Non-linear sigma models and their Q-lump solutions

We find the conditions under which the three-dimensional Kähler sigma model with a potential term has non-dissipative but time dependent solutions, called Q-lumps, which saturate a Bogomol'nyi bound. These solutions only exist if the target manifold has a Killing vector field, k^α, with at least one fixed point and if the potential is of the form V = g_αβk^αk^β. This potential arises from dimensional reduction and in the linearised theory it is just a mass term. We discuss the elementary properties of the Q-lump solutions and construct explicit examples for the CPⁿ sigma models.