Iterated integrals on Riemann surfaces of different genus

The multitude of integrals encountered in perturbative quantum field theory and string theory calls for systematic studies of function spaces that close under taking primitives. For integrations on the sphere — Riemann surfaces of genus zero — this is accomplished by multiple polylogarithms which had huge impact on Feynman integrals and string amplitudes. The more challenging iterated integrals on genus-one surfaces seen in various physics contexts were recently harnessed by means of elliptic polylogarithms. After reviewing the Brown-Levin formulation of elliptic polylogarithms, I will present a generalization to Riemann surfaces of arbitrary genus. The construction of higher-genus polylogarithms in this talk is based on a flat connection built from convolutions of Arakelov Green functions on the surface.