Communications in Number Theory and Physics

Volume 7 (2013)

Number 2

Modular forms in quantum field theory

Pages: 293 – 325



Francis Brown (IHES, Bures-sur-Yvette, France)

Oliver Schnetz (Department Mathematik, Emmy-Noether-Zentrum, FAU Erlangen-Nürenberg, Erlangen, Germany)


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 $\mathbb{F}_q$ 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 $\leq 8$ and levels $\leq 17$.

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