Gerbes, simplicial forms and invariants for families of foliated bundles
Let \(X\) be a compact oriented smooth manifold and \(G\) a Lie group with finitely many components. A family of principal \(G\)-bundles over \(X\) with connections consists of a smooth fiber bundle \(\pi:Y\to Z\) with fiber \(X\) and structure group \(\text{Diff}^+(X)\), a principal \(G\)-bundle \(p:E\to Y\), and a smooth family of connections \(A_z\) in the \(G\)-bundles \(P_z=E| _{X_z}\), \(X_z=\pi^{-1}(z)\), \(z\in Z\). This family of connections can be obtained from some global connection \(B\) in \(E\), but this global connection is not part of the structure. Let \(I_{{\mathbb Z}}^{n+1}(G)\subseteq I^{n+1}(G)\) be the set of invariant homogeneous polynomials \(Q\) of degree \(n+1\) on the Lie algebra \(\mathfrak g\) of \(G\) such that the Chern-Weil image in \(H^{2n+2}(BG,{\mathbb R})\) is an integral class, where \(BG\) is the classifying space; i.e., it is the image of a class \(u\in H^{2n+2}(BG,{\mathbb Z})\). Set \(\dim X=2n+1-\ell\) with \(0\leq\ell\leq 2n+1\), let \(F_{A_z}\) denote the curvature form of each \(A_z\), and let \(F_B\) be the curvature form of any given global extension \(B\) of \(\{A_z\;| \;z\in Z\}\). Then the authors distinguish between the following two cases. In Case~I (the `Godbillon-Vey' case), we have \(Q\in\ker(I^*(G)\to I^*(K))\) for some maximal compact Lie subgroup \(K\subseteq G\), and we can choose \(u=0\). In Case~II (the `Cheeger-Chern-Simons' case), we have \(u\neq0\). In Case~I, the authors prove that there is a natural class \([\Lambda_{Y/Z}(Q,B)]\in\Omega^\ell(Z)/d\Omega^{\ell-1}(Z)\); this class is independent of the choice of \(B\) provided \(F_{A_z}^{n+1-\ell}=0\) for all \(z\in Z\) \[ d\Lambda_{Y/Z}(Q,B)=(-1)^{\ell-1}\int_\pi Q(F_B^{n+1}) \] and, if \(F_{A_z}^{n+1-\ell}=0\) for all \(z\in Z\), then \([\Lambda_{Y/Z}(Q,B)]\) lies in \(H^\ell(BG,{\mathbb Z})\). In Case~II, the influence of \(u\) is taken into account by using cocycles in the Čech-de~Rham model for the Deligne cohomology associated to the sheaf \(\underline{U(1)}\) of smooth local functions with values in the circle group \(U(1)\subseteq{\mathbb C}\); these are called `(Hermitian line) gerbes with connection' (with a shift in degree) by the authors. Then the above statement is modified as follows. There is a natural class of \(\ell\)-gerbes \(\theta=\theta(Q,u,B)\) with connection \(\omega=(\omega^0,\dots,\omega^\ell)\) for a suitable open covering \(\mathcal U\); the class \([\theta,\omega]\) is independent of the choice of \(B\) provided \(F_{A_z}^{n+1-\ell}=0\) for all \(z\in Z\), \[ d\omega^0=(-1)^{\ell-1}\epsilon^*\int_\pi Q(F_B^{n+1})\quad\text{and}\quad \delta_*[\omega]=(-1)^{\ell-1}\,\pi_!(u(E)) \] and, if \(F_{A_z}^{n+1-\ell}=0\) for all \(z\in Z\), then \(d\omega^0=0\) and the invariant \([\theta,\omega]\) lies in \(H^\ell(BG,{\mathbb R}/{\mathbb Z})\). Here, \(\epsilon^*:\Omega^*(Z)\to\check{\Omega}^0({\mathcal U},\underline{\Omega}^*)\) is the natural inclusion of the de~Rham complex into the Čech-de~Rham bicomplex, \(u(E)\in H^{2n+2}(Y,{\mathbb Z})\) is the associated characteristic class of \(G\)-bundles \(E\to Y\), \(\pi_!:H^{2n+2}(Y,{\mathbb Z})\to H^{\ell+1}(Z,{\mathbb Z})\) is the usual transfer map, and \(\delta_*:H^\ell(Z,\underline{U(1)})\to H^{\ell+1}(Z,{\mathbb Z})\) is the usual isomorphism in Čech cohomology. The classical secondary characteristic classes are obtained when \(X\) is a singleton and \(\ell=2n+1\). The case \(\ell=1\) includes the construction of generalized Quillen line bundles. These results are applied to families of foliated \(G\)-bundles, yielding invariants of their foliations.
family of principal bundles with connections
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