[Paper Review] Constructing polylogarithms on higher-genus Riemann surfaces

Eric D’Hoker, Martijn Hidding+1 more|arXiv (Cornell University)|Jun 14, 2023

Advanced Mathematical Identities4 citations

TL;DR

This paper presents an explicit construction of homotopy-invariant iterated integrals—higher-genus polylogarithms—on compact Riemann surfaces of arbitrary genus using a flat connection built from modular tensors. The kernels are derived from convolutions of the Arakelov Green function and its derivatives with holomorphic Abelian differentials, generalizing elliptic polylogarithms and providing a tractable framework for higher-genus functions in quantum field theory and string theory.

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.

Motivation & Objective

• To close the gap in the explicit construction of polylogarithms on higher-genus Riemann surfaces beyond genus one.
• To provide a systematic, algorithmic framework for iterated integrals that remain invariant under homotopy, essential for physical amplitudes.
• To generalize the integration kernels and flat connection of Brown and Levin to arbitrary genus using modular tensor structures.
• To establish a bridge between the abstract function spaces of Enriquez-Zerbini and concrete, computable expressions for physics applications.
• To ensure the resulting polylogarithms transform covariantly under modular transformations and inherit closure under integration.

Proposed method

• Construct a flat connection valued in a freely generated Lie algebra using modular tensors derived from the Arakelov Green function and holomorphic Abelian differentials.
• Define integration kernels as convolutions of the Arakelov Green function and its derivatives with holomorphic one-forms, forming modular tensor-valued differential forms.
• Build iterated integrals via path-ordered exponentials of the flat connection, ensuring homotopy invariance through flatness.
• Use generating functions and index-raising via the matrix $Y^{IJ}$ to construct higher-weight polylogarithms from basic building blocks.
• Analyze degeneration limits of the Green function and its derivatives in separating degenerations (e.g., genus two to two tori), confirming consistency with genus-one limits.
• Verify that the resulting polylogarithms exhibit logarithmic singularities and modular transformation properties analogous to lower-genus cases.

Experimental results

Research questions

• RQ1 How can one explicitly construct homotopy-invariant iterated integrals on higher-genus Riemann surfaces using modular tensor structures?
• RQ2 What is the generalization of the elliptic polylogarithm integration kernel to arbitrary genus, and how does it maintain flatness and modularity?
• RQ3 How do the higher-genus polylogarithms behave under separating degenerations, and do they reduce consistently to genus-one or genus-zero limits?
• RQ4 Can the function space of higher-genus polylogarithms be realized concretely through a flat connection built from Arakelov Green functions and Abelian differentials?
• RQ5 What is the relationship between this construction and the abstract function space framework of Enriquez-Zerbini in the mathematics literature?

Key findings

• The paper constructs explicit higher-genus polylogarithms as iterated integrals of a flat connection built from modular tensors of the Arakelov Green function and holomorphic Abelian differentials.
• The integration kernels exhibit logarithmic singularities analogous to those in genus-one polylogarithms, ensuring physical relevance.
• In the separating degeneration limit, the higher-genus Green function derivatives reduce to genus-one expressions: $\partial_{x_1}{\cal G}^1(x_1,y_1) \to \partial_{x_1}^2 g_2(x_1 - y_1|\tau) - \frac{1}{2}\partial_{x_1}^2 g_2(x_1 - p_1|\tau)$, confirming consistency.
• The construction preserves modular transformation properties, with the resulting polylogarithms transforming as tensorial modular forms.
• The flat connection ensures homotopy invariance of the iterated integrals, enabling algorithmic integration and generalizing the genus-zero and genus-one cases.
• The method provides a concrete realization of the function space proposed by Enriquez and Zerbini, offering a path toward physical applications in higher-loop amplitudes.