Quantization of the ${\rm AdS}_3$ Superparticle on ${\rm OSP}(1|2)^2/{\rm SL}(2,\mathbb{R})$

Abstract

We analyze ${\rm AdS}_3$ superparticle dynamics on the coset ${\rm OSP}(1|2) \times {\rm OSP}(1|2)/{\rm SL}(2,\mathbb{R})$. The system is quantized in canonical coordinates obtained by gauge invariant Hamiltonian reduction. The left and right Noether charges of a massive particle are parametrized by coadjoint orbits of a timelike element of $\frak{osp}(1|2)$. Each chiral sector is described by two bosonic and two fermionic canonical coordinates corresponding to a superparticle with superpotential $W=q-m/q$, where $m$ is the particle mass. Canonical quantization then provides a quantum realization of $\frak{osp}(1|2)\oplus\frak{osp}(1|2)$. For the massless particle the chiral charges lie on the coadjoint orbit of a nilpotent element of $\frak{osp}(1|2)$ and each of them depends only on one real fermion, which demonstrates the underlying $\kappa$-symmetry. These remaining left and right fermionic variables form a canonical pair and the system is described by four bosonic and two fermionic canonical coordinates. Due to conformal invariance of the massless particle, the $\frak{osp}(1|2)\oplus\frak{osp} (1|2)$ extends to the corresponding superconformal algebra $\frak{osp}(2|4)$. Its 19 charges are given by all real quadratic combinations of the canonical coordinates, which trivializes their quantization.