Explicit computations of Bismut-Cheeger eta form for $S^{2n}$ bundles

Question

I'm interested in computing eta invariants of Dirac operators (on spinor bundles tensored with some vector bundles) on the total space of $S^{2n}$ bundles over odd-dimensional manifolds. I found the papers of Bismut-Cheeger and Dai where a formula is given in terms of Bismut-Cheeger eta forms, but I'm having a hard time actually computing them.

Are there places where they are computed explicitly for some $S^{2n}$-bundles? This MO answer gives various explicit computations of eta invariants but I didn't find $S^{2n}$ bundles there.

Edit:

To add a bit more context, I’m interested in the following restricted situation.

Let $Spin(2n)$ act on $S^{2n}$ fixing the north and the south poles. Pick a metric compatible with this action. Now pick an $Spin(2n)$ bundle $P$ with a connection over an odd dimensional base $B$. From this data we have a fibration $S^{2n}\to X\to B$. Then I should be able to compute the eta form of the Dirac operator on $X$.

What I’m confused is that, as the eta form is local on $B$, it should be a local expression of the metric on $B$ and the $Spin(2n)$ connection on $P$. But the eta form is an odd-degree form, and I can’t think of any way to write a local expression in terms of the metric and the $Spin(2n)$ connection! Does it mean that the eta form in this class of examples is simply zero?

I would also like to consider the case when we have an $Spin(2n)$ equivariant vector bundle $E$ over $X$ and compute the eta form for the Dirac operator tensored with $E$. I’m still at a loss what nonzero term I can write in this situation.

Answer 1

If you consider a fibre bundle with compact structure group that acts by isometries on the typical fibre and preserves a given Dirac operator, then the $\eta$-form can be read of from an infinitesimally equivariant $\eta$-invariant on the typical fibre, the fibre bundle curvature, and its action on the typical fibre. This is explained in this article and much better in an article by Liu and Ma. If the fibre is even-dimensional, the infinitesimally equivariant $\eta$-invariant vanishes, so your suspicion is correct.

In your special case, there is an orientation reversing, fibre preserving isometry that swaps north and south poles. If this respects the superconnection in question, then you get another argument for the vanishing of the $\eta$-form.