Modular forms in Quantum Field Theory

TL;DR

This paper explores the connection between Feynman graph amplitudes in Quantum Field Theory and modular forms, revealing that many point counts over finite fields correspond to Fourier coefficients of modular forms of specific weights and levels.

Contribution

It provides an experimental analysis linking graph hypersurface point counts to modular forms, extending understanding of their arithmetic properties in quantum field theory.

Findings

Many graph hypersurface point counts match Fourier coefficients of modular forms.

Point counts are related to modular forms of weights up to 8 and levels up to 17.

The study covers graphs up to 10 loops, revealing deep arithmetic structures.

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_q modulo q^3, 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.