The growth of $SO(9)$ super-representations in Type II string theory
We study the growth of massive $SO(9)$ super-representations in type II and type I string theories. We do this first directly by finding an empirical formula that specifies which representations appear at any level, and then compute the multiplicities from a refined partition function evaluated over finite fields up to level 501 for individual representations, and level 226 for all representations. We then derive asymptotic formulae for the growth of any representation, which are constructed by integrating about the peaks of the refined partition function. Using a Rademacher sum we can find very accurate approximations for the multiplicities of the representations. Unlike the superstring partition function, which only receives contributions from the odd Rademacher terms, the multiplicities for any representation have contributions from both even and odd terms. Finally, we apply the same techniques to the ordinary representations, where a simplified refined partition function allows for better computational speed and simpler expressions for the asymptotic approximations.