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

Authors: Eric D'Hoker, Martijn Hidding and Oliver Schlotterer

Preprint number: UUITP-21/23

Abstract: A wealth of information on multiloop string amplitudes is encoded in two-point functions of worldsheet fermions known as Szegö kernels. Cyclic products of an arbitrary number of Szegö kernels for any even spin structure δ 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 δ. The coefficients in these linear combinations are independent of δ, 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.