On the Superconformal Flatness of AdS Superspaces

TL;DR

The article investigates whether AdS_m × S^n superspaces possess superconformal flatness. It develops two complementary methods—solving Maurer–Cartan equations and embedding isometry supergroups into superconformal groups—to test superconformal flatness, and finds that conventional AdS_4, AdS_2×S^2, AdS_3×S^3, and AdS_5×S^5 cosets are not all superconformally flat, despite bosonic conformal flatness. Remarkably, certain alternative supercosets based on OSp(4^*|2) are superconformally flat for AdS_2×S^2 and AdS_3×S^3, while pure AdS_D cosets exist for D=2–5; AdS_5×S^5 remains non-flat in this sense due to embedding obstructions. The work further discusses implications for the classical and quantum dynamics of supersymmetric particles and branes in these backgrounds and hints at broader applications to higher-dimensional theories with tensorial charges. Overall, it clarifies when AdS superspaces can be linked to flat superspace via super-Weyl transformations and provides explicit constructions of the relevant conformal factors for several key backgrounds.

Abstract

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 flat, though their bosonic submanifolds are conformally flat. We give a 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 flat. We also argue that in D=2,3,4 and 5 there exist superconformally flat `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 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.