Modular forms in quantum field theory

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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Published 10 February 2014