On Schr\"odinger superalgebras

TL;DR

This paper constructs new supersymmetric extensions of the Schr"odinger algebra using the supersymplectic framework, revealing novel superalgebras in various dimensions and contexts, including connections to non-relativistic Chern-Simons theory.

Contribution

It introduces two types of supersymmetric extensions of the Schr"odinger algebra, generalizing known supersymmetries and discovering new exotic superalgebras in two dimensions.

Findings

Existence of $I$-type extensions in any dimension for any pair of integers $N_+$ and $N_-$.

Introduction of $IJ$-type exotic extensions in two dimensions for each pair of integers $ u_+$ and $ u_-$.

Reduction of symmetry for magnetic monopoles and vortices to specific 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 $\o(2)\times\osp(1/2)$.