@highergeometer Thanks! This sounds quite reasonable. Also since the curvature 2-forms on overlapping frames (say (\alpha ) and ( \beta )-frames) are related by [ F(A^{(\alpha)})=g^{-1}F(A^{(\beta)})g]it follows that
[\mbox{tr}(F(A^{(\alpha)})\wedge \ast F(A^{(\alpha)}))=\mbox{tr}(F(A^{(\beta)})\wedge \ast F(A^{(\beta)}))]So one has a global n-form for integration where n is the dimension of M. If one is integrating over M, then it seems that we also require M to be compact.