Single-valued flat connections in several variables on arbitrary Riemann surfaces
Eric D'Hoker, Oliver Schlotterer
TL;DR
This work constructs a multi-variable, single-valued flat connection ${\cal J}_{\text{DHS}}$ on the configuration space of points on a Riemann surface, capturing high-genus polylogarithms with modular invariance. It extends the prior single-variable framework to arbitrarily many variables and punctures, proving flatness and modular invariance, and shows how these multi-variable polylogarithms relate to Enriquez’s meromorphic, multi-variable connection ${\cal K}_{\text{E}}$ via a gauge transformation plus a Lie-algebra automorphism. The authors provide a detailed, self-contained treatment of the multi-variable DHS connection and its relation to Enriquez’s construction, including an alternative direct proof of flatness for ${\cal K}_{\text{E}}$ and a rigorous matching of poles and residues between the two formalisms. A companion paper is announced to prove the equivalence between flatness identities (interchange and Fay identities) and the DHS/Enriquez kernels for arbitrary genus and number of variables. Collectively, the results deepen the algebraic and analytic understanding of higher-genus polylogarithms and their flat-connection representations, with implications for motivic coaction and single-valued maps in the higher-genus setting.
Abstract
Polylogarithms on Riemann surfaces may be constructed efficiently in terms of flat connections that can enjoy various algebraic and analytic properties. In this paper, we present a single-valued and modular invariant connection ${\cal J}_\text{DHS}$ on the configuration space $\text{Cf}_n(Σ)$ of an arbitrary number $n$ of points on an arbitrary compact Riemann surface $Σ$ with or without punctures. The connection ${\cal J}_\text{DHS}$ generalizes an earlier construction for a single variable and is built out of the same integration kernels. We show that ${\cal J}_\text{DHS}$ is flat on $\text{Cf}_n(Σ)$. For the case without punctures, we relate it to the meromorphic multiple-valued Enriquez connection ${\cal K}_\text{E}$ in $n$ variables on the universal cover $\tilde Σ$ of $Σ$ by the composition of a gauge transformation and an automorphism of the Lie algebra in which ${\cal J}_\text{DHS}$ and ${\cal K}_\text{E}$ take values. In a companion paper, we shall establish the equivalence between the flatness of these connections and the corresponding interchange and Fay identities, for arbitrary compact Riemann surfaces.