Wonderful Compactifications in Quantum Field Theory

TL;DR

This paper explores the use of combinatorial models called wonderful compactifications to understand and simplify the process of renormalization in quantum field theory, focusing on the behavior of distributions under renormalization group transformations.

Contribution

It introduces a new combinatorial approach using the poset of divergent subgraphs to analyze renormalization, simplifying the process and enabling detailed study of renormalization operators and the renormalization group.

Findings

The combinatorial approach simplifies the renormalization process.

Detailed analysis of renormalization operators is possible.

Behavior of distributions under renormalization group is characterized.

Abstract

This article reviews the use of DeConcini-Procesi wonderful models in renormalization of ultraviolet divergences in position space as introduced by Bergbauer, Brunetti and Kreimer. In contrast to the exposition there we employ 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 an article by Feichtner, 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.