BPS Saturated Amplitudes and Non-perturbative String Theory’, talk at Cargese 97, hep-th/9712079

Computations of non-perturbative contributions is still a difficult task. In (supersymmetric) Field theory a direct computation requires the construction of the measure of integration over vacua containing n-instantons, which is not yet available for every n. The use of dualities is still the main tool for getting the answer. In String theory the D-instantons contributions for the R4 were conjectured thanks to the Sl(2,Z) invariance of the Type IIb theory [1], and for the Spin(32)/Z2 some ad hoc rules for the contributions for the R4 and F 4 were derived from the S-duality relation between the Heterotic and Type I strings [2]. In this proceeding I will explain how these non-perturbative contributions can be computed exactly, concentrating on the eight-dimensional case. Details and generalizations will be found in [3]. On the Heterotic side the F 4 and R4 terms are extracted from the fourth point amplitude on the torus involving four vertex operators. In order to get a non zero contribution, eight fermionic zero modes have to be soaked up. So all the fermionic zero modes from the supersymmetric side of the vertex operators have to be saturated, leaving a complete freedom on the non-supersymmetrical current algebra side. These contributions involve only states belonging to the short multiplets of the N4d = 4 supersymmetry algebra [4]. The non-perturbative contributions in eight-dimensions read