Proposition 2: Corollary 4 and Conjecture ending §4 in kiev19
  • Proposition 3
  • Remark 1
  • Proposition 4: see \ref{['AppStarAssoc6']}
  • proof : Proof scheme
  • Proposition 5
  • proof : Proof scheme
  • proof : Proof scheme (for the reduced affine star-product $\star_{\textup{aff}}^{\textup{red}}$ mod $\bar{o}(\hbar^7)$)
  • ...and 3 more