New realizations of modular forms in Calabi-Yau threefolds arising from $\phi^4$ theory

Abstract

It has been found experimentally by Brown and Schnetz that the number of points over ${\mathbb F}_p$ of a graph hypersurface is often related to the coefficients of a modular form. In this paper I prove this relation for one example of a modular form of weight $4$ and two of weight $3$, refine the statement and suggest a method of proving it for four more of weight $4$, and use the one proved example to construct two new rigid Calabi-Yau threefolds that realize Hecke eigenforms of weight $4$ (one provably and one conjecturally).