Two-loop superstrings and S-duality

The two-loop contribution to the Type IIB low energy effective action term $D^4{R}^4$, predicted by SL(2,Z) duality, is compared with that of the two-loop 4-point function derived recently in superstring perturbation theory through the method of projection onto super period matrices. For this, the precise overall normalization of the 4-point function is determined through factorization. The resulting contributions to $D^4{R}^4$ match exactly, thus providing an indirect check of SL(2,Z) duality. The two-loop Heterotic low energy term $D^2{F}^4$ is evaluated in string perturbation theory; its form is closely related to the $D^4{R}^4$ term in Type II, although its significance to duality is an open issue.