On Schr\"odinger superalgebras

Abstract

We construct, using the supersymplectic framework of Berezin, Kostant and others, two types of supersymmetric extensions of the Schr\"odinger algebra (itself a conformal extension of the Galilei algebra). An `$I$-type' extension exists in any space dimension, and for any pair of integers $N_+$ and $N_-$. It yields an $N=N_++N_-$ superalgebra, which generalizes the N=1 supersymmetry Gauntlett et al. found for a free spin-$\half$ particle, as well as the N=2 supersymmetry of the fermionic oscillator found by Beckers et al. In two space dimensions, new, `exotic' or `$IJ$-type' extensions arise for each pair of integers $\nu_+$ and $\nu_-$, yielding an $N=2(\nu_++\nu_-)$ superalgebra of the type discovered recently by Leblanc et al. in non relativistic Chern-Simons theory. For the magnetic monopole the symmetry reduces to $\o(3)\times\osp(1/1)$, and for the magnetic vortex it reduces to $\o(2)\times\osp(1/2)$.