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

Alain Connes (College de France); Matilde Marcolli (MPIM)

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

Alain Connes (College de France), Matilde Marcolli (MPIM)

High Energy Physics - Theory · Submitted 11 Nov 2004

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim 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).

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arXiv IDhep-th/0411114 (version 1)

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CitationAlain Connes (College de France), Matilde Marcolli (MPIM). "From Physics to Number Theory via Noncommutative Geometry, Part II: Renormalization, the Riemann-Hilbert correspondence, and motivic Galois theory." arXiv:hep-th/0411114 (2004). https://pith.science/paper/BOLWWAER

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What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.06 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.27 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; show every role.

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.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	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).	result 0.66	0.94	0.67	asserted 1.00
s1	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.	contribution 0.56	0.93	0.29	asserted 1.00
s2	This group is closely related to motivic Galois theory.	result 0.49	0.52	0.12	asserted 0.98
s3	We construct a universal singular frame of geometric nature, in which all divergences disappear.	contribution 0.77	0.94	0.10	asserted 1.00
s4	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.	background 0.95	0.34	0.02	asserted 0.99
s5	We give a complete account of our results announced in math.NT/0409306.	contribution 0.77	0.57	0.04	asserted 0.98

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.99.

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arXiv listing abstract v1

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C citations
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The paper

Original 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.

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References

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8 arXiv papers cite this in 15 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain `` motivic Galois group'', which is uniquely determined and universal with respect to the set of physical theories.

Renormalization and motivic Galois theory · 0.80

Papers citing it · 8 sentences without a confident label

Cites it as background 6 papers

Section two of this paper reviews the development of the tools necessary for the construction of the renormalization bundle, following [ref] , [ref] , [ref] .

A Perspective on Regularization and Curvature · Introduction · 0.95
The affine group scheme $G$ is developed in detail in [ref] .

The geometric $\beta$-function in curved space-time under operator regularization · The Hopf algebra · 0.88
Remarkably enough, these multizeta values play an important role in Connes, Kreimer, and Marcolli's Galois-theoretic reinterpretation [ref] of the classical BPS renormalization theory of Feynman integrals, in which MZVs appear ubiquitously in explicit computations.

The Stable Symplectic category and a conjecture of Kontsevich · Appendix: Grothendieck-Teichmüller groups · 0.89

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