Kontsevich graphs act on Nambu--Poisson brackets, V. Implementation
Mollie S. Jagoe Brown, Arthemy V. Kiselev
TL;DR
The work addresses the problem of whether Kontsevich graph flows yield coboundaries for Nambu--Poisson brackets by constructing a trivialising vector field $\vec{X}^{\gamma}_d(P)$ such that $Q^{\gamma}_d(P)=\llbracket P,\vec{X}^{\gamma}_d(P)\rrbracket$ in dimension $d$, focusing on the tetrahedral cocycle $\gamma_3$. It presents a detailed computational workflow in SageMath using the package $\textsf{gcaops}$ to generate graph encodings, map them to Nambu micro-graphs, extract formulas, form monomial bases, build evaluation matrices, identify linearly independent contributions, skew-symmetrise for $d\ge 4$, and solve the $(\dot{a}_i,\dot{\varrho})$ system to obtain $\vec{X}^{\gamma}_d(P)$. A key contribution is demonstrating the existence of a trivialising vector field in $d=4$ for $\gamma_3$, supported by a complete outline of the methodological steps, including isomorphism pruning and linear-algebraic resolution. The paper also discusses practical limitations, notably high memory consumption in graph-to-formula evaluation, and proposes incremental computation and skew-pair techniques as avenues for optimization, thereby providing a practical blueprint for studying Kontsevich graph deformations of Poisson structures.
Abstract
In this series of papers, we established that $Q^{γ_3}_{d=4}(P)$ is a coboundary in 4D (paper II arXiv:2409.12555), and we presented a series of experimental results about the (non)trivialisation of Kontsevich graph flows of Nambu--Poisson brackets on $\mathbb{R}^d$ (paper IV). This immediate sequel V. to I.--IV. is a guide to working with the package $\textsf{gcaops}$ (https://github.com/rburing/gcaops) ($\textbf{G}$raph $\textbf{C}$omplex $\textbf{A}$ction $\textbf{O}$n $\textbf{P}$oisson $\textbf{S}$tructures) for $\textsf{SageMath}$ by Buring (2022). Specifically, we shall explain the script used in paper II (arXiv:2409.12555) and the use of it.