Anti–de Sitter Particles and Manifest (Super)Isometries

Starting from the classical action for a spin-zero particle in a $D$-dimensional anti–de Sitter (AdS) spacetime, we recover the Breitenlohner-Freedman bound by quantization. For $D=4$, 5, 7 and using an $Sl(2;\mathbb{K})$ spinor notation for $\mathbb{K}=\mathbb{R},\mathbb{C},\mathbb{H}$, we find a bitwistor form of the action for which the AdS isometry group is linearly realized, although only for zero mass when $D=4$, 7 in agreement with previous constructions. For zero mass and $D=4$ the conformal isometry group is linearly realized. We extend these results to the superparticle in the maximally supersymmetric “$\mathrm{AdS}\times \mathrm{S}$” string or $M$-theory vacua, showing that quantization yields a $128+128$ component supermultiplet. We also extend them to the null string.