Feynman amplitudes on moduli spaces of graphs

Marko Berghoff

Published 2017-09-02Version 1

This article aims to highlight the similarity between the combinatorial structures underlying renormalization of Feynman integrals on one side and certain compactifications of moduli spaces of graphs on the other side. Both concepts are brought together by interpreting Feynman amplitudes as volume densities on these moduli spaces which decompose into disjoint unions of open cells indexed by graphs. Renormalization translates in this setting into the task of assigning to every cell a finite volume in a way that respects the boundary relations between neighboring cells. We show that this renormalization procedure can be organized systematically using Borel-Serre compactifications of these moduli spaces. The key point is that the newly added boundary components in a compactified cell have a combinatorial description that resembles the forest structure of subdivergences of the Feynman diagram associated to it.