Higher spin algebras as higher symmetries

TL;DR

This work surveys how rigid symmetries of free higher-spin fields on constant-curvature spacetimes constrain and inform the construction of higher-spin interactions. It develops a Noether-method framework for first-order deformations and identifies Killing tensors as key reducibility parameters, linking them to the space of potential gauge-algebra deformations. It then formalizes higher-spin algebras as the universal enveloping algebra of spacetime isometries, proving that free theories carry infinite-dimensional symmetry algebras and detailing their structure in examples like AdS and two-dimensional models. A gauge-principle proposal elevates these rigid symmetries to dynamical gauge symmetries via a connection valued in the higher-spin algebra, leading to a non-Abelian gauge structure that generalizes gravity and Yang–Mills at low spins and suggests a path toward Vasiliev-type theories. The paper concludes that understanding the full rigid-symmetry content of free higher-spin theories provides essential guidance for the form and consistency of interacting higher-spin gauge theories.

Abstract

The exhaustive study of the rigid symmetries of arbitrary free field theories is motivated, along several lines, as a preliminary step in the completion of the higher-spin interaction problem in full generality. Some results for the simplest example (a scalar field) are reviewed and commented along these lines.