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

TL;DR

This paper introduces and analyzes directed graph complexes with loops and wheels, developing cohomology computation techniques and applying them to deformation quantization of wheeled Poisson structures on graded manifolds.

Contribution

It develops new methods for cohomology of graph complexes with loops and applies these to deformation quantization of wheeled Poisson structures.

Findings

Cohomology techniques for graph complexes with loops and wheels.

Deformation quantization theorem for wheeled Poisson structures.

Applications to operads and dg props in deformation theory.

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.