Modular forms in quantum field theory

Abstract

The amplitude of a Feynman graph in Quantum Field Theory is related to the point-count over finite fields of the corresponding graph hypersurface. This article reports on an experimental study of point counts over F{double-struck}q modulo q3, for graphs up to loop order 10. It is found that many of them are given by Fourier coefficients of modular forms of weights ≤8 and levels ≤17.