Connes, Alain and Marcolli, Matilde (2004) Renormalization and motivic galois theory. International Mathematics Research Notices, 76 . pp. 4073-4091. ISSN 1073-7928 http://resolver.caltech.edu/CaltechAUTHORS:20110817-140437615
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We investigate the nature of divergences in quantum field theory, showing that they are organized in the structure of a certain “motivic Galois group” U*, 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 of U*. The group U* arises 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, U* is a semidirect product by the multiplicative group G_m 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 Connes-Moscovici. When working with formal Laurent series over ℚ, the data of equisingular flat vector bundles define a Tannakian category whose properties are reminiscent of a category of mixed Tate motives.
|Additional Information:||© 2004 Hindawi Publishing Corporation. Received September 12, 2004. Accepted November 2, 2004. Communicated by Yuri I. Manin.|
|Usage Policy:||No commercial reproduction, distribution, display or performance rights in this work are provided.|
|Deposited By:||Tony Diaz|
|Deposited On:||17 Oct 2011 16:53|
|Last Modified:||17 Oct 2011 16:53|
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