From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory

TL;DR

This paper reveals a deep connection between Galois theory, noncommutative geometry, and quantum field theory renormalization, identifying a universal symmetry group that unifies these concepts through a Riemann-Hilbert framework.

Contribution

It establishes a precise link between motivic Galois theory and perturbative renormalization, introducing the cosmic Galois group and a universal geometric singular frame.

Findings

Identification of the cosmic Galois group as a symmetry of physical theories

Construction of a universal singular frame where divergences vanish

Relation of the Galois group to motivic Galois theory

Abstract

We establish a precise relation between Galois theory in its motivic form with the mathematical theory of perturbative renormalization (in the minimal subtraction scheme with dimensional regularization). We identify, through a Riemann-Hilbert correspondence based on the Birkhoff decomposition and the t'Hooft relations, a universal symmetry group (the "cosmic Galois group" suggested by Cartier), which contains the renormalization group and acts on the set of physical theories. This group is closely related to motivic Galois theory. We construct a universal singular frame of geometric nature, in which all divergences disappear. The paper includes a detailed overview of the work of Connes-Kreimer and background material on the main quantum field theoretic and algebro-geometric notions involved. We give a complete account of our results announced in math.NT/0409306.