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

Abstract

We recall the construction of the Kontsevich graph orientation morphism $\gamma \mapsto \mathrm{O}\vec{\mathrm{r}}(\gamma )$ which maps cocycles $\gamma$ in the non-oriented graph complex to infinitesimal symmetries $\dot{\mathcal{P}}=\mathrm{O}\vec{\mathrm{r}}(\gamma )(\mathcal{P})$ of Poisson bi-vectors on affine manifolds. We reveal in particular why there always exists a factorization of the Poisson cocycle condition $[[\mathcal{P},\mathrm{O}\vec{\mathrm{r}}(\gamma )(\mathcal{P})]]\doteq 0$ 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 $\dot{\mathcal{P}}=\mathrm{O}\vec{\mathrm{r}}({\gamma }_3)(\mathcal{P})$, 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.

12 pages. Talk given by R.B. at Group32 (Jul 9--13, 2018; CVUT Prague, Czech Republic). Big formula in Appendix A retained from the (unpublished) Appendix in arXiv:1712.05259 [math-ph]. Signs corrected in v2