  We present two-dimensional gauged Lifshitz scalar field theories by
considering the duality relation between the source current and the Noether
current. Requiring the duality partially, we obtain a gauged model which
recovers the bosonized Schwinger model for the IR limit. For the exact duality,
however, the source current is not conserved, which means that the resulting
theory is anomalous, so that the number of degrees of freedom is increased. The
second model is consistently formulated by adding the Wess-Zumino type action
to maintain the gauge invariance.
