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

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.