ar X iv : 0 80 7 . 05 69 v 1 [ he p - th ] 3 J ul 2 00 8 Supersymmetry of the magnetic vortex

, not only the conformal generators D and K, but also the fermionic generators Q=1/ √ 2mπ ·σ and S= √ m/2 r ·σ− tQ are conserved. Thus, the spin system admits an o(3)× osp(1/1) conformal supersymmetry, yielding now an algebraic solution of the Pauli equation [2]. More recently, Jackiw [3] found that the o(2, 1) symmetry, generated by — formally — the same D and K as above, is also present for a magnetic vortex (an idealization for the Aharonov-Bohm experiment [4]), allowing for a group-theoretic treatment of the problem. In this Letter we show that the N = 2 supersymmetry of the Pauli Hamiltonian of a spin1 2 particle (present for any magnetic field in the plane [5]) combines, for a magnetic vortex, with Jackiw’s o(2)× o(2, 1) into an o(2) × osp(1/2) superalgebra. This result is to be compared with the Galilean supersymmetry discovered recently by Leblanc et al. [6] for non-relativistic Chern-Simons systems, and with the osp(1/2) found by Hughes et al. for a constant magnetic field [7]. Thus the planar system has more symmetries as its higher-dimensional counterpart. We call this supersymmetry exotic, because it is realized with two — rather then four-component — objects. Our pseudoclassical calculations in Ref. [6] indicate that such an ‘exotic’ supersymmetry is only possible in two spatial dimensions — one more indication of the particular status of two-dimensional physics. Let us start with a spin1 2 particle in a static magnetic field B= (