Higher-spin Cotton tensors and massive gauge-invariant actions in AdS₃

In a conformally flat three-dimensional spacetime, the linearised higher-spin Cotton tensor ℭ_α(n)(h) is the unique conserved conformal current which is a gauge-invariant descendant of the conformal gauge prepotential h_α(n). The explicit form of ℭ_α(n)(h) is well known in Minkowski space. Here we solve the problem of extending the Minkowskian result to the case of anti-de Sitter (AdS) space and derive a closed-form expression for ℭ_α(n)(h) in terms of the AdS Lorentz covariant derivatives. It is shown that every conformal higher-spin action SCS(n)[h]∝∫d3xehα(n)ℭα(n)(h) factorises into a product of (n − 1) first-order operators that are associated with the spin-n/2 partially massless AdS values. Our findings greatly facilitate the on-shell analysis of massive higher-spin gauge-invariant actions in AdS₃. The main results are extended to the case of N = 1 AdS supersymmetry. In particular, we derive simple expressions for the higher-spin super-Cotton tensors in AdS₃.