Embedding formalism for $$ \mathcal{N} $$-extended AdS superspace in four dimensions

The supertwistor and bi-supertwistor formulations for $$ \mathcal{N} $$ N -extended anti-de Sitter (AdS) superspace in four dimensions, $$ Ad{S}^{4\mid 4\mathcal{N}} $$ Ad S 4 ∣ 4 N , were derived two years ago in [1]. In the present paper, we introduce a novel realisation of the $$ \mathcal{N} $$ N -extended AdS supergroup OSp( $$ \mathcal{N} $$ N |4; ℝ ) and apply it to develop a coset construction for $$ {\textrm{AdS}}^{4\mid 4\mathcal{N}} $$ AdS 4 ∣ 4 N and the corresponding differential geometry. This realisation naturally leads to an atlas on $$ {\textrm{AdS}}^{4\mid 4\mathcal{N}} $$ AdS 4 ∣ 4 N (that is a generalisation of the stereographic projection for a sphere) that consists of two charts with chiral transition functions for $$ \mathcal{N} $$ N > 0. A manifestly OSp( $$ \mathcal{N} $$ N |4; ℝ ) invariant model for a superparticle in $$ {\textrm{AdS}}^{4\mid 4\mathcal{N}} $$ AdS 4 ∣ 4 N is proposed. Additionally, by employing a conformal superspace approach, we describe the most general conformally flat $$ \mathcal{N} $$ N -extended supergeometry. This construction is then specialised to the case of $$ {\textrm{AdS}}^{4\mid 4\mathcal{N}} $$ AdS 4 ∣ 4 N .