Communications in Number Theory and Physics

Volume 17 (2023)

Number 1

Graph complexes and Feynman rules

Pages: 103 – 172



Marko Berghoff (Institut für Mathematik, Humboldt-Universität zu Berlin, Germany)

Dirk Kreimer (Institut für Mathematik, Humboldt-Universität zu Berlin, Germany)


We investigate Feynman graphs and their Feynman rules from the viewpoint of graph complexes. We focus on the interplay between graph homology, Hopf-algebraic structures on Feynman graphs and the analytic structure of their associated integrals. Furthermore, we discuss the appearance of cubical complexes where the differential is formed by reducing internal edges and by putting edge-propagators on the mass-shell.


coaction, complex of graphs, Cutkosky rules, Feynman diagram, Feynman graph, Feynman rules, graph complex, graph homology, Hopf algebra, Landau equations, Landau singularities, looptree duality, moduli space of graphs, renormalization

2010 Mathematics Subject Classification

Primary 18Gxx, 57T05, 81Q30. Secondary 14D21, 81T15.

Received 11 March 2021

Accepted 18 January 2023

Published 23 February 2023