Framings for graph hypersurfaces
Francis Brown, Dzmitry Doryn
TL;DR
This work develops a method to compute the framing on the cohomology of graph hypersurfaces defined by the Feynman differential form $\omega_G$, and proves that for denominator-reducible graphs the maximal weight piece is Tate with $\mathrm{gr}^W_{2N_G-6} H^{N_G-1}({\mathbb P}^{N_G-1}\setminus X_G) \cong {\mathbb Q}(3-N_G)$, with the de Rham framing generated by $[\omega_G]$. Applying the method to Brown–Schnetz modular graphs shows the Feynman form is not Tate in general, providing a non-Tate counterexample at eight loops and thereby disproving the folklore conjecture that Feynman periods factor through mixed Tate motives. The paper also establishes a robust set of denominator-reduction tools—via Dodgson polynomials and a sequence of denominators $D_k$—that translate questions about the cohomology of $X_G$ to simpler hypersurfaces, clarifying when top-weight pieces are Tate and when non-Tate contributions arise. The results imply that, while many graph families yield Tate framings, non-Tate phenomena can appear in middle cohomology, and their presence constrains period factorization in quantum-field-theory contexts, with a corollary suggesting the top weight of certain theories could still be mixed-Tate in remote cases.
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.