The anti-de Sitter supergeometry revisited

In a supergravity framework, the $\mathcal{N}$-extended anti-de Sitter (AdS) superspace in four spacetime dimensions, ${\text{AdS}}^{4\mathrm{\mid }4\mathcal{N}}$, is a maximally symmetric background that is described by a curved superspace geometry with structure group $\mathsf{S}\mathsf{L}(2,\mathbb{C})\times \mathsf{U}(\mathcal{N})$. On the other hand, within the group-theoretic setting, ${\text{AdS}}^{4\mathrm{\mid }4\mathcal{N}}$ is realised as the coset superspace $\mathsf{O}\mathsf{S}\mathsf{p}(\mathcal{N}\mathrm{\mid }4;\mathbb{R})\mathrm{/}[\mathsf{S}\mathsf{L}(2,\mathbb{C})\times \mathsf{O}(\mathcal{N})]$, with its structure group being $\mathsf{S}\mathsf{L}(2,\mathbb{C})\times \mathsf{O}(\mathcal{N})$. Here we explain how the two frameworks are related. We give two explicit realisations of ${\text{AdS}}^{4\mathrm{\mid }4\mathcal{N}}$ as a conformally flat superspace, thus extending the $\mathcal{N}=1$ and $\mathcal{N}=2$ results available in the literature. As applications, we describe: (i) a two-parameter deformation of the ${\text{AdS}}^{4\mathrm{\mid }4\mathcal{N}}$ interval and the corresponding superparticle model; (ii) some implications of conformal flatness for superconformal higher-spin multiplets and an effective action generating the $\mathcal{N}=2$ super-Weyl anomaly; and (iii) $\kappa$-symmetry of the massless AdS superparticle.