Invariant Differential Forms on Complexes of Graphs and Feynman Integrals

Abstract

We study differential forms on an algebraic compactification of a moduli space of metric graphs. Canonical examples of such forms are obtained by pulling back invariant differentials along a tropical Torelli map. The invariant differential forms in question generate the stable real cohomology of the general linear group, as shown by Borel. By integrating such invariant forms over the space of metrics on a graph, we define canonical period integrals associated to graphs, which we prove are always finite and take the form of generalised Feynman integrals. Furthermore, canonical integrals can be used to detect the non-vanishing of homology classes in the commutative graph complex. This theory leads to insights about the structure of the cohomology of the commutative graph complex, and new connections between graph complexes, motivic Galois groups and quantum field theory.

arXiv commentContribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

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• Graph complexes and Feynman rules
• Weight two compactly supported cohomology of moduli spaces of curves
• Algebraic Interplay between Renormalization and Monodromy
• The wheel classes in the locally finite homology of ${\mathrm{GL}}_n(\mathbb{Z})$, canonical integrals and zeta values
• Tropicalized quantum field theory and global tropical sampling