Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory

• Graph complexes and Cutkosky rules.- Differential equations and dispersion relations for Feynman amplitudes with elliptic functions.- Elliptic integrals and the two-loop ttbar production in QCD.- Solutions of 2nd and 3rd order differential equations with more singularities.- Analytic continuation of Feynman diagrams with elliptic solutions.- Twisted elliptic multiple zeta values and non-planar one-loop open-string amplitudes.- Genus one superstring amplitudes and modular forms.- Difference field methods in Feynman diagram calculations.- Feynman integrals and iterated integrals of modular forms.- Iterated elliptic and hypergeometric integrals for Feynman diagrams. - Feynman integrals, L-series and Kloosterman moments.

“The book is explicitly concerned with ‘the mathematics of modular forms [as it is] of central importance for the analytic solution of Feynman diagrams,’ but modular formers should be equally interested in the material offered in this compendium.” (Michael Berg, MAA Reviews, May 19, 2019)