The Expansion ⋆ mod ō (ℏ⁴) and Computer-Assisted Proof Schemes in the Kontsevich Deformation Quantization

Abstract

The Kontsevich deformation quantization combines Poisson dynamics, noncommutative geometry, number theory, and calculus of oriented graphs. To manage the algebra and differential calculus of series of weighted graphs, we present software modules: these allow generating the Kontsevich graphs, expanding the noncommutative ⋆-product by using a priori undetermined coefficients, and deriving linear relations between the weights of graphs. Throughout this text we illustrate the assembly of the Kontsevich ⋆-product up to order 4 in the deformation parameter $\hslash .$ Already at this stage, the ⋆-product involves hundreds of graphs; expressing all their coefficients via 149 weights of basic graphs (of which 67 weights are now known exactly), we express the remaining 82 weights in terms of only 10 parameters (more specifically, in terms of only 6 parameters modulo gauge-equivalence). Finally, we outline a scheme for computer-assisted proof of the associativity, modulo $\overline{o}({\hslash }^4),$ for the newly built ⋆-product expansion.

KEYWORDS AND PHRASES:

• Associative algebra
• noncommutative geometry
• deformation quantization
• Kontsevich graph complex
• computer-assisted proof scheme
• software module
• template library

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 05C22
• 53D55
• 68R10
• also 05C31
• 16Z05
• 53D17
• 81R60
• 81Q30

Acknowledgements

The authors are grateful to the anonymous referees for their critical comments and suggestions which helped us improve this text, to prof. S. Tabachnikov (Editor-in-Chief) for persistence and constructive criticism, and B. Pym and E. Panzer for communicating the values of ten master-parameters obtained via a different technique [34]. We thank prof. M. Gerstenhaber and M. Kontsevich for their attention to our work.

This research was supported in part by JBI RUG project 106552 (Groningen, The Netherlands) and IM JGU project 5020 (Mainz, Germany). The authors also thank the Center for Information Technology of the University of Groningen for providing access to Peregrine high performance computing cluster. A part of this research was done while the authors were visiting at the IHÉS in Bures-sur-Yvette, France and AVK was visiting at the MPIM Bonn, Germany; warm hospitality and partial financial support by these institutions are gratefully acknowledged.

This text was submitted in its original form on 20 December 2017.

Notes

1 We recall that the expansion ⋆ mod $\overline{o}({\hslash }^2)$ in [32] was gauge-equivalent to the genuine one so that the two-cycle graph at ${\hslash }^2/6$ in the first line of above formula (1) was gauged out: see Example 25 on p. 23 where we explain how this is done.

2 The indication L and R for Left $\prec$ Right, respectively, matches the indices – which the pairs of edges carry – with the ordering of indices in the coefficients of the Poisson structure contained in the arrowtail vertex. Note that exactly two edges are issued from every internal vertex in every graph in formula (1); not everywhere displayed in (1), the ordering $L\prec R$ in each term is determined from same object’s expansion (2)(2) $$\begin{array}{c}f\star g=f\times g+\hslash {\mathcal{P}}^{ij}{\partial }_{i}f{\partial }_{j}g+{\hslash }^2(\frac{1}{2}{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_k{\partial }_{i}f{\partial }_{\ell }{\partial }_{j}g+\frac{1}{3}{\partial }_{\ell }{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_k{\partial }_{i}f{\partial }_{j}g-\frac{1}{3}{\partial }_{\ell }{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_{i}f{\partial }_k{\partial }_{j}g-\frac{1}{6}{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{i}f{\partial }_{k}g)+{\hslash }^3(\frac{1}{6}{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_m{\partial }_k{\partial }_{i}f{\partial }_n{\partial }_{\ell }{\partial }_{j}g\\ -\frac{1}{6}{\partial }_m{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_n{\partial }_j{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_{i}f{\partial }_{k}g-\frac{1}{6}{\mathcal{P}}^{ij}{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{1}{6}{\partial }_m{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_n{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_k{\partial }_{i}f{\partial }_{j}g-\frac{1}{6}{\partial }_m{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_n{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_{i}f{\partial }_k{\partial }_{j}g\\ +\frac{1}{6}{\partial }_n{\partial }_{\ell }{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_m{\partial }_k{\partial }_{i}f{\partial }_{j}g+\frac{1}{6}{\partial }_n{\partial }_{\ell }{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_{i}f{\partial }_m{\partial }_k{\partial }_{j}g\\ +\frac{1}{3}{\partial }_n{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_m{\partial }_k{\partial }_{i}f{\partial }_{\ell }{\partial }_{j}g-\frac{1}{3}{\partial }_n{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{\ell }{\partial }_{j}g\\ -\frac{1}{6}{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_n{\partial }_j{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_m{\partial }_{i}f{\partial }_{k}g+\frac{1}{6}{\partial }_n{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_j{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_{i}f{\partial }_m{\partial }_{k}g\\ -\frac{1}{6}{\partial }_n{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g-\frac{1}{6}{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_n{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g)+\overline{o}({\hslash }^3).\end{array}$$(2) .

3 In fact, there are many other admissible graphs, not shown in (1), in which every internal vertex is a tail for two oriented edges, but the weights of those graphs are found to be zero.

4 The values of all these ten master-parameters have recently been claimed by Panzer and Pym [34] as a result of implementation of another technique to calculate the Kontsevich weights: see Table 4 on p. 57 in Appendix A.2. In particular, the values which we conjecture in Table 3 fully agree with the exact values suggested in [34]. Based on this external input, the expansion of the Kontsevich ⋆-product becomes (13)(13) $$\begin{array}{l}-\frac{7}{240}{\partial }_n{\partial }_k{\mathcal{P}}^{ij}{\partial }_q{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_p{\partial }_{i}f{\partial }_{m}g-\frac{1}{180}{\partial }_m{\mathcal{P}}^{ij}{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_p{\partial }_{i}f{\partial }_{k}g\\ -\frac{1}{60}{\partial }_k{\mathcal{P}}^{ij}{\partial }_q{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_p{\partial }_{i}f{\partial }_{m}g+\frac{1}{90}{\mathcal{P}}^{ij}{\partial }_m{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_p{\partial }_{i}f{\partial }_{k}g\\ -\frac{1}{60}{\partial }_k{\mathcal{P}}^{ij}{\partial }_q{\partial }_n{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_p{\partial }_{i}f{\partial }_{m}g-\frac{1}{6}{\mathcal{P}}^{ij}{\partial }_p{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ +\frac{1}{72}{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_m{\partial }_{i}f{\partial }_p{\partial }_{k}g+\frac{17}{360}{\partial }_p{\partial }_m{\mathcal{P}}^{ij}{\partial }_q{\partial }_n{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_{\ell }{\partial }_{j}g\\ +\frac{1}{24}{\partial }_p{\partial }_m{\mathcal{P}}^{ij}{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_{\ell }{\partial }_{j}g+\frac{1}{180}{\partial }_m{\mathcal{P}}^{ij}{\partial }_p{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_{\ell }{\partial }_{j}g\\ +\frac{2}{45}{\partial }_p{\partial }_n{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_q{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g-\frac{2}{45}{\partial }_p{\partial }_n{\partial }_{\ell }{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{1}{30}{\partial }_n{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_p{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g+\frac{1}{8}{\partial }_p{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_q{\partial }_n{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{1}{8}{\partial }_p{\partial }_n{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g+\frac{1}{720}{\partial }_p{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{1}{720}{\partial }_p{\partial }_n{\mathcal{P}}^{ij}{\partial }_q{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g-\frac{11}{180}{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_p{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{11}{180}{\partial }_n{\mathcal{P}}^{ij}{\partial }_p{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g+\frac{1}{36}{\partial }_p{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_q{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{1}{36}{\partial }_q{\partial }_m{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{j}g+\frac{1}{90}{\partial }_p{\partial }_{\ell }{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{1}{90}{\partial }_q{\partial }_m{\mathcal{P}}^{ij}{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{j}g-\frac{1}{180}{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_p{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{1}{180}{\partial }_q{\mathcal{P}}^{ij}{\partial }_m{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{j}g+\frac{1}{18}{\partial }_p{\mathcal{P}}^{ij}{\partial }_q{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{1}{18}{\partial }_p{\mathcal{P}}^{ij}{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g+\frac{1}{144}{\partial }_p{\mathcal{P}}^{ij}{\partial }_q{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ -\frac{1}{144}{\partial }_m{\mathcal{P}}^{ij}{\partial }_q{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{j}g-\frac{1}{90}{\partial }_p{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_m{\partial }_{j}g\\ +\frac{1}{90}{\partial }_m{\mathcal{P}}^{ij}{\partial }_q{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{j}g-\frac{1}{60}{\partial }_m{\mathcal{P}}^{ij}{\partial }_n{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{j}g\\ +\frac{1}{60}{\partial }_m{\mathcal{P}}^{ij}{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{j}g-\frac{1}{240}{\partial }_q{\partial }_n{\mathcal{P}}^{ij}{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{m}g\\ -\frac{1}{240}{\partial }_n{\mathcal{P}}^{ij}{\partial }_q{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_p{\partial }_{i}f{\partial }_m{\partial }_{k}g-\frac{13}{720}{\partial }_n{\mathcal{P}}^{ij}{\partial }_q{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{m}g\\ -\frac{13}{720}{\mathcal{P}}^{ij}{\partial }_q{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_m{\partial }_{i}f{\partial }_p{\partial }_{k}g-\frac{1}{90}{\partial }_n{\mathcal{P}}^{ij}{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{m}g\\ -\frac{1}{90}{\mathcal{P}}^{ij}{\partial }_q{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{m}g+\frac{1}{60}{\partial }_q{\mathcal{P}}^{ij}{\partial }_n{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{m}g\\ +\frac{1}{60}{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_q{\partial }_j{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_m{\partial }_{i}f{\partial }_p{\partial }_{k}g+\frac{1}{30}{\mathcal{P}}^{ij}{\partial }_q{\partial }_n{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{m}g\\ +\frac{1}{30}{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_q{\partial }_n{\partial }_j{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_m{\partial }_{i}f{\partial }_p{\partial }_{k}g-\frac{1}{90}{\mathcal{P}}^{ij}{\partial }_n{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{m}g\\ +\frac{1}{360}{\partial }_q{\mathcal{P}}^{ij}{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_k{\partial }_{i}f{\partial }_p{\partial }_{m}g+\frac{13}{90}{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_n{\partial }_j{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_m{\partial }_{i}f{\partial }_p{\partial }_{k}g\\ +\frac{13}{180}{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_n{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_q{\mathcal{P}}^{mn}{\mathcal{P}}^{pq}{\partial }_m{\partial }_{i}f{\partial }_p{\partial }_{k}g+\frac{13}{180}{\partial }_q{\partial }_{\ell }{\mathcal{P}}^{ij}{\partial }_j{\mathcal{P}}^{k\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_m{\partial }_{i}f{\partial }_p{\partial }_{k}g\\ +\frac{1}{72}{\partial }_q{\mathcal{P}}^{ij}{\partial }_j{\mathcal{P}}^{k\ell }{\partial }_{\ell }{\mathcal{P}}^{mn}{\partial }_n{\mathcal{P}}^{pq}{\partial }_m{\partial }_{i}f{\partial }_p{\partial }_{k}g)+\overline{o}({\hslash }^4).\end{array}$$(13) on pp. 57–61.

5 From Theorem 12 we also assert that the associativity of Kontsevich’s ⋆-product does not carry on but it can leak at orders ${\hslash }^{⩾4}$ of the deformation parameter, should one enlarge the construction of ⋆ to an affine bundle set-up of Nⁿ-valued fields over a given affine manifold M^m and of variational Poisson brackets ${\{·,·\}}_{\mathcal{P}}$ for local functionals $F,G,H:C^{\infty }(M^m\to N^n)\to \mathbb{k},$ see [23–26] and [27].

6 In the variational set-up of Poisson field models, the affine manifold Nⁿ is realized as fiber in an affine bundle π over another affine manifold M^m equipped with a volume element. The variational Poisson brackets ${\{·,·\}}_{\mathcal{P}}$ are then defined for integral functionals that take sections of such bundle π to numbers. The encoding of variational polydifferential operators by the Kontsevich graphs now reads as follows. Decorated by an index i, every edge denotes the variation with respect to the ith coordinate along the fiber. By construction, the variations act by first differentiating their argument with respect to the fiber variables (or their derivatives along the base M^m); secondly, the integrations by parts over the underlying space M^m are performed. Whenever two or more arrows arrive at a graph vertex, its content is first differentiated the corresponding number of times with respect to the jet fiber variables in $J^{\infty }(\pi )$ and only then it can be differentiated with respect to local coordinates on the base manifold M^m. The assumption that both the manifolds M^m and Nⁿ be affine makes the construction coordinate-free, see [23, 27] and [24, 26].

7 The current version of the software does not allow specification of an arbitrary Poisson structure at runtime (e.g. input as a matrix of functions); however, in the source file util/poison_structure_examples.hpp the list of Poisson structures (as matrices) can be extended to one’s heart’s desire.

8 We omit the factor $1/k!$ that was written in [32], to make the weight multiplicative (see Lemma 5).

9 The fact that the differential order of ⋆ is positive with respect to either of its arguments should be expected, in view of the required property of the ⋆-product to be unital: $f\star 1=f=1\star f.$

10 A convenient approach to calculation of Kontsevich weights (5)(5) $$w(\Gamma )=\frac{1}{{(2\pi )}^{2k}}{\int }_{C_k(\mathbb{H})}\underset{j=1}{\overset{k}{\bigwedge }}\mathrm{d}\phi (p_j,p_{\text{Left}(j)})\wedge \mathrm{d}\phi (p_j,p_{\text{Right}(j)}),$$(5) at order 3 by using direct integration (and for that, using methods of complex analysis such as the Cauchy residue theorem) was developed in [14], see Appendix A.1 on p. 53 below. However, we note that most successful at k = 3, this method is no longer effective for all graphs at $k⩾4.$ More progress is badly needed to allow $k⩾5.$

11 Numerical approximations of two of these weights are listed in Table 3 in Appendix A.1.

12 Should a graph $\Gamma \in {\tilde{G}}_{2,\ell +1}$ be composite so that its Kontsevich weight is factorized using formula (7)(7) $$w({\Gamma }_i\overline{\times }{\Gamma }_j)=w({\Gamma }_i)\times w({\Gamma }_j),$$(7) , the resulting nonlinearity with respect to the weights would actually involve only the graphs with at most ℓ internal vertices.

13 From the factorization of associator for ⋆ via differential consequences of the Jacobi identity for a Poisson structure $\mathcal{P},$ which will be revealed in Section 3.2 below, it will be seen in hindsight that the construction of linear relations between the graph weights is overall insensitive to a choice of local coordinates in a chart within a given Poisson manifold. Indeed, the factorization will have been achieved simultaneously for all Poisson structures on all the manifolds at once, irrespective of any local coordinates.

14 If the rank of the resulting linear algebraic system is equal to the number of unknowns – and if all the coefficients coming from lower orders $⩽\ell$ within the ⋆-product expansion with respect to $\hslash$ are also rational – then all the solution components are rational numbers as well, cf. [17].

15 The values of ten master-parameters have been suggested by Pym and Panzer [34], see Table 4 on p. 57 in Appendix A.2 below. Their prediction completely agrees with our numeric data.

16 Sample outputs of specified type are contained in Table 9 in Appendix D.

17 Let us recall that the property of a parameter in a family of star-products to be removable by some gauge transformation is not the same as setting such parameter to zero (or any other value). Indeed, other graph coefficients, not depending on the parameter at hand, might get modified by that gauge transformation. However – and similarly to the removal of the loop graph at ${\hslash }^2$ in the Kontsevich ⋆-product (see Examples 24 and 25) – the trivialization of four parameters at no extra cost is the case which Theorem 14 states.

18 However, this does not imply that each individual graph in it is a skew-symmetric bi-derivation. Rather, each graph which is a bi-derivation can be skew-symmetrized, which yields either the original graph or the sum of two graphs which are mirror-reflections of each other. It is clear that this is what Willwacher intended, e.g. because the mirror-reflection of his graph D is not drawn.

19 The graphical calculation of $[a_1,b_3]$ in [38] contains errors, e.g. the first graph has a vertex with three outgoing edges and the term with coefficient β has arrows in the wrong direction.

20 The solution reported here differs from Willwacher’s, not in sign (which is left ambiguous in [38]) but in proportion: he claims $A=\pm 2\beta \text{and} B=\pm 2(\alpha +\beta ).$

21 Note that this problem is different from the computation of obstructions to Kontsevich’s Formality [31, 32]. Specifically, in “Formality Conjecture” [31], Kontsevich reports the absence of obstructions to Formality up to $n⩽6.$ Formality is now a theorem: Kontsevich’s ⋆-product exists at all orders.

22 Called a Gauss map by M. Polyak [36].

23 G. Dito used the residue method for one graph [16] at k = 2, and remarked that that it becomes unpractical for $k⩾3.$