Single-valued flat connections in several variables on arbitrary Riemann surfaces

TL;DR

This paper constructs a single-valued, modular invariant flat connection on configuration spaces of points on arbitrary Riemann surfaces, generalizing previous single-variable cases and relating to known connections.

Contribution

It introduces a new flat connection ${ m extbf{J}}_ ext{DHS}$ on configuration spaces of Riemann surfaces, extending single-variable constructions to multiple variables and arbitrary surfaces.

Findings

${ m extbf{J}}_ ext{DHS}$ is flat on configuration spaces.

Relation established between ${ m extbf{J}}_ ext{DHS}$ and Enriquez connection ${ m extbf{K}}_ ext{E}$.

Connection generalizes earlier single-variable constructions.

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 $\text{Cf}_n(\Sigma)$ of an arbitrary number $n$ of points on an arbitrary compact Riemann surface $\Sigma$ 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 $\text{Cf}_n(\Sigma)$. 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 $\tilde \Sigma$ of $\Sigma$ 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.