Constructing polylogarithms on higher-genus Riemann surfaces
Abstract
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. The integration kernels consist of modular tensors, built from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials, combined into a flat connection. Our construction thereby produces explicit formulas for polylogarithms as higher-genus modular tensors. This construction generalizes the elliptic polylogarithms of Brown-Levin, and prompts future investigations into the relation with the function spaces of higher-genus polylogarithms in the work of Enriquez-Zerbini.
Cite
@article{arxiv.2306.08644,
title = {Constructing polylogarithms on higher-genus Riemann surfaces},
author = {Eric D'Hoker and Martijn Hidding and Oliver Schlotterer},
journal= {arXiv preprint arXiv:2306.08644},
year = {2025}
}
Comments
55 pages, 2 figures; v2: references added, expanded the discussion of modular properties in sections 3 and 4; v3: restructured Theorem 6.3, small improvements throughout; version to be published in CNTP up to minor formatting-related rearrangements