Harmonic forms on principal bundles (Q1939224)

Let \((M,g_M)\) be a connected, compact, oriented Riemannian manifold without boundary, and let \(\pi:P\rightarrow M\) be a principal \(G\) bundle endowed with a \(G\)-invariant connection \(\Theta\), for \(G\) a compact semisimple Lie group. After choosing a bi-invariant metric \(g_G\) on \(G\), one can define a \(G\)-invariant metric on \(P\) by \(g_P:=\pi^*g_M\oplus g_G\). The limit as \(\delta\rightarrow 0\) of the one-parameter family of metrics \(g_\delta:=\delta^{-2}\pi^*g_M\oplus g_G\) is known as the adiabatic limit. The paper under review begins with a brief review of Hodge theory, the Leray-Serre spectral sequence, and the adiabatic spectral sequence, which is defined in terms of the asymptotic behavior of the Laplacian corresponding to \(g_\delta\) as \(\delta\rightarrow 0\), and is known to be isomorphic to the Leray-Serre spectral sequence associated to \(\pi:P\rightarrow M\), see [\textit{R. Forman}, Comm. Math. Phys. 168, No. 1, 57--116 (1995; Zbl 0827.58001)]. The Chern-Simons 1-form and 3-form corresponding to the \(G\)-invariant connection \(\Theta\) are related to the harmonic forms on \((P,g_P)\) by means of the adiabatic limit and the isomorphism between the adiabatic and Leray-Serre spectral sequences.