The tower of Kontsevich deformations for Nambu-Poisson structures on $\mathbb{R}^{d}$: dimension-specific micro-graph calculus

In Kontsevich's graph calculus, internal vertices of directed graphs are inhabited by multi-vectors, e.g., Poisson bi-vectors; the Nambu-determinant Poisson brackets are differential-polynomial in the Casimir(s) and density $\varrho$ times Levi-Civita symbol. We resolve the old vertices into subgraphs such that every new internal vertex contains one Casimir or one Levi-Civita symbol${}\times\varrho$. Using this micro-graph calculus, we show that Kontsevich's tetrahedral $\gamma_3$-flow on the space of Nambu-determinant Poisson brackets over $\mathbb{R}^3$ is a Poisson coboundary: we realize the trivializing vector field $\smash{\vec{X}}$ over $\smash{\mathbb{R}^3}$ using micro-graphs. This $\smash{\vec{X}}$ projects to the known trivializing vector field for the $\gamma_3$-flow over $\smash{\mathbb{R}^2}$.


2.
In dimension d = 2, Poisson-coboundary equation ( 3) is valid as an equality of bi-vectors with differential-polynomial coefficients (w.r.t.ϱ in bi-vector P), but its left-hand side cannot be expressed as a linear combination of zero Kontsevich graphs and Leibniz graphs (that is, differential consequences of the Jacobi identity, see [1]).
Proof.Equality (3) in Part 1 of Proposition 1 is verified in d = 2 by straightforward calculation with differential-polynomial coefficient (multi)vectors.
To explore why equality (3) is valid in d = 2, let us inspect whether it is the standard, working in all dimensions d ⩾ 2, Leibniz-graph mechanism that would ensure the vanishing, = 0 for any ϱ(x, y), by force of the Jacobi identity realized by Kontsevich graphs.(We claim that it is not only this mechanism which does the job.) For this, we list all connected directed graphs built over two sinks of in-degree 1 from one trident and two wedges, without co-directed double or triple edges, and with none or one tadpole.(The Jacobiator vertex with three outgoing edges will be expanded to the linear combination of three Kontsevich's graphs, see [1].)There are three admissible Leibniz graphs without tadpoles, (There remain four connected Leibniz graphs with two tadpoles but they are irrelevant for our present attempt to balance Eq.(3) using the topologies of Kontsevich graph expansions in the right-hand side.)When the wedge graph P acts on the 'sunflower' 1-vector graph X γ 3 , three graph topologies with one tadpole are produced, among others (the tadpole is absent from all the other topologies that appear in [[P, X In Kontsevich's nonzero graph B, vertex 3 has in-degree four (tadpole counted).But no expansion -of Leibniz graph from the above list of twelve -into Kontsevich graphs results in a graph with vertex of in-degree four.Independently, nonzero bi-vector graph C -with tadpole on vertex 2 at distance two from either sink and with double edge 3 ⇄ 4 -is not obtained in the Kontsevich graph expansion of any of the relevant bi-vector Leibniz graphs 4-7 from the above list.Thirdly, only the Leibniz graph 8 reproduces nonzero graph A, but the Kontsevich graph expansion of 8 contains five more graphs (with a tadpole at distance one from the sink), none of which appears in the linear combination ] of Kontsevich graphs.Therefore, we have that for any values of coefficients ∈ near Leibniz graphs in the right-hand side.The proof is complete.

The fact of Poisson trivialization of the tetrahedral-graph flow Ṗ
, impossible in this or any higher dimension d ⩾ 2 through the mechanism of vanishing by force of the Jacobi identity as the only obstruction, implies the existence of other analytic mechanism(s); those can work in combination with the former.
Remark 1.When all the sums over repeated indices, each running from 1 to the finite dimension d = 2, are expanded in the left-hand side of the identity aff , the arithmetic cancellation mechanism works several times: the l.-h.s.splits into sums which cancel out separately from each other.For instance, when the Poisson differential [[P(ϱ), •]] acts on ⃗ X and this 1-vector gets into a sink of the wedge graph of bi-vector, thus forming graphs A and B in particular (see Eq. ( 4)), the wedge top coefficient ϱ has no derivative(s) falling on it.Yet no such terms occur at d = 2 in the graphs of Q γ 3 with strictly positive in-degrees of internal vertices.Hence the two parts -with(out) zero-order derivative of ϱ -of the differential-polynomial coefficient in the identity's l.-h.s.vanish simultaneously.

Remark 2 (on G L(∞)-invariants parent to G L(d)-invariants). Kontsevich's construction of tensors from
Poisson bi-vectors by using graphs is well-behaved under affine coordinate changes on the underlying manifold, thanks to contraction of upper indices of Poisson tensors (P i j ) in the arrowtail vertices and of lower indices in derivations at the edge arrowheads.The Jacobians of coordinate changes then belong to the general linear group, while the affine shifts are not felt at all by the index contractions.But, uniform over the dimensions d of affine Poisson manifolds, the graph technique builds invariants of linear representations for G L(∞). Linearly independent as graph expressions, such invariants can be linearly tied after projection to a given finite dimension d, where they become tensor-valued G L(d)-invariants. 7o conclude this section, we note that the original graph language of [5] for construction of tensor-valued invariants of G L(∞) is no longer enough for the study of Poisson (non)triviality for graph-cocycle flows on spaces of Poisson brackets over affine manifolds of given dimension d.To manage the bound d < ∞, one must either take a quotient over the (unknown) new linear relations between Kontsevich's graphs, or work ab initio with G L(d)-invariants.
Our next finding in [4, Theorem 8] is that for the graph cocycle γ 3 and d = 3, the action of vector field ⃗ X (which trivializes the The usual view of Nambu-Poisson bracket (5) on d is that the Jacobian determinant is multiplied by an arbitrary factor ϱ(x 1 , . . ., x d ) which behaves appropriately under coordinate changes x ⇄ x ′ .Our viewpoint is that ϱ(x ) ∂ x 1 ∧. ..∧∂ x d is a top-degree multi-vector on d for any d ⩾ 2, so that for all dimensions greater than two, the Nambu-determinant Poisson bi-vector is derived: The Casimirs a ℓ Poisson-commute with any f ∈ C 1 ( d ); the symplectic leaves are intersections of level sets a ℓ = const(ℓ) ∈ , so that for any d ⩾ 2 these leaves are at most two-dimensional.
on 3 ) upon P(ϱ, [a]) factors through the initially known -from γ 3 -velocities of a and ϱ: having solved (1) for ⃗ X , we then verified that ȧ = [[a, ⃗ X ]] and ρ . By using this factorization -i.e. the lifting of the sought vector field's action on the elements of P(ϱ, [a]) -the other way round, we create an economical scheme to inspect the existence of trivializing vector field ⃗ X for larger problems (i.e. for bigger graph cocycles or higher dimension d ⩾ 3).When this shortcut works, so that ⃗ X is found, it saves much effort.Otherwise, to establish the (non)existence of ⃗ X one deals with a larger PDE, namely Eq. ( 1). ) on d are realized using micrographs, namely by resolving ϱ(x )•ϵ i 1 ,...,i d in one vertex against d −2 vertices with the Casimirs a 1 , . . ., a d−2 .The out-degree of vertex with ϱ(x ) • ϵ ⃗ ı equals d; the in-degree of each vertex with a Casimir equals 1 and its out-degree is zero: the Casimir vertices are terminal (not to be confused with the two sinks, of in-degree 1, for the Poisson bracket arguments).The ordered d-tuple of edges is decorated with summation indices: for the Levi-Civita symbol ϵ i 1 ,...,i d in their common arrowtail vertex, the range is 1 Example 3. In d = 3, the micro-graph expansion of Q γ 3 (P) for P(ϱ, [a]) over 3 consists of directed graphs on 2 sinks for f and g, on four terminal vertices -for copies of the Casimir awithout outgoing arrows, and on four vertices for ϱ • ϵ i jk with three ordered outgoing edges.In every micro-graph in bi-vector Q γ 3 (P) there are 12 edges, with exactly one going towards f and one to g in the sinks.To have a solution ⃗ X of the equation ), we thus need micro-graphs on one sink, three terminal vertices with a, and three trident vertices for ϱ • ϵ i jk .Of the nine edges in each micro-graph, exactly one goes to the sink, so that ⃗ X is a 1-vector.
Example 4. In d = 3, two Kontsevich's graphs of the 'sunflower' 1-vector (which trivializes the restriction of tetrahedral-graph flow (2) to the space of bi-vectors on an affine twofold) expand to a linear combination of 42 one-vector micro-graphs over one sink, three trident vertices, three terminal vertices, and 3 × 3 = 9 edges (one into the sink).A tadpole is met in 10 = 3 + 3 + 4 such micro-graphs, and the other 32 = 8 × 4 have none.
Remark 5.At the level of micro-graphs, the solution X γ 3 contains a tadpole, i.e. a 1-cycle, in the last graph.In terms of differential polynomials this means the presence of a deriative ∂ x i acting on the coefficient ϱ(x ) near the Levi-Civita symbol ϵ i jk containing the index i of the base coordinate x i in that derivative; that is, the last term in the vector field ⃗ X contains Proposition 3. Without tadpoles in the micro-graph ansatz X γ 3 , there is no solution ⃗ X to the trivialization problem Q Proof.The micro-graph expansion of the 'sunflower' graph is not enough in d = 3 because, in particular, the before-last micro-graph in X γ 3 3 , namely (−8) • [(4, 0),(4, 1),(4, 5),(5, 2),(5, 3), (5, 6),(6, 0),(6, 1), (6,4)] with trident vertices 4, 5, 6, sink 2, and terminal vertices 0, 1, 3, does not originate from either graph in the 'sunflower' X γ 3 2 .Indeed, the above micro-graph contains a 3-cycle 4 → 5 → 6 → 4 but no edge from 6 to either 5 or its Casimir 3 in a would-be expansion of P(ϱ, [a]) with 5 in the trident top. 13  Let us examine how, by which mechanism(s), Eq. ( 1) is verified for the tetrahedral cocycle γ 3 and the respective flow of Nambu brackets on 3 .Proof.Suppose for contradiction that a linear combination ◊ of Leibniz and zero micro-graphs turns the coboundary equation, 2 [[P, P]] , into an equality of bivector micro-graphs.By setting the Casimir a := z we fix the values of indices decorating the arrows which run into the terminal vertices a; now for ϱ(x, y) independent of z, the rest of each micro-graph becomes a Kontsevich graph over d = 2. Zero micro-graphs from dimension 3 remain zero over dimension 2. This dimensional reduction would yield a Leibnizgraph realization of the corresponding r.-h.s. for the two-dimensional problem from Part 2 of Proposition 1.But this is impossible; therefore, at least one new mechanism of differential polynomials' vanishing is at work.

Conclusion
Kontsevich's symmetries Ṗ = Q γ (P) of the Jacobi identity 1  2 [[P, P]] = 0 are produced from suitable graph cocycles γ as described in [2,5].We study the (non)triviality of these flows w.r.t. the Poisson differential ∂ P = [[P, •]].The original formalism of [5] yields tensorial G L(∞)invariants; but in finite dimension d of Poisson manifolds, these tensors, encoded by Kontsevich's graphs, can become linearly dependent, whence the 'incidental' trivializations (e.g., in d = 2).The graph language can be adapted to the subclass of Nambu-determinant Poisson brackets P(ϱ, [a]); Kontsevich's graph-cocycle flows do restrict to the Nambu subclass (see [4]).The new calculus of micro-graphs is good for encoding known flows and vector 13 Not all of the micro-graphs appearing in the expansion of Kontsevich's graphs X Open problem 1.Is there a dimension d + 1 < ∞ at which the tetrahedral-graph flow on the space of Nambu structures over d+1 becomes nontrivial in the second Poisson cohomology ?

Remark 4 .
If the wedge tops contain Nambu-Poisson bi-vectors P(ϱ, [a]) on d , every Kontsevich graph expands to a linear combination of micro-graphs: the arrow(s) originally in-coming to an aerial vertex with a copy of P, now work(s) by the Leibniz rule over the d − 1 vertices, with ϱ • ϵ ⃗ ı and with a 1 , . . ., a d−2 , in the subgraphs P(ϱ, [a 1 ], . . ., [a d−2 ]) of the micro-graph. 10

γ 3 d=2 3 3
mod [[P, H]] are needed for a solution X γ , see Example 4 and the eleven micro-graphs on p. 7. fields.Still the core task of this research is finding these fields ⃗ X [ϱ], [a] or proving their non-existence over d .The calculus of micro-graphs is (almost) well-behaved 14 under the dimensional reduction d → d − 1 by the loss of one Casimir a d−2 := x d and last coordinate x d in ϱ and all other Casimirs a ℓ , ℓ < d − 2. The forward move, d → d + 1, is not well defined.A priori there is no guarantee that any solution X γ d+1 exists at all: the dimensions d 0 < d 0 + 1 can mark the threshold where the G L(∞)-invariants from Kontsevich's graphs lose their linear dependence in lower dimensions d ⩽ d 0 , and the flow Ṗ [ϱ], [a 1 ], . . ., [a d−1 ] = Q γ d+1 P(ϱ, [a]) becomes Poisson-nontrivial is dimension d 0 + 1 and onwards.Lemma 7. Suppose there exists a trivializing vector field X γ d+1 for a γ-flow of P(ϱ, [a]) over d+1 , and this solution projects -when a 1 := x 3 , . .., a d−1 := x d+1 -onto the known linear combination X γ d=2 of Kontsevich's graphs.Then X γ d+1 contains (at least) those micro-graphs from the expansion of X γ d=2 over d + 1 in which all the old edges between Poisson structures P(ϱ, [a]) head to arrowtail vertices for ϱ but not to terminal vertices for the Casimirs a ℓ .Indeed, it is the differential polynomials from these micro-graphs which, staying nonzero, retract to the bottom-most solution.Remark 7. Another constraint -upon the derivative order profiles in Xi ([ϱ], [a]), hence in [[P, ⃗ X ]] -comes from the bi-vector Q γ 3 (P(ϱ, [a])) with known differential-polynomial coefficients.In effect, terms in X i can contain only those orders of derivatives which, under [[P, •]], reproduce the actually existing profiles of derivatives in Q γ 3 ([ϱ], [a]).
else zero.8Remark 3. Nambu-Poisson brackets on d⩾3 can be obtained from Nambu-Poisson brackets on d+1 by taking a d−1 = ±x d+1 on d+1 and by excluding the last Cartesian coordinate x d+1 from the list of arguments for ϱ(x ) and a 1 , ..., a d (x ).•By doing the above for d +1 = 3, one obtains a generic bi-vectorP = ϱ(x 1 , x 2 ) ∂ x 1 ∧ ∂ x 2 ,which is Poisson on 2 , and the (Nambu)

Definition 2 .
[5] the dimension d ⩾ 2. A micro-graph is a directed graph built over m ⩾ 0 sinks, over n ⩾ 0 aerial vertices with out-degree d and ordering of outgoing edges, and over n items of (d − 2)-tuples of aerial vertices with in-degree ⩾ 1 and no outgoing edges.•Thecorrespondence between micro-graphs and differential-polynomial expressions in ϱ, a 1 , ..., a d−2 and the content of sink(s) is defined in the same way as the mapping of Kontsevich's graphs to multidifferential operators on C ∞ (M d aff ), see [4, §2.2] or[5].• Same as for Kontsevich graphs, a micro-graph is zero if it admits a sign-reversing automorphism, i.e. a symmetry which acts by parity-odd permutation on the ordered set of edges.But now, in finite dimension d of M d aff , a micro-graph is vanishing if the differential polynomial (in ϱ and a 1 , . .., a d−2 ), obtained by expanding all the sums over indices that decorate the edges, vanishes identically.