Quantum fields and motives

TL;DR

This survey explores the deep connection between quantum field theory renormalization and motivic Galois groups, revealing a universal symmetry that governs divergences across all theories.

Contribution

It establishes that all quantum field theories share a universal motivic Galois symmetry, generalizing the renormalization group, and provides a mathematical framework for counterterms via Riemann-Hilbert correspondence.

Findings

Existence of a universal motivic Galois group for QFTs

The group acts as a symmetry of divergences and counterterms

Counterterms are classified by a Riemann-Hilbert correspondence

Abstract

This is a survey of our results on the relation between perturbative renormalization and motivic Galois theory. The main result is that all quantum field theories share a common universal symmetry realized as a motivic Galois group, whose action is dictated by the divergences and generalizes that of the renormalization group. The existence of such a group was conjectured by P. Cartier based on number theoretic evidence and on the Connes-Kreimer theory of perturbative renormalization. The group provides a universal formula for counterterms and is obtained via a Riemann-Hilbert correspondence classifying equivalence classes of flat equisingular bundles, where the equisingularity condition corresponds to the independence of the counterterms on the mass scale.