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To ask my question I have to write down the setup. Basically the setup is the adiabatic limit of the reduced eta invariant of Dirac operator associated to the submersion metric and connection. So if you are familiar with it you can skip the first three paragraphs (and at the same time I hope I get the background setup correct).

Let $M$ be a compact spin (or spin^c) manifold of odd dimension, and $E\to M$ a Hermitian vector bundle. Let $g^{TM}$ be the Riemannian metric on and $g^E$ a Hermitian metric on $E\to M$, $\nabla^{TM}$ the Levi-Civita connection on $TM\to M$ and $\nabla^E$ a unitary connection on $E\to M$. Thus one can form the twisted Dirac operator $D$ on the space of sections of $S(TM)\otimes E$ by the connection $\nabla^{S(TM)}\otimes\nabla^E$ composed with the Clifford multiplication, where $S(TM)\to M$ is the spinor bundle of $TM\to M$. Then one can define the eta invariant $\eta(D):=\eta(0)$ of $D$, where $$\eta(s)=\sum_{\lambda\neq0}\frac{\textrm{sign}(\lambda)}{|\lambda|^s}.$$ Define the reduced eta invariant of $D$ to be $\bar{\eta}(D):=\bar{\eta}(0)$, where $$\bar{\eta}(s)=\frac{1}{2}(\eta(s)+\dim(\ker(D)))\mod\mathbb{Z}.$$

Now consider a submersion $\pi:M\to B$ of manifolds with closed spin fibers $Z$ of even dimension. As expected, endow $TM\to M$ the submersion metric $$g^{TM}=\pi^*g^{TB}\oplus g^{TZ}$$ and pass it to the adiabatic limit $$g^{TM}_\varepsilon=\frac{1}{\varepsilon}\pi^*g^{TB}\oplus g^{TZ},$$ where $\varepsilon>0$, $g^{TB}$ and $g^{TZ}$ are Riemannian metrics on $TB\to B$ and $TZ\to Z$ respectively. Then we have a Levi-Civita connection $\nabla^{TM, \varepsilon}$ associated to the Riemannian metric $g^{TM}_\varepsilon$. Thus one can form the twisted Dirac operator and consider its reduced eta invariant $\bar{\eta}(D_\varepsilon)$.

My question is, whether $\bar{\eta}(D_\varepsilon)$ is differentiable with respect to $\varepsilon$, and if it is, what would be the formula for $$\frac{d}{d\varepsilon}\bar{\eta}(D_\varepsilon)$$

I don't know whether this question was addressed before. If so, can you give me the reference?

Many thanks.

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Consider $\varepsilon>0$ first. Here, we have the variation formula for the $\eta$-invariant from Atiyah, Patodi, and Singer, part 1, $$\frac{\eta + h}{2}(D_M^1) - \frac{\eta + h}{2}(D_M^0) \\ = \int_M \widetilde{\hat A}\bigl(TM,\nabla^{TM,\varepsilon_0},\nabla^{TM,\varepsilon_1}\bigr)\, \mathrm{ch}\bigl(E/S,\nabla^E\bigr)\in\mathbb R/\mathbb Z$$ It implies differentiability, and you can obtain from it a formula for the derivative.

At $\varepsilon=0$, you need the limit to exist. The proof I know by Dai needs that the kernels of the fibrewise Dirac operators form a vector bundle. From curvature computations by Bismut and Cheeger, section 4 one concludes that the derivative above extends continuously to $\varepsilon=0$, so you get differentiability.

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