Connes, Alain and Marcolli, Matilde (2004) Renormalization and motivic galois theory. International Mathematics Research Notices, 76 . pp. 4073-4091. ISSN 1073-7928. doi:10.1155/S1073792804143122. https://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.
Item Type: | Article | ||||||||||||
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Additional Information: | © 2004 Hindawi Publishing Corporation. Received September 12, 2004. Accepted November 2, 2004. Communicated by Yuri I. Manin. | ||||||||||||
DOI: | 10.1155/S1073792804143122 | ||||||||||||
Record Number: | CaltechAUTHORS:20110817-140437615 | ||||||||||||
Persistent URL: | https://resolver.caltech.edu/CaltechAUTHORS:20110817-140437615 | ||||||||||||
Usage Policy: | No commercial reproduction, distribution, display or performance rights in this work are provided. | ||||||||||||
ID Code: | 24916 | ||||||||||||
Collection: | CaltechAUTHORS | ||||||||||||
Deposited By: | Tony Diaz | ||||||||||||
Deposited On: | 17 Oct 2011 16:53 | ||||||||||||
Last Modified: | 09 Nov 2021 16:27 |
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