Eta invariants, differential characters and flat vector bundles
The paper addresses the heat equation method for obtaining the explicit representation of the Chern character of the index bundle of a family of Dirac operators as a differential form on the base space. In order to formulate a result of the paper let us consider the following situation. Let \(\pi: M\to B\) be a smooth bundle over a closed manifold \(B\) with a compact spin even dimensional fibre \(Z\), with a suitable connection \(\nabla^{TZ}\) on the vertical tangent bundle \(TZ\) . Let \(D^Z\) be a family of Dirac operators with coefficients in a Hermitian bundle \(\xi\) over \(M\) with connection \(\nabla^\xi\), and let \(\nabla^{\ker D^Z}\) be the Bismut superconnection on the index bundle \(\ker D^Z\) constructed in [\textit{J.-M. Bismut}, Invent. Math. 83, 91--151 (1986; Zbl 0592.58047)]. Let \(\widetilde\eta\) be the Bismut-Cheeger form such that \[ d\widetilde\eta= \pi_*[\widehat A(TZ,\nabla^{TZ})\text{ch}(\xi, \nabla^\xi)]- \text{ch}(\ker D^Z,\nabla^{\ker D^Z}). \] Finally, let \(\widehat H(M, R/Q)\) be the Cheeger-Simons ring of differential characters (a refinement of the ordinary cohomology group \(H(M,R/Q)\)), and write c\(\widehat {\text{h}}(\xi,\nabla^\xi)\) for the ``lift'' of \(\text{ch}(\xi,\nabla^\xi)\) to \(\widehat H(M,R/Q)\) and similarly for other characteristic classes. One of the main results of the paper states that in \(\widehat H(M,R/Q)\) the following equation holds: \[ \text{c}\widehat {\text{h}}(\ker D^Z, \nabla^{\ker D^Z})+ \widetilde\eta= \pi_*[\widehat{\widehat A}(TZ, \nabla^{TZ}) \text{c}\widehat {\text{h}}(\xi,\nabla^\xi)]. \] The proof of the equation is based on some earlier results concerning adiabatic limits due to the author and Cheeger and the Cheeger-Simons expression of the \(\text{mod\,}Q\) reduction of the eta-invariant as the integral over \(M\) of \(\widehat{\widehat A}(TM,\nabla^{TM}) \text{c}\widehat{\text{h}}(\xi,\nabla^\xi)\). Another result of the paper is the equality between the lift to \(\widehat H(B,C/Q)\) of the total Chern form of the direct image by \(\pi\) of a flat vector bundle \(F\) over \(M\) with connection and the direct image by \(\pi_*\) in \(\widehat H(B,C/Q)\) of the lift to \(\widehat H(M,C/Q)\) of the Euler form of \(TZ\) times the Chern form of \(F\).
scientific article; zbMATH DE number 2162857
scientific article