Cyclic products of higher-genus Szegö kernels, modular tensors and polylogarithms

TL;DR

This work develops a general descent procedure that disentangles the spin-structure dependence of cyclic Szegö kernel products $C_δ({\bf z})$ from their marked-point dependence on Riemann surfaces of arbitrary genus. By expressing $C_δ({\bf z})$ as a sum of δ-dependent modular tensors $C^{I_1\cdots I_r}_δ$ with δ-independent coefficients $F^{(r)}_{I_1\cdots I_r}({\bf z})$, the authors reduce spin-structure sums to algebraic problems in Torelli space and connect the point-structure to higher-genus polylogarithms via integration kernels ${\cal G}$, ${\cal G}^I$, and related convolutions. The framework yields explicit genus-1 and genus-2 results, including trilinear relations that constrain tensor powers and a holomorphic symmetric sector on moduli space, and it lays groundwork for extending to higher genus and for applications to multi-loop amplitudes and double-copy constructions in gravity. Overall, the method provides a unifying, modular-tensor-centric approach to organizing multi-point, higher-genus string amplitudes and their spin-structure sums, with concrete recursion relations and appendices detailing genus-specific examples and differential relations.

Abstract

A wealth of information on multiloop string amplitudes is encoded in fermionic two-point functions known as Szegö kernels. In this paper we show that cyclic products of any number of Szegö kernels on a Riemann surface of arbitrary genus may be decomposed into linear combinations of modular tensors on moduli space that carry all the dependence on the spin structure $δ$. The $δ$-independent coefficients in these combinations carry all the dependence on the marked points and are composed of the integration kernels of higher-genus polylogarithms. We determine the antiholomorphic moduli derivatives of the $δ$-dependent modular tensors.