Kontsevich–Nambu Micro-Graphs

• Kontsevich–Nambu micro-graphs are combinatorial objects that encode the structure of Nambu–Poisson brackets through determinant expressions and Casimir fields.
• They serve as templates for constructing multilinear differential operators that reveal algebraic symmetries and facilitate trivialization of Poisson deformations.
• Their systematic formulation and computational implementation, notably in SageMath’s gcaops library, streamline the analysis of graph flows and polynomial identities in various dimensions.

Kontsevich–Nambu micro-graphs are dimension-specific combinatorial objects within the graph complex calculus for universal Poisson deformations, encoding how graph cocycles—most notably the tetrahedral cocycle—act on the class of Nambu–Poisson brackets given by determinant expressions with Casimir fields and inverse densities. These micro-graphs function as templates for multilinear differential operators, making explicit the underlying algebraic structure and symmetries of induced flows, while also revealing a tower of polynomial identities and essential features in the trivialization of Kontsevich graph flows in finite dimensions.

1. Kontsevich Graph Complex and Cocycle Deformations

The Kontsevich graph complex, denoted $G^\bullet$, is a graded vector space of finite, undirected graphs $\gamma$ with $k$ vertices and $\ell$ edges, equipped with a bracket $[\cdot, \cdot]$ defined by grafting graphs along designated edges, and a differential $d = [e, \cdot]$ via insertion of a single-edge graph $e$ (Brown et al., 19 Sep 2024). Cocycle elements satisfy $d\gamma = 0$ and encode universal infinitesimal deformations acting on Poisson bivector fields $P$. The smallest nontrivial cocycle is $\gamma_3$, the tetrahedron on four vertices (six edges).

This formalism produces, for each cocycle (e.g., the wheel series $\gamma_{2\ell+1}$), universal bidifferential operators $Q^\gamma(P)$ acting on Poisson tensors in the Lichnerowicz–Poisson cohomology $H^2_P$ (Kiselev et al., 27 Sep 2024). The key problem addressed is to determine, for each $Q^\gamma$, whether the induced deformation is trivial—i.e., a Poisson coboundary $Q^\gamma(P) = [[P, X^\gamma(P)]]$ for some vector field $X^\gamma(P)$.

2. Nambu–Poisson Brackets and Dimension-Specific Structure

Nambu–Poisson brackets of order $d$ on $\mathbb{R}^d$ are defined by a nondegenerate density $\rho(x)$ and $(d-2)$ Casimir functions $a^1(x), \dots, a^{d-2}(x)$: $$\{f, g\}_d(x) = \rho(x) \det\left(\begin{matrix} \partial_{x_1} f & \partial_{x_1} g & \partial_{x_1} a^1 & \cdots & \partial_{x_1} a^{d-2} \ \vdots & \vdots & \vdots & \ddots & \vdots \ \partial_{x_d} f & \partial_{x_d} g & \partial_{x_d} a^1 & \cdots & \partial_{x_d} a^{d-2} \end{matrix}\right) \tag{1}$$ or, equivalently, in index notation: $$P^{ij}(x) = \rho(x) \varepsilon^{ij i_1 \dots i_{d-2}} \partial_{i_1} a^1 \cdots \partial_{i_{d-2}} a^{d-2}$$ with the Jacobi identity equivalent to the vanishing of the Schouten bracket $[[P,P]] = 0$. The class of Nambu–Poisson structures is closed under the Kontsevich flows induced by cocycles, exhibiting special symmetry and tensorial properties dictated by the determinant construction (Brown et al., 19 Sep 2024, Kiselev et al., 27 Sep 2024).

3. Definition and Construction of Kontsevich–Nambu Micro-Graphs

Kontsevich–Nambu micro-graphs are combinatorial encodings designed to leverage the particular multilinear structure of the Nambu–Poisson bracket in each dimension (Buring et al., 2022, Brown et al., 8 Nov 2025). Each "Nambu block" represents a copy of the bracket, realized as a vertex carrying the Levi-Civita symbol $\varepsilon^{i_1 \dots i_d}$ and the density $\rho$, with $d$ ordered outgoing edges. These edges act as derivations: two target the function arguments $f$, $g$, while the remaining $d-2$ target the Casimir fields $a^1, \dots, a^{d-2}$.

A micro-graph on $m$ blocks consists of $m d$ directed edges. The admissibility conditions prescribe that every block has exactly $d$ outgoing arrows (no loops or coincident arrows on the same factor), enforcing the Leibniz rule and permutation symmetry. Encodings of such graphs are tuples of vertex labels, typically of the form $(0,1;\,1,3;\,1,2)$ in $d=2$, generalizing to $(0,2,4;\,1,3,5;\,1,2,6)$, etc., in higher $d$.

The orientation-to-formula map $\varphi$ sends micro-graphs to explicit multilinear differential operators, with the weight function $\phi(\Gamma)[P]$ summing over all index contractions generated by the graph's edge pattern (Brown et al., 8 Nov 2025).

4. Tetrahedral Flow, Trivialization, and Uniqueness

The tetrahedral cocycle $\gamma_3$ induces a flow $Q_d^{\gamma_3}(P)$ via a universal sum over oriented tetrahedron graphs, yielding a quartic differential operator in $P$ (Brown et al., 19 Sep 2024, Schipper et al., 24 Sep 2024): $$Q_d^{\gamma_3}(P) = w_3 \sum_{\Gamma} \varphi(\Gamma)[P,P,P,P]$$ where the sum is over all oriented tetrahedral micro-graphs in dimension $d$. In coordinates, this expression involves high-order derivatives and multiple $\varepsilon$-tensor contractions.

Trivialization seeks a 1-vector field $X_d(P)$ such that $Q_d^{\gamma_3}(P) = [[P,X_d(P)]]$, meaning the deformation corresponds to an infinitesimal coordinate transformation and is Poisson trivial. In all cases $d=2,3,4$, explicit trivializers have been constructed as linear combinations of micro-graphs derived from the sunflower graph, subject to dimension-dependent symmetry such as skew-symmetrization under Casimir swaps in $d=4$ (Brown et al., 19 Sep 2024, Schipper et al., 24 Sep 2024):

• In $d=2$, the solution is the unique "sunflower" graph.
• In $d=3$, the trivializer is a sum of descendants of the 2D sunflower, requiring solution of a finite linear system.
• In $d=4$, due to $a^1\leftrightarrow a^2$ symmetry, skew-pairs are formed and a further reduced linear system is solved computationally.

Uniqueness of $X_d^{\gamma_3}$ is established: the trivializing field is unique up to addition of Hamiltonian vector fields in the graph complex, with the representatives themselves not unique at the micro-graph level, due to the existence of "synonyms," distinct graphs with identical images under $\varphi$ (Schipper et al., 24 Sep 2024). Synonym selection affects which trivializations persist under dimension shifts.

5. Dimensional Shift and Resilience of the Micro-Graph Calculus

A key structural property in higher-dimensional settings is the "resilience" of the micro-graph calculus under the dimension shift $d \mapsto d+1$ (Brown et al., 13 Mar 2025). The following features are salient:

• Vanishing Propagation: If a micro-graph vanishes in dimension $d$ (its weight sum is zero), its canonical embedding by addition of one new Casimir per internal vertex also vanishes in $d+1$ (Proposition 14).
• Automorphism-Induced Vanishing: Micro-graphs with nontrivial automorphism groups acting on Casimir legs are forced to vanish by termwise cancellation (Proposition 15).
• Linear Relations Lifting: Linear dependencies among micro-graph formulas in $d$ persist in their $d+1$ descendants, facilitating systematic construction of trivializers by pruning the search space to core sunflower descendants plus the vanishing hub.

In practical terms, this results in drastic reduction of computational complexity: the exponential explosion in the number of possible micro-graphs is heavily pruned by restricting to a small set of dimensionally "resilient" candidates (Brown et al., 13 Mar 2025).

6. Implementation: SageMath and the gcaops Library

Algorithmic construction and verification of Kontsevich–Nambu micro-graph calculus is implemented in the $\mathsf{gcaops}$ SageMath package (“Graph Complex Action On Poisson Structures”) (Brown et al., 13 Mar 2025), which automates the generation, encoding, and evaluation of micro-graphs:

• Micro-graph Generation: Encodings are produced as tuples aligned with the graph’s combinatorial pattern.
• Evaluation Routines: Each micro-graph is mapped to superfunctions in polynomial rings with explicit management of $\varepsilon$-symbols and permutations. Full $(d!)^p$ index sums are processed.
• Linear Algebra: Sparse matrices are assembled, monomial bases extracted, isomorphism reductions performed, and skew-symmetrizations under Casimir swaps applied.
• Triviality Verification: Systematic solving of the reduced $(\dot a_i, \dot\rho)$ system certifies that computed $X_d$ trivializes $\gamma_3$ in each dimension.
• Benchmarks: For $d=4$, 324 candidate graphs are pruned to $64$ skew-pairs and $27$ independent solutions via high-performance linear algebra.

The $\mathsf{gcaops}$ library is the reference implementation for this calculus, and its routines generalize to other cocycles and higher dimensions (Brown et al., 13 Mar 2025).

7. Identities, Symmetries, and Open Problems

Kontsevich–Nambu micro-graphs underpin a tower of differential-polynomial identities among Jacobian determinants in arbitrary dimension, due to their vanishing properties and symmetry mechanisms (Brown et al., 8 Nov 2025, Kiselev et al., 27 Sep 2024):

• Identity Families: Every vanishing micro-graph produces a polynomial identity for $d\times d$ Jacobians and their minors, stable under embedding to higher dimensions. This yields an infinite family of such identities conjecturally spanning all $d$.
• Collapse via $\varepsilon$-symbols: Each term in the unreduced polydifferential sum can be characterized by its differential profile; upon full skew-symmetrization, these profiles collapse to a handful of marker-monomials.
• Discrete Hidden Symmetry: The direct product $S_d^{n-1}$ of permutation groups acts on the set of differential profiles, ensuring that nonzero orbits yield identical types of polynomial invariants.
• Open Questions: No uniform formula for the trivializer $X_d^{\gamma_3}$ exists for all $d$, nor is the web of identities and their relation to graph complex cohomology fully classified (Brown et al., 8 Nov 2025).

The micro-graph calculus thus reveals rich structure in both the combinatorics and algebraic invariants of Nambu–Poisson brackets under Kontsevich-type deformations, forming a core methodology for both theoretical exploration and computational verification in Poisson geometry and universal deformation quantization.