The anti-de Sitter supergeometry revisited

Abstract

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

Keywords

• Extended Supersymmetry
• Superspaces