Cyclic Products of Higher-Genus Szegö Kernels, Modular Tensors, and Polylogarithms

A wealth of information on multiloop string amplitudes is encoded in fermionic two-point functions known as Szegö kernels. Here 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.