A random question on the local family index theorem A random question.
Say we have a complex vector bundle $E\to M$ that is isomorphic to a trivial bundle, and $d$ is a trivial connection on it. Let $M\to B$ be a fiber bundle with even dimensional closed connected oriented spin fibers. If we run the process of the local family index theorem, what happens to the kernel bundle (assume it exists) and what about the connection on it? Would the $k$-fold direct sum of the kernel bundle be trivial for some large $k$?
In the literature we see some secondary index theorems but I never know what is the (primary) family index theorem for trivial (or trivialized) vector bundle.
 A: The $k$-fold direct sum of the kernel bundle $H\to B$ (assuming it exists) is trivial as a topological bundle for some $k$ if and only if $\mathrm{ch}(H)=0\in H^\bullet(B)$ because the Chern character is an isomorphism $K^0(B)\otimes_{\mathbb Z}\mathbb Q\to H^{\mathrm{even}}(B;\mathbb Q)$. The primary cohomological index theorem takes the form $$\mathrm{ch}(H)=\int_{M/B}\hat A(TX)\,\mathrm{ch}(E)=\mathrm{rk} E\int_{M/B}\hat A(TX)\;,$$
see e.g. the books by Lawson-Michelsohn or Berline-Getzler-Vergne.
On the other hand, if you put the natural connection $\nabla^H$ on $H$, then its $k$-fold direct sum can be trivial as a bundle with connection if and only if $(H,\nabla^H)$ itself is trivial as a bundle with connection. However, I don't know a form of the family index theorem that is strong enough to detect this. The best I know is a geometric (or, if you like, secondary) index theorem that computes the Cheeger-Simons Chern character $\widehat{\mathrm{ch}}(H)$. But I doubt that $\widehat{\mathrm{ch}}(H,\nabla^H)=0$ already implies triviality of $(H,\nabla^H)$ as a vector bundle with connection.
