Growth of so (9) massive representations in string theory

Sammanfattning

We investigate the spectrum of type II string theory with the aim of addressing two related open problems: the organization of string states into irreducible representations of so(9) at fixed mass levels and the determination of the density of states beyond its known asymptotic behavior. Building on the notion of the "refined" partition function introduced in previous works, we reformulate it in terms of Jacobi Theta functions and compute the complete decomposition of the left-moving spectrum up to level 100. The resulting data reveal previously unseen patterns, an empirical rule, and allow the derivation of numerical formulas with strong predictive power at higher mass levels. We further revisit the problem of state counting by employing techniques inspired by Hardy, Ramanujan and Rademacher's work. After illustrating the method in the context of the integer partition function, we extend it to the type II string partition function and derive a convergent series representation for the density of states valid at arbitrary level. Remarkably, the series converges so rapidly that its leading contribution alone accurately reproduces the spectrum,providing an efficient approximation across all levels. We analyze the convergence and predictive power of this expression and demonstrate its consistency with the standard asymptotic formula for the string density of states. Finally, we discuss the implications of these results and outline directions for future investigations of string spectra that that further integrate the computational and analytical approaches developed in this work.