Complex structure-induced deformations of sigma-models

We describe a deformation of the principal chiral model (with an even-dimensional target space G) by a B-field proportional to the K"ahler form on the target space. The equations of motion of the deformed model admit a zero-curvature representation. As a simplest example, we consider the case of G=S^1 x S^3. We also apply a variant of the construction to a deformation of the AdS_3 x S^3 x S^1 (super-)sigma-model.