Renormalization and motivic Galois theory
Abstract
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 RiemannHilbert 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 semidirect product by the multiplicative group of a prounipotent 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 ConnesMoscovici. 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 September 2004
 DOI:
 10.48550/arXiv.math/0409306
 arXiv:
 arXiv:math/0409306
 Bibcode:
 2004math......9306C
 Keywords:

 Mathematics  Number Theory;
 Mathematical Physics;
 Mathematics  Algebraic Geometry;
 Mathematics  Mathematical Physics;
 Mathematics  Quantum Algebra;
 High Energy Physics  Theory;
 58B34;
 11S20;
 34M50;
 81T15;
 81T16
 EPrint:
 15 pages, LaTeX, 1 eps figure. To appear in IMRN