Ricardo Buring (Johannes Gutenberg-Universität Mainz, Germany)

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

It is known that cocycles in the non-oriented graph complex with the vertex-expanding differential $d$ yield universal flows on the spaces of Poisson structures on affine manifolds. We present an explicit and relatively elementary proof explaining how and why this ``orientation'' process works. The presentation is based on the proof by Jost (2013), which in turn is based on the outline by Willwacher (2010), itself based on the seminal work by Kontsevich (1996). As an illustration, for the tetrahedral flow $\mathcal{Q}_{\text{tetra}}(\mathcal{P})$ this explains the vanishing of its Poisson differential $\partial_{\mathcal{P}}(\mathcal{Q}_{\text{tetra}}(\mathcal{P})) = [\![\mathcal{P}, \mathcal{Q}_{\text{tetra}}(\mathcal{P})]\!]$ via graphical differential consequences of the Jacobi identity \![\mathcal{P},\mathcal{P}\!] = 0$ (for a Poisson structure $\mathcal{P}$). We had found that factorization before using brute force, jointly with A. Bouisaghouane (2017). Our present reasoning also shows how $d$-coboundaries are mapped to universally $\partial_{\mathcal{P}}$-trivial deformations $\mathcal{Q}(\mathcal{P}) = \partial_\mathcal{P}(\mathfrak{X})$. The results along the way will be illustrated by examples.

date: 11 September 2018

time: 15:00

room: 5161.0105 (Bernoulliborg)