Framings for graph hypersurfaces

TL;DR

This paper introduces a method to compute the framing on the cohomology of graph hypersurfaces, revealing that Feynman integrals do not always factor through mixed Tate motives, challenging previous conjectures.

Contribution

The paper provides a new method for computing framings on graph hypersurfaces and demonstrates that Feynman integrals are not universally of Tate type, disproving a longstanding folklore conjecture.

Findings

Framings can be computed for an infinite class of graphs, confirming they are Tate motives.

Feynman differential forms are not always of Tate type, especially for modular graphs.

The folklore conjecture that Feynman periods factor through mixed Tate motives is disproved.

Abstract

We present a method for computing the framing on the cohomology of graph hypersurfaces defined by the Feynman differential form. This answers a question of Bloch, Esnault and Kreimer in the affirmative for an infinite class of graphs for which the framings are Tate motives. Applying this method to the modular graphs of Brown and Schnetz, we find that the Feynman differential form is not of Tate type in general. This finally disproves a folklore conjecture stating that the periods of Feynman integrals of primitive graphs in phi^4 theory factorise through a category of mixed Tate motives.