Kontsevich graphs act on Nambu--Poisson brackets, IV. When the invisible becomes crucial
Mollie S. Jagoe Brown, Arthemy V. Kiselev
TL;DR
The paper analyzes how Kontsevich graph cocycles, particularly the tetrahedron γ_3, act on Nambu–Poisson brackets on affine space by seeking coboundaries of the form $Q^{\gamma_3}(P)=\llbracket P, \vec{X}^{\gamma_3}(P)\rrbracket$ for $2\le d<5$. Using graph calculus with sunflowers and their $d$-descendants, the authors study existence and structure of trivialising vector fields $\vec{X}^{\gamma_3}_d(P)$ for Nambu–Poisson brackets on $\mathbb{R}^d$, revealing that higher-dimensional coboundaries depend on graphs that vanish in lower dimensions, i.e., invisible graphs. They show that in 4D a coboundary cannot be obtained solely from 3D sunflower data, but a 4D trivialising vector field does exist when considering the 4D-descendants of seventeen specific 3D sunflower graphs, with linear independence preserved across the dimension shift. This work highlights the dimension-specific nature of Kontsevich graph actions and guides efficient graph-selection strategies by demonstrating the necessity of certain 3D graphs and their higher-dimensional descendants to capture 4D coboundaries.
Abstract
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 $γ$ acts (non)trivially -- in the second Poisson cohomology -- on the space of Nambu brackets, taking a vector field solution $\smash{\vec{X}^γ_d}$ from dimension $d$ does not work in $d+1$. For $2 \leqslant d \leqslant 4$, the action of tetrahedron $γ_3$ on Nambu brackets is known to be a Poisson coboundary, $\dot{P} = [[ P,\smash{\vec{X}^{γ_3}_d} (P)]]$. 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 $\smash{\vec{X}^{γ_3}_{d+1}}$ 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 $d=4$.