Renormalization and motivic Galois theory

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dc.creatorConnes, Alain
dc.creatorMarcolli, Matilde
dc.date2004-09-17
dc.date.accessioned2026-07-07T05:12:15Z
dc.date.available2026-07-07T05:12:15Z
dc.descriptionWe 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.
dc.description15 pages, LaTeX, 1 eps figure. To appear in IMRN
dc.identifierhttps://arxiv.org/abs/math/0409306
dc.identifierhttp://arxiv.org/abs/math/0409306
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/72513
dc.subjectNumber Theory
dc.subjectHigh Energy Physics - Theory
dc.subjectMathematical Physics
dc.subjectAlgebraic Geometry
dc.subjectQuantum Algebra
dc.subject58B34, 11S20, 34M50, 81T15, 81T16
dc.titleRenormalization and motivic Galois theory
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