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.