Elliptic monopoles and (4,0)-supersymmetric sigma models with torsion

We explicitly construct the metric and torsion couplings of two-dimensional (4,0)-supersymmetric sigma models with target space a four-manifold that are invariant under a U(1) symmetry generated by a tri-holomorphic Killing vector field that leaves in addition the torsion invariant. We show that the metric couplings arise from magnetic monopoles on the three-sphere which is the space of orbits of the group action generated by the tri-holomorphic Killing vector field on the sigma model target manifold. We also examine the global structure of a subclass of these metrics that are in addition SO(3)-invariant and find that the only non-singular one, for models with non-zero torsion, is that of SU(2)× U(1) WZW model.