Algebro-geometric Feynman rules

Understand

We give a general procedure to construct algebro-geometric Feynman rules, that is, characters of the Connes-Kreimer Hopf algebra of Feynman graphs that factor through a Grothendieck ring of immersed conical varieties, via the class of the complement of the affine graph hypersurface.

• In particular, this maps to the usual Grothendieck ring of varieties, defining motivic Feynman rules.
• We also construct an algebro-geometric Feynman rule with values in a polynomial ring, which does not factor through the usual Grothendieck ring, and which is defined in terms of characteristic classes of singular varieties.
• This invariant recovers, as a special value, the Euler characteristic of the projective graph hypersurface complement.