On the superconformal flatness of AdS superspaces -: art. no. 040

The superconformal structure of coset superspaces with AdS(m) x S-n geometry of bosonic subspaces is studied. It is shown, in particular, that the conventional superspace extensions of the coset manifolds AdS(2) x S-2, AdS(3) x S-3 and AdS(5) x S-5, 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 at, though their bosonic submanifolds are conformally at. We give group-theoretical reasoning for this fact. We find that in the AdS(2) x S-2 and AdS(3) x S-3 cases there exist different supercosets based on the supergroup OSp(4*|2) which are superconformally at. We also argue that in D = 2, 3, 4 and 5 there exist superconformally at "pure" AdS(D) 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 AdS(m) x S-n superspace into superconformal group in (m + n) dimensional Minkowski space. Finally, we discuss some applications of the above results to the description of supersymmetric dynamical systems.