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Published June 2010 | Published
Journal Article Open

Parametric Feynman integrals and determinant hypersurfaces

Abstract

The purpose of this paper is to show that, under certain combinatorial conditions on the graph, parametric Feynman integrals can be realized as periods on the complement of the determinant hypersurface in an affine space depending on the number of loops of the Feynman graph. The question of whether the Feynman integrals are periods of mixed Tate motives can then be reformulated (modulo divergences) as a question on a relative cohomology being a realization of a mixed Tate motive. This is the cohomology of the pair of the determinant hypersurface complement and a normal crossings divisor depending only on the number of loops and the genus of the graph. We show explicitly that this relative cohomology is a realization of a mixed Tate motive in the case of three loops and we give alternative formulations of the main question in the general case, by describing the locus of intersection of the divisor with the determinant hypersurface complement in terms of intersections of unions of Schubert cells in flag varieties. We also discuss different methods of regularization aimed at removing the divergences of the Feynman integral.

Additional Information

© 2010 International Press. The first author acknowledges partial support by NSA grant H98230-07-1-0024. The second author is partially supported by NSF grants DMS-0651925, DMS-0901221, and DMS-1007207. This work was partly carried out during stays of the authors at the MPI and MSRI. The first author also thanks the California Institute of Technology, where part of this work was done.

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