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Peer-reviewedCC BYSciPost Physics

Fay identities for polylogarithms on higher-genus Riemann surfaces

  • Eric D’hoker A1
  • Oliver Schlotterer2
  1. 1Mani L. Bhaumik Institute for Theoretical Physics
  2. 2Uppsala University
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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.

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Cite

BibTeXRIS
BibTeX
@article{dhoker2026identities,
  title = {{Fay identities for polylogarithms on higher-genus Riemann surfaces}},
  author = {Eric D’hoker A and Oliver Schlotterer},
  journal = {SciPost Physics},
  year = {2026},
  month = jul,
  doi = {10.21468/scipostphys.21.1.021},
  url = {https://doi.org/10.21468/scipostphys.21.1.021},
  note = {CC BY},
}
RIS
TY  - JOUR
TI  - Fay identities for polylogarithms on higher-genus Riemann surfaces
AU  - Eric D’hoker A
AU  - Oliver Schlotterer
PY  - 2026
DA  - 2026/07/27
JO  - SciPost Physics
DO  - 10.21468/scipostphys.21.1.021
UR  - https://doi.org/10.21468/scipostphys.21.1.021
AB  - 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.
SN  - 2542-4653
ER  -