Harmonic forms on principal bundles

Abstract

We show a relationship between Chern–Simons 1- and 3-forms and harmonic forms on a principal bundle. Doing so requires one to consider an adiabatic limit. For the 3-form case, assume that $G$ is simple and the corresponding Chern–Weil 4-form is exact. Then, the Chern–Simons 3-form on the princpal bundle $G$-bundle, minus a canonical term from the base, is harmonic in the adiabatic limit.