Doryn, Dzmitry:

### Cohomology of graph hypersurfaces associad to certain Feynman graphs

Duisburg, Essen (2008), 117 Bl.
Dissertation / Fach: Mathematik
Fakultät für Mathematik
Esnault, Hélène (Doktorvater, Betreuerin)
Viehweg, Eckart (GutachterIn)
Dissertation
##### 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.

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