2004/09/17 by Alain Connes, Matilde Marcolli, Connes, Alain +1
Mathematics · Physics and Astronomy · #11S20 #34M50 #58B34 #81T15 #81T16 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #FOS: Physical sciences #High Energy Physics - Theory (hep-th) #Homotopy and Cohomology in Algebraic Topology #Mathematical Physics (math-ph) #Number Theory (math.NT) #Quantum Algebra (math.QA) #hep-th #math-ph #math.AG #math.MP #math.NT #math.QA #msc:11S20 #msc:34M50 #msc:58B34 #msc:81T15 #msc:81T16
paper · pdf · doi:10.48550/arxiv.math/0409306
15 pages, LaTeX, 1 eps figure. To appear in IMRN
arxiv created 2004/09/17 · openalex publication_date 2004/09/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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. The renormalization group can be identified canonically with a one parameter subgroup. The group is obtained through a Riemann-Hilbert correspondence. Its representations classify equisingular flat vector bundles, where the equisingularity condition is a geometric formulation of the fact that in quantum field theory the counterterms are independent of the choice of a unit of mass. As an algebraic group scheme, it is a semi-direct product by the multiplicative group of a pro-unipotent group scheme whose Lie algebra is freely generated by one generator in each positive integer degree. There is a universal singular frame in which all divergences disappear. When computed as iterated integrals, its coefficients are certain rational numbers that appear in the local index formula of Connes-Moscovici. When working with formal Laurent series over the field of rational numbers, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.