Two-loop statsum of superstrings

We discuss, whether there is a choice of odd moduli on super-Riemann surfaces of genus p ≥ 2, which leads to the vanishing of statistical sums of superstring before integration over the space of even moduli. The answer is shown to be positive at least for p = 2, when odd moduli are localized at ramification points. The relation between various definitions of many-loop statistical sums in superstring theory is discussed. (orig.)