Chern-simons forms on associated bundles, and boundary terms

Abstract

Let $E$ be a principle bundle over a compact manifold $M$ with compact structural group $G$. For any $G$-invariant polynomial $P$, The transgressive forms $TP(\omega)$ defined by Chern and Simons are shown to extend to forms $\Phi P(\omega)$ on associated bundles $B$ with fiber a quotient $F=G/H$ of the group. These forms satisfy a heterotic formula $$d\Phi P(\omega)=P(\Omega)-P(\Psi),$$ relating the characteristic form $P(\Omega)$ to a fiber-curvature characteristic form. For certain natural bundles $B$, $P(\Psi)=0$, giving a true transgressive form on the associated bundle, which leads to the standard obstruction properties of characteristic classes as well as natural expressions for boundary terms.