The orientation morphism: from graph cocycles to deformations of Poisson structures

Abstract

We recall the construction of the Kontsevich graph orientation morphism $\gamma \mapsto {\rm O\vec{r}}(\gamma)$ which maps cocycles $\gamma$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma)(\mathcal{P})$ of Poisson bi-vectors on affine manifolds. We reveal in particular why there always exists a factorization of the Poisson cocycle condition $[\![\mathcal{P},{\rm O\vec{r}}(\gamma)(\mathcal{P})]\!] \doteq 0$ through the differential consequences of the Jacobi identity $[\![\mathcal{P},\mathcal{P}]\!]=0$ for Poisson bi-vectors $\mathcal{P}$. To illustrate the reasoning, we use the Kontsevich tetrahedral flow $\dot{\mathcal{P}} = {\rm O\vec{r}}(\gamma_3)(\mathcal{P})$, as well as the flow produced from the Kontsevich--Willwacher pentagon-wheel cocycle $\gamma_5$ and the new flow obtained from the heptagon-wheel cocycle $\gamma_7$ in the unoriented graph complex.