Lie graph homology model for $\mathfrak{grt}_1$

TL;DR

This paper introduces a new chain model for the commutative graph complex based on Lie graph homology, revealing connections to the structure of the Grothendieck-Teichmüller Lie algebra and modular forms.

Contribution

It develops a novel chain model linking Lie graph homology with graph complexes and identifies key relations in genus 2 related to the conjectural generators of rf1.

Findings

Identification of a contractible complex of tadpoles and higher genus vertices

Relation between Lie graph homology in genus 2 and rf1 generators

Unification of modular cusp forms in graph homology study

Abstract

This paper develops a new chain model for the commutative graph complex $\mathsf{GC}_2$ which takes Lie graph homology as an input. Our main technical result is the identification of a large contractible complex of (certain) tadpoles and higher genus vertices of the Feynman transform of Lie graph homology. Using this result we identify the anti-invarints of Lie graph homology in genus $2$ with relations between bracketings of conjectural generators of $\mathfrak{grt}_1$ in depth 2 modulo depth 3, unifying two a priori disparate appearances of the space of modular cusp forms in the study of graph homology.