The last project in this programme and perhaps the most technically involved is the determination of quotients of the Freund-Rubin backgrounds of the form AdS1+p x Sq.
There are several motivations for the study of quotients of AdS backgrounds. First, it is the next natural step in the classification of supersymmetric space forms, of which the results with George Papadopoulos on classification of maximally supersymmetric backgrounds are a first step. Second, it is well-known that a certain ℤ-quotient of AdS3 gives rise to the celebrated BTZ black hole. It is therefore natural to try to construct higher-dimensional analogues of the BTZ black hole via quotients of higher-dimensional AdS spaces. Third, there is a possible connection with holography: thinking of the AdS background as the near-horizon geometry of a brane configuration.
Remarkably the techniques we developed for the flat space quotients still apply in principle. The reason is that both the sphere and (locally) AdS embed in flat space. The novelty is that in the case of AdS1+p, the embedding space has signature (2,p) and we had to determine the adjoint orbits of SO(2,p), which have a more complicated taxonomy than those of the Poincaré or Lorentz groups. This required a refinement of results of Charles Boubel.
The paper Supersymmetric Kaluza-Klein reductions of AdS backgrounds contains the results of our classification for the M-theory backgrounds AdS4 x S7 and AdS7 x S4, as well as the IIB background AdS5 x S5 and also the six-dimensional (1,0) and (2,0) backgrounds AdS3 x S3 and their lifts to IIB as AdS3 x S3 x ℝ4.
Simon Ross and his student Owen Madden were also looking into the problem of AdS quotients, although not in the context of supergravity theories. We decided to pull our efforts together and wrote Quotients of AdS x S: causally well-behaved spaces and black holes, where we concentrate on discrete quotients.