New supersymmetry of the monopole

Abstract

The non-relativistic dynamics of a spin-1/2 particle in a monopole field possesses a rich supersymmetry structure. One supersymmetry, uncovered by d'Hoker and Vinet, is of the standard type: it squares to the Hamiltonian. In this paper we show the presence of another supersymmetry which squares to the Casimir invariant of the full rotation group. The geometrical origin of this supersymmetry is traced, and its relationship with the constrained dynamics of a spinning particle on a sphere centered at the monopole is described. * On leave of absence from DAMTP,

On the nature of fermion-monopole supersymmetry

“…The nonlinear supersymmetry of the similar form was observed also under investigation of the space-time symmetries in terms of the motion of pseudoclassical spinning point particles [3,4,5], and was revealed in the 3D P, T -invariant systems of relativistic fermions [6] and Chern-Simons fields [7]. In this letter we show that the generators of the nonstandard [1] and standard [2] supersymmetries have to be supplemented by their product operator to be treated as independent supercharge. As a result, the fermion-monopole system possesses the nonlinear N = 3/2 supersymmetry having the nature of the 3D spin-1/2 free particle's supersymmetry generated by the supercharges represented in a scalar form.…”

“…A supersymmetric quantum mechanical system [9] is characterized by the HamiltonianĤ = 1 2 (p 2 +W 2 (x)+σ 3 W ′ (x)) witĥ p = −id/dx, and by the superchargesQ 1 …”

Killing tensors and a new geometric duality

“…These second-rank Killing tensors correspond to constants of motion which are quadratic in the momenta and play an important role in the complete solution of the problem of geodesic motion in these spaces [7]. Similar results concerning the monopole with spin and 3-D fermion systems have been presented in [8,9].…”

Hidden supersymmetries in supersymmetric quantum mechanics

“…The simple ansatz F ij = a(Z)f ijk x k where Z = x k x k , satisfies (48) when the algebra is su(2), in which case a(Z) ∝ Z −3/2 , as is expected for the monopole [8]. But for su(n), e.g.…”

“…We shall consider three models: one with bosonic superfields, and two with fermionic superfields (with N = 1 and N = 2 respectively) for which the bosonic component variables are auxiliary [7]. A typical example of the bosonic superfield case, in which the Lie algebra G is su (2), is that of the non-relativistic motion of a spin- 1 2 particle in the background field of a Dirac monopole [8] (see also [9]). The case G = SU(2) is, however, rather exceptional since it is the only group for which the structure constants of its algebra coincide with the fully antisymmetric tensor (ǫ ijk ) of a (dim G)-dimensional space.…”