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 $\dot {\mathcal{P}} = {{\text{O}\vec{\text{r}}}}(\gamma )(\mathcal{P})$ preserves the space of Poisson structures if γ is a cocycle with respect to the vertex-expanding differential d in the graph complex. A class of such cocycles ${{\gamma }_{{2\ell + 1}}}$ 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 ${{\gamma }_{3}}$ itself is a cocycle; at $\ell = 2$ the Kontsevich–Willwacher pentagon-wheel cocycle ${{\gamma }_{5}}$ consists of two graphs. We reconstruct the symmetry ${{\mathcal{Q}}_{5}}(\mathcal{P}) = {{\text{O}\vec{\text{r}}}}({{\gamma }_{5}})(\mathcal{P})$ and verify that ${{\mathcal{Q}}_{5}}$ is a Poisson cocycle indeed: $\left[\kern-0.15em\left[ {\mathcal{P},{{\mathcal{Q}}_{5}}(\mathcal{P})} \right]\kern-0.15em\right] \doteq 0$ via $\left[\kern-0.15em\left[ {\mathcal{P},\mathcal{P}} \right]\kern-0.15em\right] = 0.$