Constructing polylogarithms on higher-genus Riemann surfaces

Eric D'Hoker; Martijn Hidding; Oliver Schlotterer

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arXiv · 2306.08644

Constructing polylogarithms on higher-genus Riemann surfaces

Eric D'Hoker·Martijn Hidding·Oliver Schlotterer

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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.

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Eric D'Hoker, Martijn Hidding, Oliver Schlotterer. 2023-06-14. Constructing polylogarithms on higher-genus Riemann surfaces. https://arxiv.org/abs/2306.08644

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