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

TL;DR

This paper reviews the historical context of AdS/CFT duality, discusses recent work on specific AdS spaces and spheres, and introduces a novel Hopf duality that reveals BPS states in dual descriptions.

Contribution

It introduces an unconventional Hopf duality along U(1) fibres in AdS spaces, leading to the phenomenon of BPS without BPS states.

Findings

U(1) bundle structure of odd-dimensional spheres enables Hopf duality.

Discovery of BPS without BPS phenomenon in dual AdS descriptions.

Enhanced understanding of branes, singletons, and superconformal theories in AdS contexts.

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.