Yoann Sohnle: Feynman integrals and modular forms

Abstract

In this thesis we discuss mathematical methods to expand Feynman integrals in dimensional regularisation with a particular focus on integrals related to periods of elliptic curves and Calabi-Yau manifolds. As a motivating example we apply the discussed methods to the running example of Gauss's hypergeometric function. We show how expansions in the dimensional-regularisation parameter introduce generating series of multiple polylogarithms as well as their elliptic generalisations which are closely related to (non-holomorphic) modular forms. Finally we motivate and summarise the construction of single valued and equivariant versions of elliptic multiple polylogarithms.