Modular forms in Quantum Field Theory

Francis Brown; Oliver Schnetz

Modular forms in Quantum Field Theory

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arxiv 1304.5342 v2 pith:SFH6CLN3 submitted 2013-04-19 math.AG hep-th

Francis Brown , Oliver Schnetz This is my paper · ORCID

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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.

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Quasi-finite Feynman integrals produce sparse Fano and reflexive polytopes that encode degenerate Calabi-Yau varieties and link to del Pezzo surfaces, K3 surfaces, and Calabi-Yau threefolds.

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