Current Exchanges and Unconstrained Higher Spins

TL;DR

The paper investigates how relaxing the trace constraints in higher-spin gauge theories leads to non-local formulations based on higher-spin curvatures and how a minimal local unconstrained description, involving a pair of auxiliary fields, can reproduce the correct current exchanges. It constructs one-parameter families of Lagrangians for both bosonic and fermionic fully symmetric fields, extends the results to AdS backgrounds, and demonstrates an exact equivalence with constrained local theories at the level of current exchanges. A central result is the identification of a unique non-local Einstein-like tensor that reproduces the same current exchange as the local theories, providing a precise link between local and non-local unconstrained formulations. The work clarifies the role of gauge-invariant building blocks and how currents couple consistently across flat and AdS spacetimes, establishing a consistent geometric picture for unconstrained higher spins and laying groundwork for future exploration of interactions and string-theoretic connections.

Abstract

The (Fang-)Fronsdal formulation for free fully symmetric (spinor-) tensors rests on (gamma-)trace constraints on gauge fields and parameters. When these are relaxed, glimpses of the underlying geometry emerge: the field equations extend to non-local expressions involving the higher-spin curvatures, and with only a pair of additional fields an equivalent ``minimal'' local formulation is also possible. In this paper we complete the discussion of the ``minimal'' formulation for fully symmetric (spinor-) tensors, constructing one-parameter families of Lagrangians and extending them to (A)dS backgrounds. We then turn on external currents, that in this setting are subject to conventional conservation laws and, by a close scrutiny of current exchanges in the various formulations, we clarify the precise link between the local and non-local versions of the theory. To this end, we first show the equivalence of the constrained and unconstrained local formulations, and then identify a unique set of non-local Lagrangian equations which behave in exactly the same fashion in current exchanges.