The oriented graph complexes

TL;DR

This paper demonstrates that oriented graph complexes without cycles are quasi-isomorphic to known graph complexes, providing new insights into Lie bialgebra quantization and formality morphisms.

Contribution

It establishes a quasi-isomorphism between oriented graph complexes and ordinary graph complexes, enabling combinatorial descriptions of algebraic actions and rational constructions.

Findings

Oriented graph complex GC^or_n is quasi-isomorphic to GC_{n-1}.

Provides a combinatorial description of the Grothendieck-Teichmüller Lie algebra action.

Constructs cycle-free formality morphisms rationally without configuration space integrals.

Abstract

Oriented graph complexes, in which graphs are not allowed to have oriented cycles, govern for example the quantization of Lie bialgebras and infinite dimensional deformation quantization. It is shown that the oriented graph complex GC^or_n is quasi-isomorphic to the ordinary commutative graph complex GC_{n-1}, up to some known classes. This yields in particular a combinatorial description of the action of the Grothendieck-Teichm\"uller Lie algebra on Lie bialgebras, and shows that a cycle-free formality morphism in the sense of Shoikhet can be constructed rationally without reference to configuration space integrals.