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Higher-genus Fay-like identities from meromorphic generating functions

Konstantin Baune, Johannes Broedel, Egor Im, Artyom Lisitsyn, Yannis Moeckli

TL;DR

This work develops a higher-genus formalism for polylogarithms built from Enriquez's meromorphic generating function by introducing component forms and a Hopf-algebra framework to manage quasi-periodicities and poles. It derives a complete set of Fay-like quadratic identities, including index-swapping and $z$-reduction techniques, and proves the uniqueness of these kernel identities up to trivial linear relations. The authors connect these kernel identities to functional relations among higher-genus polylogarithms and demonstrate consistency with genus-one Fay identities, while outlining methods to remove $z$ from polylogarithm labels to produce genus-zero MZV-type relations. They also discuss implications for polylogarithm closure, potential links to Fay trisecant equations, and future directions toward higher-genus associators and MZV structures.

Abstract

A possible way of constructing polylogarithms on Riemann surfaces of higher genera facilitates integration kernels, which can be derived from generating functions incorporating the geometry of the surface. Functional relations between polylogarithms rely on identities for those integration kernels. In this article, we derive identities for Enriquez' meromorphic generating function and investigate the implications for the associated integration kernels. The resulting identities are shown to be exhaustive and therefore reproduce all identities for Enriquez' kernels conjectured in arXiv:2407.11476 recently.

Higher-genus Fay-like identities from meromorphic generating functions

TL;DR

This work develops a higher-genus formalism for polylogarithms built from Enriquez's meromorphic generating function by introducing component forms and a Hopf-algebra framework to manage quasi-periodicities and poles. It derives a complete set of Fay-like quadratic identities, including index-swapping and -reduction techniques, and proves the uniqueness of these kernel identities up to trivial linear relations. The authors connect these kernel identities to functional relations among higher-genus polylogarithms and demonstrate consistency with genus-one Fay identities, while outlining methods to remove from polylogarithm labels to produce genus-zero MZV-type relations. They also discuss implications for polylogarithm closure, potential links to Fay trisecant equations, and future directions toward higher-genus associators and MZV structures.

Abstract

A possible way of constructing polylogarithms on Riemann surfaces of higher genera facilitates integration kernels, which can be derived from generating functions incorporating the geometry of the surface. Functional relations between polylogarithms rely on identities for those integration kernels. In this article, we derive identities for Enriquez' meromorphic generating function and investigate the implications for the associated integration kernels. The resulting identities are shown to be exhaustive and therefore reproduce all identities for Enriquez' kernels conjectured in arXiv:2407.11476 recently.
Paper Structure (37 sections, 9 theorems, 91 equations)

This paper contains 37 sections, 9 theorems, 91 equations.

Table of Contents

  1. Introduction
  2. Enriquez' connection
  3. Linear 3-point identities
  4. Component forms
  5. Hopf algebra
  6. First component form
  7. Second component form
  8. Hopf algebra operations on component forms
  9. Antipode.
  10. Coproduct.
  11. Product and permutation.
  12. Contractions of component forms.
  13. Identities for component forms
  14. Differential 2-point identities
  15. Quadratic 3-point identities
  16. ...and 22 more sections

Key Result

Proposition 2

Any linear 3-point identity between integration kernels can be written as a linear combination of identities eq. (eqn:non-matching ) and eq. (eqn:matching ).

Theorems & Definitions (14)

  • Definition 1: Enriquez--ZerbiniEZ1
  • Proposition 2: Triviality of linear identities
  • Definition 3
  • Definition 4
  • Definition 5
  • Proposition 6
  • Lemma 7
  • Lemma 8
  • Theorem 9: Fay-like identity
  • Definition 10
  • ...and 4 more