On the Superconformal Flatness of AdS Superspaces

TL;DR

This paper investigates the superconformal structure of AdS superspaces, revealing that certain well-known superspaces are not superconformally flat, but alternative supercosets with superconformal flatness exist in specific dimensions, with methods for checking this property.

Contribution

It demonstrates that standard AdS superspaces are not superconformally flat and identifies alternative supercosets that are, providing new insights into their geometric structure.

Findings

Standard AdS superspaces are not superconformally flat.

Existence of superconformally flat supercosets in certain dimensions.

Two methods proposed for checking superconformal flatness.

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.