标题： Kontsevich 图作用于Nambu--泊松括号，第四部分。 当不可见的东西变得至关重要 显示英文标题

标题： Kontsevich graphs act on Nambu--Poisson brackets, IV. When the invisible becomes crucial

摘要： Kontsevich图可以用于编码系数为给定仿射实流形上泊松括号系数的微分多项式的多向量。 通过有向图进行编码公式适用于Nambu行列式泊松括号类，但图的拓扑结构变得与维度相关。 为了检查给定的Kontsevich图上循环$\gamma$在第二个泊松上同调中是否对Nambu括号空间起（非）平凡作用，从维度$d$中取一个向量场解$\smash{\vec{X}^\gamma_d}$在维度$d+1$中不起作用。 对于$2 \leqslant d \leqslant 4$，四面体$\gamma_3$对Nambu括号的作用已知为泊松上边界，$\dot{P} = [[ P,\smash{\vec{X}^{\gamma_3}_d} (P)]]$。 我们探索哪些最小的图集（子集），在$\mathbb{R}^d_{\text{aff}}$上编码（非）消失的对象，能够生成足够的拓扑数据以使解$\smash{\vec{X}^{\gamma_3}_{d+1}}$出现。 我们发现，在没有随着公式在$d=3$中消失的不可见图的情况下，高维中不可能有解，但这些不可见图的后代并非都在$d=4$上消失。 显示英文摘要

摘要： Kontsevich's graphs allow encoding multi-vectors whose coefficients are differential-polynomial in the coefficients of a given Poisson bracket on an affine real manifold. Encoding formulas by directed graphs adapts to the class of Nambu-determinant Poisson brackets, yet the graph topology becomes dimension-specific. To inspect whether a given Kontsevich graph cocycle $\gamma$ acts (non)trivially -- in the second Poisson cohomology -- on the space of Nambu brackets, taking a vector field solution $\smash{\vec{X}^\gamma_d}$ from dimension $d$ does not work in $d+1$. For $2 \leqslant d \leqslant 4$, the action of tetrahedron $\gamma_3$ on Nambu brackets is known to be a Poisson coboundary, $\dot{P} = [[ P,\smash{\vec{X}^{\gamma_3}_d} (P)]]$. We explore which minimal (sub)sets of graphs, encoding (non)vanishing objects over $\mathbb{R}^d_{\text{aff}}$, generate the topological data that suffice for a solution $\smash{\vec{X}^{\gamma_3}_{d+1}}$ to appear. We detect that there can be no solution in higher dimension without invisible graphs that vanish as formulas in $d=3$, but whose descendants do not all vanish over $d=4$.