Hidden symmetry and (super)conformal mechanics in a monopole background

We study classical and quantum hidden symmetries of a particle with electric charge e in the background of a Dirac monopole of magnetic charge g subjected to an additional central potential V (r) = U (r) + (eg)(2)/2mr(2) with U (r) = 12 m omega(2)r(2), similar to that in the one-dimensional conformal mechanics model of de Alfaro, Fubini and Furlan (AFF). By means of a non-unitary conformal bridge transformation, we establish a relation of the quantum states and of all symmetries of the system with those of the system without harmonic trap, U (r) = 0. Introducing spin degrees of freedom via a very special spin-orbit coupling, we construct the sop(2|2) superconformal extension of the system with unbroken N = 2 Poincare supersymmetry and show that two different superconformal extensions of the one-dimensional AFF model with unbroken and spontaneously broken supersymmetry have a common origin. We also show a universal relationship between the dynamics of a Euclidean particle in an arbitrary central potential U (r) and the dynamics of a charged particle in a monopole background subjected to the potential V (r).