Current exchanges and unconstrained higher spins

The (Fang–) Fronsdal formulation for free fully symmetric (spinor-) tensors rests on (γ  -) 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$(A)\mathit{dS}$ 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 the same fashion in current exchanges.