Associativity certiﬁcates for Kontsevich’s star-product ⋆ mod ¯ o ( ~ k ) : k 6 6 unlike k > 7

. The formula ⋆ mod ¯ o ( ~ k ) of Kontsevich’s star-product with harmonic propagators was known in full at ~ k 6 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 ¯ o ( ~ 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 ¯ o ( ~ 6 ), whereas at order ~ 7 and higher, some of the necessary diﬀerential consequences are reached from the Kontsevich graphs in the associator in strictly more than one step.

Introduction.Deformation quantization extends the commutative associative unital product × in the algebra A := C ∞ (M ) of smooth functions on a manifold M to an associative product ⋆ on the space of formal power series A[[ ]]; the skew-symmetric part of the bi-derivation in the leading deformation term at 1 in ⋆ is readily seen to be a Poisson bracket {•, •} P on the algebra A.
Theorem 1 ([9]).For every Poisson bi -vector P on a finite-dimensional affine real manifold M and an infinitesimal deformation × → × + {•, •} P + ō( ) towards the respective Poisson bracket, there exists a system of weights w(Γ), uniformly given by an integral formula, such that the R[[ ]]-bilinear star -product, is associative; here Ĝn m ⊂ G n m is the subset of Kontsevich graphs built of wedges (with each aerial vertex having exactly two outgoing edges) in the set G n m of all Formality graphs with m ground vertices and n aerial vertices. 2 Convention.To the edges L and R of the wedge graph Λ we ascribe independent indices i and j respectively, and with this graph Λ we associate the operator Λ(P )(f, g) = P ij • ∂ i f • ∂ j g which is the Poisson bracket.More generally, for a Formality graph Γ ∈ G n m we ascribe independent indices to all edges; the multi-linear multi-differential operator Γ(J 0 , . .., J n−1 )(f 0 , . .., f m−1 ) associated with the graph Γ is then a sum over those indices, with each summand being a product over the (differentiated) contents of vertices, the ground vertex k containing the argument f k of the operator, and the aerial vertex m + ℓ containing the component of the multi-vector field J ℓ specified by indices of the ordered outgoing edges; here the content of each vertex is differentiated with respect to the local affine coordinates specified by the incoming edges (if any).
Elementary properties of the graph weights w(Γ) are summarized in [4, Lemmas 1-5 and Remark 8]; the Shoikhet-Felder-Willwacher cyclic weight relations from [7, App.E] are recalled in [4, Proposition 7].(These relations are not enough to determine the weights completely.)Another ample source of relations between weights is the associativity of ⋆; this can be exploited as in [4,.
The Kontsevich star-product with harmonic propagators (as in [9]) was known at orders 1 , . .., 4 in 2017 from [4].The weights w(Γ) of all Kontsevich graphs at 5 and 6 in ⋆ were obtained by the end of 2018 in [1]; the Riemann zeta value ζ(3) 2 /π 6 starts appearing in the weights w(Γ) for some Kontsevich graphs Γ ∈ Ĝn 2 from n = 6 onwards.In [3] from 2022 we found the weights w(Γ) of all Kontsevich graphs Γ ∈ Ĝn=7 m=2 with in-degree 1 of aerial vertices, that is the weights of the graphs which are relevant for the case of affine Poisson brackets {•, •} P and thus, as n 7, for the seventh order expansion ⋆ aff mod ō( 7) of the affine star-product.We then establish in [3] that the entire coefficient of ζ(3) 2 /π 6 , which does show up in ⋆ aff at 6 and 7 , equals a linear combination of differential consequences of the Jacobi identity (for affine Poisson brackets {•, •} P ) because the respective linear combination of Kontsevich graphs near ζ(3) 2 /π 6 assimilates into a linear combination of Leibniz graphs on m = 2 ground vertices and ñ = n − 1 aerial vertices.Definition 1.A Leibniz graph is a Formality graph containing at least one aerial vertex with three outgoing edges, such that those three edges have three distinct targets, and none of those three edges are tadpoles.The other aerial vertices (if any) have two outgoing edges, and the ground vertices are as usual.These graphs will be evaluated with the Jacobiator 1 2 [[P, P ]] of the Poisson structure P in the vertex with three outgoing edges, hence representing a differential operator that is identically zero whenever P is Poisson. 3  We recall from the breakthrough paper [9] the guaranteed existence of a factorization of the star-product associator, Assoc(⋆ Leibniz graphs (here M is an affine real manifold of finite dimension d).
Proposition 2 (Corollary 4 and Conjecture ending §4 in [5]).The operator ♦ that solves the factorization problem is given by where F k is the k-ary component of the Formality L ∞ -morphism, and where we claim that the constants c n are equal to n/6.
3 Homogeneous components of differential consequences of the Jacobi identity (now realized by using Leibniz graphs) vanish separately thanks to the following lemma: A tri-differential operator |I|,|J |,|K| 0 c IJ K ∂I ⊗∂J ⊗∂K vanishes identically iff all its coefficients vanish: c IJ K = 0 for every triple (I, J, K) of multi-indices; here for a multi-index L = (α1, . . ., αn).Moreover, the sums |I|=i,|J |=j,|K|=k c IJ K ∂I ⊗ ∂J ⊗ ∂K are then zero for all homogeneity orders (i, j, k).
The number of graphs which actually show up at order k in the left-and right-hand sides of factorization problem (2) is reported in Table 1.For example, in [5, §5] we inspect many graphs Table 1.The number of graphs in either side of the associator's factorization.LHS: # Kontsevich graphs, 3 (Jac) 39 740 12464 290305 ?coeff = 0 RHS: # Leibniz graphs, 1 (Jac) 13 241 4609 ??coeff = 0 of different orders, and establish the equality of sums of Kontsevich graphs in the associator and sums of Leibniz graphs -in the factorizing operator -after they are expanded into the Kontsevich graphs.
In the right-hand side of the associator for ⋆, there are Leibniz graphs: at k 2 , such Leibniz graphs have 3 sinks, k − 1 aerial vertices (of which one vertex, the Jacobiator, has three outgoing edges, and the remaining k−2 vertices (if any) each have two outgoing edges), and, by the above, 3 + (k − 2) • 2 = 2k − 1 edges; tadpoles are not allowed, graphs with multiple edges are discarded.For each k = 2, 3, 4, 5 we generate all such admissible Leibniz graphs (those can be zero graphs with a parity-reversing automorphism, cf.[4]); the respective number of such Leibniz graphs at each order k is in Table 2.At every order k, we generate the entire set of the cyclic weight relations (cf.[7]); every cyclic weight relation is a linear constraint upon the weights of several Leibniz graphs; all those weights are given by the Kontsevich integral formula from [9].The number of these linear relations and the (co)rank of this linear algebraic system follow in Table 2. Banks-Panzer-Pym in [1] do not list the weights of Leibniz graphs (as in Table 2 above), for these graphs do not show up in the ⋆-product itself where the vertex-edge valency is different for the Kontsevich graphs.We use the software kontsevint by Panzer (cf.[1]) to calculate the Kontsevich weights of all the Leibniz graphs which are admissible for the right-hand side of starproduct's associator.(Some weights can -and actually do -vanish because either the graph is zero, or the weight integrand is identically zero, or the weight formula integrates to a zero number.)The count of admissible Leibniz graphs with nonzero weights is in the fourth line of Table 2: the corresponding line in Table 1 is reproduced verbatim.(The Leibniz graphs with zero weights do nominally show up in the cyclic weight relations for Leibniz graphs, but in fact stay invisible in the formulas.)Proposition 3. The numeric values of the Kontsevich weights w(L) of Leibniz graphs with k aerial vertices on 3 sinks, which we calculated using Panzer's software kontsevint, do satisfy 4 the system of linear algebraic equations given by the cyclic weight relations for k = 1,2,3,4.
From now on in this paper, we study the associativity of Kontsevich's ⋆-product from a different perspective, because at k 6 the number of Leibniz graphs -to realize the associator of ⋆ at k -is too big for the weight w(L) to be computed for every such graph L (either before or after the system of cyclic weight relations is formed at k 6).
Let us recall that the associator naturally splits into homogeneous orders with respect to the three sinks; so does the set of relevant Leibniz graphs.(The cyclic weight relations correlate the weights w(L) of Leibniz graphs for different tri-differential orders in the associator.)We say that finding the values of Leibniz graph coefficients dictated by the Kontsevich integral formula yields the solution of the strong factorization problem for the associator of the star-product.Yet, to certify the associativity it suffices to find a realization of each tri-differential component as a weighted sum of Leibniz graphs regardless of any such realizations for other tri-differential orders, that is without imposing the known constraints upon the Leibniz graph weights.One big problem thus splits into many small subproblems, which are solved independently.The result is a solution to the weak factorization problem, which we report in this paper.Let us remember that the found values of Leibniz graph coefficients are then not necessarily equal to the Kontsevich integrals w(L) (times the rational factors which count the multiplicities).

The layers of Leibniz graphs: contract and expand edges in the Kontsevich graphs
The idea which we start with is to not consider those Leibniz graphs whose expansion -of Jacobiators into sums of Kontsevich's graphs, and of all the derivations acting on the Jacobiators by the Leibniz rule -does not reproduce any of the Kontsevich graphs in the associator itself.
By definition, the 0th layer of Leibniz graphs is obtained -for a given linear combination of Kontsevich graphs -by contracting one internal edge in every Kontsevich graph in all possible ways.By expanding the 0th layer Leibniz graphs back to Kontsevich's graphs, one reproduces their original set, but new Kontsevich's graphs can be obtained.The coefficients of these new Kontsevich graphs, not initially present in the given linear combination, either cancel out or do not all vanish identically.If not, then by repeating for those new Kontsevich graphs the above contraction-expansion procedure, one reproduces (part of) the 0th layer but also produces the 1st layer of new Leibniz graphs and from them, possibly a still larger set of Kontsevich's graphs (whose number is finite for a given number of aerial vertices).The construction of layers is iterated until saturation (e.g., see Table 1 in [6]).Even if the saturation requires layers to achieve, the resulting number of Leibniz graphs at hand is much smaller than the number of Leibniz graphs on equally many vertices. 5Moreover, in this paper running the algorithm until saturation is not obligatory; the first layer of Leibniz graphs is already enough.Remark 1.To avoid repetitions, we use the normal form of Leibniz graphs; it refers to the encoding of directed graphs in nauty within SageMath (see Appendix A for further discussion).
Proposition 4 (see Appendix B).The Kontsevich ⋆-product with the harmonic graph weights, known up to ō( 6 ) from Banks-Panzer-Pym [1], is associative modulo ō( 6): every tri-differential homogeneous component of the associator admits some realization by Leibniz graphs; to find such solution, the 0th layer of Leibniz graphs suffices for each of the tri-differential orders.
There are 105 homogeneous tri-differential order components at 6 in the associator Assoc(⋆) mod ō( 6).We import the harmonic graph weights at 5 and 6 in ⋆ mod ō( 6) from the kontsevint repository of E. Panzer (Oxford).At order 6 , the weights of Kontsevich graphs in ⋆ are expressed as Q-linear combinations of 1 and ζ(3) 2 /π 6 .In consequence, the coefficients of Kontsevich graphs in the associator at order 6 are also Q-linear combinations of that kind.Every tri-differential homogeneous component of the associator is thus split into the rationaland ζ(3) 2 /π 6 -slice: either of the slices is a linear combination of Kontsevich's graphs with rational coefficients.The rational slices are met in all of the 105 tri-differential orders; we detect that in every such slice the Kontsevich graphs provide the 0th layer of Leibniz graphs which suffices to realize that sum of Kontsevich graphs as a linear combination of these Leibniz graphs.The ζ(3) 2 /π 6 -slice is nontrivial in 28 tri-differential orders of the associator at 6 ; here the Formality mechanism works as follows.For all but 6 tri-differential orders, the Kontsevich graphs from the linear combination near ζ(3) 2 /π 6 suffice to provide the set of 0th layer Leibniz graphs which are enough for a solution of the factorization problem.The tri-differential orders {(1, 1, 3), (3, 1, 1), (2, 1, 2), (1, 2, 2), (2, 2, 1), (1, 3, 1)} are special: for a solution to appear, the sets of Kontsevich graphs from the rational and ζ(3) 2 /π 6 -slices within that tri-differential order must be merged and then the union set is enough to provide a factorization of the ζ(3) 2 /π 6 -slice by the 0th layer of Leibniz graphs.The corresponding computations are presented in Appendix B. We conclude that at order 6 for the full Kontsevich star-product, Kontsevich's Formality mechanism works as expected.
The seventh order expansion of the Kontsevich star-product for arbitrary Poisson brackets is unknown (see Table 1 in [3] for the count of 2,814,225 Kontsevich graphs at 7 ).So far, in [3] we have obtained the affine star-product ⋆ aff mod ō( 7) under the assumption that the coefficients of the Poisson bracket are affine functions (e.g., linear on the affine base manifold).For example such are the Kirillov-Kostant Poisson brackets on the duals g * of Lie algebras.We discover that the associativity mechanism for the affine star-product at 7 differs from the mechanism which worked at lower orders of expansion in for the full star-product ⋆ mod ō( 6 ).Moreover, the new mechanism of associativity for ⋆ aff mod ō( 7) forces a new mechanism of associativity for the full star-product ⋆ mod ō( 7 ) starting at order seven.The difference is the necessity of Leibniz graph layers beyond the 0th layer, which itself was enough at lower orders to build a solution of the weak problem for associator's factorization via the Jacobi identity.

Proof scheme.
With not yet specified undetermined coefficients of Kontsevich graphs at 7 in the affine star-product ⋆ aff mod ō( 7), its associator's part at 7 expands to 203 tri-differential order components.As soon as the weights of all the new Kontsevich graphs on n = 7 aerial vertices are fixed (see [3]), the number of tri-differential orders (d 0 , d 1 , d 2 ) actually showing up at 7 in the associator A for ⋆ aff mod ō( 7) drops to 161.For all but four tri-differential order components A d 0 d 1 d 2 in the associator A, the 0th layer of Leibniz graphs, which are obtained by contracting6 one edge between aerial vertices in the Kontsevich graphs of every such tridifferential component A d 0 d 1 d 2 , is enough to provide a solution for the factorization problem, ), expressing that component by using differential consequences of the Jacobi identity (encoded by Leibniz graphs).We detect that for the tri-differential orders (d 0 , d 1 , d 2 ) in the set {(3, 3, 2), (2,3,3), (3,2,3), (2, 4, 2)}, the Leibniz graphs from the 0th layer are not enough to reach a solution ♦ d 0 d 1 d 2 ; still a solution ♦ d 0 d 1 d 2 appears in each of these four exceptional cases after we add the Leibniz graphs from the 1st layer (i.e.those graphs obtained by contraction of edges in the Kontsevich graph expansion of Leibniz graphs from the previous layer).(There are 2294 Kontsevich graphs in A 2,3,3 , producing 3584 Leibniz graphs in the respective 0th layer immediately after the edge contractions; the component A 3,3,2 contains equally many Kontsevich graphs and the same number of Leibniz graphs in the 0th layer; the largest component A 3,2,3 contains 2331 Kontsevich graphs and gives 3603 Leibniz graphs in the 0th layer; and finally A 2,4,2 contains 1246 Kontsevich graphs and produces 2041 Leibniz graphs in the 0th layer.)In [2, Part I, §3.7.8] we generate a Leibniz graph factorization of all tridifferential components in the associator for ⋆ aff mod ō( 7) and we provide the data files of Leibniz graphs and their coefficients: see Appendix C on p. 11 below.
The reduced affine star-product ⋆ red aff mod ō( 7) is obtained from the affine star-product ⋆ aff mod ō( 7) by realizing the coefficient of ζ(3) 2 /π 6 as the Kontsevich graph expansion of a linear combination of Leibniz graphs with rational coefficients and, now that this combination does not contribute to either the star-product or its associator when restricted to any affine Poisson structure, by discarding this part of ⋆ aff mod ō( 7) proportional to ζ(3) 2 /π 6 .The same applies to many terms in the rational part of ⋆ aff mod ō( 7) which also assimilate to Leibniz graphs, see [3].In the reduced affine star-product ⋆ red aff mod ō( 7) there remain only 326 nonzero rational coefficients of Kontsevich graphs at k for k = 0, . . ., 7 (in contrast with 1423 nonzero (ir)rational coefficients at orders up to 7 in ⋆ aff mod ō( 7)).
Proof scheme (for the reduced affine star -product ⋆ red aff mod ō( 7)).The associator for ⋆ red aff contains 95 tri-differential orders at 6 and 161 tri-differential orders at 7 .We see that the associator Assoc(⋆ red aff ) mod ō( 7) becomes much smaller than Assoc(⋆ aff ) mod ō( 7), now containing only 29371 Kontsevich graphs instead of 59905.But the work of the associativity mechanism for ⋆ red aff requires the use of the 1st layer of Leibniz graphs much more often than it already was for the affine star-product ⋆ aff mod ō( 7 ) before the reduction.Now, at orders 7 in , new Leibniz graphs from the layer(s) beyond the 0th are indispensable for the factorization of 114 out of 336 homogeneous tri-differential order components of the associator, see Appendix D where we list all these exceptional orders.
Remark 2. We observe that the number ζ(3) 2 /π 6 , not showing up in any restriction of the affine star-product f ⋆ aff g mod ō( 7) to an affine Poisson structure and any arguments f, g ∈ A[[ ]], acts in effect as a placeholder of the Kontsevich graphs which, by contributing to the associator and then creating the Leibniz graphs by edge contraction, provide almost all of the Leibniz graphs needed for a factorization of the associator for ⋆ aff mod ō( 7 ) via the Jacobi identity.When the ζ(3) 2 /π 6 -part of ⋆ aff mod ō( 7 ) itself is eliminated by using the Jacobi identity for affine Poisson structures, the remaining ⋆ red aff mod ō( 7) and its associator rely heavily on the use of higher layer(s) of Leibniz graphs for a factorization solution to be achieved.Proposition 6.The 0th layer of Leibniz graphs is not enough to provide a factorization of the associator for the (either affine or full) Kontsevich star-product at order 7 , whereas, according to Proposition 4 above, the 0th layer of Leibniz graphs was enough at order 6 to factor the associator for the full star-product.

Conclusion.
The above iterative scheme gives us a solution to the weak factorization problem: each tri-differential component A d 0 d 1 d 2 is factorized independently from the others, so that the coefficients of the Leibniz graphs are not yet constrained overall -over different componentsby the Shoikhet-Felder-Willwacher cyclic weight relations and other relations.In particular, the above scheme does not guarantee that the found coefficients of Leibniz graphs are equal (up to the multiplicity and constants c n ) to the genuine Kontsevich weights of those Leibniz graphs.The above scheme provides the necessary minimum number of layers of Leibniz graphs, whereas the calculation of Kontsevich's genuine weights of Leibniz graphs is sufficient to build a solution (the canonical one) for the associator factorization problem.We remember that there exist identities, i.e. sums of Leibniz graphs which expand to zero sums of Kontsevich graphs (here, in the associator); such identities could make unnecessary the use of a Leibniz graph with nonzero genuine weight from a (high number, in particular the last) layer.Hypothetically it might be that any solution needs the 0th and 1st layers, hence they are "necessary", but Kontsevich's canonical solution stretches over the 0th, 1st and 2nd layers, thus they are "sufficient".The above scheme does not guarantee that the genuine Kontsevich weight of a Leibniz graph in the known associator's factorization at order 7 will definitely be equal (up to the multiplicity and constants c n ) to this Leibniz graph's coefficient in a solution found using the last necessary layer.We conclude that the 1st layer of Leibniz graphs becomes necessary at k 7 for any factorization of the associator (with harmonic propagators for the Kontsevich graph weights in ⋆ in its authentic gauge from [9]).Such use of the 1st and higher layers could start earlier, at orders k < 7, for the factorization problem's canonical solution given by the Kontsevich weights w(L) of Leibniz graphs.
Open problem 1.Over how many layers do the canonical Kontsevich solutions of associator's factorization problem stretch ?In particular what is the factorization guaranteed by the Formality theorem for orders 3, 4, 5, 6, 7, which we have considered so far using solutions of the weak factorization problem ?Czech Republic.A part of this research was done while the authors were visiting at the IH ÉS in Bures-sur-Yvette, France.R.B. thanks E. Panzer for granting access to kontsevint software; A.K. thanks G. Dito and M. Kontsevich for helpful discussions. 8his paper is extracted in part from the text [3] by the same authors; the authors thank colleagues and anonymous experts who acted as referees of this work.
Appendix A. Corrigendum: The encoding and use of Leibniz graphs Normal forms for Leibniz graphs with one Jacobiator were introduced in [6, Definition 5]: the idea was to re-use the normal form for Kontsevich graphs.Namely, the Jacobiator was expanded into the sum of three Kontsevich graphs (built of wedges), all the incoming arrows (to the top of the tripod) were formally directed to the top of the lower wedge in each Kontsevich graph, and then we found the normal forms of the resulting three Kontsevich graphs, while also remembering where the internal edge of the Jacobiator is located in those normal forms.The normal form of the Leibniz graph then was: choose the minimal (w.r.t.base-(m + n) numbers) Kontsevich graph encoding, supplemented with the indication of the internal Jacobiator edge.(Besides, it is necessary to pay attention to whether the internal Jacobiator edge is labeled Left or Right, in order to expand the Leibniz graph with the correct sign ±1.) This definition, i.e. the pair (Kontsevich graph, marked edge) is unfortunately not a true normal form of the Leibniz graph.Namely, it can happen that the resulting Kontsevich graph has an automorphism that maps the marked edge elsewhere, to a new place in the graph.Consequently, two isomorphic Leibniz graphs could have different "normal forms" (differing only by the marking where the internal Jacobiator edge is).This led to a visible pathology, namely to redundant parameters in the systems of equations: one and the same Leibniz graph, encoded differently, acquired two unrelated coefficients.Fortunately, the effect disappeared when Leibniz graphs were expanded to sums of Kontsevich graphs and similar terms were collected.
In consequence, that normal form was abandoned in favor of inambiguous (and fast) description of Leibniz graphs by using the nauty software [10].

Table 2 .
The count of admissible Leibniz graphs in the associator for Kontsevich's ⋆.