Fay identities for polylogarithms on higher-genus Riemann surfaces

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.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of polylogarithms on compact Riemann surfaces of arbitrary genus (h). Building on the recent construction of higher‑genus polylogarithms (arXiv:2306.08644), the authors introduce a flat connection whose building blocks are single‑valued, non‑holomorphic integration kernels (f^{I_{1}\dots I_{r}}_{J}(x,y)) that depend on two points (x,y) on the surface and transform as tensors under the modular group (Sp(2h,\mathbb Z)). These kernels are obtained by iterated convolutions of the Arakelov Green function with holomorphic Abelian differentials and their complex conjugates.

The central achievement is the systematic derivation of infinite families of bilinear relations—generalizations of the classical Fay trisecant identity—among these kernels. The authors first prove scalar Fay identities (Theorem 4.1) that involve three points but remain invariant under the modular group. They then develop tensorial “interchange identities” (Theorem 5.2) that allow one to swap the order of two‑point kernels while preserving the tensor structure, and finally establish full-fledged three‑point Fay identities (Theorems 6.2 and 6.3) for arbitrary tensor rank. The essential feature of all these identities is that any bilinear expression depending on three points (x,y,z) can be rewritten so that each factor involves at most one of the three points, thereby eliminating repeated point dependence. This property is crucial for integrating over a chosen point and obtaining primitives that stay within the space of higher‑genus polylogarithms.

The paper also analyses the coincident‑point limits (y\to x). In this limit the kernels collapse to purely modular tensors that depend only on the period matrix (\Omega) of the surface. These tensors generalize (almost) holomorphic Eisenstein series to higher genus and appear in Theorems 8.3 and 8.4.

Beyond the single‑valued kernels, the authors consider the multi‑valued meromorphic kernels (g^{I_{1}\dots I_{r}}_{J}(x,y)) introduced by Enriquez. They prove analogous interchange identities for (g) (Theorem 9.2) and conjecture that the same three‑point Fay identities hold (Conjectures 9.6, 9.7) together with their coincident‑point limits (Conjectures 9.10, 9.11). While full proofs are pending, the conjectures are strongly supported by the structural similarity to the (f)‑kernels and by recent progress reported in subsequent works.

With these identities in hand, the authors demonstrate how to construct primitives of multivariable higher‑genus polylogarithms (Section 7). The interchange identities generate “exchange” relations that reduce products of kernels, while the Fay identities provide the necessary reshuffling to keep the resulting expressions within the allowed tensor space. This machinery ensures that the space of polylogarithms is closed under both differentiation and integration, a property essential for applications in quantum field theory and string perturbation theory where iterated integrals on moduli spaces of punctured Riemann surfaces frequently appear.

The paper concludes with a discussion of potential applications: evaluation of higher‑genus string amplitudes, computation of higher‑genus multiple‑zeta values, and connections to modular graph forms. It also outlines future directions, notably the rigorous proof of the meromorphic Fay identities, the exploration of algebraic structures (e.g., Hopf algebras) underlying the kernel algebra, and the implementation of these results in concrete physical calculations.

In summary, this work provides the first comprehensive set of Fay‑type identities for the integration kernels that generate polylogarithms on arbitrary‑genus Riemann surfaces, thereby establishing a solid algebraic foundation for the study of higher‑genus polylogarithmic functions and their role in modern mathematical physics.