The Kontsevich tetrahedral flow in 2D: a toy model

Abstract

In the paper "Formality conjecture" (1996) Kontsevich designed a universal flow $\dot{\mathcal{P}}=\mathcal{Q}_{a:b}(\mathcal{P})=a\Gamma_{1}+b\Gamma_{2}$ on the spaces of Poisson structures $\mathcal{P}$ on all affine manifolds of dimension $n \geqslant 2$. We prove a claim from $\textit{loc. cit.}$ stating that if $n=2$, the flow $\mathcal{Q}_{1:0}=\Gamma_{1}(\mathcal{P})$ is Poisson-cohomology trivial: $\Gamma_{1}(\mathcal{P})$ is the Schouten bracket of $\mathcal{P}$ with $\mathcal{X}$, for some vector field $\mathcal{X}$; we examine the structure of the space of solutions $\mathcal{X}$. Both the construction of differential polynomials $\Gamma_{1}(\mathcal{P})$ and $\Gamma_{2}(\mathcal{P})$ and the technique to study them remain valid in higher dimensions $n \geqslant 3$, but neither the trivializing vector field $\mathcal{X}$ nor the setting $b:=0$ survive at $n\geqslant 3$, where the balance is $a:b=1:6$.