Feynman motives of banana graphs

TL;DR

This paper analyzes the algebraic and geometric properties of banana graph Feynman diagrams, computing their classes in the Grothendieck ring and characteristic classes, revealing structural patterns and proposing conjectures.

Contribution

It provides explicit computations of graph hypersurface classes and characteristic classes for banana graphs, introducing new formulas and conjectures in the algebraic geometry of Feynman diagrams.

Findings

Explicit formulas for classes of banana graph hypersurfaces

Identification of structural similarities in graph operations

Proposal of a positivity conjecture for characteristic classes

Abstract

We consider the infinite family of Feynman graphs known as the "banana graphs" and compute explicitly the classes of the corresponding graph hypersurfaces in the Grothendieck ring of varieties as well as their Chern-Schwartz-MacPherson classes, using the classical Cremona transformation and the dual graph, and a blowup formula for characteristic classes. We outline the interesting similarities between these operations and we give formulae for cones obtained by simple operations on graphs. We formulate a positivity conjecture for characteristic classes of graph hypersurfaces and discuss briefly the effect of passing to noncommutative spacetime.