PROP profile of deformation quantization and graph complexes with loops and wheels

Abstract

Motivated by the problem of deformation quantization we introduce and study directed graph complexes with oriented loops and wheels. We develop some technique for computing cohomology of such graph complexes and apply it to several concrete examples such as wheeled completion of the operad of strongly homotopy Lie algebras and the wheeled completion of the dg prop of Poisson structures. We prove also a deformation quantization theorem of wheeled Poisson structures on arbitrary formal graded manifolds.

The final version. The original text has grown into two stories -- one about graph complexes with wheels and another about propic approach to deformation quantization. Though interrelated, these stories can be read independently, and will be published independently