Wess-Zumino-Witten model based on a nonsemisimple group

We present a conformal field theory which describes a homogeneous four dimensional Lorentz-signature space-time. The model is an ungauged Wess-Zumino-Witten model based on a central extension of the Poincar\'e algebra. The central charge of this theory is exactly four, just like four dimensional Minkowski space. The model can be interpreted as a four dimensional monochromatic plane wave. As there are three commuting isometries, other interesting geometries are expected to emerge via O(3,3) duality.