Title: 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 $a:b=1:6$.