Higher-spin Cotton tensors and massive gauge-invariant actions in AdS$_3$

In a conformally flat three-dimensional spacetime, the linearised higher-spin Cotton tensor $\mathfrak{C}_{\alpha(n)}(h)$ is the unique conserved conformal current which is a gauge-invariant descendant of the conformal gauge prepotential $h_{\alpha(n)}$. The explicit form of $\mathfrak{C}_{\alpha(n)}(h)$ 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 $\mathfrak{C}_{\alpha(n)}(h)$ in terms of the AdS Lorentz covariant derivatives. It is shown that every conformal higher-spin action $S_{\text{CS}}^{(n)}[h]\propto \int\text{d}^3x\, e \, h^{\alpha(n)}\mathfrak{C}_{\alpha(n)}(h)$ 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 AdS$_3$. The main results are extended to the case of $\mathcal{N}=1$ AdS supersymmetry. In particular, we derive simple expressions for the higher-spin super-Cotton tensors in AdS$_3$.