Motives Associated to Sums of Graphs

In quantum field theory, the path integral is interpreted perturbatively as a sum indexed by graphs. The coefficient (Feynman amplitude) associated to a graph Γ is a period associated to the motive given by the complement of a certain hypersurface XΓ in projective space. Based on considerable numerical evidence, Broadhurst and Kreimer suggested [4] that the Feynman amplitudes should be sums of multizeta numbers. On the other hand, Belkale and Brosnan [2] showed that the motives of the XΓ were not in general mixed Tate. A recent paper of Aluffi and Marcolli [1] studied the images [XΓ] of graph hypersurfaces in the Grothendieck ring K0(V ark) of varieties over a field k. Let Z[Ak] ⊂ K0(V ark) be the subring generated by 1 = [Spec k] and [Ak]. It follows from [2] that [XΓ] 6∈ Z[Ak] for many graphs Γ. Let n ≥ 3 be an integer. In this note we consider a sum Sn ∈ K0(V ark) of [XΓ] over all connected graphs Γ with n vertices, no multiple edges, and no tadpoles (edges with just one vertex). (There are some subtleties here. Each graph Γ appears with multiplicity n!/|Aut(Γ)|. For a precise definition of Sn see (5.1) below.) Our main result is