Unraveling the Spectrum of the Open String

TL;DR

Unraveling the Spectrum of the Open String develops a light-cone quantization framework to construct a large portion of the open bosonic string's massive spectrum, delivering explicit oscillator realizations for single-particle states and complete Regge trajectories. The authors show how transverse oscillators assemble into irreps of $SO(25)$ by combining $SO(24)$ structures and introduce an algorithm to climb Regge trajectories by adding boxes to the Young tableau, with level-dependent coefficients admitting closed forms. They establish that the spectrum has a recursive, highly ordered structure and that the number of oscillator structures along a trajectory saturates to a finite set, with the Lorentz-action matrices depending linearly on the level, enabling infinite families to be solved from finite data. They further develop group-theoretic methods to relate trajectories, compute spectrum-depth decompositions, and reveal regularities across generations, setting groundwork for three-point amplitudes of arbitrary massive states. The work provides a computational foundation for analyzing string interactions and the organization of high-spin spectra in a UV-complete framework.

Abstract

We construct a large portion of the massive spectrum of the open bosonic string using light-cone quantization, providing explicit oscillator realizations for individual single-particle states as well as for full Regge trajectories. We show how combinations of transverse oscillators organize into irreducible SO(25) representations, and provide an algorithm for constructing them level by level. We then develop a general method to "climb" the spectrum-adding oscillators in a controlled way that generates entire Regge trajectories from a finite set of seed states. Remarkably, the coefficients determining each state's oscillator composition depend on the level in a simple way, allowing closed-form expressions for infinitely many states. Beyond individual trajectories, we explore internal regularities of the spectrum and establish relations among families of trajectories, extending the concept of a Regge trajectory to more general constructions. Our results expose a highly ordered and recursive structure underlying the open-string spectrum, suggesting that its massive excitations form an algorithmically constructible network. The framework presented here lays the groundwork for computing three-point amplitudes of arbitrary massive states, the essential building blocks of string interactions, which we tackle in upcoming work.

Figures (2)

• Figure 1: The whole spectrum, divided into Regge trajectories. The top blue one is the leading Regge trajectory. The one right below is the empty trajectory which does not contain any particles. Except for the leading Regge trajectory, we only draw the seed of each trajectory, and we leave it implicit that, climbing the trajectory, new particles are found with greater mass and number of boxes in the first row of the Young tableau. For example, at level 4 there is a $\ydiagram{2,1}$ (implicit) originating from the $\ydiagram{1,1}$ (explicit) at level 3. Lastly, notice that at level 6 there is not only the (red) $\ydiagram{2}$ coming from the scalar at level 4, but also a new (gray) $\ydiagram{2}$. This is an example of a degeneracy, which we will see is ubiquitous in the string spectrum.
• Figure 2: Action of the Lorentz generator $J^{-i}$ on fully symmetric $SO(24)$ structures at level 4. When acting on lower-spinning components of a single-particle state it increases the number of free indices by one, while it annihilates maximally spinning components.