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Fay identities for polylogarithms on higher-genus Riemann surfaces

Eric D'Hoker, Oliver Schlotterer

TL;DR

The paper develops Fay identities for polylogarithms on higher-genus Riemann surfaces by constructing a modularly covariant flat connection from higher-genus integration kernels. It proves infinite families of bilinear and tensorial Fay identities among the kernels, enabling closure of polylogarithm spaces under integration and change of fibration bases. Scalar and tensor Fay identities, along with their coincident and meromorphic extensions, are derived and conjectured where explicit proofs are challenging, including the Enriquez meromorphic kernels. The framework yields a structured approach to primitives of higher-genus polylogarithms, with significant implications for multi-variable polylogarithms, modular tensors, Eisenstein-series-like objects, and potential applications to string amplitudes and high-energy computations.

Abstract

A recent construction of polylogarithms on Riemann surfaces of arbitrary genus in arXiv:2306.08644 is based on a flat connection assembled from single-valued non-holomorphic integration kernels that depend on two points on the Riemann surface. In this work, we construct and prove infinite families of bilinear relations among these integration kernels that are necessary for the closure of the space of higher-genus polylogarithms under integration over the points on the surface. Our bilinear relations generalize the Fay identities among the genus-one Kronecker-Eisenstein kernels to arbitrary genus. The multiple-valued meromorphic kernels in the flat connection of Enriquez are conjectured to obey higher-genus Fay identities of exactly the same form as their single-valued non-holomorphic counterparts. We initiate the applications of Fay identities to derive functional relations among higher-genus polylogarithms involving either single-valued or meromorphic integration kernels.

Fay identities for polylogarithms on higher-genus Riemann surfaces

TL;DR

The paper develops Fay identities for polylogarithms on higher-genus Riemann surfaces by constructing a modularly covariant flat connection from higher-genus integration kernels. It proves infinite families of bilinear and tensorial Fay identities among the kernels, enabling closure of polylogarithm spaces under integration and change of fibration bases. Scalar and tensor Fay identities, along with their coincident and meromorphic extensions, are derived and conjectured where explicit proofs are challenging, including the Enriquez meromorphic kernels. The framework yields a structured approach to primitives of higher-genus polylogarithms, with significant implications for multi-variable polylogarithms, modular tensors, Eisenstein-series-like objects, and potential applications to string amplitudes and high-energy computations.

Abstract

A recent construction of polylogarithms on Riemann surfaces of arbitrary genus in arXiv:2306.08644 is based on a flat connection assembled from single-valued non-holomorphic integration kernels that depend on two points on the Riemann surface. In this work, we construct and prove infinite families of bilinear relations among these integration kernels that are necessary for the closure of the space of higher-genus polylogarithms under integration over the points on the surface. Our bilinear relations generalize the Fay identities among the genus-one Kronecker-Eisenstein kernels to arbitrary genus. The multiple-valued meromorphic kernels in the flat connection of Enriquez are conjectured to obey higher-genus Fay identities of exactly the same form as their single-valued non-holomorphic counterparts. We initiate the applications of Fay identities to derive functional relations among higher-genus polylogarithms involving either single-valued or meromorphic integration kernels.
Paper Structure (107 sections, 23 theorems, 356 equations)

This paper contains 107 sections, 23 theorems, 356 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Results obtained after the first archive version of this paper
  4. Acknowledgments
  5. Motivation: Fay identities at genus zero and one
  6. Partial fraction decomposition at genus zero
  7. Kronecker-Eisenstein series at genus one
  8. Three-point Fay identities at genus one
  9. Three-point Fay identities in terms of $f^{(r)}$
  10. Three-point Fay identities in terms of $g^{(r)}$
  11. Higher-point Fay identities at genus one
  12. The Arakelov Green function and polylogarithms
  13. Homology, cohomology and $Sp(2h,\mathbb Z)$ basics
  14. Higher-genus integration kernels
  15. Anti-holomorphic derivatives
  16. ...and 92 more sections

Key Result

Theorem 4.1

The three-point Fay identity that is a scalar under modular transformations states that the following combination, which is a $(1,0)$ for in $x,y$ and a $(0,0)$ form in $z$, vanishes identically on a Riemann surface $\Sigma$ of arbitrary genus, Recall that the ingredients of (3.a.2) were defined in section sec:2, and we will see in section sec:3 that the last term $\partial_x \partial_y {\cal G}

Theorems & Definitions (30)

  • Theorem 4.1
  • Theorem 4.2
  • Theorem 4.3
  • Lemma 5.1
  • Theorem 5.2
  • Corollary 5.3
  • Lemma 6.1
  • Theorem 6.2
  • Theorem 6.3
  • Lemma 6.4
  • ...and 20 more