  Wave maps (i.e. nonlinear sigma models) with torsion are considered in 2+1
dimensions. Global existence of smooth solutions to the Cauchy problem is
proven for certain reductions under a translation group action: invariant wave
maps into general targets, and equivariant wave maps into Lie group targets. In
the case of Lie group targets (i.e. chiral models), a geometrical
characterization of invariant and equivariant wave maps is given in terms of a
formulation using frames.
