Algebro-geometric Feynman rules

TL;DR

This paper develops a framework for algebro-geometric Feynman rules using Grothendieck rings and characteristic classes, connecting Feynman graph invariants with algebraic geometry and motivic measures.

Contribution

It introduces a general method to construct algebro-geometric Feynman rules via Grothendieck rings and characteristic classes, extending the usual motivic Feynman rules.

Findings

Constructed a new motivic Feynman rule not factoring through the usual Grothendieck ring.

Derived a formula for characteristic classes of joins of projective varieties.

Connected Feynman rules with algebraic geometry and motivic zeta functions.

Abstract

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. The main result underlying the construction of this invariant is a formula for the characteristic classes of the join of two projective varieties. We discuss the BPHZ renormalization procedure in this algebro-geometric context and some motivic zeta functions arising from the partition functions associated to motivic Feynman rules.