On the Superconformal Flatness of AdS Superspaces

The superconformal structure of coset superspaces with AdSm × Sn geometry of bosonic subspaces is studied. It is shown, in particular, that the conventional superspace extensions of the coset manifolds AdS2 × S2, AdS3 × S3 and AdS5 × S5, which arise as solutions of corresponding D = 4, 6, 10 supergravities and have been extensively studied in connection with AdS/CFT correspondence, are not superconformally flat, though their bosonic submanifolds are conformally flat. We give a group–theoretical reasoning for this fact. We find that in the AdS2 × S2 and AdS3 × S3 cases there exist different supercosets based on the supergroup OSp(4∗|2) which are superconformally flat. We also argue that in D = 2, 3, 4 and 5 there exist superconformally flat ‘pure’ AdSD supercosets. Two methods of checking the superconformal flatness are proposed. One of them consists in solving the Maurer–Cartan structure equations and the other is based on embedding the isometry supergroup of the AdSm×S superspace into a superconformal group in (m+n)–dimensional Minkowski space. Finally, we discuss some applications of the above results to the description of supersymmetric dynamical systems. PACS: 11.15. q; 11.17. + y