Anti-de Sitter space, branes, singletons, superconformal field theories and all that

TL;DR

The paper surveys the historical and modern connections between anti-de Sitter space, branes, and boundary superconformal theories, linking singleton/doubleton/tripleton representations with brane dynamics and AdS/CFT. It develops the Hopf-fibration framework to generate unconventional dualities (Hopf dualities) that yield phenomena such as BPS without BPS, and analyzes near-horizon geometries of branes to reveal an aristocracy of AdS spacetimes with conformal supergroup symmetries. The work bridges the old membrane/singleton bootstrap with Maldacena's AdS/CFT, showcasing how dilatonic vs non-dilatonic backgrounds, orbifolds, lens spaces, and squashed spheres affect supersymmetry and dual boundary theories. Overall, it highlights the intricate interplay between bulk supergravity, M-theory, and boundary conformal field theories, offering tools to explore non-perturbative and less-supersymmetric holographic dualities.

Abstract

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 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 BPS without BPS whereby states which appear to be non-BPS in one picture are seen to be BPS in the dual picture.