Kontsevich graphs act on Nambu--Poisson brackets, IV. Resilience of the graph calculus in the dimensional shift $d\mapsto d+1$

Abstract: We examine whether the Kontsevich flows $\dot{P}=Q^\gamma_d(P)$ of Nambu--Poisson structures $P$ on $\mathbb{R}^d$ are Poisson coboundaries, for $\gamma$ some suitable cocycle in the Kontsevich graph complex. That is, we inspect the existence of a vector field $\vec{X}^\gamma_d(P)$ such that $Q^\gamma_d(P)=[[ P,\vec{X}^\gamma_d(P)]]$, where $[[\cdot,\cdot]]$ is the Schouten bracket of multivector fields (the generalised Lie bracket). To tackle this class of problems in dimensions $d\geq3$, we introduced a series of simplications in paper II (arXiv:2409.12555); here, we present a series of results regarding the down-up behaviour of solutions $\vec{X}^\gamma_d(P)$ and vanishing micro-graphs in the course of dimension shift $d\mapsto d+1$.