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

Eric D'Hoker, Martijn Hidding, Oliver Schlotterer

Published 2023-08-09Version 1

A wealth of information on multiloop string amplitudes is encoded in two-point functions of worldsheet fermions known as Szeg\"o kernels. Cyclic products of an arbitrary number of Szeg\"o kernels for any even spin structure $\delta$ on a Riemann surface of arbitrary genus are decomposed into linear combinations of modular tensors on moduli space that carry all the dependence on the spin structure $\delta$. The coefficients in these linear combinations are independent of $\delta$, carry all the dependence on the marked points and are composed of the integration kernels of higher-genus polylogarithms constructed in arXiv:2306.08644. The conditions under which these modular tensors are locally holomorphic on moduli space are determined and explicit formulas for the special case of genus two are presented.