Fay identities for polylogarithms on higher-genus Riemann surfaces

TL;DR

This paper extends Fay identities to higher-genus Riemann surfaces, establishing bilinear relations among polylogarithm kernels that generalize genus-one identities, crucial for understanding higher-genus polylogarithms.

Contribution

It constructs and proves infinite families of Fay identities for higher-genus polylogarithms, generalizing known genus-one relations to arbitrary genus surfaces.

Findings

Established bilinear relations among integration kernels

Generalized Fay identities to arbitrary genus

Initiated applications to functional relations among polylogarithms

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.