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New D(2,1;alpha) Mechanics with Spin Variables

We elaborate on a novel superconformal mechanics model possessing D(2,1;alpha) symmetry and involving extra U(2) spin variables. It is the one-particle case of the N=4 superconformal matrix model recently proposed in arXiv:0812.4276 [hep-th], and it generalizes to arbitrary alpha\neq0 the OSp(4|2) superconformal mechanics of arXiv:0905.4951 [hep-th]. As in the latter case, the U(2) spin variables are described by a Wess-Zumino action and define the first Hopf map S^3 -> S^2 in the target space. Upon quantization, they represent a fuzzy sphere. We find the classical and quantum generators of the D(2,1;alpha) superalgebra and their realization on the physical states. The super wavefunction encompasses various multiplets of the SU(2)_R and SU(2)_L subgroups of D(2,1;alpha), with fixed isospins. The conformal potential is determined by the external magnetic field in the Wess-Zumino term, whose amplitude is quantized like in the OSp(4|2) case. As a byproduct, we reveal new invariant subspaces in the enveloping algebra of D(2,1;alpha) for our quantum realization.