Anti-de Sitter Space, Branes, Singletons, Superconformal Field Theories and All That
Michael Duff
There has recently been a revival of interest in anti de-Sitter space (AdS) brought about by the conjectured duality beteeen physics in the bulk of AdS and a conformal field theory on the boundary. Since the whole subject of branes, singletons and superconformal field theories on the AdS boundary was an active area of research about ten years ago, I begin with a historical review, including the ``Membrane at the end of the universe'' idea. Next I discuss two recent papers with Lu and Pope on $AdS_{5} \times S^{5}$ and on $AdS_{3} \times S^{3}$, respectively. In each case we note that odd-dimensional spheres $S^{{2n+1}}$ may be regarded as $U(1)$ bundles over $CP^{n}$ and that this permits an unconventional ``Hopf'' duality along the $U(1)$ fibre. This leads in particular to the phenomenon of {\it BPS without BPS} whereby states which appear to be non-BPS in one picture are seen to be BPS in the dual picture.

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