We review the gauge and ghost cyle graph complexes as defined by Kreimer, Sars and van Suijlekom in "Quantization of gauge fields, graph polynomials and graph homology" and compute their cohomology. These complexes are generated by labelings on the edges or cycles of graphs and the differentials act by exchanging these labels. We show that both cases are instances of a more general construction of double complexes associated with graphs. Furthermore, we describe a universal model for these kinds of complexes which allows to treat all of them in a unified way.
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. arXiv:1809.09954v3 [math.AT]
In [BBK10] it was shown how so-called wonderful compactifications can be used for renormalization in the position space formulation of quantum field theory. This article aims to continue this idea, using a slightly different approach; instead of the subspaces in the arrangement of divergent loci, we use the poset of divergent subgraphs as the main tool to describe the whole renormalization process. This is based on [Fei05] where wonderful models were studied from a purely combinatorial viewpoint. The main motivation for this approach is the fact that both, renormalization and the model construction, are governed by the combinatorics of this poset. Not only simplifies this the exposition considerably, but also allows to study the renormalization operators in more detail. Moreover, we explore the renormalization group in this setting by studying how the renormalized distributions behave under a change of renormalization points.
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