Abstract:

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.