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Feynman motives and deletion-contraction relations

Aluffi, Paolo and Marcolli, Matilde (2011) Feynman motives and deletion-contraction relations. In: Topology of Algebraic Varieties and Singularities. Contemporary Mathematics . No.538. American Mathematical Society , Providence, RI, pp. 21-64. ISBN 978-0-8218-4890-6.

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We prove a deletion-contraction formula for motivic Feynman rules given by the classes of the affine graph hypersurface complement in the Grothendieck ring of varieties. We derive explicit recursions and generating series for these motivic Feynman rules under the operation of multiplying edges in a graph and we compare it with similar formulae for the Tutte polynomial of graphs, both being specializations of the same universal recursive relation. We obtain similar recursions for outerplanar graphs (given in full for chains of polygons) and for graphs obtained by replacing an edge by a chain of triangles. We show that the deletion-contraction relation can be lifted to the level of the category of mixed motives in the form of a distinguished triangle, similarly to what happens in categorfications of graph invariants.

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Additional Information:© 2011 American Mathematical Society. The results in this article were catalyzed by a question posed to the first author by Michael Falk during the Jaca 'Lib60ber' conference, in June 2009. We thank him and the organizers of the conference, and renew our best wishes to Anatoly Libgober. We thank Friedrich Hirzebruch for pointing out Remark 5.4 to us, thereby triggering the thoughts leading to much of the material in §6. We also thank Don Zagier for a useful conversation on divisibility sequences and generating functions. The second author is partially supported by NSF grants DMS-0651925 and DMS-0901221. This work was carried out during a stay of the authors at the Max Planck Institute in July 2009.
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Series Name:Contemporary Mathematics
Issue or Number:538
Record Number:CaltechAUTHORS:20110613-094812678
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:23983
Deposited By: Tony Diaz
Deposited On:15 Jun 2011 17:27
Last Modified:03 Oct 2019 02:52

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