# Wonderful Compactifications in Quantum Field Theory

@article{Berghoff2014WonderfulCI, title={Wonderful Compactifications in Quantum Field Theory}, author={Marko Berghoff}, journal={arXiv: Mathematical Physics}, year={2014} }

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…

## 9 Citations

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The structure of overlapping subdivergences, which appear in the perturbative expansions of quantum field theory, is analyzed using algebraic lattice theory. It is shown that for specific QFTs the…

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This thesis provides an extension of the work of Dirk Kreimer and Alain Connes on the Hopf algebra structure of Feynman graphs and renormalization to general graphs. Additionally, an algebraic…

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We present the lattice structure of Feynman diagram renormalization in physical QFTs from the viewpoint of Dyson-Schwinger-Equations and the core Hopf algebra of Feynman diagrams. The lattice…

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The content of this chapter is partially based on the author’s article (Borinsky, Lett Math Phys 106(7):879–911, 2016) [1].

## 34 References

### RENORMALIZATION AND RESOLUTION OF SINGULARITIES

- 2009

Mathematics

Since the seminal work of Epstein and Glaser it is well established that perturbative renormalization of ultravi olet divergences in position space amounts to extension of distributions onto…

### Rota – Baxter Algebras in Renormalization of Perturbative Quantum Field Theory

- 2006

Mathematics, Physics

Recently, the theory of renormalization in perturbative quantum field theory underwent some exciting new developments. Kreimer discovered an organization of Feynman graphs into combinatorial Hopf…

### Microlocal Analysis and¶Interacting Quantum Field Theories:¶Renormalization on Physical Backgrounds

- 2000

Mathematics

Abstract:We present a perturbative construction of interacting quantum field theories on smooth globally hyperbolic (curved) space-times. We develop a purely local version of the…

### Chern-Simons perturbation theory. II

- 1994

Physics

We study the perturbation theory for three dimensional Chern--Simons quantum field theory on a general compact three manifold without boundary. We show that after a simple change of variables, the…

### Renormalization in Quantum Field Theory and the Riemann--Hilbert Problem II: The β-Function, Diffeomorphisms and the Renormalization Group

- 1999

Mathematics

Abstract: We showed in Part I that the Hopf algebra ℋ of Feynman graphs in a given QFT is the algebra of coordinates on a complex infinite dimensional Lie group G and that the renormalized theory is…

### Wonderful models of subspace arrangements

- 1995

Mathematics

The motivation stems from our attempt to understand Drinfeld's construction (el. [Dr2]) of special solutions of the Khniznik-Zamolodchikov equation (of. [K-Z]) with some prescribed asymptotic…

### The Hopf Algebra of Rooted Trees in Epstein-Glaser Renormalization

- 2005

Physics

Abstract.We show how the Hopf algebra of rooted trees encodes the combinatorics of Epstein-Glaser renormalization and coordinate space renormalization in general. In particular, we prove that the…

### On the periods of some Feynman integrals

- 2009

Mathematics

We study the related questions: (i) when Feynman amplitudes in massless $\phi^4$ theory evaluate to multiple zeta values, and (ii) when their underlying motives are mixed Tate. More generally, by…

### Polydiagonal compactification of configuration spaces

- 1999

Mathematics

A smooth compactification X〈n〉 of the configuration space of n distinct labeled points in a smooth algebraic variety X is constructed by a natural sequence of blowups, with the full symmetry of the…

### Quantum periods : A census of φ 4-transcendentals

- 2010

Mathematics

Perturbative quantum field theories frequently feature rational linear combinations of multiple zeta values (periods). In massless φ4theory we show that the periods originate from certain “primitive”…