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

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.

💡 Research Summary

The paper develops a single‑valued, modular‑invariant flat connection 𝒥_DHS on the configuration space Cfₙ(Σ) of n points on an arbitrary compact Riemann surface Σ, possibly with punctures. Building on the previously known single‑variable DHS connection, the authors extend the construction to several variables using the same integration kernels f_{I₁…I_r J}(x,y) derived from the Arakelov Green function and holomorphic Abelian differentials ω_I. The connection takes values in the completed Lie algebra 𝔱̂_{h,n} (or its punctured version 𝔱̂_{h,n,p}), a non‑freely generated algebra introduced by Enriquez, whose defining relations are crucial for the flatness proof.

The flatness condition d𝒥_DHS + 𝒥_DHS∧𝒥_DHS = 0 is proved in two steps. First, the differential equations satisfied by the DHS kernels (∂̄_x f = …, ∂̄_y f = …) are used to expand the exterior derivative of 𝒥_DHS. Second, the Lie‑algebraic relations of 𝔱̂_{h,n} are invoked to show that all commutator terms vanish, in particular that