The anti-de Sitter supergeometry revisited

Abstract

In a supergravity framework, the \( \mathcal{N} \)-extended anti-de Sitter (AdS) superspace in four spacetime dimensions, \( {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} \), is a maximally symmetric background that is described by a curved superspace geometry with structure group SL(2, ℂ) × \( \textrm{U}\left(\mathcal{N}\right) \). On the other hand, within the group-theoretic setting, \( {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} \) is realised as the coset superspace \( \textrm{O}\textrm{Sp}\left(\left.\mathcal{N}\right|4;\mathbb{R}\right)/\left[\textrm{SL}\left(2,\mathbb{C}\right)\times \textrm{O}\left(\mathcal{N}\right)\right] \), with its structure group being SL(2, ℂ) × \( \textrm{O}\left(\mathcal{N}\right) \). Here we explain how the two frameworks are related. We give two explicit realisations of \( {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} \) 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 \( {\textrm{AdS}}^{4\left|4\mathcal{N}\right.} \) 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) κ-symmetry of the massless AdS superparticle.