Semi-invariant terms for gauged non-linear σ-models

We determine all the terms that are gauge-invariant up to a total spacetime derivative (“semi-invariant terms”) for gauged non-linear sigma models. Assuming that the isotropy subgroup H of the gauge group is compact or semi-simple, we show that (non-trivial) such terms exist only in odd dimensions and are equivalent to the familiar Chern-Simons terms for the subgroup H. Various applications are mentioned, including one to the gauging of the Wess-Zumino-Witten terms in even spacetime dimensions. Our approach is based on the analysis of the descent equation associated with semi-invariant terms.