Spin-(s, j) projectors and gauge-invariant spin-s actions in maximally symmetric backgrounds

A bstract Given a maximally symmetric d -dimensional background with isometry algebra g g , a symmetric and traceless rank- s field ϕ a ( s ) satisfying the massive Klein-Gordon equation furnishes a collection of massive g g -representations with spins j ∈ {0, 1, · · · , s }. In this paper we construct the spin-( s , j ) projectors, which are operators that isolate the part of ϕ a ( s ) that furnishes the representation from this collection carrying spin j . In the case of an (anti-)de Sitter ((A)dS d ) background, we find that the poles of the projectors encode information about (partially-)massless representations, in agreement with observations made earlier in d = 3, 4. We then use these projectors to facilitate a systematic derivation of two-derivative actions with a propagating massless spin- s mode. In addition to reproducing the massless spin- s Fronsdal action, this analysis generates new actions possessing higher-depth gauge symmetry. In (A)dS d we also derive the action for a partially-massless spin- s depth- t field with 1 ≤ t ≤ s . The latter utilises the minimum number of auxiliary fields, and corresponds to the action originally proposed by Zinoviev after gauging away all Stückelberg fields. Some higher-derivative actions are also presented, and in d = 3 are used to construct (i) generalised higher-spin Cotton tensors in (A)dS 3 ; and (ii) topologically-massive actions with higher-depth gauge symmetry. Finally, in four-dimensional N N = 1 Minkowski superspace, we provide closed-form expressions for the analogous superprojectors.