Higher genus polylogarithms on families of Riemann surfaces

TL;DR

This paper develops higher genus polylogarithms on families of Riemann surfaces, enabling the study of monodromies of meromorphic connections and providing a method to compute these polylogarithms as power series in deformation parameters.

Contribution

It introduces a construction of higher genus polylogarithms on Riemann surface families, linking them to monodromies of meromorphic connections and offering a computational approach.

Findings

Polylogarithms describe monodromies of meromorphic connections.

Polylogarithms are computable as power series in deformation parameters.

The construction applies to families of Riemann surfaces of any genus.

Abstract

We construct polylogarithms on families of pointed Riemann surfaces of any genus which describe monodromies of meromorphic connections with simple poles. Furthermore, we show that the polylogaritms are computable as power series in deformation parameters and their logarithms associated with the families.