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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## 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” 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.

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Additional Information: | © 2004 Hindawi Publishing Corporation. Received September 12, 2004. Accepted November 2, 2004. Communicated by Yuri I. Manin. | |||||||||

Record Number: | CaltechAUTHORS:20110817-140437615 | |||||||||

Persistent URL: | http://resolver.caltech.edu/CaltechAUTHORS:20110817-140437615 | |||||||||

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Deposited By: | Tony Diaz | |||||||||

Deposited On: | 17 Oct 2011 16:53 | |||||||||

Last Modified: | 17 Oct 2011 16:53 |

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