Spin field correlators on an arbitrary genus Riemann surface and non-renormalization theorems in string theories

We calculate the n-point correlation functions of spin fields on an arbitrary genus Riemann surface. We also calculate the corresponding correlators for the spin fields associated with the local supersymmetry ghosts using a specific ansatz for screening the background ghost charge. Using these results we show by explicit calculation that to all orders in the string perturbation theory all n-point amplitudes involving massless fields in the superstring and the heterotic string theory vanish identically for 0≤n≤3