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Algebro-Geometric Feynman Rules

Aluffi, Paolo and Marcolli, Matilde (2011) Algebro-Geometric Feynman Rules. International Journal of Geometric Methods in Modern Physics, 8 (1). pp. 203-237. ISSN 0219-8878. https://resolver.caltech.edu/CaltechAUTHORS:20111205-113131774

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Abstract

We give a general procedure to construct "algebro-geometric Feynman rules", that is, characters of the Connes–Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface. In particular, this maps to the usual Grothendieck ring of varieties, defining "motivic Feynman rules". We also construct an algebro-geometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties. This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebro-geometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.1142/S0219887811005099DOIArticle
http://www.worldscinet.com/ijgmmp/08/0801/S0219887811005099.htmlPublisherArticle
https://arxiv.org/abs/0811.2514arXivDiscussion Paper
Additional Information:© 2011 World Scientific Publishing Co. Received 24 September 2010. Accepted 26 September 2010.
Subject Keywords:Feynman rules; parametric Feynman integrals; graph hypersurfaces; Grothendieck ring of varieties; Chern–Schwartz–MacPherson classes.
Issue or Number:1
Record Number:CaltechAUTHORS:20111205-113131774
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20111205-113131774
Official Citation:ALGEBRO-GEOMETRIC FEYNMAN RULES PAOLO ALUFFI and MATILDE MARCOLLI DOI No: 10.1142/S0219887811005099 Page: 203-237
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:28295
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:05 Dec 2011 21:01
Last Modified:03 Oct 2019 03:31

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