Constructing polylogarithms on higher-genus Riemann surfaces

Authors: Eric D'Hoker, Martijn Hidding, and Oliver Schlotterer

Preprint number: UUITP-17/23

An explicit construction is presented of homotopy-invariant iterated integrals on a Riemann surface of arbitrary genus in terms of a flat connection valued in a freely generated Lie algebra. Our construction generalizes the generating series of elliptic polylogarithms in the work of Brown and Levin and thereby leads to a concrete proposal for polylogarithms at higher genus. The integration kernels are built from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials, combined into a flat connection.