An introduction to string states and their interactions

TL;DR

The paper addresses the open bosonic string spectrum and interactions, focusing on how to access the infinite tower of deep Regge trajectories and their tree-level amplitudes. It introduces a covariant, efficient spectrum-generation framework based on a spectrum-generating symplectic algebra and Howe duality, enabling the construction of Weinberg states and their depth w clones across all levels with finite Virasoro constraint Ansatzes. It also presents a generating functional approach to amplitudes that exponentiates vertex operator polynomials and assembles subleading trajectories via auxiliary variables, recovering YM-type interactions for massless vectors and gravity in the closed-string sector, while clarifying the role of the intercept a and critical dimension D = 26. This framework illuminates the connection between the deep string spectrum and the good UV behavior of string amplitudes, offering a unified, covariant route to higher-spin content and potential links to higher-spin gravity and nonperturbative objects in string theory.

Abstract

The subject matter of these lecture notes is the open bosonic critical string, its perturbative spectrum and interactions. We begin with a brief review of classical string propagation, quantization, as well as the level-by-level construction of physical string states. We then review a new, covariant and efficient technology of constructing entire trajectories of infinitely many physical states deeper in the string spectrum. Finally, elements of the calculation of string scattering amplitudes, including aspects of the application of the technology for the efficient calculation of tree-level amplitudes of deeper trajectories, are also covered. The material is based on three invited lectures delivered by the author at the 2024 Modave Summer School in Mathematical Physics.