Introduction to the Classical Theory of Higher Spins

TL;DR

The work surveys the landscape of higher spin field theory, tracing the development from free massless HS fields described by symmetric tensors and spin-tensors to geometric curvature formulations and nonlocal/compensator approaches. It highlights the central obstacles in constructing consistent interactions, notably in flat space, and explains how AdS backgrounds and the unfolded formalism (with infinite HS towers) enable progress toward a non-linear theory. The discussion covers both bosonic and fermionic HS fields, the role of generalized curvatures and Damour–Deser-type identities, and the tension between locality and gauge invariance in HS dynamics. It also surveys alternate frameworks, including tensorial spaces and tensionless limits of string theory, which aim to unify HS states within a broader quantum gravity context. Overall, the notes underscore the deep connections between HS theory, higher-derivative gauge structures, and holographic dualities, while acknowledging that a complete interacting HS theory in flat space remains elusive.

Abstract

We review main features and problems of higher spin field theory and flash some ways along which it has been developed over last decades.