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Supermanifolds from Feynman graphs

Marcolli, Matilde and Rej, Abhijnan (2008) Supermanifolds from Feynman graphs. Journal of Physics A: Mathematical and Theoretical, 41 (31). p. 315402. ISSN 1751-8113. doi:10.1088/1751-8113/41/31/315402.

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We generalize the computation of Feynman integrals of log divergent graphs in terms of the Kirchhoff polynomial to the case of graphs with both fermionic and bosonic edges, to which we assign a set of ordinary and Grassmann variables. This procedure gives a computation of the Feynman integrals in terms of a period on a supermanifold, for graphs admitting a basis of the first homology satisfying a condition generalizing the log divergence in this context. The analog in this setting of the graph hypersurfaces is a graph supermanifold given by the divisor of zeros and poles of the Berezinian of a matrix associated with the graph, inside a superprojective space. We introduce a Grothendieck group for supermanifolds and identify the subgroup generated by the graph supermanifolds. This can be seen as a general procedure for constructing interesting classes of supermanifolds with associated periods.

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Additional Information:Copyright © Institute of Physics and IOP Publishing Limited 2008. Received 4 April 2008, in final form 10 June 2008. Published 4 July 2008. The first author is partially supported by NSF grant DMS-0651925. The second author is supported as a Marie Curie Early Stage Researcher at Durham University and by the Clay Mathematical Institute.
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European CommissionUNSPECIFIED
Clay Mathematical InstituteUNSPECIFIED
Issue or Number:31
Record Number:CaltechAUTHORS:MARjpa08
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:11978
Deposited By: Archive Administrator
Deposited On:15 Oct 2008 22:19
Last Modified:12 Jul 2022 19:44

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