Supersymmetric monopole quantum mechanics on a sphere

We study $N=2$ supersymmetric quantum mechanics of a charged particle on sphere in the background of Dirac magnetic monopole. We adopt $CP(1)$ model approach in which the monopole interaction is free of singularity. It turns out that this approach admits a compact $N=2$ superspace formulation. In order to exploit manifest U(1) covariance in the superspace formalism, we introduce a gauged chiral superfield which is annihilated by the gauge covariant superderivative instead of the usual superderivative. We carry out the Dirac quantization of the resulting system and compute the quantum mechanical spectrum. We obtain the condition for the spontaneous breaking of supersymmetry explicitly in terms of the monopole charge and a parameter which characterizes the operator ordering ambiguity. We find that the supersymmetry is spontaneously broken unless a certain combination of these quantities satisfies some quantization condition.

Figure 1

Diagram for $k$ versus $\tilde{g}$.