Cohomology of graph hypersurfaces associated to certain Feynman graphs

Details Cohomology of graph hypersurfaces associated to certain Feynman graphs Cohomology of graph hypersurfaces associated to certain Feynman graphs

2008-11-03

arxiv.org/abs/0811.0402v1

Mathematics

Algebraic Geometry

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

Scientific paper

To any Feynman graph (with 2n edges) we can associate a hypersurface X\subset\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\geq 3. Using the same technique, we introduce the generalized zigzag graphs and prove that W_5(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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