Cyclic products of higher-genus Szeg\"o kernels, modular tensors and polylogarithms

TL;DR

This paper demonstrates how cyclic products of Szeg"o kernels on higher-genus Riemann surfaces can be decomposed into modular tensors and polylogarithmic kernels, revealing detailed structure of multiloop string amplitudes.

Contribution

It introduces a novel decomposition of Szeg"o kernel products into modular tensors and polylogarithmic kernels, elucidating their dependence on spin structures and moduli.

Findings

Decomposition of Szeg"o kernel products into modular tensors.

Identification of coefficients involving higher-genus polylogarithms.

Derivation of antiholomorphic moduli derivatives of modular tensors.

Abstract

A wealth of information on multiloop string amplitudes is encoded in fermionic two-point functions known as Szeg\"o kernels. In this paper we show that cyclic products of any number of Szeg\"o 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 $\delta$. The $\delta$-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 $\delta$-dependent modular tensors.