A Note on Multi-Oriented Graph Complexes and Deformation Quantization of Lie Bialgebroids

Abstract

Universal solutions to deformation quantization problems can be conveniently classified by the cohomology of suitable graph complexes. In particular, the deformation quantizations of (finite-dimensional) Poisson manifolds and Lie bialgebras are characterised by an action of the Grothendieck-Teichm\"uller group via one-colored directed and oriented graphs, respectively. In this note, we study the action of multi-oriented graph complexes on Lie bialgebroids and their "quasi" generalisations. Using results due to T. Willwacher and M. Zivkovi\'c on the cohomology of (multi)-oriented graphs, we show that the action of the Grothendieck-Teichm\"uller group on Lie bialgebras and quasi-Lie bialgebras can be generalised to quasi-Lie bialgebroids via graphs with two colors, one of them being oriented. However, this action generically fails to preserve the subspace of Lie bialgebroids. By resorting to graphs with two oriented colors, we instead show the existence of an obstruction to the quantization of a generic Lie bialgebroid in the guise of a new $\mathsf{Lie}_\infty$-algebra structure non-trivially deforming the "big bracket" for Lie bialgebroids. This exotic $\mathsf{Lie}_\infty$-structure can be interpreted as the equivalent in $d=3$ of the Kontsevich-Shoikhet obstruction to the quantization of infinite-dimensional Poisson manifolds (in $d=2$). We discuss the implications of these results with respect to a conjecture due to P. Xu regarding the existence of a quantization map for Lie bialgebroids.