Cohomology of graph hypersurfaces associated to certain Feynman graphs

2008/11/03 by Dmitry Doryn, Dzmitry Doryn, Doryn, Dzmitry · 1 citation
Mathematics · #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #math.AG

paper · pdf · doi:10.48550/arxiv.0811.0402

111 pages, Ph.D. thesis at the University of Duisburg-Essen

arxiv created 2008/11/03 · openalex publication_date 2008/11/03 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

To any Feynman graph (with 2n edges) we can associate a hypersurface X⊂\PP^2n-1. We study the middle cohomology H^2n-2(X) of such hypersurfaces. S. Bloch, H. Esnault, and D. Kreimer (Commun. Math. Phys. 267, 2006) have computed this cohomology for the first series of examples, the wheel with spokes graphs WS_n, n≥ 3. Using the same technique, we introduce the generalized zigzag graphs and prove that W₅(H^2n-2(X))=\QQ(-2) for all of them (with W_* the weight filtration). Next, we study primitively log divergent graphs with small number of edges and the behavior of graph hypersurfaces under the gluing of graphs.

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• Feynman Amplitudes and Cosmic Galois group

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