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

97 pages LaTeX, 17 eps figures

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• p-Adic Mathematical Physics
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• Shuffle relations for regularised integrals of symbols
• Quantum fields and motives
• The Hopf algebra of Feynman graphs in QED
• The algebra of integro-differential operators on a polynomial algebra
• Feynman integrals and motives
• Algebraic Birkhoff decomposition and its applications
• Matrix Representation of Renormalization in Perturbative Quantum Field Theory
• Renormalization of multiple zeta values
• Feynman integrals and motives of configuration spaces
• Differential Birkhoff decomposition and the renormalization of multiple zeta values
• The combinatorics of Bogoliubov's recursion in renormalization
• Representations of Rota-Baxter algebras and regular singular decompositions
• Toward a fundamental groupoid for the stable homotopy category
• Noncommutative geometry and motives: the thermodynamics of endomotives
• Double shuffle relations and renormalization of multiple zeta values
• Cohomologies, extensions and deformations of differential algebras with any weights
• Universal Hopf Algebra of Renormalization and Hopf Algebras of Rooted Trees
• On Differential Rota-Baxter Algebras
• Motives: an introductory survey for physicists
• Rota-Baxter operators on generalized power series rings
• Dynkin operators and renormalization group actions in pQFT