  We study a functional, whose critical points couple Dirac-harmonic maps from
surfaces with a two form. The critical points can be interpreted as coupling
the prescribed mean curvature equation to spinor fields. On the other hand,
this functional also arises as part of the supersymmetric sigma model in
theoretical physics. In two dimensions it is conformally invariant. We call
critical points of this functional magnetic Dirac-harmonic maps. We study
geometric and analytic properties of magnetic Dirac-harmonic maps including
their regularity and the removal of isolated singularities.
