Poisson brackets symmetry from the pentagon-wheel cocycle in the graph complex

Abstract

Kontsevich designed a scheme to generate infinitesimal symmetries $\dot{\mathcal{P}} = \mathcal{Q}(\mathcal{P})$ of Poisson brackets $\mathcal{P}$ on all affine manifolds $M^r$; every such deformation is encoded by oriented graphs on $n+2$ vertices and $2n$ edges. In particular, these symmetries can be obtained by orienting sums of non-oriented graphs $γ$ on $n$ vertices and $2n-2$ edges. The bi-vector flow $\dot{\mathcal{P}} = \text{Or}(γ)(\mathcal{P})$ preserves the space of Poisson structures if $γ$ is a cocycle with respect to the vertex-expanding differential in the graph complex. A class of such cocycles $\boldsymbolγ_{2\ell+1}$ is known to exist: marked by $\ell \in \mathbb{N}$, each of them contains a $(2\ell+1)$-gon wheel with a nonzero coefficient. At $\ell=1$ the tetrahedron $\boldsymbolγ_3$ itself is a cocycle; at $\ell=2$ the Kontsevich--Willwacher pentagon-wheel cocycle $\boldsymbolγ_5$ consists of two graphs. We reconstruct the symmetry $\mathcal{Q}_5(\mathcal{P}) = \text{Or}(\boldsymbolγ_5)(\mathcal{P})$ and verify that $\mathcal{Q}_5$ is a Poisson cocycle indeed: $[\![\mathcal{P},\mathcal{Q}_5(\mathcal{P})]\!]\doteq 0$ via $[\![\mathcal{P},\mathcal{P}]\!]=0$.