Constructing polylogarithms on higher-genus Riemann surfaces

TL;DR

The paper extends the genus-zero and genus-one polylogarithm program to arbitrary genus by constructing a flat, modularly invariant connection valued in a freely generated Lie algebra, built from convolutions of the Arakelov Green function and holomorphic Abelian differentials. This yields a generating series of higher-genus polylogarithms as homotopy-invariant iterated integrals with tensorial modular properties, unifying the genus-one Brown–Levin framework with higher-genus function theory on Torelli space. The authors develop a detailed toolkit of modular tensors, generating functions, and endpoint regularization, and demonstrate the machinery through low-order examples, multi-variable extensions, and separating degenerations, while outlining strong evidence for closure under primitives and proposing higher-genus associators. The work promises applications to string perturbation theory and quantum field theory, providing a concrete analytic handle on polylogarithmic structures on complex curves beyond genus one.

Abstract

An explicit construction is presented of homotopy-invariant iterated integrals on a Riemann surface of arbitrary genus in terms of a flat connection valued in a freely generated Lie algebra. The integration kernels consist of modular tensors, built from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials, combined into a flat connection. Our construction thereby produces explicit formulas for polylogarithms as higher-genus modular tensors. This construction generalizes the elliptic polylogarithms of Brown-Levin, and prompts future investigations into the relation with the function spaces of higher-genus polylogarithms in the work of Enriquez-Zerbini.

Figures (2)

• Figure 1: A choice of canonical homology basis on a compact genus-two Riemann surface $\Sigma$.
• Figure 2: Funnel construction of a family of genus-two Riemann surfaces $\Sigma$ near the separating divisor in terms of genus-one surfaces $\Sigma _1$ and $\Sigma _2$. The circles $\partial \mathfrak{D}_1$ and $\partial \mathfrak{D}_2$ are centered at the punctures $p_1$ and $p_2$ and bound the discs $\mathfrak{D}_1$ and $\mathfrak{D}_2$, respectively. The surface $\Sigma$ is constructed from the surfaces $\Sigma _1 \setminus \mathfrak{D}_1$ and $\Sigma _2 \setminus \mathfrak{D}_2$ by identifying $\partial \mathfrak{D}_1$ and $\partial \mathfrak{D}_2$.