The anti-de Sitter supergeometry revisited

In a supergravity framework, the 𝑁-extended anti-de Sitter (AdS) superspace in four spacetime dimensions, AdS4|4𝑁, is a maximally symmetric background that is described by a curved superspace geometry with structure group SL(2, ℂ) × U(𝑁). On the other hand, within the group-theoretic setting, AdS4|4𝑁 is realised as the coset superspace OSp(𝑁|4;𝑅)/[SL(2,𝐶)×O(𝑁)], with its structure group being SL(2, ℂ) × O(𝑁). Here we explain how the two frameworks are related. We give two explicit realisations of AdS4|4𝑁 as a conformally flat superspace, thus extending the 𝑁 = 1 and 𝑁 = 2 results available in the literature. As applications, we describe: (i) a two-parameter deformation of the AdS4|4𝑁 interval and the corresponding superparticle model; (ii) some implications of conformal flatness for superconformal higher-spin multiplets and an effective action generating the 𝑁 = 2 super-Weyl anomaly; and (iii) κ-symmetry of the massless AdS superparticle.