2004/11/11 by Alain Connes, Matilde Marcolli, Connes, Alain +1
Mathematics · Physics and Astronomy · #Algebraic Geometry (math.AG) #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Mathematical Physics (math-ph) #Number Theory (math.NT) #Quantum Algebra (math.QA) #hep-th #math-ph #math.AG #math.MP #math.NT #math.QA
paper · pdf · doi:10.48550/arxiv.hep-th/0411114
97 pages LaTeX, 17 eps figures
arxiv created 2004/11/11 · arxiv updated 2009/12/01
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.