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.

15 pages, no figures, AMS-LaTeX

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