N=4 chiral supermultiplet interacting with a magnetic field
We couple N=4 chiral supermultiplet with an auxiliary N=4 fermionic supermutiplet containing on-shell four physical fermions and four auxiliary bosons. The latter ones play the role of isospin variables. We choose the very specific coupling which results in a component action containing only time derivatives of fermionic components presented in the auxiliary supermultiplet, which therefore may be dualized into auxiliary ones. The resulting component action describes the interaction of the chiral supermultiplet with a magnetic field constant on the pseudo-sphere $SU(1,1)/U(1)$. Then we specify the prepotential of our theory to get, in the bosonic sector, the action for the particle moving over the pseudo-sphere -- Lobachevsky space. We provided also the Hamiltonian formulation of this system and show that the full symmetry group of our system is $SU(1,1)\times U(1)$. The currents forming the $su(1,1)$ algebra are modified, as compared to the bosonic case, by the fermionic and isospin terms, while the additional u(1) current contains only isospin variables. One of the most important features of our construction is the presence in the Hamiltonian and supercharges of all currents of the isospin group SU(2). Despite the fact that two of the su(2) currents ${T,{\bar{T}}}$ enter the Hamiltonian only through the Casimir operator of the SU(2) group, they cannot be dropped out, even after fixing the total isospin of the system, because these currents themselves enter into the supercharges. We also present the Hamiltonian and supercharges describing the motion of a particle over the sphere $S^2$ in the background of constant magnetic field. In this case the additional isospin currents form the $su(1,1)$ algebra.