Moduli spaces of colored graphs

TL;DR

This paper introduces moduli spaces of colored graphs to analyze Feynman integrals in quantum field theory, exploring their combinatorial, topological, and homological properties.

Contribution

It defines new moduli spaces of edge-colored graphs with applications to quantum field theory, connecting combinatorics, topology, and physics.

Findings

Spaces are cell complexes with rich combinatorial structures

Detailed examples illustrate topological and homological properties

Connections to Feynman diagrams in quantum field theories

Abstract

We introduce moduli spaces of colored graphs, defined as spaces of non-degenerate metrics on certain families of edge-colored graphs. Apart from fixing the rank and number of legs these families are determined by various conditions on the coloring of their graphs. The motivation for this is to study Feynman integrals in quantum field theory using the combinatorial structure of these moduli spaces. Here a family of graphs is specified by the allowed Feynman diagrams in a particular quantum field theory such as (massive) scalar fields or quantum electrodynamics. The resulting spaces are cell complexes with a rich and interesting combinatorial structure. We treat some examples in detail and discuss their topological properties, connectivity and homology groups.