Linearised actions for N $$ \mathcal{N} $$ -extended (higher-spin) superconformal gravity

Abstract The off-shell actions for N $$ \mathcal{N} $$ -extended conformal supergravity theories in three dimensions were formulated in [1, 2] for 1 ≤ N $$ \mathcal{N} $$ ≤ 6 using a universal approach. Each action is generated by a closed super three-form which is constructed in terms of the constrained geometry of N $$ \mathcal{N} $$ -extended conformal superspace. In this paper we initiate a program to recast these actions (and to formulate their higher-spin counterparts) in terms of unconstrained gauge prepotentials as integrals over the full superspace. We derive transverse projection operators in N $$ \mathcal{N} $$ -extended Minkowski superspace and then use them to construct linearised rank-n super-Cotton tensors and off-shell N $$ \mathcal{N} $$ -extended superconformal actions. We also propose off-shell gauge-invariant actions to describe massive higher-spin super-multiplets in N $$ \mathcal{N} $$ -extended supersymmetry. Our analysis leads to general expressions for identically conserved higher-spin current multiplets in N $$ \mathcal{N} $$ -extended supersymmetry.