Spanning Forest Polynomials and the Transcendental Weight of Feynman Graphs

Modular forms in quantum field theory

“…By combining the results of [6,9], we deduce that the piece of maximal (generic) Hodge-theoretic weight of the cohomology of the graph hypersurface is invariant under double-triangle reduction. In particular, a graph has 'weight drop' in the sense of [9] if and only if its double-triangle reductions do also.…”

“…Furthermore, there is not a single graph with c 2 = −z 3 or −z 4 whose period is known. What we do have is the conjectured period of two completed primitive graphs at loop order 7 (graphs P 7,8 and P 7,9 in [20]) with c 2 -invariant −z 2 . They are weight 11 multiple zeta values, namely [4] To summarize the results for quasi-constant graphs, the 10 loop data are consistent with the following, rather surprising, conjecture.…”

“…In particular, a graph has 'weight drop' in the sense of [9] if and only if its double-triangle reductions do also.…”

“…Using the techniques of [8,9] Every completed primitive graph with at least five vertices has a c 2 -invariant and the first two examples are…”

“…There are several other combinatorial criteria for a graph to have vanishing c 2 -invariant which are discussed in [9].…”