Summability of Superstring Theory

Several arguments are given for the summability of the superstring perturbation series. Whereas the Schottky group coordinatization of moduli space may be used to provide refined estimates of large-order bosonic string amplitudes, the super-Schottky group variables define a measure for the supermoduli space integral which leads to upper bounds on superstring scattering amplitudes. The genus-dependence of superstring amplitudes have been estimated recently using the super-Schottky coordinatization of supermoduli space. The N-point g-loop amplitudes in Type IIB superstring theory have been found to grow exponentially with the genus, (4π(g − 1))N−1f(BK , B′ K, BH , B′ H , BB, B′ B), where BK , B′ K , BH , B′ H , BB and B′ B are the bounds of the integrals over Schottky group parameters which represent degenerating and non-degenerating moduli respectively [1]. As this genus-dependence differs significantly from the large-order growth of field theory amplitudes, several arguments in support of the conclusion shall be put forward. The advantage of using the Schottky parametrization of moduli space in the study of the growth of integrals representing the scattering amplitudes is that the dependence of these integrals on the genus is directly linked to the limits for each of the Schottky group variables. An accurate estimate of the amplitudes can be achieved if the integration region in the Schottky parameter space is that subset of the fundamental region consistent with the cut-off on the length of closed geodesics. It has been shown that the sources of infrared divergences and large-order divergences are identical, as they both arise from the genusdependence of the |Kn| limit, |Kn| ∼ 1 g [2][3]. For the superstring, this cut-off is no longer necessary, and the entire fundamental region is required for the supermoduli space integrals. The introduction of supersymmetry eliminates the infrared divergences because of the absence of the tachyon in the superstring spectrum, and therefore, the large-order divergences are eliminated simultaneously. This is a consequence of the tachyon being the source of the divergences, rather than the instanton, which could remain in the theory even after the introduction of supersymmetry. The validity of estimates based on the super-Schottky group measure depends on the range of the integration. The use of the super-Schottky group measure [4] 1 dVABC g