Higher Spin Theory - Part I

TL;DR

This work surveys the construction of free higher-spin theories and the fundamental obstructions that arise when turning on interactions. It highlights key no-go theorems for massless HS fields in flat space, the Velo–Zwanziger acausality for massive HS in backgrounds, and how string theory and AdS-based frameworks offer viable routes around these obstacles. The notes develop a BRST–antifield approach to generate consistent cubic couplings, demonstrating both a concrete 1–3/2–3/2 vertex and the obstruction to higher-order consistency without introducing additional dynamical fields or nonlocality. A central message is that HS interactions demand either nonlocal structure, extended spectra, or curved backgrounds (e.g., AdS) to be consistent, with concrete bounds on EFT cutoffs and a clear connection to open string dynamics. The material lays a rigorous foundation for understanding HS gauge theories and their connections to string theory and holography.

Abstract

These notes comprise a part of the introductory lectures on Higher Spin Theory presented in the Eighth Modave Summer School in Mathematical Physics. We construct free higher-spin theories and turn on interactions to find that inconsistencies show up in general. Interacting massless fields in flat space are in tension with gauge invariance and this leads to various no-go theorems. While massive fields exhibit superluminal propagation, appropriate non-minimal terms may cure such pathologies as they do in String Theory--a fact that we demonstrate. Given that any interacting massive higher-spin particle is described by an effective field theory, we compute a model independent upper bound on the ultraviolet cutoff in the case of electromagnetic coupling in flat space and discuss its implications. Finally, we consider various possibilities of evading the no-go theorems for massless fields, among which Vasiliev's higher-spin gauge theory is one. We employ the BRST-antifield method for a simple but non-trivial gauge system in flat space to find a non-abelian cubic coupling and to explore its higher-order consistency.