# Local index theory and the Riemann-Roch-Grothendieck theorem for complex
flat vector bundles II

Man-Ho Ho<EMAIL_ADDRESS>Hong Kong

###### Abstract.

In this paper, we define an analytic index
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}$ in differential $K$-theory
associated to families of generalized twisted de Rham operators for any given
submersion $\pi:X\to B$ with closed fibers. We prove that the analytic index
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}$ of the generator associated
to any given complex flat vector bundle $(F,\nabla^{F})$ is equal to the
generator associated to the cohomology bundle
$(H(Z,F|_{Z}),\nabla^{H(Z,F|_{Z})})$ of $(F,\nabla^{F})$. Moreover, we prove
the real part of the Riemann–Roch–Grothendieck theorem for complex flat vector
bundles at the differential form level. This gives a new proof of the real
part of the Riemann–Roch–Grothendieck theorem for complex flat vector bundles
previously proved by Ma–Zhang, and also generalizes our previous work on the
$\mathbb{Z}_{2}$-graded version of the same theorem in the even dimensional
fiber case.

###### Key words and phrases:

Riemann–Roch–Grothendieck theorem, analytic index, complex flat vector
bundles, Bismut–Cheeger eta form, differential $K$-theory

###### 2020 Mathematics Subject Classification:

Primary 19K56, 19L50, 19L10

_Dedicated to the memory of Connie_

###### Contents

  1. 1 Introduction
    1. 1.1 Background
    2. 1.2 The Main Results
  2. 2 Preliminaries
    1. 2.1 Notations and conventions
    2. 2.2 Some secondary characteristic forms
  3. 3 Analytic index for twisted de Rham operator
    1. 3.1 Local index theory for generalized twisted de Rham operator
    2. 3.2 A variational formula for the Bismut–Cheeger eta form ${\widehat{\eta}}^{\Lambda}$
    3. 3.3 Some properties of the Bismut–Cheeger eta form ${\widehat{\eta}}^{\Lambda}$
    4. 3.4 An analytic index in differential $K$-theory
  4. 4 Complex flat vector bundles and differential $K$-theory
    1. 4.1 Local index theory for twisted de Rham operator
    2. 4.2 Complex flat vector bundles and the analytic index $\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}$
    3. 4.3 RRG for complex flat vector bundles

##  1\. Introduction

###  1.1. Background

Let $\pi:X\to B$ be a submersion with closed fibers $Z$, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. The Riemann–Roch–Grothendieck
(RRG) theorem for complex flat vector bundles [16, Theorem 1.1] states that
for any complex flat vector bundle $(F,\nabla^{F})$ over $X$,

$\operatorname{CCS}(H(Z,F|_{Z}),\nabla^{H(Z,F|_{Z})})=\int_{X/B}e(T^{V}X)\cup\operatorname{CCS}(F,\nabla^{F})$
(1.1.1)

in $H^{\operatorname{odd}}(B;\mathbb{C}/\mathbb{Q})$, where
$\operatorname{CCS}(F,\nabla^{F})$ and $(H(Z,F|_{Z}),\nabla^{H(Z,F|_{Z})})$
are the Cheeger–Chern–Simons class and the cohomology bundle of
$(F,\nabla^{F})$, respectively.

Bismut–Lott prove a refinement of the imaginary part of (1.1.1) at the
differential form level [6, Theorem 3.23] by constructing the analytic torsion
form $\mathscr{T}$ [6, Definition 3.22]. Bismut then proves the real part of
(1.1.1), namely,

$\operatorname{Re}(\operatorname{CCS}(H(Z,F|_{Z}),\nabla^{H(Z,F|_{Z})}))=\int_{X/B}e(T^{V}X)\cup\operatorname{Re}(\operatorname{CCS}(F,\nabla^{F}))$
(1.1.2)

in $H^{\operatorname{odd}}(B;\mathbb{R}/\mathbb{Q})$ under the assumption that
the fibers are fiberwise orientable [5, Theorem 3.2]. On the other hand,
Ma–Zhang prove (1.1.1) by giving another proof of the imaginary part of
(1.1.1) and prove (1.1.2) in full generality.

The strategy of the proof of (1.1.2) by Ma–Zhang is roughly as follows. They
first prove that

$\begin{split}&\operatorname{Re}(\operatorname{CCS}(H(Z,F|_{Z}),\nabla^{H(Z,F|_{Z})}))-\operatorname{Re}(\operatorname{CCS}(H(Z,\mathbb{C}^{\ell}|_{Z})))\\\
&=\int_{X/B}e(T^{V}X)\cup\operatorname{Re}(\operatorname{CCS}(F,\nabla^{F}))\end{split}$
(1.1.3)

in $H^{\operatorname{odd}}(B;\mathbb{R}/\mathbb{Q})$ [16, (3.98)], where
$\ell=\operatorname{rank}(F)$. Then they show that

$\operatorname{Re}(\operatorname{CCS}(H(Z,\mathbb{C}^{\ell}|_{Z})))=0$ (1.1.4)

[16, p.613-614] by considering the parity of $\dim(Z)$ separately. For
$\dim(Z)$ odd, they prove (1.1.4) by an application of Poincaré duality; for
$\dim(Z)$ even, they prove (1.1.4) by applying a result of Bismut [5, Theorem
3.12], which is essentially an application of the special case of (1.1.2)
proved by Bismut.

Furthermore, under the additional assumptions that the fibers $Z$ are even-
dimensional, oriented and spinc, Ma–Zhang lift (1.1.3) to the level of
$\mathbb{R}/\mathbb{Z}$ $K$-theory [16, Theorem 3.14], i.e.

$\operatorname{ind}^{t}_{\mathbb{R}/\mathbb{Z}}(S(T^{V}X)^{*}\otimes\mathcal{F})=\mathcal{H}(Z,F|_{Z})-\mathcal{H}(Z,\mathbb{C}^{\ell}|_{Z})$
(1.1.5)

in the $\mathbb{R}/\mathbb{Z}$ $K$-theory group
$K^{-1}(B;\mathbb{R}/\mathbb{Z})$, where $\mathcal{F}$ and
$\mathcal{H}(Z,F|_{Z})$ are the generators of $\mathbb{R}/\mathbb{Z}$
$K$-theory group associated to $(F,\nabla^{F})$ and its cohomology bundle,
respectively, and
$\operatorname{ind}^{t}_{\mathbb{R}/\mathbb{Z}}:K^{-1}(X;\mathbb{R}/\mathbb{Z})\to
K^{-1}(B;\mathbb{R}/\mathbb{Z})$ is the topological index in
$\mathbb{R}/\mathbb{Z}$ $K$-theory. Here we use the geometric model of
$\mathbb{R}/\mathbb{Z}$ $K$-theory defined by Lott [14].

In our previous work, we prove the $\mathbb{Z}_{2}$-graded version of the real
part of the RRG theorem for complex flat vector bundles for $\dim(Z)$ even at
the differential form level [11, Theorem 1], which implies

$\operatorname{Re}(\operatorname{CCS}(H(Z,\mathsf{F}|_{Z}),\nabla^{H(Z,\mathsf{F}|_{Z})}))=\int_{X/B}e(T^{V}X)\cup\operatorname{Re}(\operatorname{CCS}(\mathsf{F},\nabla^{\mathsf{F}})).$
(1.1.6)

Here $(\mathsf{F},\nabla^{\mathsf{F}})$ is a $\mathbb{Z}_{2}$-graded complex
flat vector bundle of virtual rank zero. For any given complex flat vector
bundle $(F,\nabla^{F})$ over $X$ of rank $\ell$, by taking
$(\mathsf{F},\nabla^{\mathsf{F}})$ in (1.1.6) to be

$(\mathsf{F}^{+},\nabla^{\mathsf{F},+})=(F,\nabla^{F}),\qquad(\mathsf{F}^{-},\nabla^{\mathsf{F},-})=(\mathbb{C}^{\ell},d^{\ell}),$
(1.1.7)

we obtain (1.1.3).

###  1.2. The Main Results

The purpose of this paper is to prove the real part of the RRG theorem for
complex flat vector bundles (1.1.2) at the differential form level, and to
prove a generalization of the analytical counterpart of (1.1.5).

For the generalization of the analytical counterpart of (1.1.5), for any given
submersion $\pi:X\to B$ with closed fibers we define an analytic index in
differential $K$-theory (Definition 3.1)

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}:{\widehat{K}}(X)\to{\widehat{K}}(B)$
(1.2.1)

associated to families of twisted de Rham operators $\mathsf{D}^{\Lambda}$,
whose local index theory is developed by Bismut–Lott [6, §III]. Here
${\widehat{K}}$ is the geometric model of differential $K$-theory defined by
Freed–Lott [7]. Since the de Rham operators considered in [6] are twisted by
complex flat vector bundles (with Hermitian metrics) and the de Rham operators
in the definition of the analytic index (1.2.1) are twisted by Hermitian
bundles with unitary connections, we call the latter generalized twisted de
Rham operators. We do not assume the family of generalized twisted de Rham
operators $\mathsf{D}^{\Lambda}$ satisfies the kernel bundle assumption. After
establishing several expected properties of the corresponding Bismut–Cheeger
eta form ${\widehat{\eta}}^{\Lambda}$ and proving that the analytic index
(1.2.1) is a well defined group homomorphism (Proposition 3.7), we prove the
following theorem (Theorem 4.1).

######  Theorem 1.1.

Let $\pi:X\to B$ be a submersion with closed fibers $Z$, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. For any complex flat vector bundle
$(F,\nabla^{F})$,

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{F})=\mathcal{H}(Z,F|_{Z})$

in ${\widehat{K}}(B)$, where $\mathcal{F}$ is the generator of
${\widehat{K}}(X)$ associated to $(F,\nabla^{F})$.

Theorem 1.1 actually holds for generators $\mathcal{F}$ associated to any
$\mathbb{Z}_{2}$-graded complex flat vector bundle
$(\mathsf{F},\nabla^{\mathsf{F}})$ of virtual rank zero. In this case, both
$\mathcal{F}$ and $\mathcal{H}(Z,\mathsf{F}|_{Z})$ are also generators of the
$\mathbb{R}/\mathbb{Z}$ $K$-theory group. Thus Theorem 1.1 implies that

$\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}(\mathcal{F})=\mathcal{H}(Z,\mathsf{F}|_{Z})$
(1.2.2)

in $K^{-1}(B;\mathbb{R}/\mathbb{Z})$, where
$\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}:K^{-1}(X;\mathbb{R}/\mathbb{Z})\to
K^{-1}(B;\mathbb{R}/\mathbb{Z})$ is the analytic index in
$\mathbb{R}/\mathbb{Z}$ $K$-theory associated to families of twisted de Rham
operators, which is obtained by restricting
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}$ to
$K^{-1}(X;\mathbb{R}/\mathbb{Z})$.

We prove the RRG theorem for
$\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}$ in
$\mathbb{R}/\mathbb{Z}$ $K$-theory (Theorem 3.3), whose spinc analog is first
proved by Lott [14, Corollary 4] (cf. [10, Theorem 1.3] for an independent
proof). As an immediate consequence of Theorems 1.1 and 3.3, we derive the
$\mathbb{Z}_{2}$-graded version of the real part of the RRG theorem for
complex flat vector bundles (1.1.6) ((b) of Corollary 4.1).

As noted in the proof of [16, Theorem 3.14], it is not easy to compare both
sides of (1.2.2) directly. Moreover, as pointed out in [16, p.572], (1.1.5)
should hold without assuming the fibers $Z$ are even dimensional, oriented and
spinc. Thus we regard Theorem 1.1 as a generalization of the analytical
counterpart of (1.1.5).

Freed–Lott define an analytic index
$\operatorname{ind}^{a}_{{\widehat{K}}}:{\widehat{K}}(X)\to{\widehat{K}}(B)$
in differential $K$-theory [7, Definition 7.27] associated to families of
twisted spinc Dirac operators. The relationship between the analytic indexes
(1.2.1) and $\operatorname{ind}^{a}_{{\widehat{K}}}$ is given by the following
proposition (Proposition 3.8).

######  Proposition 1.1.

Let $\pi:X\to B$ be a submersion with closed, oriented and spinc fibers of
even dimension, equipped with a Riemann and differential spinc structure
$(T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda})$. For any
$\mathbb{Z}_{2}$-graded generator $\mathcal{E}$ of ${\widehat{K}}(X)$,

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E})=\operatorname{ind}^{a}_{{\widehat{K}}}(S(T^{V}X)^{*}\otimes\mathcal{E}).$

The second main result is the real part of the RRG theorem for complex flat
vector bundles (1.1.2) at the differential form level (Theorem 4.2).

######  Theorem 1.2.

Let $\pi:X\to B$ be a submersion with closed fibers $Z$, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. For any complex flat vector bundle
$(F,\nabla^{F})$ over $X$, there exist sufficiently large $k,N\in\mathbb{N}$,
a smooth isometric isomorphism
$\alpha:(kF,kg^{F})\to(\mathbb{C}^{k\ell},g^{k\ell})$, where
$\ell=\operatorname{rank}(F)$, and a smooth $\mathbb{Z}_{2}$-graded isometric
isomorphism

$h:(kH(Z,F|_{Z})\oplus{\underline{\mathbb{C}^{N}}},kg^{H(Z,F|_{Z})}\oplus{\underline{g^{N}}})\to(\mathbb{C}^{km}\oplus{\underline{\mathbb{C}^{N}}},g^{km}\oplus{\underline{g^{N}}})$

such that

$\operatorname{CS}\big{(}h^{*}(d^{km_{+}}\oplus{\underline{d^{N}}}),k\nabla^{H(Z,F|_{Z}),u}\oplus{\underline{d^{N}}}\big{)}\equiv\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{CS}(\alpha^{*}d^{k\ell},k\nabla^{F,u}).$
(1.2.3)

Although the left-hand side of (1.2.3) does not look like a differential form
representative of
$\operatorname{Re}(\operatorname{CCS}(H(Z,F|_{Z}),\nabla^{H(Z,F|_{Z})}))$, in
Theorem 4.3 we justify the claim that Theorem 1.2 is a refinement of (1.1.2)
at the differential form level. On the other hand, by taking $(F,\nabla^{F})$
to be a trivial bundle $(\mathbb{C}^{\ell},d^{\ell})$ in Theorem 1.2 and
noting that a differential form representative of
$\operatorname{CCS}(\mathbb{C}^{\ell},d^{\ell})$ is given by odd Chern
character, which implies $\operatorname{CCS}(\mathbb{C}^{\ell},d^{\ell})=0$ in
$H^{\operatorname{odd}}(X;\mathbb{C}/\mathbb{Q})$, we obtain a unified proof
of (1.1.4) at the differential form level regardless of the parity of
$\dim(Z)$.

The proof of Theorem 1.2 is similar to that of [11, Theorem 1] in spirit.
However, there are two notable differences.

  * •

In proving Theorem 1.2 we work with a perturbed generalized twisted de Rham
operator, in contrast to the perturbed twisted spinc Dirac operator used in
[11, Theorem 1]. The former allows us to treat the even and odd dimensional
fiber cases simultaneously, while the latter only allows us to treat the even
dimensional fiber case.

  * •

In proving Theorem 1.2 we refine several parts of the argument in the proof of
[11, Theorem 1]. Notably, we take a more suitable pair of unitary connections
on the “index bundle” over $B\times[0,1]$ when defining a certain Chern–Simons
form. This is the main reason why we can directly consider complex flat vector
bundles instead of $\mathbb{Z}_{2}$-graded ones of virtual rank zero, as we
did in [11].

While Theorem 1.2 is a generalization of [11, Theorem 1], we would like to
emphasize that [11, Theorem 1] is much weaker than Theorem 1.2. First, [11,
Theorem 1] only holds for the complex flat vector bundles that are
$\mathbb{Z}_{2}$-graded and have virtual rank zero. Second, and more
importantly, we have come to realize the difficulty of proving (1.1.4)
independently when we try to obtain (1.1.2) from (1.1.3). Although (1.1.4) is
now derived as a consequence of Theorem 1.2 as mentioned above, we still
regard proving (1.1.4) independently as a difficult problem, and the proof of
(1.1.4) given by Ma–Zhang and Bismut is more intricate than we expect.

This paper is organized as follows. In §2 we set up the notations and
conventions that are used in this paper and recall the construction and some
properties of some secondary characteristic forms. In §3 we define the
analytic index (1.2.1) and establish some of its properties. In §4 we prove
the main results of this paper.

## Acknowledgement

The author is indebted to Steve Rosenberg for his comments and suggestions for
this paper. Some of the ideas in this paper are inspired during a visit to the
Institute of Mathematics of the Academia Sinica, Taiwan in July, 2019. The
author would like to thank Jih-Hsin Cheng for the invitation, Yi-Sheng Wang
for many enlightening discussions, and the Academia Sinica for the
hospitality.

The author would like to dedicate this paper to his beloved chihuahua, Connie,
who was a cherished member of his family. We are lucky to have each other for
many years. Her memory will always be with us.

##  2\. Preliminaries

In §2.1 we set up the notations and conventions that are used in this paper.
In §2.2 we recall the constructions and some properties of some secondary
characteristic forms, and refer to [15, Appendix B.5] for the details.

###  2.1. Notations and conventions

In this paper $X$ and $B$ are closed manifolds and $I$ is the closed interval
$[0,1]$. Given a manifold $X$, define ${\widetilde{X}}=X\times I$. Given $t\in
I$, define a map $i_{X,t}:X\to{\widetilde{X}}$ by $i_{X,t}(x)=(x,t)$. Denote
by $p_{X}:{\widetilde{X}}\to X$ the standard projection map. For $k\geq 0$,
denote by $\Omega^{k}_{\mathbb{Q}}(X,\mathbb{C})$ the set of all complex-
valued closed $k$-forms on $X$ with periods in $\mathbb{Q}$, and write
$\Omega^{k}_{\mathbb{Q}}(X)$ for $\Omega^{k}_{\mathbb{Q}}(X,\mathbb{R})$. For
any differential forms $\omega$ and $\eta$, we write $\omega\equiv\eta$ if
$\omega-\eta\in\operatorname{Im}(d)$.

Let $\pi:M\to B$ a smooth fiber bundle with compact fibers of dimension $n$
which satisfies certain orientability assumptions. Then

$\int_{M/B}\pi^{*}\alpha\wedge\beta=\alpha\wedge\bigg{(}\int_{M/B}\beta\bigg{)}$

for any $\alpha\in\Omega^{\bullet}(B)$ and $\beta\in\Omega^{\bullet}(M)$. If
$M$ has nonempty boundary, then Stokes’ theorem for integration along the
fibers [9, Problem 4 of Chapter VII] states that for any
$\omega\in\Omega^{k}(M)$,

$(-1)^{k-n+1}\int_{\partial
M/B}i^{*}\omega=\int_{M/B}d^{M}\omega-d^{B}\int_{M/B}\omega,$ (2.1.1)

where $i:\partial M\hookrightarrow M$ is the inclusion map.

Let $E\to X$ be a complex vector bundle. If $E\to X$ is
$\mathbb{Z}_{2}$-graded, denote by $E^{\operatorname{op}}\to X$ the
$\mathbb{Z}_{2}$-graded complex vector bundle whose $\mathbb{Z}_{2}$-grading
is the opposite of $E\to X$. We will also use the notation $\operatorname{op}$
for other $\mathbb{Z}_{2}$-graded objects. A triple $(E,g^{E},\nabla^{E})$
consisting of a complex vector bundle with a Hermitian metric and a unitary
connection is said to be $\mathbb{Z}_{2}$-graded if $E\to X$ is
$\mathbb{Z}_{2}$-graded, and $g^{E}$ and $\nabla^{E}$ preserve the
$\mathbb{Z}_{2}$-grading (which are also said to be $\mathbb{Z}_{2}$-graded).
A $\mathbb{Z}_{2}$-graded triple $(E,g^{E},\nabla^{E})$ is said to be balanced
if $E^{+}=E^{-}$, $g^{E,+}=g^{E,-}$ and $\nabla^{E,+}=\nabla^{E,-}$.

Denote by $\mathbb{C}^{\ell}\to X$ the trivial complex vector bundle of rank
$\ell$. Write $g^{\ell}$ for the standard Hermitian metric and $d^{\ell}$ the
standard trivial connection on $\mathbb{C}^{\ell}\to X$.

Let $(E,g^{E},\nabla^{E})$ be a $\mathbb{Z}_{2}$-graded triple and
$s\in\Gamma(X,\operatorname{End}(E)^{-})$. Recall from [4, Definition 2.9]
that the quadruple $(E,g^{E},\nabla^{E},s)$ is said to split if

  1. (a)

$E=\ker(s)\oplus\operatorname{Im}(s)$,

  2. (b)

$\nabla^{E}$ preserves $\ker(s)\to X$ and $\operatorname{Im}(s)\to X$, and

  3. (c)

$s$ is a unitary odd section of $\operatorname{End}(\operatorname{Im}(s))\to
X$ preserving $\nabla^{\operatorname{Im}(s)}$.

######  Example 1.

An example of split quadruple that will be used in this paper is given by
$(W,g^{W},\nabla^{W},s_{W})$, where $(W,g^{W},\nabla^{W})$ is a balanced
$\mathbb{Z}_{2}$-graded triple and $s=\begin{pmatrix}0&\operatorname{id}\\\
\operatorname{id}&0\end{pmatrix}$.

######  Remark 2.1.

Let $E\to X$ and $F\to X$ be complex vector bundles.

  1. (a)

Write $E\otimes F\to X$ for the tensor product if at most one of $E\to X$ and
$F\to X$ is $\mathbb{Z}_{2}$-graded, and $E\mathbin{{\widehat{\otimes}}}F\to
X$ if both of $E\to X$ and $F\to X$ are $\mathbb{Z}_{2}$-graded.

  2. (b)

For any connections $\nabla^{E}$ and $\nabla^{F}$ on $E\to X$ and $F\to X$, if
no confusion arises we write $\nabla^{E\otimes F}$ for the tensor product of
$\nabla^{E}$ and $\nabla^{F}$. In terms of our notation convention, the
associativity of tensor product of connections becomes $\nabla^{(E\otimes
F)\otimes H}=\nabla^{E\otimes(F\otimes H)}$.

######  Remark 2.2.

Let $E\to X$ and $F\to X$ be complex vector bundles, and suppose there exists
a smooth bundle isomorphism $\alpha:E\to F$.

  1. (a)

We use the same symbol to denote the resulting $C^{\infty}(X)$-module
isomorphism $\Gamma(X,E)\to\Gamma(X,F)$ and some others, for example,
$\Gamma(X,T^{*}X\otimes E)\to\Gamma(X,T^{*}X\otimes F)$.

  2. (b)

Let $g^{E}$ and $g^{F}$ be Hermitian metrics on $E\to X$ and $F\to X$,
respectively. Since $g^{E}$ and $\alpha^{*}g^{F}$ are Hermitian metrics on
$E\to X$, it follows from [13, Theorem 8.8 of Chapter I] that there exists a
unique $f\in\operatorname{Aut}(E)$ such that
$g^{E}=f^{*}(\alpha^{*}g^{F})=(\alpha\circ f)^{*}g^{F}$.

Henceforth, once Hermitian metrics are put on $E\to X$ and $F\to X$, we always
assume a given smooth bundle isomorphism $\alpha:E\to F$ to be isometric.

  3. (c)

Let $\nabla^{F}$ be a connection on $F\to X$. Write $\alpha^{*}\nabla^{F}$ for
the connection on $E\to X$ defined by

$\alpha^{*}\nabla^{F}:=\alpha^{-1}\circ\nabla^{F}\circ\alpha.$

If $\nabla^{F}$ is compatible with $g^{F}$, it follows from our convention
that $\alpha^{*}\nabla^{F}$ is compatible with $g^{E}$.

###  2.2. Some secondary characteristic forms

Let $(E,g^{E},\nabla^{E})$ be a triple. Denote by $R^{E}$ the curvature of
$\nabla^{E}$. The Chern character form of $\nabla^{E}$ is defined to be

$\operatorname{ch}(\nabla^{E})=\operatorname{tr}(e^{-\frac{1}{2\pi
i}R^{E}})\in\Omega^{\operatorname{even}}_{\mathbb{Q}}(X).$

Let $(g^{E}_{0},\nabla^{E}_{0})$ and $(g^{E}_{1},\nabla^{E}_{1})$ be two pairs
of Hermitian metrics and unitary connections on $E\to X$. Define a complex
vector bundle $\mathcal{E}\to{\widetilde{X}}$ by $\mathcal{E}=p_{X}^{*}E$. Let
$f_{k}:E\to{\overline{E}}^{*}$ be the canonical smooth bundle isomorphism
associated to $g^{E}_{k}$, where $k\in\\{0,1\\}$. Then $f_{1}^{-1}\circ
f_{0}\in\operatorname{Aut}(E)$ is positive and self-adjoint with respect to
$g^{E}_{1}$ on each fiber. Denote by $f$ the unique positive self-adjoint
square root of $f_{1}^{-1}\circ f_{0}$. By the proof of [13, Theorem 8.8 of
Chapter 1] we have $g^{E}_{0}=f^{*}g^{E}_{1}$. Thus $f^{*}\nabla^{E}_{1}$ is
unitary with respect to $g^{E}_{0}$. For each $t\in I$, let
$f_{t}=(1-t)\operatorname{id}_{E}+tf^{-1}$. Then

$g^{E}_{t}=f_{t}^{*}g^{E}_{0},\qquad\nabla^{E}_{t}=f_{t}^{*}\big{(}(1-t)\nabla^{E}_{0}+tf^{*}\nabla^{E}_{1}\big{)}$
(2.2.1)

are smooth paths of Hermitian metrics and unitary connections from $g^{E}_{0}$
to $g^{E}_{1}$ and from $\nabla^{E}_{0}$ to $\nabla^{E}_{1}$, respectively.
Note that $\nabla^{E}_{t}$ is unitary with respect to $g^{E}_{t}$ for each
$t\in I$. Then

$g^{\mathcal{E}}:=p_{X}^{*}g^{E}_{t},\qquad\nabla^{\mathcal{E}}:=dt\wedge\bigg{(}\frac{\partial}{\partial
t}+\frac{1}{2}(g^{E}_{t})^{-1}\frac{\partial}{\partial
t}g^{E}_{t}\bigg{)}+\nabla^{E}_{t}$ (2.2.2)

is a Hermitian metric and a unitary connection on
$\mathcal{E}\to{\widetilde{X}}$, respectively. Note that
$i_{X,k}^{*}\nabla^{\mathcal{E}}=\nabla^{E}_{k}$ for $k\in\\{0,1\\}$. The
Chern–Simons form
$\displaystyle\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})\in\frac{\Omega^{\operatorname{odd}}(X)}{\operatorname{Im}(d)}$
is defined by

$\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1}):=-\int_{{\widetilde{X}}/X}\operatorname{ch}(\nabla^{\mathcal{E}})\mod\operatorname{Im}(d).$

Note that $\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})$ is independent of
the choice of $\nabla^{\mathcal{E}}$ satisfying
$i_{X,k}^{*}\nabla^{\mathcal{E}}=\nabla^{E}_{k}$ for $k\in\\{0,1\\}$ and
satisfies the following transgression formula

$d\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})=\operatorname{ch}(\nabla^{E}_{1})-\operatorname{ch}(\nabla^{E}_{0}).$

Equivalently, the Chern–Simons form can be defined as

$\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})=-\frac{1}{2\pi
i}\int^{1}_{0}\operatorname{tr}\bigg{(}\frac{d\nabla^{E}_{t}}{dt}e^{-\frac{1}{2\pi
i}R^{E}_{t}}\bigg{)}dt.$ (2.2.3)

The choices of 0 and 1 are immaterial. If $t<T$ are two fixed positive real
numbers, then one can replace 0 by $t$ and 1 by $T$ in (2.2.3).

Let $(E,g^{E}_{k},\nabla^{E}_{k})$ and $(F,g^{F}_{k},\nabla^{F}_{k})$ be
triples, where $k\in\\{0,1\\}$. The Chern–Simons form satisfies the following
properties:

$\displaystyle\operatorname{CS}(\nabla^{E}_{1},\nabla^{E}_{0})$
$\displaystyle\equiv-\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1}),$
(2.2.4) $\displaystyle\operatorname{CS}(\nabla^{E}_{1},\nabla^{E}_{0})$
$\displaystyle\equiv\operatorname{CS}(\nabla^{E}_{1},\nabla^{E}_{2})+\operatorname{CS}(\nabla^{E}_{2},\nabla^{E}_{0}),$
(2.2.5)
$\displaystyle\operatorname{CS}(\nabla^{E}_{1}\oplus\nabla^{F}_{1},\nabla^{E}_{0}\oplus\nabla^{F}_{0})$
$\displaystyle\equiv\operatorname{CS}(\nabla^{E}_{1},\nabla^{E}_{0})+\operatorname{CS}(\nabla^{F}_{1},\nabla^{F}_{0}).$
(2.2.6)

If $(E,g^{E})$ is a $\mathbb{Z}_{2}$-graded Hermitian bundle with two
$\mathbb{Z}_{2}$-graded unitary connections $\nabla^{E}_{0}$ and
$\nabla^{E}_{1}$, then

$\operatorname{CS}(\nabla_{0}^{E},\nabla^{E}_{1})\equiv\operatorname{CS}(\nabla_{0}^{E,+},\nabla_{1}^{E,+})-\operatorname{CS}(\nabla_{0}^{E,-},\nabla_{1}^{E,-}).$
(2.2.7)

Let $(H,g^{H},\nabla^{H})$ be a Euclidean bundle of rank $k$ with a Euclidean
connection. Denote by $o(H)\to X$ the orientation bundle of $H\to X$. It is a
flat real line bundle, which is trivial if and only if $H\to X$ is oriented.
The Euler form of $\nabla^{H}$ is defined to be

$e(\nabla^{H})=\left\\{\begin{array}[]{ll}\displaystyle\operatorname{Pf}\bigg{(}\frac{R^{H}}{2\pi}\bigg{)},&\textrm{
if }k\textrm{ is even }\\\ 0,&\textrm{ if }k\textrm{ is odd
}\end{array}\right.,$

where $\operatorname{Pf}$ is the Pfaffian. Note that
$e(\nabla^{H})\in\Omega^{k}(X,o(H))$. Write $e(H)=[e(\nabla^{H})]\in
H^{k}(X;o(H))$ for the Euler class of $H\to X$. Given two pairs
$(g^{H}_{0},\nabla^{H}_{0})$ and $(g^{H}_{1},\nabla^{H}_{1})$ of Euclidean
metrics and Euclidean connections on $H\to X$, define a real vector bundle
$\mathcal{H}\to{\widetilde{X}}$ by $\mathcal{H}=p_{X}^{*}H$. Define a
Euclidean metric $g^{\mathcal{H}}$ and a Euclidean connection
$\nabla^{\mathcal{H}}$ on $\mathcal{H}\to{\widetilde{X}}$ in the way analogous
to (2.2.2). The transgressed Euler form
$\displaystyle{\widetilde{e}}(\nabla^{H}_{0},\nabla^{H}_{1})\in\frac{\Omega^{k-1}(X,o(H))}{\operatorname{Im}(d)}$
is defined to be

${\widetilde{e}}(\nabla^{H}_{0},\nabla^{H}_{1})=-\int_{{\widetilde{X}}/X}e(\nabla^{\mathcal{H}})\mod\operatorname{Im}(d).$
(2.2.8)

Note that ${\widetilde{e}}(\nabla^{H}_{0},\nabla^{H}_{1})=0$ when $k$ is odd,
and satisfies

$d{\widetilde{e}}(\nabla^{H}_{0},\nabla^{H}_{1})=e(\nabla^{H}_{1})-e(\nabla^{H}_{0}).$

##  3\. Analytic index for twisted de Rham operator

In §3.1 we recall the local index theory for generalized twisted de Rham
operators under and without the kernel bundle assumption, respectively, and
refer to [2, Chapter 10] and [6, §III] for the details. In §3.2 we prove a
variational formula for the Bismut–Cheeger eta form
${\widehat{\eta}}^{\Lambda}$ for isomorphic Hermitian bundles (Theorem 3.1).
In §3.3 we prove the $\mathbb{Z}_{2}$-graded additivity of the Bismut–Cheeger
eta form ${\widehat{\eta}}^{\Lambda}$ (Theorem 3.2). These results on the
Bismut–Cheeger eta form ${\widehat{\eta}}^{\Lambda}$ are used to the show the
analytic index
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}:{\widehat{K}}(X)\to{\widehat{K}}(B)$
associated to families of generalized twisted de Rham operators (Definition
3.1) defined in §3.4 is a well defined group homomorphism (Proposition 3.7).
Moreover, we prove the RRG theorem for the analytic index
$\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}$ in
$\mathbb{R}/\mathbb{Z}$ $K$-theory (Theorem 3.3).

###  3.1. Local index theory for generalized twisted de Rham operator

Let $\pi:X\to B$ be a submersion with closed fibers $Z$ of dimension $n$.
Denote by $T^{V}X\to X$ the vertical tangent bundle. Recall from [7, p.918]
that a Riemannian structure on $\pi:X\to B$ consists of a horizontal
distribution $T^{H}X\to X$, i.e. $TX=T^{V}X\oplus T^{H}X$, and a metric
$g^{T^{V}X}$ on $T^{V}X\to X$. Equip $\pi:X\to B$ with a Riemannian structure
$(T^{H}X,g^{T^{V}X})$. Denote by $P^{T^{V}X}:TX\to T^{V}X$ the projection map.
Put a Riemannian metric $g^{TB}$ on $TB\to B$. Define a metric $g^{TX}$ on
$TX\to X$ by

$g^{TX}=g^{T^{V}X}\oplus\pi^{*}g^{TB}.$

Denote by $\nabla^{TX}$ and $\nabla^{TB}$ the Levi-Civita connections on
$TX\to X$ and $TB\to X$ associated to $g^{TX}$ and $g^{TB}$, respectively.
Then $\nabla^{T^{V}X}:=P^{T^{V}X}\circ\nabla^{TX}\circ P^{T^{V}X}$ is a
Euclidean connection on $T^{V}X\to X$ with respect to $g^{T^{V}X}$.

Define a connection ${\widetilde{\nabla}}^{TX}$ on $TX\to X$ by

${\widetilde{\nabla}}^{TX}=\nabla^{T^{V}X}\oplus\pi^{*}\nabla^{TB}.$

Let
$S=\nabla^{TX}-{\widetilde{\nabla}}^{TX}\in\Omega^{1}(X,\operatorname{End}(TX))$.
By [3, Theorem 1.9] the $(3,0)$ tensor $g^{TX}(S(\cdot)\cdot,\cdot)$ depends
only on the Riemannian structure $(T^{H}X,g^{T^{V}X})$. Let
$\\{e_{1},\ldots,e_{n}\\}$ be a local orthonormal frame for $T^{V}X\to X$. For
any $U\in\Gamma(B,TB)$, denote by $U^{H}\in\Gamma(X,T^{H}X)$ its horizontal
lift. Define a horizontal one-form $k$ on $X$ by

$k(U^{H})=-\sum_{k=1}^{n}g^{TX}(S(e_{k})e_{k},U^{H}).$ (3.1.1)

For any $U,V\in\Gamma(B,TB)$,

$T(U,V):=-P^{T^{V}X}[U^{H},V^{H}]$ (3.1.2)

is a horizontal two-form with values in $T^{V}X$, and is called the curvature
of $\pi:X\to B$.

The complexified exterior bundle $\Lambda(T^{V}X)^{*}\to X$ has a natural
$\mathbb{Z}_{2}$-grading and is a Clifford module with Clifford multiplication
$c(Y)=\varepsilon(Y)-i(Y)$, where $Y\in\Gamma(X,T^{V}X)$, and $\varepsilon$
and $i$ are the exterior and interior multiplications, respectively. Denote by
$g^{\Lambda(T^{V}X)^{*}}$ and $\nabla^{\Lambda(T^{V}X)^{*}}$ the extensions of
$g^{T^{V}X}$ and $\nabla^{T^{V}X}$ to $\Lambda(T^{V}X)^{*}\to X$,
respectively.

Let $(E,g^{E},\nabla^{E})$ be a triple. The Hermitian metric
$g^{\Lambda(T^{V}X)^{*}\otimes E}:=g^{\Lambda(T^{V}X)^{*}}\otimes g^{E}$ and
the unitary connection $\nabla^{\Lambda(T^{V}X)^{*}\otimes E}$ on
$\Lambda(T^{V}X)^{*}\otimes E\to X$ are $\mathbb{Z}_{2}$-graded. The
generalized twisted de Rham operator $\mathsf{D}^{\Lambda\otimes E}$ is given
by

$\mathsf{D}^{\Lambda\otimes
E}=\sum_{k=1}^{n}c(e_{k})\nabla^{\Lambda(T^{V}X)^{*}\otimes E}_{e_{k}}.$

It acts on $\Gamma(X,\Lambda(T^{V}X)^{*}\otimes E)$ and is odd self-adjoint,
i.e. $\mathsf{D}^{\Lambda\otimes E}_{-}=(\mathsf{D}^{\Lambda\otimes
E}_{+})^{*}$.

Define an infinite rank $\mathbb{Z}_{2}$-graded complex vector bundle
$\pi^{\Lambda}_{*}E\to B$ whose fiber over $b\in B$ is

$(\pi^{\Lambda}_{*}E)_{b}=\Gamma(Z_{b},(\Lambda(T^{V}X)^{*}\otimes
E)|_{Z_{b}}).$

Note that $\Omega(X,E)\cong\Omega(B,\pi^{\Lambda}_{*}E)$. Denote by $\ast$ the
fiberwise Hodge star operator associated to $g^{T^{V}X}$, and extend it from
$\Gamma(X,\Lambda(T^{V}X)^{*})$ to $\Gamma(X,\Lambda(T^{V}X)^{*}\otimes
E)\cong\Gamma(B,\pi^{\Lambda}_{*}E)$. Define an $L^{2}$-metric on
$\pi^{\Lambda}_{*}E\to B$ by

$g^{\pi^{\Lambda}_{*}E}(s_{1},s_{2})(b)=\int_{Z_{b}}g^{E}(s_{1}(b)\wedge\ast
s_{2}(b)).$

The connection on $\pi^{\Lambda}_{*}E\to B$ defined by

$\nabla^{\pi^{\Lambda}_{*}E}_{U}s=\nabla^{\Lambda(T^{V}X)^{*}\otimes
E}_{U^{H}}s,$ (3.1.3)

where $s\in\Gamma(B,\pi^{\Lambda}_{*}E)$ and $U\in\Gamma(B,TB)$, is
$\mathbb{Z}_{2}$-graded. The connection on $\pi^{\Lambda}_{*}E\to B$ defined
by

$\nabla^{\pi^{\Lambda}_{*}E,u}:=\nabla^{\pi^{\Lambda}_{*}E}+\frac{1}{2}k,$
(3.1.4)

where $k$ is given by (3.1.1), is also $\mathbb{Z}_{2}$-graded and unitary
with respect to $g^{\pi^{\Lambda}_{*}E}$.

Suppose $\mathsf{D}^{\Lambda\otimes E}$ satisfies the kernel bundle
assumption, i.e. the family of complex vector spaces
$\\{\ker(\mathsf{D}^{\Lambda\otimes E}_{b})\\}_{b\in B}$ form a
($\mathbb{Z}_{2}$-graded) complex vector bundle, which is denoted by
$\ker(\mathsf{D}^{\Lambda\otimes E})\to B$. In this case, the family of
complex vector spaces $\\{\operatorname{Im}(\mathsf{D}^{\Lambda\otimes
E}_{b})\\}_{b\in B}$ form an infinite rank $\mathbb{Z}_{2}$-graded subbundle
of $\pi^{\Lambda}_{*}E\to B$, denoted by
$\operatorname{Im}(\mathsf{D}^{\Lambda\otimes E})\to B$. Their
$\mathbb{Z}_{2}$-gradings are given by

$\ker(\mathsf{D}^{\Lambda\otimes E})^{\pm}=\ker(\mathsf{D}^{\Lambda\otimes
E}_{\pm}),\qquad\operatorname{Im}(\mathsf{D}^{\Lambda\otimes
E})^{\pm}=\operatorname{Im}(\mathsf{D}^{\Lambda\otimes E}_{\mp}).$

Note that the direct sum decompositions

$(\pi^{\Lambda}_{*}E)^{+}=\operatorname{Im}(\mathsf{D}^{\Lambda\otimes
E})^{+}\oplus\ker(\mathsf{D}^{\Lambda\otimes
E})^{+},\quad(\pi^{\Lambda}_{*}E)^{-}=\operatorname{Im}(\mathsf{D}^{\Lambda\otimes
E})^{-}\oplus\ker(\mathsf{D}^{\Lambda\otimes E})^{-}$

are orthogonal. Denote by $P^{\ker(\mathsf{D}^{\Lambda\otimes
E})}:\pi^{\Lambda}_{*}E\to\ker(\mathsf{D}^{\Lambda\otimes E})$ the orthogonal
projection. Let $g^{\ker(\mathsf{D}^{\Lambda\otimes E})}$ be the
$\mathbb{Z}_{2}$-graded Hermitian metric on $\ker(\mathsf{D}^{\Lambda\otimes
E})\to B$ inherited from $g^{\pi^{\Lambda}_{*}E}$. The connection on
$\ker(\mathsf{D}^{\Lambda\otimes E})\to B$ defined by

$\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E})}=P^{\ker(\mathsf{D}^{\Lambda\otimes
E})}\circ\nabla^{\pi^{\Lambda}_{*}E,u}\circ P^{\ker(\mathsf{D}^{\Lambda\otimes
E})}$

is $\mathbb{Z}_{2}$-graded and unitary.

Define a Bismut superconnection on $\pi^{\Lambda}_{*}E\to B$ by

$\mathbb{B}^{\Lambda,E}=\mathsf{D}^{\Lambda\otimes
E}+\nabla^{\pi^{\Lambda}_{*}E,u}-\frac{c(T)}{4},$

where $T$ is given by (3.1.2). The rescaled Bismut superconnection is given by

$\mathbb{B}^{\Lambda,E}_{t}=\sqrt{t}\mathsf{D}^{\Lambda\otimes
E}+\nabla^{\pi^{\Lambda}_{*}E,u}-\frac{c(T)}{4\sqrt{t}}.$

By [6, (3.76)], we have

$\lim_{t\to
0}\operatorname{ch}(\mathbb{B}^{\Lambda,E}_{t})=\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{E}).$
(3.1.5)

######  Remark 3.1.

In [6], (3.1.5) is proved for complex flat vector bundles. Indeed, it holds
for any triple $(E,g^{E},\nabla^{E})$ as the argument does not use the
flatness of $\nabla^{E}$.

Roughly speaking, (3.1.5) is proved by first assuming the fibers are even
dimensional, oriented and spin. As (3.1.5) is local on $B$, it holds even if
the fibers are not oriented and spin. For the odd dimensional fiber case, as
remarked in [6, p.334], a similar argument shows that the left-hand side of
(3.1.5) is zero.

By [2, Theorem 9.19], which also holds for $n$ odd, we have

$\lim_{t\to\infty}\operatorname{ch}(\mathbb{B}^{\Lambda,E}_{t})=\operatorname{ch}(\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E})}).$ (3.1.6)

The Bismut–Cheeger eta form associated to $\mathbb{B}^{\Lambda,E}$ is defined
to be

${\widetilde{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X})=\frac{1}{\sqrt{\pi}}\int^{\infty}_{0}\operatorname{str}\bigg{(}\frac{d\mathbb{B}^{\Lambda,E}_{t}}{dt}e^{-\frac{1}{2\pi
i}(\mathbb{B}^{\Lambda,E}_{t})^{2}}\bigg{)}dt.$

By (3.1.5) and (3.1.6), the local family index theorem (FIT) for
$\mathsf{D}^{\Lambda\otimes E}$ under the kernel bundle assumption is

$d{\widetilde{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X})=\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{E})-\operatorname{ch}(\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E})}).$

Now suppose $\mathsf{D}^{\Lambda\otimes E}$ does not satisfy the kernel bundle
assumption. Miščenko–Fomenko [17, p.96-97] (cf. [7, Lemma 7.13]) prove that
there exist finite rank subbundles $L^{\pm}\to B$ and complementary closed
subbundles $K^{\pm}\to B$ of $(\pi^{\Lambda}_{*}E)^{\pm}\to B$ such that

$(\pi^{\Lambda}_{*}E)^{+}=K^{+}\oplus
L^{+},\qquad(\pi^{\Lambda}_{*}E)^{-}=K^{-}\oplus L^{-},$ (3.1.7)

$\mathsf{D}^{\Lambda\otimes
E}_{+}:(\pi^{\Lambda}_{*}E)^{+}\to(\pi^{\Lambda}_{*}E)^{-}$ is block diagonal
as a map with respect to (3.1.7), and $\mathsf{D}^{\Lambda\otimes
E}_{+}|_{K^{+}}:K^{+}\to K^{-}$ is a smooth bundle isomorphism.

Given $L^{\pm}\to B$ satisfying the above conditions, we say the
$\mathbb{Z}_{2}$-graded complex vector bundle $L\to B$, defined by
$L=L^{+}\oplus L^{-}$, satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes E}$. If $L\to B$ satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes E}$, then the analytic index of $[E]\in K(X)$
associated to $\mathsf{D}^{\Lambda\otimes E}$ is defined to be

$\operatorname{ind}^{a,\Lambda}([E])=[L^{+}]-[L^{-}]\in K(B).$

It is proved in [17, p.96-97] that $\operatorname{ind}^{a,\Lambda}([E])$ is
independent of the choice of $L\to B$ satisfying the MF property for
$\mathsf{D}^{\Lambda\otimes E}$.

Let $g^{L}$ be the $\mathbb{Z}_{2}$-graded Hermitian metric on $L\to B$
inherited from $g^{\pi^{\Lambda}_{*}E}$. Denote by $P:\pi^{\Lambda}_{*}E\to L$
the $\mathbb{Z}_{2}$-graded projection map with respect to (3.1.7). Then the
connection on $L\to B$ defined by

$\nabla^{L}:=P\circ\nabla^{\pi^{\Lambda}_{*}E,u}\circ P$

is $\mathbb{Z}_{2}$-graded and unitary with respect to $g^{L}$. Henceforth,
whenever $(L,g^{L},\nabla^{L})$ is a $\mathbb{Z}_{2}$-graded triple and $L\to
B$ satisfies the MF property for $\mathsf{D}^{\Lambda\otimes E}$, $g^{L}$ and
$\nabla^{L}$ are obtained as above unless otherwise specified.

Let $i^{-}:L^{-}\to(\pi^{\Lambda}_{*}E)^{-}$ be the inclusion map. For any
$z\in\mathbb{C}$, define a map ${\widetilde{\mathsf{D}}}^{\Lambda\otimes
E}_{+}(z):(\pi^{\Lambda}_{*}E\oplus
L^{\operatorname{op}})^{+}\to(\pi^{\Lambda}_{*}E\oplus
L^{\operatorname{op}})^{-}$ by

${\widetilde{\mathsf{D}}}^{\Lambda\otimes
E}_{+}(z)=\begin{pmatrix}\mathsf{D}^{\Lambda\otimes E}_{+}&zi^{-}\\\
zP^{+}&0\end{pmatrix}.$

Note that ${\widetilde{\mathsf{D}}}^{\Lambda\otimes E}_{+}(z)$ is invertible
for any $z\neq 0$ (cf. [7, Lemma 7.20]). Define a map
${\widetilde{\mathsf{D}}}^{\Lambda\otimes E}(z):\pi^{\Lambda}_{*}E\oplus
L^{\operatorname{op}}\to\pi^{\Lambda}_{*}E\oplus L^{\operatorname{op}}$ by

${\widetilde{\mathsf{D}}}^{\Lambda\otimes
E}(z):=\begin{pmatrix}0&({\widetilde{\mathsf{D}}}^{\Lambda\otimes
E}_{+}(z))^{*}\\\ {\widetilde{\mathsf{D}}}^{\Lambda\otimes
E}_{+}(z)&0\end{pmatrix}.$

Choose and fix $a\in(0,1)$ and a smooth function $\alpha:[0,\infty)\to I$
satisfying $\alpha(t)=0$ for all $t\leq a$ and $\alpha(t)=1$ for all $t\geq
1$. Define a rescaled Bismut superconnection on $\pi^{\Lambda}_{*}E\oplus
L^{\operatorname{op}}\to B$ by

${\widehat{\mathbb{B}}}^{\Lambda,E}_{t}=\sqrt{t}{\widetilde{\mathsf{D}}}^{\Lambda\otimes
E}(\alpha(t))+\nabla^{\pi^{\Lambda}_{*}E,u}\oplus\nabla^{L,\operatorname{op}}-\frac{c(T)}{4\sqrt{t}}.$

The Bismut superconnection is recovered by taking $t=1$, i.e.

${\widehat{\mathbb{B}}}^{\Lambda,E}={\widetilde{\mathsf{D}}}^{\Lambda\otimes
E}(1)+\nabla^{\pi^{\Lambda}_{*}E,u}\oplus\nabla^{L,\operatorname{op}}-\frac{c(T)}{4}.$
(3.1.8)

Since ${\widetilde{\mathsf{D}}}^{\Lambda\otimes E}(\alpha(t))$ is invertible
for any $t\geq 1$, it follows that

$\lim_{t\to\infty}\operatorname{ch}({\widehat{\mathbb{B}}}^{\Lambda,E}_{t})=0.$
(3.1.9)

On the other hand, for $t\leq a$, ${\widehat{\mathbb{B}}}^{\Lambda,E}_{t}$
decouples, i.e.

${\widehat{\mathbb{B}}}^{\Lambda,E}_{t}=\bigg{(}\sqrt{t}\mathsf{D}^{\Lambda\otimes
E}+\nabla^{\pi^{\Lambda}_{*}E,u}-\frac{c(T)}{4\sqrt{t}}\bigg{)}\oplus\nabla^{L,\operatorname{op}}=\mathbb{B}^{\Lambda,E}_{t}\oplus\nabla^{L,\operatorname{op}}.$

By (3.1.5), we have

$\begin{split}\lim_{t\to
0}\operatorname{ch}({\widehat{\mathbb{B}}}^{\Lambda,E}_{t})&=\lim_{t\to
0}\operatorname{ch}(\mathbb{B}^{\Lambda,E}_{t})-\operatorname{ch}(\nabla^{L})\\\
&=\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{E})-\operatorname{ch}(\nabla^{L}).\end{split}$
(3.1.10)

The Bismut–Cheeger eta form ${\widehat{\eta}}^{\Lambda,E}$ associated to
${\widehat{\mathbb{B}}}^{\Lambda,E}$ is defined to be

${\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L)=\frac{1}{\sqrt{\pi}}\int^{\infty}_{0}\operatorname{str}\bigg{(}\frac{d{\widehat{\mathbb{B}}}^{\Lambda,E}_{t}}{dt}e^{-\frac{1}{2\pi
i}({\widehat{\mathbb{B}}}^{\Lambda,E}_{t})^{2}}\bigg{)}dt.$

By (3.1.9) and (3.1.10), the local FIT for $\mathsf{D}^{\Lambda\otimes E}$
without the kernel bundle assumption is

$d{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L)=\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{E})-\operatorname{ch}(\nabla^{L}).$
(3.1.11)

###  3.2. A variational formula for the Bismut–Cheeger eta form
${\widehat{\eta}}^{\Lambda}$

The proof of the following lemma is virtually the same as that of [11, Lemma
1].

######  Lemma 3.1.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. Let $(E,g^{E},\nabla^{E})$ be a
triple over $X$ and $(L,g^{L},\nabla^{L})$ a $\mathbb{Z}_{2}$-graded triple
over $B$, where $L\to B$ satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes E}$. Let $(W,g^{W},\nabla^{W},s_{W})$ be a split
quadruple over $B$. Define a rescaled superconnection $\mathbb{A}^{W}_{t}$ on
$W\to B$ by

$\mathbb{A}^{W}_{t}=\sqrt{t}\alpha(t)s_{W}+\nabla^{W},$

where $\alpha:[0,\infty)\to I$ is the smooth function in §3.1. Then

${\widehat{\eta}}^{\Lambda,E;W}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L)\equiv{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L),$

where the Bismut–Cheeger eta form on the left-hand side is associated to
${\widehat{\mathbb{B}}}^{\Lambda,E}_{t}\oplus\mathbb{A}^{W}_{t}$.

The following proposition is the variational formula for the Bismut–Cheeger
eta form ${\widehat{\eta}}^{\Lambda}$.

######  Proposition 3.1.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with two
Riemannian structures $(T^{H}_{0}X,g^{T^{V}X}_{0})$ and
$(T^{H}_{1}X,g^{T^{V}X}_{1})$. Let $E\to X$ be a complex vector bundle, and
$(g^{E}_{0},\nabla^{E}_{0})$ and $(g^{E}_{1},\nabla^{E}_{1})$ are two pairs of
Hermitian metrics and unitary connections on $E\to X$. For $j\in\\{0,1\\}$,
write $\mathsf{D}^{\Lambda\otimes E}_{j}$ for the twisted de Rham operator
defined in terms of

$(g^{E}_{j},\nabla^{E}_{j},T^{H}_{j}X,g^{T^{V}X}_{j}).$

If $(L_{j},g^{L_{j}},\nabla^{L_{j}})$ is a $\mathbb{Z}_{2}$-graded triple so
that $L_{j}\to B$ satisfies the MF property for $\mathsf{D}^{\Lambda\otimes
E}_{j}$, then there exist balanced $\mathbb{Z}_{2}$-graded triples
$(W_{j},g^{W_{j}},\nabla^{W_{j}})$ and a $\mathbb{Z}_{2}$-graded isometric
isomorphism

$h:(L_{0}\oplus W_{0},g^{L_{0}}\oplus g^{W_{0}})\to(L_{1}\oplus
W_{1},g^{L_{1}}\oplus g^{W_{1}})$ (3.2.1)

such that

$\begin{split}&{\widehat{\eta}}^{\Lambda,E}(g^{E}_{1},\nabla^{E}_{1},T^{H}_{1}X,g^{T^{V}X}_{1},L_{1})-{\widehat{\eta}}^{\Lambda,E}(g^{E}_{0},\nabla^{E}_{0},T^{H}_{0}X,g^{T^{V}X}_{0},L_{0})\\\
&\equiv\int_{X/B}{\widetilde{e}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge\operatorname{ch}(\nabla^{E}_{0})+\int_{X/B}e(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})\\\
&\qquad-\operatorname{CS}\big{(}\nabla^{L_{0}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{1}}\oplus\nabla^{W_{1}})\big{)}.\end{split}$
(3.2.2)

Note that if $\dim(X/B)$ is odd, then the first two terms of the right-hand
side of (3.2.2) vanish.

###### Proof.

Since the space of splitting maps is affine, there exists a smooth path of
horizontal distributions $\\{T^{H}_{t}X\to X\\}_{t\in I}$ joining
$T^{H}_{0}X\to X$ and $T^{H}_{1}X\to X$. Let $(g^{E}_{t},\nabla^{E}_{t})$ be
the smooth path of Hermitian metrics and unitary connections on $E\to X$
joining $(g^{E}_{0},\nabla^{E}_{0})$ and $(g^{E}_{1},\nabla^{E}_{1})$ defined
by (2.2.1). Define a smooth path $g^{T^{V}X}_{t}$ of Euclidean metrics joining
$g^{T^{V}X}_{0}$ and $g^{T^{V}X}_{1}$ in a similar way. Then

$c(t)=(g^{E}_{t},\nabla^{E}_{t},T^{H}_{t}X,g^{T^{V}X}_{t}),$

where $t\in I$, is a smooth path joining
$c(0)=(g^{E}_{0},\nabla^{E}_{0},T^{H}_{0}X,g^{T^{V}X}_{0})$ and
$c(1)=(g^{E}_{1},\nabla^{E}_{1},T^{H}_{1}X,g^{T^{V}X}_{1})$. Define a new path
${\widetilde{c}}(t)$ by

${\widetilde{c}}(t)=\left\\{\begin{array}[]{ll}(g^{E}_{0},\nabla^{E}_{0},T^{H}_{2t}X,g^{T^{V}X}_{2t}),&\textrm{
for }t\in[0,\frac{1}{2}]\\\
(g^{E}_{2t-1},\nabla^{E}_{2t-1},T^{H}_{1}X,g^{T^{V}X}_{1}),&\textrm{ for
}t\in[\frac{1}{2},1]\end{array}\right..$

Consider the submersion ${\widetilde{\pi}}:{\widetilde{X}}\to{\widetilde{B}}$,
where ${\widetilde{\pi}}:=\pi\times\operatorname{id}_{I}$. Define a complex
vector bundle $\mathcal{E}\to{\widetilde{X}}$ by
$\mathcal{E}=p_{X}^{*}\mathcal{E}$. Define the pair of Hermitian metric and
unitary connection $(g^{\mathcal{E}},\nabla^{\mathcal{E}})$ on
$\mathcal{E}\to{\widetilde{X}}$ and the Riemannian structure
$(T^{H}{\widetilde{X}},g^{T^{V}{\widetilde{X}}})$ on
${\widetilde{\pi}}:{\widetilde{X}}\to{\widetilde{B}}$ so that for each $t\in
I$, the restriction of

$(g^{\mathcal{E}},\nabla^{\mathcal{E}},T^{H}{\widetilde{X}},g^{T^{V}{\widetilde{X}}})$
(3.2.3)

to $X\times\\{t\\}$ is given by ${\widetilde{c}}(t)$. Denote by
$\mathsf{D}^{\Lambda\otimes\mathcal{E}}$ the twisted de Rham operator defined
in terms of (3.2.3). Let $(\mathcal{L},g^{\mathcal{L}},\nabla^{\mathcal{L}})$
be the $\mathbb{Z}_{2}$-graded triple so that $\mathcal{L}\to{\widetilde{B}}$
satisfies the MF property for $\mathsf{D}^{\Lambda\otimes\mathcal{E}}$, i.e.
there exists a $\mathbb{Z}_{2}$-graded complementary closed subbundle
$\mathcal{K}\to{\widetilde{B}}$ of
${\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E}\to{\widetilde{B}}$ such that

${\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E}=\mathcal{K}\oplus\mathcal{L},$
(3.2.4)

$\mathsf{D}^{\Lambda\otimes\mathcal{E}}_{+}:({\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E})^{+}\to({\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E})^{-}$
is block diagonal as a map with respect to (3.2.4), and
$\mathsf{D}^{\Lambda\otimes\mathcal{E}}_{+}|_{\mathcal{K}^{+}}:\mathcal{K}^{+}\to\mathcal{K}^{-}$
is a smooth bundle isomorphism. There exist smooth bundle isomorphisms

$i_{B,0}^{*}({\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E})^{\pm}\cong
i_{B,1}^{*}({\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E})^{\pm}\cong(\pi^{\Lambda}_{*}E)^{\pm}$
(3.2.5)

and

$i_{B,0}^{*}\mathcal{L}^{\pm}\cong i_{B,1}^{*}\mathcal{L}^{\pm},\qquad
i_{B,0}^{*}\mathcal{K}^{\pm}\cong i_{B,1}^{*}\mathcal{K}^{\pm}.$

Write $L^{\pm}\to B$ for $i_{B,0}^{*}\mathcal{L}^{\pm}\to B$ and $K^{\pm}\to
B$ for $i_{B,0}^{*}\mathcal{K}^{\pm}\to B$. By (3.2.4) and (3.2.5) we have

$(\pi^{\Lambda}_{*}E)^{+}=K^{+}\oplus
L^{+},\qquad(\pi^{\Lambda}_{*}E)^{-}=K^{-}\oplus L^{-}.$ (3.2.6)

Let $j\in\\{0,1\\}$. Since

$\mathsf{D}^{\Lambda\otimes\mathcal{E}}|_{i_{B,j}^{*}{\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E}}=\mathsf{D}^{\Lambda\otimes
E}_{j}$

it follows that $\mathsf{D}^{\Lambda\otimes
E}_{j,+}:(\pi^{\Lambda}_{*}E)^{+}\to(\pi^{\Lambda}_{*}E)^{-}$ is block
diagonal with respect to (3.2.6) and the restriction
$\mathsf{D}^{\Lambda\otimes E}_{j,+}:K^{+}\to K^{-}$ is an isomorphism. Thus
$L\to B$ satisfies the MF property for $\mathsf{D}^{\Lambda\otimes E}_{j}$.
Since $L_{j}\to B$ satisfies the MF property for $\mathsf{D}^{\Lambda\otimes
E}_{j}$, there exist balanced $\mathbb{Z}_{2}$-graded triples
$(H,g^{H},\nabla^{H})$ and $(W_{j},g^{W_{j}},\nabla^{W_{j}})$ and
$\mathbb{Z}_{2}$-graded smooth isometric isomorphisms $f_{j}:(L\oplus
H,g^{L}\oplus g^{H})\to(L_{j}\oplus W_{j},g^{L_{j}}\oplus g^{W_{j}})$ for
$j\in\\{0,1\\}$. The map $h:(L_{0}\oplus W_{0},g^{L_{0}}\oplus
g^{W_{0}})\to(L_{1}\oplus W_{1},g^{L_{1}}\oplus g^{W_{1}})$ defined by

$h=f_{1}\circ f_{0}^{-1}$

is a $\mathbb{Z}_{2}$-graded isometric isomorphism, and the following diagram
commutes.

${(L\oplus H,g^{L}\oplus g^{H})}$${(L_{0}\oplus W_{0},g^{L_{0}}\oplus
g^{W_{0}})}$${(L_{1}\oplus W_{1},g^{L_{1}}\oplus
g^{W_{1}})}$$\scriptstyle{f_{0}}$$\scriptstyle{f_{1}}$$\scriptstyle{h}$

Thus (3.2.1) holds.

Let $j\in\\{0,1\\}$. The connection on $L\oplus H\to B$ defined by

$\nabla^{L\oplus H}_{j}:=f_{j}^{*}(\nabla^{L_{j}}\oplus\nabla^{W_{j}})$
(3.2.7)

is $\mathbb{Z}_{2}$-graded and unitary. Define a complex vector bundle
$\mathcal{H}\to{\widetilde{B}}$ by $\mathcal{H}=p_{B}^{*}H$. Define a
$\mathbb{Z}_{2}$-graded Hermitian metric $g^{\mathcal{L}\oplus\mathcal{H}}$
and a $\mathbb{Z}_{2}$-graded unitary connection
$\nabla^{\mathcal{L}\oplus\mathcal{H}}$ on
$\mathcal{L}\oplus\mathcal{H}\to{\widetilde{B}}$ by (2.2.2), which satisfies

$i_{B,j}^{*}\nabla^{\mathcal{L}\oplus\mathcal{H}}=\nabla^{L\oplus H}_{j}.$

Define a rescaled Bismut superconnection
${\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{t}$ on
${\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E}\oplus(\mathcal{L}\oplus\mathcal{H})^{\operatorname{op}}\to{\widetilde{B}}$
defined by

${\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{t}=\sqrt{t}\big{(}{\widetilde{\mathsf{D}}}^{\Lambda\otimes\mathcal{E}}(\alpha(t))\oplus\alpha(t)s_{\mathcal{H}}\big{)}+(\nabla^{{\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E},u}\oplus\nabla^{\mathcal{L}\oplus\mathcal{H},\operatorname{op}})-\frac{1}{\sqrt{t}}\bigg{(}\frac{c({\widetilde{T}})}{4}\oplus
0\bigg{)},$

where $\displaystyle s_{\mathcal{H}}=\begin{pmatrix}0&\operatorname{id}\\\
\operatorname{id}&0\end{pmatrix}\in\Gamma({\widetilde{B}},\operatorname{End}(\mathcal{H})^{-})$
and ${\widetilde{T}}$ is the curvature 2-form on
${\widetilde{\pi}}:{\widetilde{X}}\to{\widetilde{B}}$. By the definition of
$\alpha$, ${\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{t}$
decouples for $t\leq a$, i.e.

${\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{t}=\mathbb{B}^{\Lambda,\mathcal{E}}_{t}\oplus\nabla^{\mathcal{L}\oplus\mathcal{H},\operatorname{op}}.$

By (3.1.5) we have

$\begin{split}\lim_{t\to
0}\operatorname{ch}({\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E}}_{t})&=\lim_{t\to
0}\operatorname{ch}(\mathbb{B}^{\Lambda,\mathcal{E}}_{t})-\operatorname{ch}(\nabla^{\mathcal{L}\oplus\mathcal{H}})\\\
&=\int_{{\widetilde{X}}/{\widetilde{B}}}e(\nabla^{T^{V}{\widetilde{X}}})\wedge\operatorname{ch}(\nabla^{\mathcal{E}})-\operatorname{ch}(\nabla^{\mathcal{L}\oplus\mathcal{H}}).\end{split}$
(3.2.8)

On the other hand, since
$({\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{t})_{[0]}$ is
invertible for every $t\geq 1$, it follows that

$\lim_{t\to\infty}\operatorname{ch}({\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{t})=0.$
(3.2.9)

Write
${\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}(g^{\mathcal{E}},\nabla^{\mathcal{E}},T^{H}{\widetilde{X}},g^{T^{V}{\widetilde{X}}},\mathcal{L})$
for the Bismut–Cheeger eta form associated to
${\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{t}$. We temporarily
suppress the data defining the Bismut–Cheeger eta form to shorten the
notation. By (3.2.8) and (3.2.9) we have

$d{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}=\int_{{\widetilde{X}}/{\widetilde{B}}}e(\nabla^{T^{V}{\widetilde{X}}})\wedge\operatorname{ch}(\nabla^{\mathcal{E}})-\operatorname{ch}(\nabla^{\mathcal{L}\oplus\mathcal{H}}).$
(3.2.10)

Denote by $i:\partial{\widetilde{B}}\to{\widetilde{B}}$ the inclusion map. By
(2.1.1) we have

$-(i_{B,1}^{*}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}-i_{B,0}^{*}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}})=-\int_{\partial{\widetilde{B}}/B}i^{*}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}=\int_{{\widetilde{B}}/B}d^{{\widetilde{B}}}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}-d^{B}\int_{{\widetilde{B}}/B}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}.$
(3.2.11)

By (3.2.10), (3.2.11) becomes

$\begin{split}i_{B,1}^{*}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}-i_{B,0}^{*}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}&\equiv-\int_{{\widetilde{B}}/B}d^{{\widetilde{B}}}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}\\\
&\equiv-\int_{{\widetilde{B}}/B}\bigg{(}\int_{{\widetilde{X}}/{\widetilde{B}}}e(\nabla^{T^{V}{\widetilde{X}}})\wedge\operatorname{ch}(\nabla^{\mathcal{E}})-\operatorname{ch}(\nabla^{\mathcal{L}\oplus\mathcal{H}})\bigg{)}\\\
&\equiv\int_{{\widetilde{B}}/B}\int_{{\widetilde{X}}/{\widetilde{B}}}-(e(\nabla^{T^{V}{\widetilde{X}}})\wedge\operatorname{ch}(\nabla^{\mathcal{E}}))-\operatorname{CS}(\nabla^{L\oplus
H}_{0},\nabla^{L\oplus H}_{1}).\end{split}$ (3.2.12)

First note that

$\begin{split}\operatorname{CS}(\nabla^{L\oplus H}_{0},\nabla^{L\oplus
H}_{1})&=\operatorname{CS}\big{(}f_{0}^{*}(\nabla^{L_{0}}\oplus\nabla^{W_{0}}),f_{1}^{*}(\nabla^{L_{1}}\oplus\nabla^{W_{1}})\big{)}\\\
&\equiv\operatorname{CS}\big{(}\nabla^{L_{0}}\oplus\nabla^{W_{0}},(f_{0}^{-1})^{*}f_{1}^{*}(\nabla^{L_{1}}\oplus\nabla^{W_{1}})\big{)}\\\
&=\operatorname{CS}\big{(}\nabla^{L_{0}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{1}}\oplus\nabla^{W_{1}})\big{)}.\end{split}$
(3.2.13)

On the other hand, since
$\displaystyle\int_{X/B}\circ\int_{{\widetilde{X}}/X}=\int_{{\widetilde{X}}/B}=\int_{{\widetilde{B}}/B}\circ\int_{{\widetilde{X}}/{\widetilde{B}}}$,
it follows from (2.2.8) and (3.2.3) that

$\begin{split}&\int_{{\widetilde{B}}/B}\int_{{\widetilde{X}}/{\widetilde{B}}}-(e(\nabla^{T^{V}{\widetilde{X}}})\wedge\operatorname{ch}(\nabla^{\mathcal{E}}))\\\
&=\int_{X/B}\circ\int_{{\widetilde{X}}/X}-(e(\nabla^{T^{V}{\widetilde{X}}})\wedge\operatorname{ch}(\nabla^{\mathcal{E}}))\\\
&\equiv\int_{X/B}{\widetilde{e}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge\operatorname{ch}(\nabla^{E}_{0})+\int_{X/B}e(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1}).\end{split}$
(3.2.14)

By (3.2.13) and (3.2.14), (3.2.12) becomes

$\begin{split}i_{B,1}^{*}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}-i_{B,0}^{*}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}&\equiv\int_{X/B}{\widetilde{e}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge\operatorname{ch}(\nabla^{E}_{0})+\int_{X/B}e(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})\\\
&\qquad-\operatorname{CS}\big{(}\nabla^{L_{0}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{1}}\oplus\nabla^{W_{1}})\big{)}.\end{split}$
(3.2.15)

By (3.2.15), to prove (3.2.2) it remains to show that

$i_{B,j}^{*}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}(g^{\mathcal{E}},\nabla^{\mathcal{E}},T^{H}{\widetilde{X}},g^{T^{V}{\widetilde{X}}},\mathcal{L})\equiv{\widehat{\eta}}^{\Lambda,E}(g^{E}_{j},\nabla^{E}_{j},T^{H}_{k}X,g^{T^{V}X}_{j},L_{j})$
(3.2.16)

for $j\in\\{0,1\\}$. First note that
$i_{B,j}^{*}{\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{t}={\widehat{\mathbb{B}}}^{\Lambda,E;H}_{j,t}$,
which is the rescaled Bismut superconnection on
$\pi^{\Lambda}_{*}E\oplus(L\oplus H)^{\operatorname{op}}\to B$ given by

${\widehat{\mathbb{B}}}^{\Lambda,E;H}_{j,t}=\sqrt{t}\big{(}{\widetilde{\mathsf{D}}}^{\Lambda\otimes
E}_{j}(\alpha(t))\oplus\alpha(t)s_{H}\big{)}+\big{(}\nabla^{\pi^{\Lambda}_{*}E,u}\oplus\nabla^{L\oplus
H,\operatorname{op}}_{j}\big{)}+\frac{1}{\sqrt{t}}\bigg{(}\frac{c(T)}{4}\oplus
0\bigg{)},$ (3.2.17)

where $s_{H}=\begin{pmatrix}0&\operatorname{id}\\\
\operatorname{id}&0\end{pmatrix}\in\Gamma(B,\operatorname{End}(H)^{-})$. By
(3.2.7), and the fact that the $\mathbb{Z}_{2}$-graded triple
$(W_{j},g^{W_{j}},\nabla^{W_{j}})$ is balanced, we have

$\nabla^{L\oplus
H,\operatorname{op}}_{j}=(f_{j}^{\operatorname{op}})^{*}(\nabla^{L_{j},\operatorname{op}}\oplus\nabla^{W_{j}}).$
(3.2.18)

Consider the split quadruple $(W_{j},g^{W_{j}},\nabla^{W_{j}},s_{W_{j}})$
given in Example 1. Denote by $\mathbb{A}^{W_{j}}_{t}$ the rescaled
superconnection on $W_{j}\to B$ given in Lemma 3.1. Since
$\operatorname{id}_{\pi^{\Lambda}_{*}E}\oplus(f_{j}^{-1})^{\operatorname{op}}:\pi^{\Lambda}_{*}E\oplus(L_{j}\oplus
W_{j})^{\operatorname{op}}\to\pi^{\Lambda}_{*}E\oplus(L\oplus
H)^{\operatorname{op}}$ is a $\mathbb{Z}_{2}$-graded isometric isomorphism, it
follows from (3.2.17) and (3.2.18) that

$\begin{split}&(\operatorname{id}_{\pi^{\Lambda}_{*}E}\oplus(f_{j}^{-1})^{\operatorname{op}})^{*}{\widehat{\mathbb{B}}}^{\Lambda,E;H}_{j,t}\\\
&=\sqrt{t}\big{(}{\widetilde{\mathsf{D}}}_{j}^{\Lambda\otimes
E}(\alpha(t))\oplus\alpha(t)s_{W_{j}}\big{)}+\big{(}\nabla^{\pi^{\Lambda}_{*}E,u}\oplus(\nabla^{L_{j},\operatorname{op}}\oplus\nabla^{W_{j}})\big{)}-\frac{1}{\sqrt{t}}\bigg{(}\frac{c(T)}{4}\oplus
0\bigg{)}\\\
&={\widehat{\mathbb{B}}}^{\Lambda,E}_{j,t}\oplus\mathbb{A}^{W_{j}}_{t}.\end{split}$

Since
$\operatorname{id}_{\pi^{\Lambda}_{*}E}\oplus(f_{j}^{-1})^{\operatorname{op}}$
covers $\operatorname{id}_{B}$, it follows that for any $t<T\in(0,\infty)$,

$\begin{split}\operatorname{CS}(i_{B,j}^{*}{\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{T},i_{B,j}^{*}{\widehat{\mathbb{B}}}^{\Lambda,\mathcal{E};\mathcal{H}}_{t})&\equiv\operatorname{CS}({\widehat{\mathbb{B}}}^{\Lambda,E;H}_{j,T},{\widehat{\mathbb{B}}}^{\Lambda,E;H}_{j,t})\\\
&\equiv\operatorname{CS}\big{(}(\operatorname{id}_{\pi^{\Lambda}_{*}E}\oplus(f_{j}^{-1})^{\operatorname{op}})^{*}{\widehat{\mathbb{B}}}^{\Lambda,E;H}_{j,T},(\operatorname{id}_{\pi^{\Lambda}_{*}E}\oplus(f_{j}^{-1})^{\operatorname{op}})^{*}{\widehat{\mathbb{B}}}^{\Lambda,E;H}_{j,t}\big{)}\\\
&\equiv\operatorname{CS}({\widehat{\mathbb{B}}}^{\Lambda,E}_{j,T}\oplus\mathbb{A}^{W_{j}}_{T},{\widehat{\mathbb{B}}}^{\Lambda,E}_{j,t}\oplus\mathbb{A}^{W_{j}}_{t}).\end{split}$

By letting $t\to 0$ and $T\to\infty$ in above, Lemma 3.1 implies that

$\begin{split}i_{B,j}^{*}{\widehat{\eta}}^{\Lambda,\mathcal{E};\mathcal{H}}(g^{\mathcal{E}},\nabla^{\mathcal{E}},T^{H}{\widetilde{X}},g^{T^{V}{\widetilde{X}}},\mathcal{L})&\equiv{\widehat{\eta}}^{\Lambda,E;W_{j}}(g^{E}_{j},\nabla^{E}_{j},T^{H}X,g^{T^{V}X},L_{j})\\\
&\equiv{\widehat{\eta}}^{\Lambda,E}(g^{E}_{j},\nabla^{E}_{j},T^{H}X,g^{T^{V}X},L_{j}).\end{split}$

Thus (3.2.16) holds. ∎

######  Remark 3.2.

As in [10, Remark 3.1], if $L_{0}\cong L_{1}$ as $\mathbb{Z}_{2}$-graded
complex vector bundles in Proposition 3.1, then one can take $W_{0}\to B$ and
$W_{1}\to B$ to be the zero bundles.

The proof of the following proposition follows verbatim from [10, Proposition
3.2].

######  Proposition 3.2.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. Let $(E,g^{E})$ be a Hermitian
bundle and $(F,g^{F},\nabla^{F})$ a triple over $X$. If there exists an
isometric isomorphism $\alpha:(E,g^{E})\to(F,g^{F})$ and
$(L_{E,\alpha},g^{L_{E,\alpha}},\nabla^{L_{E,\alpha}})$ is a
$\mathbb{Z}_{2}$-graded triple so that $L_{E,\alpha}\to B$ satisfies the MF
property for the twisted de Rham operator $\mathsf{D}^{\Lambda\otimes
E,\alpha}$ defined in terms of $\alpha^{*}\nabla^{F}$, then there exist a
unique $\mathbb{Z}_{2}$-graded triple
$(L_{F,\alpha},g^{L_{F,\alpha}},\nabla^{L_{F,\alpha}})$ and a unique
$\mathbb{Z}_{2}$-graded isometric isomorphism
${\widetilde{\alpha}}^{L}:(L_{E,\alpha},g^{L_{E,\alpha}})\to(L_{F,\alpha},g^{L_{F,\alpha}})$
such that

  1. (a)

$L_{F,\alpha}\to B$ satisfies the MF property for the twisted de Rham operator
$\mathsf{D}^{S\otimes F}$, which is defined in terms of $\nabla^{F}$, and

  2. (b)

${\widehat{\eta}}^{\Lambda,E}(g^{E},\alpha^{*}\nabla^{F},T^{H}X,g^{T^{V}X},L_{E,\alpha})={\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}X,g^{T^{V}X},L_{F,\alpha})$.

The following theorem is the variational formula for the Bismut–Cheeger eta
form ${\widehat{\eta}}^{\Lambda}$ for isomorphic Hermitian bundles.

######  Theorem 3.1.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with two
Riemannian structures $(T^{H}_{0}X,g^{T^{V}X}_{0})$ and
$(T^{H}_{1}X,g^{T^{V}X}_{1})$. Let $(E,g^{E},\nabla^{E})$ and
$(F,g^{F},\nabla^{F})$ be triples over $X$, and denote by
$\mathsf{D}^{\Lambda\otimes E}$ and $\mathsf{D}^{\Lambda\otimes F}$ the
twisted de Rham operators defined in terms of

$(T^{H}_{0}X,g^{T^{V}X}_{0},g^{E},\nabla^{E})\qquad\textrm{ and
}\qquad(T^{H}_{1}X,g^{T^{V}X}_{1},g^{F},\nabla^{F}),$

respectively. Let $(L_{E},g^{L_{E}},\nabla^{L_{E}})$ and
$(L_{F},g^{L_{F}},\nabla^{L_{F}})$ be $\mathbb{Z}_{2}$-graded triples so that
$L_{E}\to B$ and $L_{F}\to B$ satisfy the MF properties for
$\mathsf{D}^{\Lambda\otimes E}$ and $\mathsf{D}^{\Lambda\otimes F}$,
respectively.

If there exists an isometric isomorphism $\alpha:(E,g^{E})\to(F,g^{F})$, then
there exist balanced $\mathbb{Z}_{2}$-graded triples
$(W_{E},g^{W_{E}},\nabla^{W_{E}})$ and $(W_{F},g^{W_{F}},\nabla^{W_{F}})$ and
a $\mathbb{Z}_{2}$-graded isometric isomorphism

${\widetilde{h}}:(L_{E}\oplus W_{E},g^{L_{E}}\oplus g^{W_{E}})\to(L_{F}\oplus
W_{F},g^{L_{F}}\oplus g^{W_{F}}),$

which depends on $\alpha$, such that

$\begin{split}&{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{F})-{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}_{0}X,g^{T^{V}X}_{0},L_{E})\\\
&\equiv\int_{X/B}{\widetilde{e}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge\operatorname{ch}(\nabla^{E})+\int_{X/B}e(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E},\alpha^{*}\nabla^{F})\\\
&\qquad-\operatorname{CS}\big{(}\nabla^{L_{E}}\oplus\nabla^{W_{E}},{\widetilde{h}}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F}})\big{)}.\end{split}$
(3.2.19)

###### Proof.

Write $\mathsf{D}^{\Lambda\otimes E,\alpha}$ for the twisted de Rham operator
defined in terms of

$(g^{E},\alpha^{*}\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1}).$

Let $(L_{E,\alpha},g^{L_{E,\alpha}},\nabla^{L_{E,\alpha}})$ be a
$\mathbb{Z}_{2}$-graded triple so that $L_{E,\alpha}\to B$ satisfies the MF
property for $\mathsf{D}^{\Lambda\otimes E,\alpha}$. By Proposition 3.1, there
exist balanced $\mathbb{Z}_{2}$-graded triples
$(W_{0},g^{W_{0}},\nabla^{W_{0}})$ and $(W_{1},g^{W_{1}},\nabla^{W_{1}})$ and
a $\mathbb{Z}_{2}$-graded isometric isomorphism

$h:(L_{E}\oplus W_{0},g^{L_{E}}\oplus g^{W_{0}})\to(L_{E,\alpha}\oplus
W_{1},g^{L_{E,\alpha}}\oplus g^{W_{1}})$ (3.2.20)

such that

$\begin{split}&{\widehat{\eta}}^{\Lambda,E}(g^{E},\alpha^{*}\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{E,\alpha})-{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}_{0}X,g^{T^{V}X}_{0},L_{E})\\\
&\equiv\int_{X/B}{\widetilde{e}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge\operatorname{ch}(\nabla^{E})+\int_{X/B}e(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E},\alpha^{*}\nabla^{F})\\\
&\qquad-\operatorname{CS}\big{(}\nabla^{L_{E}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}})\big{)}.\end{split}$
(3.2.21)

By applying Proposition 3.2 to the isometric isomorphism $\alpha$, there exist
a unique $\mathbb{Z}_{2}$-graded triple
$(L_{F,\alpha},g^{L_{F,\alpha}},\nabla^{L_{F,\alpha}})$ and a unique
$\mathbb{Z}_{2}$-graded isometric isomorphism

${\widetilde{\alpha}}^{L}:(L_{E,\alpha},g^{L_{E,\alpha}})\to(L_{F,\alpha},g^{L_{F,\alpha}})$
(3.2.22)

such that $L_{F,\alpha}\to B$ satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes F}$ and

${\widehat{\eta}}^{\Lambda,E}(g^{E},\alpha^{*}\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{E,\alpha})={\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{F,\alpha}).$
(3.2.23)

Since $(L_{F},g^{L_{F}},\nabla^{L_{F}})$ and
$(L_{F,\alpha},g^{L_{F,\alpha}},\nabla^{L_{F,\alpha}})$ are
$\mathbb{Z}_{2}$-graded triples so that $L_{F}\to B$ and $L_{F,\alpha}\to B$
satisfy the MF property for $\mathsf{D}^{\Lambda\otimes F}$, by Proposition
3.1 there exist balanced $\mathbb{Z}_{2}$-graded triples
$(W_{F,0},g^{W_{F,0}},\nabla^{W_{F,0}})$ and
$(W_{F,1},g^{W_{F,1}},\nabla^{W_{F,1}})$ and a $\mathbb{Z}_{2}$-graded
isometric isomorphism

$h_{F}:(L_{F,\alpha}\oplus W_{F,0},g^{L_{F,\alpha}}\oplus
g^{W_{F,0}})\to(L_{F}\oplus W_{F,1},g^{L_{F}}\oplus g^{W_{F,1}})$ (3.2.24)

such that

$\begin{split}&{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{F})-{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{F,\alpha})\\\
&\equiv-\operatorname{CS}\big{(}\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{F,0}},h_{F}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F,1}})\big{)}.\end{split}$
(3.2.25)

By (3.2.20), (3.2.22) and (3.2.24), the map

${\widetilde{h}}:(L_{E}\oplus W_{0}\oplus W_{F,0},g^{L_{E}}\oplus
g^{W_{0}}\oplus g^{W_{F,0}})\to(L_{F}\oplus W_{1}\oplus
W_{F,1},g^{L_{F}}\oplus g^{W_{1}}\oplus g^{W_{F,1}})$

defined to be the composition

${L_{E}\oplus W_{0}\oplus W_{F,0}}$${L_{E,\alpha}\oplus W_{1}\oplus
W_{F,0}}$${L_{F,\alpha}\oplus W_{1}\oplus W_{F,0}}$${L_{F}\oplus W_{1}\oplus
W_{F,1}}$$\scriptstyle{h\oplus\operatorname{id}_{W_{F,0}}}$$\scriptstyle{{\widetilde{\alpha}}^{L}\oplus\operatorname{id}_{W_{1}}\oplus\operatorname{id}_{W_{F,0}}}$$\scriptstyle{h_{F}\oplus\operatorname{id}_{W_{1}}}$

is a $\mathbb{Z}_{2}$-graded isometric isomorphism. On the other hand,
(3.2.23) and (3.2.25) imply

$\begin{split}&{\widehat{\eta}}^{\Lambda,E}(g^{E},\alpha^{*}\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{E,\alpha})\\\
&={\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{F,\alpha})\\\
&\equiv{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{F})+\operatorname{CS}\big{(}\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{F,0}},h_{F}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F,1}})\big{)}.\end{split}$
(3.2.26)

By (3.2.26), (3.2.21) becomes

$\begin{split}&{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},L_{F})-{\widehat{\eta}}^{\Lambda,F}(g^{E},\nabla^{E},T^{H}_{0}X,g^{T^{V}X}_{0},L_{E})\\\
&\equiv\int_{X/B}{\widetilde{e}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge\operatorname{ch}(\nabla^{E})+\int_{X/B}e(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E},\alpha^{*}\nabla^{F})\\\
&\quad-\operatorname{CS}\big{(}\nabla^{L_{E}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}})\big{)}-\operatorname{CS}\big{(}\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{F,0}},h_{F}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F,1}})\big{)}.\end{split}$
(3.2.27)

As in the proof of [10, Theorem 3.1], we have
$\nabla^{L_{E,\alpha}}=({\widetilde{\alpha}}^{L})^{*}\nabla^{L_{F,\alpha}}$
(cf. [10, (3.2.11)]). Thus

$\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}=({\widetilde{\alpha}}^{L}\oplus\operatorname{id}_{W_{1}}\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}).$

Thus

$\begin{split}&\operatorname{CS}\big{(}\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}},(h_{F}\oplus\operatorname{id}_{W_{1}})^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}})\big{)}\\\
&\equiv\operatorname{CS}\big{(}(h\oplus\operatorname{id}_{W_{F,0}})^{*}({\widetilde{\alpha}}^{L}\oplus\operatorname{id}_{W_{1}}\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}),\\\
&\qquad(h\oplus\operatorname{id}_{W_{F,0}})^{*}({\widetilde{\alpha}}^{L}\oplus\operatorname{id}_{W_{1}}\oplus\operatorname{id}_{W_{F,0}})^{*}(h_{F}\oplus\operatorname{id}_{W_{1}})^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}})\big{)}\\\
&\equiv\operatorname{CS}\big{(}(h\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}),{\widetilde{h}}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}})\big{)}.\end{split}$
(3.2.28)

Since
$\operatorname{CS}(\nabla^{W_{F,0}},\operatorname{id}_{W_{F,0}}^{*}\nabla^{W_{F,0}})\equiv
0$ and
$\operatorname{CS}(\nabla^{W_{F,1}},\operatorname{id}_{W_{F,1}}^{*}\nabla^{W_{F,1}})\equiv
0$, it follows from (3.2.28), (2.2.6) and (2.2.5) that the sum of the last two
terms of the right-hand side of (3.2.27) is equal to

$\begin{split}&\operatorname{CS}\big{(}\nabla^{L_{E}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}})\big{)}+\operatorname{CS}\big{(}\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{F,0}},h_{F}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F,1}})\big{)}\\\
&\equiv\operatorname{CS}\big{(}\nabla^{L_{E}}\oplus\nabla^{W_{0}}\oplus\nabla^{W_{F,0}},(h\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}})\big{)}\\\
&\qquad+\operatorname{CS}\big{(}\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}},(h_{F}\oplus\operatorname{id}_{W_{1}})^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}})\big{)}\\\
&\equiv\operatorname{CS}\big{(}\nabla^{L_{E}}\oplus\nabla^{W_{0}}\oplus\nabla^{W_{F,0}},(h\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}})\big{)}\\\
&\qquad+\operatorname{CS}\big{(}(h\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}),{\widetilde{h}}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}})\big{)}\\\
&\equiv\operatorname{CS}\big{(}\nabla^{L_{E}}\oplus\nabla^{W_{0}}\oplus\nabla^{W_{F,0}},{\widetilde{h}}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}})\big{)}.\end{split}$
(3.2.29)

By taking $W_{E}=W_{0}\oplus W_{F,0}$ and $W_{F}=W_{1}\oplus W_{F,1}$, and
similarly for $g^{W_{E}}$, $g^{W_{F}}$ and $\nabla^{W_{E}}$, $\nabla^{W_{F}}$,
(3.2.27) and (3.2.29) show that (3.2.19) holds. ∎

###  3.3. Some properties of the Bismut–Cheeger eta form
${\widehat{\eta}}^{\Lambda}$

All the results in this subsection are under the following setup. Let
$\pi:X\to B$ be a submersion with closed fibers, equipped with a fixed
Riemannian structure $(T^{H}X,g^{T^{V}X})$.

The proofs of Propositions 3.2, 3.4 and Lemmata 3.2 and 3.3 follow verbatim
from [10, Propositions 4.1, Proposition 4.2] and [10, Lemma 4.1, Lemma 4.2],
respectively.

The following proposition is the additivity of the Bismut–Cheeger eta form
${\widehat{\eta}}^{\Lambda}$.

######  Proposition 3.3.

Let $(E,g^{E},\nabla^{E})$ and $(F,g^{F},\nabla^{F})$ be two
$\mathbb{Z}_{2}$-graded triples over $X$. If
$(L_{E},g^{L_{E}},\nabla^{L_{E}})$ and $(L_{F},g^{L_{F}},\nabla^{L_{F}})$ are
$\mathbb{Z}_{2}$-graded triples so that $L_{E}\to B$ and $L_{F}\to B$ satisfy
the MF properties for $\mathsf{D}^{\Lambda\otimes E}$ and
$\mathsf{D}^{\Lambda\otimes F}$, respectively, then there exists a
$\mathbb{Z}_{2}$-graded triple

$(L_{E\oplus F},g^{L_{E\oplus F}},\nabla^{L_{E\oplus F}})$

such that $L_{E\oplus F}\to B$ satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes(E\oplus F)}$, and

$\begin{split}&{\widehat{\eta}}^{\Lambda,E\oplus F}(g^{E}\oplus
g^{F},\nabla^{E}\oplus\nabla^{F},T^{H}X,g^{T^{V}X},L_{E\oplus F})\\\
&\equiv{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E})+{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}X,g^{T^{V}X},L_{F}).\end{split}$

As in the proof of [10, Proposition 4.1], the $\mathbb{Z}_{2}$-graded triple
$(L_{E\oplus F},g^{L_{E\oplus F}},\nabla^{L_{E\oplus F}})$ in Proposition 3.3
is given by

$L_{E\oplus F}=L_{E}\oplus L_{F},\qquad g^{L_{E\oplus F}}=g^{L_{E}}\oplus
g^{L_{F}},\qquad\nabla^{L_{E\oplus F}}=\nabla^{L_{E}}\oplus\nabla^{L_{F}}.$
(3.3.1)

######  Lemma 3.2.

Let $(E,g^{E},\nabla^{E})$ be a $\mathbb{Z}_{2}$-graded triple over $X$. If
$L_{E^{+}}\to B$ and $L_{E^{-}}\to B$ are $\mathbb{Z}_{2}$-graded complex
vector bundles satisfying the MF property for $\mathsf{D}^{\Lambda\otimes
E^{+}}$ and $\mathsf{D}^{\Lambda\otimes E^{-}}$, respectively, then
$L_{E^{+}}\oplus L_{E^{-}}^{\operatorname{op}}\to B$ is a
$\mathbb{Z}_{2}$-graded complex vector bundle satisfying the MF property for
$\mathsf{D}^{\Lambda\otimes E}$.

######  Lemma 3.3.

Let $(E,g^{E},\nabla^{E})$ be a $\mathbb{Z}_{2}$-graded triple over $X$. If
$(L,g^{L},\nabla^{L})$ is a $\mathbb{Z}_{2}$-graded triple so that $L\to B$
satisfies the MF property for $\mathsf{D}^{\Lambda\otimes E}$, then
$(L^{\operatorname{op}},g^{L,\operatorname{op}},\nabla^{L,\operatorname{op}})$
is a $\mathbb{Z}_{2}$-graded triple so that $L^{\operatorname{op}}\to B$
satisfies the MF property for $\mathsf{D}^{\Lambda\otimes
E^{\operatorname{op}}}$, and

${\widehat{\eta}}^{\Lambda,E^{\operatorname{op}}}(g^{E,\operatorname{op}},\nabla^{E,\operatorname{op}},T^{H}X,g^{T^{V}X},L^{\operatorname{op}})\equiv-{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L).$

######  Proposition 3.4.

Let $(V,g^{V},\nabla^{V})$ be a balanced $\mathbb{Z}_{2}$-graded triple over
$X$. If $(L_{+},g^{L_{+}},\nabla^{L_{+}})$ is a $\mathbb{Z}_{2}$-graded triple
so that $L_{+}\to B$ satisfies the MF property for $\mathsf{D}^{\Lambda\otimes
V^{+}}$, then there exists a balanced $\mathbb{Z}_{2}$-graded triple
$(L,g^{L},\nabla^{L})$ such that $L\to B$ satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes V}$ and

${\widehat{\eta}}^{\Lambda,V}(g^{V},\nabla^{V},T^{H}X,g^{T^{V}X},L)\equiv 0.$

The following theorem is the $\mathbb{Z}_{2}$-graded additivity of the
Bismut–Cheeger eta form ${\widehat{\eta}}^{\Lambda}$.

######  Theorem 3.2.

Let $(E,g^{E},\nabla^{E})$ be a $\mathbb{Z}_{2}$-graded triple over $X$. If
$(L_{E^{+}},g^{L_{E^{+}}},\nabla^{L_{E_{+}}})$ and
$(L_{E^{-}},g^{L_{E^{-}}},\nabla^{L_{E^{-}}})$ are $\mathbb{Z}_{2}$-graded
triples so that $L_{E^{+}}\to B$ and $L_{E^{-}}\to B$ satisfy the MF property
for $\mathsf{D}^{\Lambda\otimes E^{+}}$ and $\mathsf{D}^{\Lambda\otimes
E^{-}}$, respectively, then

$\begin{split}&{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\\\
&\equiv{\widehat{\eta}}^{\Lambda,E^{+}}(g^{E,+},\nabla^{E,+},T^{H}X,g^{T^{V}X},L_{E^{+}})-{\widehat{\eta}}^{\Lambda,E^{-}}(g^{E,-},\nabla^{E,-},T^{H}X,g^{T^{V}X},L_{E^{-}}).\end{split}$
(3.3.2)

###### Proof.

By Lemma 3.2, $L_{E^{+}}\oplus L_{E^{+}}^{\operatorname{op}}\to B$ satisfies
the MF property for $\mathsf{D}^{\Lambda\otimes E}$. Thus the left-hand side
of (3.3.2) makes sense.

Consider the (ungraded) triple $(E\oplus E^{-},g^{E}\oplus
g^{E,-},\nabla^{E}\oplus\nabla^{E,-})$. By Proposition 3.3,

$\big{(}(L_{E^{+}}\oplus L_{E^{-}}^{\operatorname{op}})\oplus
L_{E^{-}},(g^{L_{E^{+}}}\oplus g^{L_{E^{-}},\operatorname{op}})\oplus
g^{L_{E^{-}}},(\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-},\operatorname{op}}})\oplus\nabla^{L_{E^{-}}}\big{)}$

is a $\mathbb{Z}_{2}$-graded triple so that $(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{E^{-}}\to B$ satisfies the MF property
for $\mathsf{D}^{\Lambda\otimes(E\oplus E^{-})}$, which is defined in terms of
$\nabla^{E}\oplus\nabla^{E,-}$.

On the other hand, define a balanced $\mathbb{Z}_{2}$-graded triple
$(H,g^{H},\nabla^{H})$ by $H^{\pm}:=E^{-}$, and similarly for $g^{H}$ and
$\nabla^{H}$. Consider the (ungraded) triple $(E^{+}\oplus H,g^{E,+}\oplus
g^{H},\nabla^{E,+}\oplus\nabla^{H})$. The map

$\alpha:(E\oplus E^{-},g^{E}\oplus g^{E,-})\to(E^{+}\oplus H,g^{E,+}\oplus
g^{H})$

defined by

$\alpha((a,b),c)=(a,(c,b))$

is obviously an isometric isomorphism and satisfies

$\alpha^{*}(\nabla^{E,+}\oplus\nabla^{H})=\nabla^{E}\oplus\nabla^{E,-}.$
(3.3.3)

By applying Proposition 3.2 to the isometric isomorphism $\alpha$ and noting
that $(L_{E^{+}}\oplus L_{E^{-}}^{\operatorname{op}})\oplus L_{E^{-}}\to B$
satisfies the MF property for $\mathsf{D}^{\Lambda\otimes(E\oplus E^{-})}$,
which is defined in terms of
$\nabla^{E}\oplus\nabla^{E,-}=\alpha^{*}(\nabla^{E,+}\oplus\nabla^{H})$ by
(3.3.3), there exists a unique $\mathbb{Z}_{2}$-graded triple
$(L_{\alpha},g^{L_{\alpha}},\nabla^{L_{\alpha}})$ and a unique
$\mathbb{Z}_{2}$-graded isometric isomorphism

${\widetilde{\alpha}}^{L}:\big{(}(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{E^{-}},(g^{L_{E^{+}}}\oplus
g^{L_{E^{-}},\operatorname{op}})\oplus
g^{L_{E^{-}}}\big{)}\to(L_{\alpha},g^{L_{\alpha}})$

such that $L_{\alpha}\to B$ satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes(E^{+}\oplus H)}$, which is defined in terms of
$\nabla^{E,+}\oplus\nabla^{H}$, and

$\begin{split}&{\widehat{\eta}}^{\Lambda,E\oplus E^{-}}(g^{E}\oplus
g^{E,-},\alpha^{*}(\nabla^{E,+}\oplus\nabla^{H}),T^{H}X,g^{T^{V}X},(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{E^{-}})\\\
&\equiv{\widehat{\eta}}^{\Lambda,E^{+}\oplus H}(g^{E,+}\oplus
g^{H},\nabla^{E,+}\oplus\nabla^{H},T^{H}X,g^{T^{V}X},L_{\alpha}).\end{split}$
(3.3.4)

By the definition of $\alpha$, the $\mathbb{Z}_{2}$-graded triple
$(L_{\alpha},g^{L_{\alpha}},\nabla^{L_{\alpha}})$ is given by

$\big{(}L_{E^{+}}\oplus(L_{E^{-}}\oplus
L_{E^{-}}^{\operatorname{op}}),g^{L_{E^{+}}}\oplus(g^{L_{E^{-}}}\oplus
g^{L_{E^{-}},\operatorname{op}}),\nabla^{L_{E^{+}}}\oplus(\nabla^{L_{E^{-}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})\big{)}.$
(3.3.5)

By (3.3.3) and (3.3.5), (3.3.4) becomes

$\begin{split}&{\widehat{\eta}}^{\Lambda,E\oplus E^{-}}(g^{E}\oplus
g^{E,-},\nabla^{E}\oplus\nabla^{E,-},T^{H}X,g^{T^{V}X},(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{E^{-}})\\\
&={\widehat{\eta}}^{\Lambda,E^{+}\oplus H}(g^{E,+}\oplus
g^{H},\nabla^{E,+}\oplus\nabla^{H},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus(L_{E^{-}}\oplus
L_{E^{-}}^{\operatorname{op}})).\end{split}$ (3.3.6)

By Propositions 3.3 and 3.4 and (3.3.6), we have

$\begin{split}&{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})+{\widehat{\eta}}^{\Lambda,E^{-}}(g^{E,-},\nabla^{E,-},T^{H}X,g^{T^{V}X},L_{E^{-}})\\\
&\equiv{\widehat{\eta}}^{\Lambda,E\oplus E^{-}}(g^{E}\oplus
g^{E,-},\nabla^{E}\oplus\nabla^{E,-},T^{H}X,g^{T^{V}X},(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{E^{-}})\\\
&={\widehat{\eta}}^{\Lambda,E^{+}\oplus H}(g^{E,+}\oplus
g^{H},\nabla^{E,+}\oplus\nabla^{H},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus(L_{E^{-}}\oplus
L_{E^{-}}^{\operatorname{op}}))\\\
&\equiv{\widehat{\eta}}^{\Lambda,E^{+}}(g^{E,+},\nabla^{E,+},T^{H}X,g^{T^{V}X},L_{E^{+}})+{\widehat{\eta}}^{\Lambda,H}(g^{H},\nabla^{H},T^{H}X,g^{T^{V}X},L_{E^{-}}\oplus
L_{E^{-}}^{\operatorname{op}})\\\
&\equiv{\widehat{\eta}}^{\Lambda,E^{+}}(g^{E,+},\nabla^{E,+},T^{H}X,g^{T^{V}X},L_{E^{+}}).\end{split}$

Thus (3.3.2) holds. ∎

###  3.4. An analytic index in differential $K$-theory

The Freed–Lott differential $K$-group ${\widehat{K}}(X)$ [7, Definition 2.16]
can be described in terms of $\mathbb{Z}_{2}$-graded generators, which are of
the form $\mathcal{E}=(E,g^{E},\nabla^{E},\omega)$, where
$(E,g^{E},\nabla^{E})$ is a $\mathbb{Z}_{2}$-graded triple and
$\displaystyle\omega\in\frac{\Omega^{\operatorname{odd}}(X)}{\operatorname{Im}(d)}$.
Two $\mathbb{Z}_{2}$-graded generators $\mathcal{E}$ and $\mathcal{F}$ are
equal in ${\widehat{K}}(X)$ if and if only there exist balanced
$\mathbb{Z}_{2}$-graded triples $(V_{E},g^{V_{E}},\nabla^{V_{E}})$ and
$(V_{F},g^{V_{F}},\nabla^{V_{F}})$ and a $\mathbb{Z}_{2}$-graded isometric
isomorphism $\alpha:(E\oplus V_{E},g^{E}\oplus g^{V_{E}})\to(F\oplus
V_{F},g^{F}\oplus g^{V_{F}})$ such that

$\omega_{E}-\omega_{F}\equiv\operatorname{CS}\big{(}\nabla^{E}\oplus\nabla^{V_{E}},\alpha^{*}(\nabla^{F}\oplus\nabla^{V_{F}})\big{)}.$
(3.4.1)

We define the analytic index $\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}$
associated to families of generalized twisted de Rham operators.

######  Definition 3.1.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. If $\mathcal{E}$ is a
$\mathbb{Z}_{2}$-graded generator $\mathcal{E}$ of ${\widehat{K}}(X)$ so that
$\mathsf{D}^{\Lambda\otimes E}$ satisfies the kernel bundle assumption, define

$\begin{split}\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E})&=\bigg{(}\ker(\mathsf{D}^{\Lambda\otimes
E}),g^{\ker(\mathsf{D}^{\Lambda\otimes
E})},\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E})},\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega\\\
&\qquad+{\widetilde{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X})\bigg{)}.\end{split}$

If $\mathcal{E}$ is a $\mathbb{Z}_{2}$-graded generator $\mathcal{E}$ of
${\widehat{K}}(X)$ so that $\mathsf{D}^{\Lambda\otimes E}$ does not satisfy
the kernel bundle assumption, define

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E};L)=\bigg{(}L,g^{L},\nabla^{L},\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega+{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L)\bigg{)},$

where $(L,g^{L},\nabla^{L})$ is a $\mathbb{Z}_{2}$-graded triple so that $L\to
B$ satisfies the MF property for $\mathsf{D}^{\Lambda\otimes E}$.

The following proposition says the analytic index
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E})$ does not depend
on the choice of the $\mathbb{Z}_{2}$-graded triple $(L,g^{L},\nabla^{L})$ so
that $L\to B$ satisfies the MF property for $\mathsf{D}^{\Lambda\otimes E}$.

######  Proposition 3.5.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. If $\mathcal{E}$ is a
$\mathbb{Z}_{2}$-graded generator of ${\widehat{K}}(X)$ so that
$\mathsf{D}^{\Lambda\otimes E}$ does not satisfy the kernel bundle assumption,
and $(L_{0},g^{L_{0}},\nabla^{L_{0}})$ and $(L_{1},g^{L_{1}},\nabla^{L_{1}})$
are $\mathbb{Z}_{2}$-graded triples so that $L_{0}\to B$ and $L_{1}\to B$
satisfy the MF property for $\mathsf{D}^{\Lambda\otimes E}$, then

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E};L_{0})=\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E};L_{1}).$
(3.4.2)

###### Proof.

By Proposition 3.1, there exist balanced $\mathbb{Z}_{2}$-graded triples
$(W_{0},g^{W_{0}},\nabla^{W_{0}})$ and $(W_{1},g^{W_{1}},\nabla^{W_{1}})$ and
a $\mathbb{Z}_{2}$-graded isometric isomorphism

$h:(L_{0}\oplus W_{0},g^{L_{0}}\oplus g^{W_{0}})\to(L_{1}\oplus
W_{1},g^{L_{1}}\oplus g^{W_{1}})$

such that

$\begin{split}&{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{1})-{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{0})\\\
&\equiv-\operatorname{CS}\big{(}\nabla^{L_{0}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{1}}\oplus\nabla^{W_{1}})\big{)}.\end{split}$

It follows from (3.4.1) that (3.4.2) holds. ∎

The following proposition says if $\mathcal{E}$ is a $\mathbb{Z}_{2}$-graded
triple of ${\widehat{K}}(X)$ so that $\mathsf{D}^{\Lambda\otimes E}$ satisfies
the kernel bundle assumption, then the analytic index
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E})$ defined under
and without the kernel bundle assumption coincide.

######  Proposition 3.6.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. If $\mathcal{E}$ is a
$\mathbb{Z}_{2}$-graded generator of ${\widehat{K}}(X)$ so that
$\mathsf{D}^{\Lambda\otimes E}$ satisfies the kernel bundle assumption and
$(L,g^{L},\nabla^{L})$ is a $\mathbb{Z}_{2}$-graded triple so that $L\to B$
satisfies the MF property for $\mathsf{D}^{\Lambda\otimes E}$, then

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E};L)=\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E}).$

###### Proof.

Since the kernel bundle $\ker(\mathsf{D}^{\Lambda\otimes E})\to B$ exists, it
satisfies the MF property for $\mathsf{D}^{\Lambda\otimes E}$. By Proposition
3.5 we have
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E};L)=\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E};\ker(\mathsf{D}^{\Lambda\otimes
E}))$. Thus it suffices to show that

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E};\ker(\mathsf{D}^{\Lambda\otimes
E}))=\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E}).$ (3.4.3)

To show (3.4.3), it suffices to show that

${\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},\ker(\mathsf{D}^{\Lambda\otimes
E}))\equiv{\widetilde{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X}).$
(3.4.4)

The Bismut–Cheeger eta form on the left-hand side of (3.4.4) is associated to
the Bismut superconnection ${\widehat{\mathbb{B}}}^{\Lambda,E}$ (3.1.8) whose
degree 1 part is given by

${\widehat{\mathbb{B}}}^{\Lambda,E}_{[1]}=\nabla^{\pi^{\Lambda}_{*}E,u}\oplus\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}}.$

Choose and fix $t,T\in(0,\infty)$ that satisfy $t<a$ and $T\geq 1$. Since
${\widehat{\mathbb{B}}}^{\Lambda,E}_{t}=\mathbb{B}^{\Lambda,E}_{t}\oplus\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}}$, it follows from (2.2.6), (2.2.4) and (2.2.5) that

$\begin{split}&\operatorname{CS}({\widehat{\mathbb{B}}}^{\Lambda,E}_{T},{\widehat{\mathbb{B}}}^{\Lambda,E}_{t})-\operatorname{CS}(\mathbb{B}^{\Lambda,E}_{T},\mathbb{B}^{\Lambda,E}_{t})\\\
&\equiv\operatorname{CS}({\widehat{\mathbb{B}}}^{\Lambda,E}_{T},{\widehat{\mathbb{B}}}^{\Lambda,E}_{t})-\operatorname{CS}(\mathbb{B}^{\Lambda,E}_{T},\mathbb{B}^{\Lambda,E}_{t})-\operatorname{CS}(\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}},\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}})\\\
&\equiv\operatorname{CS}({\widehat{\mathbb{B}}}^{\Lambda,E}_{t},{\widehat{\mathbb{B}}}^{\Lambda,E}_{t})-\operatorname{CS}(\mathbb{B}^{\Lambda,E}_{T}\oplus\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}},\mathbb{B}^{\Lambda,E}_{t}\oplus\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}})\\\
&\equiv\operatorname{CS}({\widehat{\mathbb{B}}}^{\Lambda,E}_{T},{\widehat{\mathbb{B}}}^{\Lambda,E}_{t})+\operatorname{CS}(\mathbb{B}^{\Lambda,E}_{t}\oplus\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}},\mathbb{B}^{\Lambda,E}_{T}\oplus\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}})\\\
&\equiv\operatorname{CS}({\widehat{\mathbb{B}}}^{\Lambda,E}_{T},\mathbb{B}^{\Lambda,E}_{T}\oplus\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}}).\end{split}$

By letting $T\to\infty$ and $t\to 0$ in above, we have

$\begin{split}&{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},\ker(\mathsf{D}^{\Lambda\otimes
E}))-{\widetilde{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X})\\\
&\equiv\lim_{T\to\infty}\operatorname{CS}({\widehat{\mathbb{B}}}^{\Lambda,E}_{T},\mathbb{B}^{\Lambda,E}_{T}\oplus\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}}).\end{split}$

By the estimates in [2, §9.3], we have

$\lim_{T\to\infty}\operatorname{CS}({\widehat{\mathbb{B}}}^{\Lambda,E}_{T},\mathbb{B}^{\Lambda,E}_{T}\oplus\nabla^{\ker(\mathsf{D}^{\Lambda\otimes
E}),\operatorname{op}})=0.$

Thus (3.4.4) holds. ∎

By Propositions 3.5 and 3.6, we can drop the notation $L$ in
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E};L)$.

######  Proposition 3.7.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. The analytic index

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}:{\widehat{K}}(X)\to{\widehat{K}}(B)$

in differential $K$-theory associated to families of twisted de Rham operator
is a well defined group homomorphism.

###### Proof.

We first show that $\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}$ passes to
a well defined map ${\widehat{K}}(X)\to{\widehat{K}}(B)$. Let $\mathcal{E}$
and $\mathcal{F}$ be $\mathbb{Z}_{2}$-graded generators of ${\widehat{K}}(X)$
so that $\mathsf{D}^{\Lambda\otimes E}$ and $\mathsf{D}^{\Lambda\otimes F}$ do
not satisfy the kernel bundle assumption and $\mathcal{E}=\mathcal{F}$. Then
there exist balanced $\mathbb{Z}_{2}$-graded triples
$(V_{E},g^{V_{E}},\nabla^{V_{E}})$ and $(V_{F},g^{V_{F}},\nabla^{V_{F}})$ and
a $\mathbb{Z}_{2}$-graded isometric isomorphism

$\alpha:(E\oplus V_{E},g^{E}\oplus g^{V_{E}})\to(F\oplus V_{F},g^{F}\oplus
g^{V_{E}})$ (3.4.5)

such that (3.4.1) holds.

Let $(L_{E^{+}},g^{L_{E^{+}}},\nabla^{L_{E^{+}}})$ and
$(L_{E^{-}},g^{L_{E^{-}}},\nabla^{L_{E^{-}}})$ be $\mathbb{Z}_{2}$-graded
triples so that $L_{E^{+}}\to B$ and $L_{E^{-}}\to B$ satisfy the MF property
for $\mathsf{D}^{\Lambda\otimes E^{+}}$ and $\mathsf{D}^{\Lambda\otimes
E^{-}}$, respectively. By Lemma 3.2, $(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}},g^{L_{E^{+}}}\oplus
g^{L_{E^{-}},\operatorname{op}},\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})$
is a $\mathbb{Z}_{2}$-graded triple so that $L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}}\to B$ satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes E}$. By Definition 3.1 and Proposition 3.5 we have

$\begin{split}\operatorname{ind}^{a,\Lambda}(\mathcal{E})&=\bigg{(}L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}},g^{L_{E^{+}}}\oplus
g^{L_{E^{-}},\operatorname{op}},\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}},\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega_{E}\\\
&\qquad+{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\bigg{)}.\end{split}$

Let $(L_{F^{+}},g^{L_{F^{+}}},\nabla^{L_{F^{+}}})$ and
$(L_{F^{-}},g^{L_{F^{-}}},\nabla^{L_{F^{-}}})$ be $\mathbb{Z}_{2}$-graded
triples so that $L_{F^{+}}\to B$ and $L_{F^{-}}\to B$ satisfy the MF property
for $\mathsf{D}^{\Lambda\otimes F^{+}}$ and $\mathsf{D}^{\Lambda\otimes
F^{-}}$, respectively. Similarly, we have

$\begin{split}\operatorname{ind}^{a,\Lambda}(\mathcal{F})&=\bigg{(}L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}},g^{L_{F^{+}}}\oplus
g^{L_{F^{-}},\operatorname{op}},\nabla^{L_{F^{+}}}\oplus\nabla^{L_{F^{-}},\operatorname{op}},\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega_{F}\\\
&\qquad+{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}X,g^{T^{V}X},L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})\bigg{)}.\end{split}$

By Proposition 3.4, there exist balanced $\mathbb{Z}_{2}$-graded triples
$(L_{V_{E}},g^{L_{V_{E}}},\nabla^{L_{V_{E}}})$ and
$(L_{V_{F}},g^{L_{V_{F}}},\nabla^{L_{V_{F}}})$ such that $L_{V_{E}}\to B$ and
$L_{V_{F}}\to B$ satisfy the MF property for $\mathsf{D}^{\Lambda\otimes
V_{E}}$ and $\mathsf{D}^{\Lambda\otimes V_{F}}$, respectively, and

${\widehat{\eta}}^{\Lambda,V_{E}}(g^{V_{E}},\nabla^{V_{E}},T^{H}X,g^{T^{V}X},L_{V_{E}})\equiv
0\equiv{\widehat{\eta}}^{\Lambda,V_{F}}(g^{V_{F}},\nabla^{V_{F}},T^{H}X,g^{T^{V}X},L_{V_{F}}).$
(3.4.6)

By Proposition 3.3,

$\big{(}(L_{E^{+}}\oplus L_{E^{-}}^{\operatorname{op}})\oplus
L_{V_{E}},(g^{L_{E^{+}}}\oplus g^{L_{E^{-}},\operatorname{op}})\oplus
g^{L_{V_{E}}},(\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{E}}}\big{)}$

and

$\big{(}(L_{F^{+}}\oplus L_{F^{-}}^{\operatorname{op}})\oplus
L_{V_{F}},(g^{L_{F^{+}}}\oplus g^{L_{F^{-}},\operatorname{op}})\oplus
g^{L_{V_{F}}},(\nabla^{L_{F^{+}}}\oplus\nabla^{L_{F^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{F}}}\big{)}$

are $\mathbb{Z}_{2}$-graded triples so that $(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{V_{E}}\to B$ and $(L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})\oplus L_{V_{F}}\to B$ satisfy the MF property
for $\mathsf{D}^{\Lambda\otimes(E\oplus V_{E})}$ and
$\mathsf{D}^{\Lambda\otimes(F\oplus V_{F})}$, respectively.

By applying Theorem 3.1 to the $\mathbb{Z}_{2}$-graded isometric isomorphism
$\alpha$ in (3.4.5), there exist balanced $\mathbb{Z}_{2}$-graded triples
$(W_{E},g^{W_{E}},\nabla^{W_{E}})$ and $(W_{F},g^{W_{F}},\nabla^{W_{F}})$ and
a $\mathbb{Z}_{2}$-graded isometric isomorphism

${\big{(}(L_{E^{+}}\oplus L_{E^{-}}^{\operatorname{op}})\oplus L_{V_{E}}\oplus
W_{E},(g^{L_{E^{+}}}\oplus g^{L_{E^{-}},\operatorname{op}})\oplus
g^{L_{V_{E}}}\oplus g^{W_{E}}\big{)}}$${\big{(}(L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})\oplus L_{V_{F}}\oplus
W_{F},(g^{L_{F^{+}}}\oplus g^{L_{F^{-}},\operatorname{op}})\oplus
g^{L_{V_{F}}}\oplus g^{W_{F}}\big{)}}$$\scriptstyle{h}$ (3.4.7)

such that

$\begin{split}&{\widehat{\eta}}^{\Lambda,F\oplus V_{F}}(g^{F}\oplus
g^{V_{F}},\nabla^{F}\oplus\nabla^{V_{F}},T^{H}X,g^{T^{V}X},(L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})\oplus L_{V_{F}})\\\
&\qquad-{\widehat{\eta}}^{\Lambda,E\oplus V_{E}}(g^{E}\oplus
g^{V_{E}},\nabla^{E}\oplus\nabla^{V_{E}},T^{H}X,g^{T^{V}X},(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{V_{E}})\\\
&\equiv\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{CS}\big{(}\nabla^{E}\oplus\nabla^{V_{E}},\alpha^{*}(\nabla^{F}\oplus\nabla^{V_{F}})\big{)}\\\
&\qquad-\operatorname{CS}\big{(}(\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{E}}}\oplus\nabla^{W_{E}},h^{*}((\nabla^{L_{F^{+}}}\oplus\nabla^{L_{F^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{F}}}\oplus\nabla^{W_{F}})\big{)}.\end{split}$
(3.4.8)

By Proposition 3.3 and (3.4.6),

$\begin{split}&{\widehat{\eta}}^{\Lambda,E\oplus V_{E}}(g^{E}\oplus
g^{V_{E}},\nabla^{E}\oplus\nabla^{V_{E}},T^{H}X,g^{T^{V}X},(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{V_{E}})\\\
&\equiv{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}}),\end{split}$ (3.4.9)

and

$\begin{split}&{\widehat{\eta}}^{\Lambda,F\oplus V_{F}}(g^{F}\oplus
g^{V_{F}},\nabla^{F}\oplus\nabla^{V_{F}},T^{H}X,g^{T^{V}X},(L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})\oplus L_{V_{F}})\\\
&\equiv{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}X,g^{T^{V}X},L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}}).\end{split}$ (3.4.10)

By (3.4.1), (3.4.9) and (3.4.10), (3.4.8) becomes

$\begin{split}&{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F},T^{H}X,g^{T^{V}X},L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})-{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\\\
&\equiv\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{CS}\big{(}\nabla^{E}\oplus\nabla^{V_{E}},\alpha^{*}(\nabla^{F}\oplus\nabla^{V_{F}})\big{)}\\\
&\qquad-\operatorname{CS}\big{(}(\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{E}}}\oplus\nabla^{W_{E}},h^{*}((\nabla^{L_{F^{+}}}\oplus\nabla^{L_{F^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{F}}}\oplus\nabla^{W_{F}})\big{)}.\end{split}$
(3.4.11)

Since the $\mathbb{Z}_{2}$-graded triples $(L_{V_{E}}\oplus
W_{E},g^{L_{V_{E}}}\oplus g^{W_{E}},\nabla^{L_{V_{E}}}\oplus\nabla^{W_{E}})$
and $(L_{V_{F}}\oplus W_{F},g^{L_{V_{F}}}\oplus
g^{W_{F}},\nabla^{L_{V_{F}}}\oplus\nabla^{W_{F}})$ are balanced, it follows
from (3.4.7) and (3.4.11) that
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E})=\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{F})$.
Thus
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}:{\widehat{K}}(X)\to{\widehat{K}}(B)$
is a well defined map.

It follows from Proposition 3.3 that
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}$ is a group homomorphism. ∎

We now prove Proposition 1.1. In the following proposition and its proof, we
refer to [10] for the notions and notations that are related to spinc
structures.

######  Proposition 3.8.

Let $\pi:X\to B$ be a submersion with closed, oriented and spinc fibers of
even dimension, equipped with a Riemannian and differential spinc structure
$(T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda})$. For any
$\mathbb{Z}_{2}$-graded generator $\mathcal{E}$ of ${\widehat{K}}(X)$ we have

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E})=\operatorname{ind}^{a}_{{\widehat{K}}}(S(T^{V}X)^{*}\otimes\mathcal{E}),$
(3.4.12)

where, by [7, (2.22)],

$S(T^{V}X)^{*}\otimes\mathcal{E}=(S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E,g^{S(T^{V}X)^{*}}\otimes
g^{E},\nabla^{S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E},\operatorname{ch}(\nabla^{S(T^{V}X)^{*}})\wedge\omega).$

Here, the $\mathbb{Z}_{2}$-graded triple
$(S(T^{V}X)^{*},g^{S(T^{V}X)^{*}},\nabla^{S(T^{V}X)^{*}})$ consists of the
dual of the spinor bundle of $T^{V}X\to X$ with the dual metric and the dual
connection.

###### Proof.

By [7, Definition 7.27],
$\operatorname{ind}^{a}_{{\widehat{K}}}(S(T^{V}X)^{*}\otimes\mathcal{E})$ is
given by

$\begin{split}\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{S}(T^{V}X)^{*}\otimes\mathcal{E})&=\bigg{(}H,g^{H},\nabla^{H},\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{S(T^{V}X)^{*}})\wedge\omega\\\
&\qquad+{\widehat{\eta}}^{S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E}(g^{S(T^{V}X)^{*}}\otimes
g^{E},\nabla^{S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E},T^{H}X,g^{T^{V}X},H)\bigg{)},\end{split}$
(3.4.13)

where $(H,g^{H},\nabla^{H})$ is a $\mathbb{Z}_{2}$-graded triple so that $H\to
X$ satisfies the MF property for
$\mathsf{D}^{S\otimes(S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E)}$.

Recall that a differential spinc structure [7, p.918] on $\pi:X\to B$ consists
of a topological spinc structure on $T^{V}X\to X$, which fixes a complex line
bundle $\lambda\to X$ satisfying $w_{2}(T^{V}X)=c_{1}(\lambda)$, a Hermitian
metric $g^{\lambda}$ and a unitary connection $\nabla^{\lambda}$ on
$\lambda\to X$. The spinor bundle $S(T^{V}X)\to X$ is defined to be

$S(T^{V}X)=S_{0}(T^{V}X)\otimes\lambda^{\frac{1}{2}},$ (3.4.14)

where $S_{0}(T^{V}X)$ is the spinor bundle for the locally existing spin
structure of $T^{V}X\to X$, and $\lambda^{\frac{1}{2}}$ is the locally
existing square root of $\lambda\to X$. By (3.4.14), we have

$S(T^{V}X)\mathbin{{\widehat{\otimes}}}S(T^{V}X)\mathbin{{\widehat{\otimes}}}E\cong
S_{0}(T^{V}X)\mathbin{{\widehat{\otimes}}}S_{0}(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E\cong\Lambda(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E$
(3.4.15)

as $\mathbb{Z}_{2}$-graded complex vector bundles. Under the isomorphism
(3.4.15), the Clifford multiplication on $S(T^{V}X)\to X$ corresponds to that
on $\Lambda(T^{V}X)^{*}\to X$. Since

$\nabla^{S(T^{V}X)\mathbin{{\widehat{\otimes}}}(S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E)}=\nabla^{(S(T^{V}X)\mathbin{{\widehat{\otimes}}}S(T^{V}X)^{*})\mathbin{{\widehat{\otimes}}}E}=\nabla^{\Lambda(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E}$

(cf. (b) of Remark 2.1), it follows that

$\mathsf{D}^{\Lambda\otimes
E}=\mathsf{D}^{S\otimes(S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E)}.$
(3.4.16)

Since (3.4.15) implies

$\pi^{\Lambda}_{*}E\cong\pi_{*}(S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E)$
(3.4.17)

as $\mathbb{Z}_{2}$-graded complex vector bundles, it follows from (3.4.16)
and (3.4.17) that $H\to B$ also satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes E}$. By Proposition 3.5,
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E})$ is given by

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{E})=\bigg{(}H,g^{H},\nabla^{H},\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega+{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},H)\bigg{)}.$
(3.4.18)

By (3.4.13) and (3.4.18), to prove (3.4.12) it suffices to show that

$\begin{split}&\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega+{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},H)\\\
&\equiv\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{S(T^{V}X)^{*}})\wedge\omega\\\
&\qquad+{\widehat{\eta}}^{S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E}(g^{S(T^{V}X)^{*}}\otimes
g^{E},\nabla^{S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E},T^{H}X,g^{T^{V}X},H).\end{split}$
(3.4.19)

Under the isomorphism (3.4.17), we have
$g^{\pi^{\Lambda}_{*}E}=g^{\pi_{*}(S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E)}$.
Moreover, since

$\nabla^{S(T^{V}X)\mathbin{{\widehat{\otimes}}}S(T^{V}X)^{*}}=\nabla^{\Lambda(T^{V}X)^{*}},$
(3.4.20)

it follows from [10, (3.1.6) and (3.1.7)], (3.1.3), (3.1.4) and (3.4.20) that

$\nabla^{\pi_{*}(S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E),u}=\nabla^{\pi^{\Lambda}_{*}E,u}.$

Thus

$\nabla^{\pi_{*}(S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E),u}\oplus\nabla^{H,\operatorname{op}}=\nabla^{\pi^{\Lambda}_{*}E,u}\oplus\nabla^{H,\operatorname{op}}.$
(3.4.21)

Therefore, (3.4.16) and (3.4.21) show that
${\widehat{\mathbb{B}}}^{\Lambda,E}={\widehat{\mathbb{B}}}^{S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E}$.
Hence

$\begin{split}&{\widehat{\eta}}^{S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E}(g^{S(T^{V}X)^{*}}\otimes
g^{E},\nabla^{S(T^{V}X)^{*}\mathbin{{\widehat{\otimes}}}E},T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda},H)\\\
&\equiv{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},H).\end{split}$
(3.4.22)

On the other hand, by the definition of $\operatorname{Todd}(\nabla^{T^{V}X})$
[11, (3.1.3)] and [1, (8.30)], we have

$\begin{split}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{S(T^{V}X)^{*}})&={\widehat{A}}(\nabla^{T^{V}X})\wedge
e^{\frac{1}{2}c_{1}(\nabla^{\lambda})}\wedge\operatorname{ch}(\nabla^{S_{0}(T^{V}X)^{*}})\wedge
e^{-\frac{1}{2}c_{1}(\nabla^{\lambda})}\\\
&={\widehat{A}}(\nabla^{T^{V}X})\wedge\frac{e(\nabla^{T^{V}X})}{{\widehat{A}}(\nabla^{T^{V}X})}\\\
&=e(\nabla^{T^{V}X}).\end{split}$ (3.4.23)

By (3.4.22) and (3.4.23), (3.4.19) holds. ∎

Define a map
$\operatorname{ch}_{{\widehat{K}}}:{\widehat{K}}(X)\to\Omega^{\operatorname{even}}_{\mathbb{Q}}(X)$
by
$\operatorname{ch}_{{\widehat{K}}}(\mathcal{E})=\operatorname{ch}(\nabla^{E})+d\omega$.
The geometric model of the $\mathbb{R}/\mathbb{Z}$ $K$-group
$K^{-1}(X;\mathbb{R}/\mathbb{Z})$ by Lott [14, Definition 7] (cf. [7, (2.20)])
is defined to be
$K^{-1}(X;\mathbb{R}/\mathbb{Z})=\ker(\operatorname{ch}_{{\widehat{K}}})$.
Thus a $\mathbb{Z}_{2}$-graded generator $\mathcal{E}$ of
$K^{-1}(X;\mathbb{R}/\mathbb{Z})$ is a $\mathbb{Z}_{2}$-graded generator of
${\widehat{K}}(X)$ satisfying

$\operatorname{ch}(\nabla^{E,+})-\operatorname{ch}(\nabla^{E,-})=-d\omega.$
(3.4.24)

Note that (3.4.24) implies that
$\operatorname{rank}(E^{+})=\operatorname{rank}(E^{-})$.

The following corollary is an immediate consequence of Propositions 3.7 and
3.8.

######  Corollary 3.1.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$.

  1. (a)

The analytic index
$\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}:K^{-1}(X;\mathbb{R}/\mathbb{Z})\to
K^{-1}(B;\mathbb{R}/\mathbb{Z})$, obtained by restricting
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}$ to
$K^{-1}(X;\mathbb{R}/\mathbb{Z})$, is a well defined group homomorphism.

  2. (b)

Suppose, furthermore, that the fibers of $\pi:X\to B$ are oriented, spinc and
even dimensional. Equip $\pi:X\to B$ with a differential spinc structure
$(g^{\lambda},\nabla^{\lambda})$. Then for any $\mathbb{Z}_{2}$-graded
generator $\mathcal{E}$ of $K^{-1}(X;\mathbb{R}/\mathbb{Z})$, we have

$\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}(\mathcal{E})=\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}(S(T^{V}X)^{*}\otimes\mathcal{E}).$

The $\mathbb{R}/\mathbb{Q}$ Chern character
$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}:K^{-1}(X;\mathbb{R}/\mathbb{Z})\to
H^{\operatorname{odd}}(X;\mathbb{R}/\mathbb{Q})$ is defined as follows. Let
$\mathcal{E}=(E,g^{E},\nabla^{E},\omega)$ be a generator of
$K^{-1}(X;\mathbb{R}/\mathbb{Z})$ with $\operatorname{rank}(E)=\ell$. Since
$\operatorname{ch}(\nabla^{E})-\ell=-d\omega$, there exists a $k\in\mathbb{N}$
such that $kE\cong k\mathbb{C}^{\ell}$ (see [11, Remark 1] for a proof). Let
$\alpha:(kE,kg^{E})\to(\mathbb{C}^{k\ell},g^{k\ell})$ be an isometric
isomorphism. Define

$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E})=\bigg{[}\frac{1}{k}\operatorname{CS}(\alpha^{*}d^{k\ell},k\nabla^{E})+\omega\bigg{]}\mod\mathbb{Q}.$
(3.4.25)

If $\mathcal{E}$ is $\mathbb{Z}_{2}$-graded, then define

$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E})=\bigg{[}\frac{1}{k}\operatorname{CS}(\alpha^{*}(k\nabla^{E,-}),k\nabla^{E,+})+\omega\bigg{]}\mod\mathbb{Q},$
(3.4.26)

where $\alpha:(kE^{+},kg^{E,+})\to(kE^{-},kg^{E,-})$ is an isometric
isomorphism. Note that
$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E})$ is independent of the
choices of $k$ and $\alpha$ [14, p.289].

The following theorem is the RRG theorem for
$\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}$ in
$\mathbb{R}/\mathbb{Z}$ $K$-theory.

######  Theorem 3.3.

Let $\pi:X\to B$ be a submersion with closed fibers, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. The following diagram commutes

${K^{-1}(X;\mathbb{R}/\mathbb{Z})}$${H^{\operatorname{odd}}(X;\mathbb{R}/\mathbb{Q})}$${K^{-1}(B;\mathbb{R}/\mathbb{Z})}$${H^{\operatorname{odd}}(B;\mathbb{R}/\mathbb{Q})}$$\scriptstyle{\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}}$$\scriptstyle{\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}}$$\scriptstyle{\int_{X/B}e(T^{V}X)\cup(\cdot)}$$\scriptstyle{\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}}$
(3.4.27)

That is, for every $\mathbb{Z}_{2}$-graded generator $\mathcal{E}$ of
$K^{-1}(X;\mathbb{R}/\mathbb{Z})$,

$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}(\mathcal{E}))=\int_{X/B}e(T^{V}X)\cup\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E}).$
(3.4.28)

As noted in [10], (3.4.28) implies diagram (3.4.27) commutes only if (a) of
Corollary 3.1 holds.

###### Proof.

Let $\mathcal{E}$ be a $\mathbb{Z}_{2}$-graded generator of
$K^{-1}(X;\mathbb{R}/\mathbb{Z})$. By (3.4.24), there exists a
$k_{1}\in\mathbb{N}$ such that $k_{1}E^{+}\cong k_{1}E^{-}$.

Let $(L_{E^{+}},g^{L_{E^{+}}},\nabla^{L_{E^{+}}})$ and
$(L_{E^{-}},g^{E^{-}},\nabla^{L_{E^{-}}})$ be $\mathbb{Z}_{2}$-graded triples
so that $L_{E^{+}}\to B$ and $L_{E^{-}}\to B$ satisfy the MF property for
$\mathsf{D}^{\Lambda\otimes E^{+}}$ and $\mathsf{D}^{\Lambda\otimes E^{-}}$,
respectively. By Lemma 3.2,

$(L_{E^{+}}\oplus L_{E^{-}}^{\operatorname{op}},g^{L_{E^{+}}}\oplus
g^{L_{E^{-}},\operatorname{op}},\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})$

is a $\mathbb{Z}_{2}$-graded triple so that $L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}}\to B$ satisfies the MF property for
$\mathsf{D}^{\Lambda\otimes E}$. By applying Proposition 3.5 to
$\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}$, we have

$\begin{split}\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}(\mathcal{E})&=\bigg{(}L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}},g^{L_{E^{+}}}\oplus
g^{L_{E^{-}},\operatorname{op}},\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}},\\\
&\qquad\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega+{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\bigg{)}.\end{split}$ (3.4.29)

On the other hand, by (3.1.11), (3.4.24) and Theorem 3.2, we have

$\begin{split}&\operatorname{ch}(\nabla^{L_{E^{+}}})-\operatorname{ch}(\nabla^{L_{E^{-}}})\\\
&=\int_{X/B}e(\nabla^{T^{V}X})\wedge\big{(}\operatorname{ch}(\nabla^{E,+})-\operatorname{ch}(\nabla^{E,-})\big{)}-d{\widehat{\eta}}^{\Lambda,E^{+}}(g^{E,+},\nabla^{E,+},T^{H}X,g^{T^{V}X},L_{E^{+}})\\\
&\qquad+d{\widehat{\eta}}^{\Lambda,E^{-}}(g^{E,-},\nabla^{E,-},T^{H}X,g^{T^{V}X},L_{E^{-}})\\\
&=\int_{X/B}e(\nabla^{T^{V}X})\wedge(-d\omega)-d{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\\\
&=-d\bigg{(}\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega+{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\bigg{)}.\end{split}$

Thus there exists a $k_{2}\in\mathbb{N}$ such that $k_{2}L_{E^{+}}\cong
k_{2}L_{E^{-}}$.

Let $k$ be the least common multiple of $k_{1}$ and $k_{2}$. Denote by
$\alpha:(kE^{+},kg^{E,+})\to(kE^{-},kg^{E,-})$ the resulting isometric
isomorphism. By Theorem 3.1, there exist balanced $\mathbb{Z}_{2}$-graded
triples $(W_{+},g^{W_{+}},\nabla^{W_{+}})$ and
$(W_{-},g^{W_{-}},\nabla^{W_{-}})$ and a $\mathbb{Z}_{2}$-graded isometric
isomorphism $h:(L_{kE^{+}}\oplus W_{+},g^{L_{kE^{+}}}\oplus
g^{W_{+}})\to(L_{kE^{-}}\oplus W_{-},g^{L_{kE^{-}}}\oplus g^{W_{-}})$ such
that

$\begin{split}&{\widehat{\eta}}^{\Lambda,kE^{-}}(kg^{E,-},k\nabla^{E,-},T^{H}X,g^{T^{V}X},L_{kE^{-}})-{\widehat{\eta}}^{\Lambda,kE^{+}}(kg^{E,+},k\nabla^{E,+},T^{H}X,g^{T^{V}X},L_{kE^{+}})\\\
&\equiv\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{CS}(k\nabla^{E,+},\alpha^{*}(k\nabla^{E,-}))-\operatorname{CS}\big{(}\nabla^{L_{kE^{+}}}\oplus\nabla^{W_{+}},h^{*}(\nabla^{L_{kE^{-}}}\oplus\nabla^{W_{-}})\big{)}.\end{split}$
(3.4.30)

By (3.3.1), the $\mathbb{Z}_{2}$-graded triples
$(L_{kE^{\pm}},g^{L_{kE^{\pm}}},\nabla^{L_{kE^{\pm}}})$ are given by

$L_{kE^{\pm}}=kL_{E^{\pm}},\qquad
g^{L_{kE^{\pm}}}=kg^{L_{E^{\pm}}},\qquad\nabla^{L_{kE^{\pm}}}=k\nabla^{L_{E^{\pm}}}.$
(3.4.31)

By applying Remark 3.2 to $kL_{E^{+}}\cong kL_{E^{-}}$, we take $W_{+}\to B$
and $W_{-}\to B$ to be the zero bundles. By (3.4.31), (3.4.30) becomes

$\begin{split}&{\widehat{\eta}}^{\Lambda,kE^{-}}(kg^{E,-},k\nabla^{E,-},T^{H}X,g^{T^{V}X},kL_{E^{-}})-{\widehat{\eta}}^{\Lambda,kE^{+}}(kg^{E,+},k\nabla^{E,+},T^{H}X,g^{T^{V}X},kL_{E^{+}})\\\
&\equiv\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{CS}(k\nabla^{E,+},\alpha^{*}(k\nabla^{E,-}))-\operatorname{CS}(k\nabla^{L_{E^{+}}},h^{*}(k\nabla^{L_{E^{-}}})).\end{split}$
(3.4.32)

By Proposition 3.3 and Theorem 3.2,

$\begin{split}&{\widehat{\eta}}^{\Lambda,kE^{-}}(kg^{E,-},k\nabla^{E,-},T^{H}X,g^{T^{V}X},kL_{E^{-}})-{\widehat{\eta}}^{\Lambda,kE^{+}}(kg^{E,+},k\nabla^{E,+},T^{H}X,g^{T^{V}X},kL_{E^{+}})\\\
&\equiv
k\big{(}{\widehat{\eta}}^{\Lambda,E^{-}}(g^{E,-},\nabla^{E,-},T^{H}X,g^{T^{V}X},L_{E^{-}})-{\widehat{\eta}}^{\Lambda,E^{+}}(g^{E,+},\nabla^{E,+},T^{H}X,g^{T^{V}X},L_{E^{+}})\\\
&\equiv-k{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}}).\end{split}$ (3.4.33)

On the other hand, denote by
$h_{+}:(kL_{E^{+}}^{+},kg^{L_{E^{+}},+})\to(kL_{E^{-}}^{+},kg^{L_{E^{-}},+})$
and
$h_{-}:(kL_{E^{+}}^{-},kg^{L_{E^{+}},-})\to(kL_{E^{-}}^{-},kg^{L_{E^{-}},-})$
the even and the odd part of $h$, respectively. Then

$h_{+}\oplus h_{-}^{-1}:(kL_{E^{+}}^{+}\oplus
kL_{E^{-}}^{-},kg^{L_{E^{+}},+}\oplus
kg^{L_{E^{-}},-})\to(kL_{E^{-}}^{+}\oplus
kL_{E^{+}}^{-},kg^{L_{E^{-}},+}\oplus kg^{L_{E^{+}},-})$

is an isometric isomorphism. By (2.2.7), (2.2.4) and (2.2.6),

$\displaystyle\operatorname{CS}\big{(}k\nabla^{L_{E^{+}}},h^{*}(k\nabla^{L_{E^{-}}})\big{)}$
$\displaystyle=\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+}\oplus
k\nabla^{L_{E^{+}},-},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\oplus
h_{-}^{*}(k\nabla^{L_{E^{-}},-})\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\big{)}-\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},-},h_{-}^{*}(k\nabla^{L_{E^{-}},-})\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\big{)}-\operatorname{CS}\big{(}(h_{-}^{-1})^{*}(k\nabla^{L_{E^{+}},-}),k\nabla^{L_{E^{-}},-}\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\big{)}+\operatorname{CS}\big{(}k\nabla^{L_{E^{-}},-},(h_{-}^{-1})^{*}(k\nabla^{L_{E^{+}},-})\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+}\oplus
k\nabla^{L_{E^{-}},-},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\oplus(h_{-}^{-1})^{*}(k\nabla^{L_{E^{+}},-})\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k(\nabla^{L_{E^{+}},+}\oplus\nabla^{L_{E^{-}},-}),(h_{+}\oplus
h_{-}^{-1})^{*}(k(\nabla^{L_{E^{-}},+}\oplus\nabla^{L_{E^{+}},-}))\big{)}.$
(3.4.34)

By (3.4.33) and (3.4.34), (3.4.32) becomes

$\begin{split}&-k{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\equiv\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{CS}(k\nabla^{E,+},\alpha^{*}(k\nabla^{E,-}))\\\
&\qquad-\operatorname{CS}\big{(}k(\nabla^{L_{E^{+}},+}\oplus\nabla^{L_{E^{-}},-}),(h_{+}\oplus
h_{-}^{-1})^{*}(k(\nabla^{L_{E^{-}},+}\oplus\nabla^{L_{E^{+}},-}))\big{)}.\end{split}$

By (2.2.4) it becomes

$\begin{split}&k{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\equiv\int_{X/B}e(\nabla^{T^{V}X})\wedge\operatorname{CS}(\alpha^{*}(k\nabla^{E,-}),k\nabla^{E,+})\\\
&\qquad-\operatorname{CS}\big{(}(h_{+}\oplus
h_{-}^{-1})^{*}(k(\nabla^{L_{E^{-}},+}\oplus\nabla^{L_{E^{+}},-})),k(\nabla^{L_{E^{+}},+}\oplus\nabla^{L_{E^{-}},-})\big{)}.\end{split}$
(3.4.35)

By first dividing both sides of (3.4.35) by $k$, and adding
$\displaystyle\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega$ to both sides of
(3.4.35), (3.4.35) becomes

$\begin{split}&\frac{1}{k}\operatorname{CS}\big{(}(h_{+}\oplus
h_{-}^{-1})^{*}(k(\nabla^{L_{E^{-}},+}\oplus\nabla^{L_{E^{+}},-})),k(\nabla^{L_{E^{+}},+}\oplus\nabla^{L_{E^{-}},-})\big{)}\\\
&\qquad+\int_{X/B}e(\nabla^{T^{V}X})\wedge\omega+{\widehat{\eta}}^{\Lambda,E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\\\
&\equiv\int_{X/B}e(\nabla^{T^{V}X})\wedge\bigg{(}\frac{1}{k}\operatorname{CS}(\alpha^{*}(k\nabla^{E,-}),k\nabla^{E,+})+\omega\bigg{)}.\end{split}$
(3.4.36)

By (3.4.26), the left-hand side of (3.4.36) is a differential form
representative of
$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\operatorname{ind}^{a,\Lambda}_{\mathbb{R}/\mathbb{Z}}(\mathcal{E}))$
and the right-hand side of (3.4.36) is that of
$\displaystyle\int_{X/B}e(\nabla^{T^{V}X})\cup\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E})$.
Thus (3.4.28) holds. ∎

Note that (3.4.36) is a refinement of (3.4.28) at the differential form level.

##  4\. Complex flat vector bundles and differential $K$-theory

In §4.1 we recall the local index theory for twisted de Rham operators [6,
§III]. In §4.2 we prove the analytical counterpart of (1.1.5) (Theorem 4.1).
In §4.3 we prove that real part of the RRG theorem for complex flat vector
bundles (1.1.2) at the differential form level (Theorem 4.2).

###  4.1. Local index theory for twisted de Rham operator

Let $\pi:X\to B$ be a submersion with closed fibers $Z$ of dimension $n$,
equipped with a Riemannian structure $(T^{H}X,g^{T^{V}X})$. Let $F\to X$ be a
complex flat vector bundle with flat connection $\nabla^{F}$. Put a Hermitian
metric $g^{F}$ on $F\to X$. Define

$\omega(F,g^{F}):=(g^{F})^{-1}(\nabla^{F}g^{F})\in\Omega^{1}(X,\operatorname{End}(F)).$

The connection on $F\to X$ defined by

$\nabla^{F,u}=\nabla^{F}+\frac{1}{2}\omega(F,g^{F})$ (4.1.1)

is unitary with respect to $g^{F}$ and has curvature

$(\nabla^{F,u})^{2}=-\frac{1}{4}\omega(F,g^{F})^{2}.$

Since $\operatorname{tr}(\omega(F,g^{F})^{2k})=0$ for any $k\in\mathbb{N}$, it
follows that

$\operatorname{ch}(\nabla^{F,u})=\operatorname{rank}(F).$ (4.1.2)

Note that while only the $\mathbb{Z}_{2}$-grading of the complexified exterior
algebra bundle $\Lambda(T^{V}X)^{*}\to X$ is considered in §3.1, we consider
the $\mathbb{Z}$-grading of $\Lambda(T^{V}X)^{*}\to X$ in this section. As in
§3.1, one has the twisted de Rham operator $\mathsf{D}^{\Lambda\otimes F}$ and
the infinite rank complex vector bundle $\pi^{\Lambda}_{*}F\to B$ with its
$L^{2}$-metric $g^{\pi^{\Lambda}_{*}F}$, which are now $\mathbb{Z}$-graded.

Denote by $d^{Z}$ the fiberwise de Rham operator coupled with $\nabla^{F}$
acting on $\pi^{\Lambda}_{*}F\to B$. The connection on $\pi^{\Lambda}_{*}F\to
B$ defined by

${\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F}_{U}s:=\mathcal{L}_{U^{H}}s,$

where $\mathcal{L}_{U^{H}}:\Gamma(X,\Lambda(T^{V}X)^{*}\otimes
F)\to\Gamma(X,\Lambda(T^{V}X)^{*}\otimes F)$ is the Lie derivative, is
$\mathbb{Z}$-graded.

The exterior differential $d^{X}:\Omega(X,F)\to\Omega(X,F)$ coupled with
$\nabla^{F}$ can be regarded as a flat superconnection on
$\pi^{\Lambda}_{*}F\to B$ of total degree 1 and can be decomposed as

$d^{X}=d^{Z}+{\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F}+i_{T},$

where $T$ is given by (3.1.2). Consider $d^{Z}$ as an element in
$\Gamma(B,\operatorname{Hom}((\pi^{\Lambda}_{*}F)^{\bullet},(\pi^{\Lambda}_{*}F)^{\bullet+1}))$.
For each $b\in B$,

$\setcounter{MaxMatrixCols}{11}\begin{CD}0@>{}>{}>(\pi^{\Lambda}_{*}F)_{b}^{0}@>{d^{Z_{b}}}>{}>(\pi^{\Lambda}_{*}F)_{b}^{1}@>{d^{Z_{b}}}>{}>\cdots
@>{d^{Z_{b}}}>{}>(\pi^{\Lambda}_{*}F)_{b}^{n}@>{}>{}>0\end{CD}$

is a cochain complex. Denote by $H^{k}(Z_{b},F|_{Z_{b}})$ the associated
$k$-th cohomology group and define $\displaystyle
H(Z_{b},F|_{Z_{b}}):=\bigoplus_{k=0}^{n}H^{k}(Z_{b},F|_{Z_{b}})$. Define a
$\mathbb{Z}$-graded complex vector bundle $H(Z,F|_{Z})\to B$ whose fiber over
$b\in B$ is given by $H(Z,F|_{Z})_{b}:=H(Z_{b},F|_{Z_{b}})$. Denote by
$\psi:\ker(d^{Z})\to H(Z,F|_{Z})$ the quotient map. For
$s\in\Gamma(B,H^{k}(Z,F|_{Z}))$, let
$e\in\Gamma(B,(\pi^{\Lambda}_{*}F)^{k}\cap\ker(d^{Z}))$ satisfy $\psi(e)=s$.
The connection on $H(Z,F|_{Z})\to B$ defined by

$\nabla^{H(Z,F|_{Z})}_{U}s=\psi({\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F}_{U}e)$

is a well defined $\mathbb{Z}$-graded flat connection.

Denote by $d^{Z\ast}$ the formal adjoint of $d^{Z}$ with respect to
$g^{\pi_{*}^{\Lambda}F}$. Define the twisted de Rham–Hodge operator by

$\mathsf{D}^{Z,\operatorname{dR}}=d^{Z}+d^{Z\ast}.$

It acts on $\Gamma(X,\Lambda(T^{V}X)^{*}\otimes F)$ and is odd self-adjoint.
Denote by $({\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F})^{*}$ the formal adjoint
of ${\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F}$ with respect to
$g^{\pi_{*}^{\Lambda}F}$. The connection on $\pi^{\Lambda}_{*}F\to B$ defined
by

${\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F,u}=\frac{1}{2}\big{(}{\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F}+({\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F})^{*}\big{)}$

is unitary with respect to $g^{\pi_{*}^{\Lambda}F}$. By [6, Propositions 3.5
and 3.7] we have

${\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F,u}=\nabla^{\pi^{\Lambda}_{*}F,u}.$
(4.1.3)

By Hodge theory we have

$H(Z_{b},F|_{Z_{b}})\cong\ker(\mathsf{D}^{Z_{b},\operatorname{dR}}).$

Define a $\mathbb{Z}$-graded complex vector bundle
$\ker(\mathsf{D}^{Z,\operatorname{dR}})\to B$ whose fiber over $b\in B$ is
given by
$\ker(\mathsf{D}^{Z,\operatorname{dR}})_{b}:=\ker(\mathsf{D}^{Z_{b},\operatorname{dR}})$.
Then $\ker(\mathsf{D}^{Z,\operatorname{dR}})\to B$ is a finite rank subbundle
of $\pi^{\Lambda}_{*}F\to B$ and

$H(Z,F|_{Z})\cong\ker(\mathsf{D}^{Z,\operatorname{dR}})$ (4.1.4)

as $\mathbb{Z}$-graded complex vector bundles. Let
$h_{0}:H(Z,F|_{Z})\to\ker(\mathsf{D}^{Z,\operatorname{dR}})$ be a
$\mathbb{Z}$-graded smooth bundle isomorphism. Note that
$\ker(\mathsf{D}^{Z,\operatorname{dR}})\to B$ inherits a $\mathbb{Z}$-graded
Hermitian metric, which will be denoted by
$g^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}$. Let
$P^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}:\pi^{\Lambda}_{*}F\to\ker(\mathsf{D}^{Z,\operatorname{dR}})$
be the orthogonal projection. Define a $\mathbb{Z}$-graded unitary connection
on $\ker(\mathsf{D}^{Z,\operatorname{dR}})\to B$ by

$\nabla^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}=P^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}\nabla^{\pi^{\Lambda}_{*}F,u}.$

Define $g^{H(Z,F|_{Z})}=h_{0}^{*}g^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}$.
The connection on $H(Z,F|_{Z})\to B$ defined by

$\nabla^{H(Z,F|_{Z}),u}=\frac{1}{2}\big{(}\nabla^{H(Z,F|_{Z})}+(\nabla^{H(Z,F|_{Z})})^{*}\big{)}$

is unitary with respect to $g^{H(Z,F|_{Z})}$. Since
$\nabla^{H(Z,F|_{Z}),u}=P^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}{\widetilde{\nabla}}^{\pi^{\Lambda}_{*}F,u}$
by [6, Proposition 3.14], it follows from (4.1.3) that

$\nabla^{H(Z,F|_{Z}),u}=h_{0}^{*}\nabla^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}.$

In other words,

$h_{0}:(H(Z,F|_{Z}),g^{H(Z,F|_{Z})})\to(\ker(\mathsf{D}^{Z,\operatorname{dR}}),g^{\ker(\mathsf{D}^{Z,\operatorname{dR}})})$
(4.1.5)

is a $\mathbb{Z}$-graded isometric isomorphism preserving the
$\mathbb{Z}$-graded unitary connections.

Set ${\widehat{c}}(Y)=\varepsilon(Y)+i(Y)$. By [6, Propositions 3.5 and 3.7]
we have

$\mathsf{D}^{Z,\operatorname{dR}}=\mathsf{D}^{\Lambda\otimes F}+V,$ (4.1.6)

where

$V=-\frac{1}{2}\sum_{k=1}^{n}{\widehat{c}}(e_{k})\omega(F,g^{F})(e_{k}).$
(4.1.7)

Note that $V$ is odd self-adjoint and anti-commutes with the $c(X)$’s.

######  Remark 4.1.

Note that Proposition 3.1 extends to the odd self-adjoint operators
$\mathsf{D}^{\Lambda\otimes\mathcal{E}}+{\widetilde{V}}$ and
$\mathsf{D}^{\Lambda\otimes\mathcal{E}}+t{\widetilde{V}}$, where $t\in I$, on
${\widetilde{\pi}}^{\Lambda}_{*}\mathcal{E}\to{\widetilde{B}}$, with virtually
the same proof.

Define a Bismut superconnection on $\pi^{\Lambda}_{*}F\to B$ by

$\mathbb{B}^{\operatorname{dR}}:=\mathsf{D}^{Z,\operatorname{dR}}+\nabla^{\pi^{\Lambda}_{*}F,u}-\frac{c(T)}{4}.$

The rescaled Bismut superconnection is given by

$\mathbb{B}^{\operatorname{dR}}_{t}:=\sqrt{t}\mathsf{D}^{Z,\operatorname{dR}}+\nabla^{\pi^{\Lambda}_{*}F,u}-\frac{c(T)}{4\sqrt{t}}.$

The Bismut–Cheeger eta form associated to $\mathbb{B}^{\operatorname{dR}}$ is
defined by

${\widetilde{\eta}}^{\operatorname{dR}}:=\frac{1}{\sqrt{\pi}}\int^{\infty}_{0}\operatorname{str}\bigg{(}\frac{d\mathbb{B}^{\operatorname{dR}}_{t}}{dt}e^{-\frac{1}{2\pi
i}(\mathbb{B}^{\operatorname{dR}}_{t})^{2}}\bigg{)}dt.$ (4.1.8)

By [5, Theorem 3.7] (cf. [18, Proposition 2.3]) we have

${\widetilde{\eta}}^{\operatorname{dR}}=0.$ (4.1.9)

###  4.2. Complex flat vector bundles and the analytic index
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}$

Given the setup in §4.1, write

$\mathcal{F}=(F,g^{F},\nabla^{F,u},0)$ (4.2.1)

for the generator of ${\widehat{K}}(X)$ associated to $(F,\nabla^{F})$, and
write

$\mathcal{H}(Z,F|_{Z})=(H(Z,F|_{Z}),g^{H(Z,F|_{Z})},\nabla^{H(Z,F|_{Z}),u},0)$
(4.2.2)

for $\mathbb{Z}_{2}$-graded generator of ${\widehat{K}}(B)$ associated to
$(H(Z,F|_{Z}),\nabla^{H(Z,F|_{Z})})$.

We now prove Theorem 1.1.

######  Theorem 4.1.

Let $\pi:X\to B$ be a submersion with closed fibers $Z$, equipped with a
Riemannian structure $(T^{H}X,g^{T^{V}X})$. For any generator $\mathcal{F}$ of
${\widehat{K}}(X)$ the form (4.2.1), we have

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{F})=\mathcal{H}(Z,F|_{Z}).$
(4.2.3)

###### Proof.

Define a $\mathbb{Z}_{2}$-graded triple
$(\mathscr{F},g^{\mathscr{F}},\nabla^{\mathscr{F},u})$ over ${\widetilde{X}}$
by

$\mathscr{F}:=p_{X}^{*}F,\qquad
g^{\mathscr{F}}:=p_{X}^{*}g^{F},\qquad\nabla^{\mathscr{F},u}:=p_{X}^{*}\nabla^{F,u}.$

Denote by $(T^{H}{\widetilde{X}},g^{T^{V}{\widetilde{X}}})$ the Riemannian
structure on ${\widetilde{\pi}}:{\widetilde{X}}\to{\widetilde{B}}$ obtained by
pulling back $(T^{H}X,g^{T^{V}X})$. Write
$\nabla^{\Lambda(T^{V}{\widetilde{X}})^{*}\otimes\mathscr{F},u}$ for the
tensor product of $\nabla^{\Lambda(T^{V}{\widetilde{X}})^{*}}$ and
$\nabla^{\mathscr{F},u}$. Denote by $\mathsf{D}^{\Lambda\otimes\mathscr{F}}$
the twisted de Rham operator defined in terms of
$(g^{\mathscr{F}},\nabla^{\mathscr{F},u},T^{H}{\widetilde{X}},g^{T^{V}{\widetilde{X}}})$.
Define

$\mathsf{D}=\mathsf{D}^{\Lambda\otimes\mathscr{F}}+t{\widetilde{V}},$ (4.2.4)

where $t\in I$ and ${\widetilde{V}}$ is given by (4.1.7). Note that
$\mathsf{D}$ is an odd self-adjoint operator acting on
$\Gamma({\widetilde{X}},\Lambda(T^{V}{\widetilde{X}})^{*}\otimes\mathscr{F})$.

Let $(\mathcal{L},g^{\mathcal{L}},\nabla^{\mathcal{L}})$ be a
$\mathbb{Z}_{2}$-graded triple so that $\mathcal{L}\to{\widetilde{B}}$
satisfies the MF property for $\mathsf{D}$, i.e. there exist complementary
closed subbundles $\mathcal{K}^{\pm}\to{\widetilde{B}}$ of
$(\pi^{\Lambda}_{*}\mathscr{F})^{\pm}\to{\widetilde{B}}$ such that

$(\pi^{\Lambda}_{*}\mathscr{F})^{+}=\mathcal{K}^{+}\oplus\mathcal{L}^{+},\qquad(\pi^{\Lambda}_{*}\mathscr{F})^{-}=\mathcal{K}^{-}\oplus\mathcal{L}^{-},$
(4.2.5)

$\mathsf{D}_{+}$ is block diagonal as a map with respect to (4.2.5), and
$\mathsf{D}_{+}|_{\mathcal{K}^{+}}:\mathcal{K}^{+}\to\mathcal{K}^{-}$ is an
isomorphism. Let $j\in\\{0,1\\}$. Note that

$i_{B,j}^{*}(\pi^{\Lambda}_{*}\mathscr{F})\cong\pi^{\Lambda}_{*}F$

as $\mathbb{Z}_{2}$-graded complex vector bundles over $B$. Write
$(L_{j},g^{L_{j}},\nabla^{L_{j}})$ for the $\mathbb{Z}_{2}$-graded triple
$(i_{B,j}^{*}\mathcal{L},i_{B,j}^{*}g^{\mathcal{L}},i_{B,j}^{*}\nabla^{\mathcal{L}})$.
Note that

$L_{0}\cong L_{1}$ (4.2.6)

as $\mathbb{Z}_{2}$-graded complex vector bundles over $B$. Since
$\mathcal{L}\to{\widetilde{B}}$ satisfies the MF property for $\mathsf{D}$, it
follows that

$\displaystyle L_{0}\to B$ $\displaystyle\textrm{ satisfies the MF property
for }\mathsf{D}^{\Lambda\otimes F},\textrm{ and }$ (4.2.7) $\displaystyle
L_{1}\to B$ $\displaystyle\textrm{ satisfies the MF property for
}\mathsf{D}^{\Lambda\otimes F}+V.$ (4.2.8)

By (4.2.7) and Proposition 3.5,
$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{F})$ is given by

$\operatorname{ind}^{a,\Lambda}_{{\widehat{K}}}(\mathcal{F})=\bigg{(}L_{0},g^{L_{0}},\nabla^{L_{0}},{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},L_{0})\bigg{)}.$
(4.2.9)

By (4.2.9) and (4.2.2), to prove (4.2.3) it suffices to show that there exist
balanced $\mathbb{Z}_{2}$-graded triples $(W_{0},g^{W_{0}},\nabla^{W_{0}})$
and $(W_{1},g^{W_{1}},\nabla^{W_{1}})$ and a $\mathbb{Z}_{2}$-graded isometric
isomorphism

$h:(L_{0}\oplus W_{0},g^{L_{0}}\oplus g^{W_{0}})\to(H(Z,F|_{Z})\oplus
W_{1},g^{H(Z,F|_{Z})}\oplus g^{W_{1}})$ (4.2.10)

such that

${\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},L_{0})\equiv\operatorname{CS}\big{(}\nabla^{L_{0}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{H(Z,F|_{Z}),u}\oplus\nabla^{W_{1}})\big{)}.$
(4.2.11)

Recall from Remark 4.1 that Proposition 3.1 applies to (4.2.4) as well. By
noting (4.2.6) and Remark 3.2, there exists a $\mathbb{Z}_{2}$-graded
isometric isomorphism $h_{1}:(L_{0},g^{L_{0}})\to(L_{1},g^{L_{1}})$ such that

$\begin{split}&{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},L_{1})-{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},L_{0})\\\
&\equiv-\operatorname{CS}(\nabla^{L_{0}},h_{1}^{*}\nabla^{L_{1}}).\end{split}$
(4.2.12)

Since $\mathsf{D}^{\Lambda\otimes F}+V=\mathsf{D}^{Z,\operatorname{dR}}$ by
(4.1.6), it follows from (4.1.4) that the kernel bundle
$\ker(\mathsf{D}^{\Lambda\otimes F}+V)\to B$ exists and is given by

$\ker(\mathsf{D}^{\Lambda\otimes
F}+V)=\ker(\mathsf{D}^{Z,\operatorname{dR}}).$

Thus the $\mathbb{Z}_{2}$-graded triple

$(\ker(\mathsf{D}^{Z,\operatorname{dR}}),g^{\ker(\mathsf{D}^{Z,\operatorname{dR}})},\nabla^{\ker(\mathsf{D}^{Z,\operatorname{dR}})})$

satisfies the MF property for $\mathsf{D}^{\Lambda\otimes F}+V$. By noting
(4.2.8) and Remark 4.1 that Proposition 3.1 applies to
$\mathsf{D}^{\Lambda\otimes F}+V$, there exist balanced
$\mathbb{Z}_{2}$-graded triples $(W_{0},g^{W_{0}},\nabla^{W_{0}})$ and
$(W_{1},g^{W_{1}},\nabla^{W_{1}})$ and a $\mathbb{Z}_{2}$-graded isometric
isomorphism

$h_{2}:(L_{1}\oplus W_{0},g^{L_{1}}\oplus
g^{W_{0}})\to(\ker(\mathsf{D}^{Z,\operatorname{dR}})\oplus
W_{1},g^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}\oplus g^{W_{1}})$

such that

$\begin{split}&{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},\ker(\mathsf{D}^{Z,\operatorname{dR}}))-{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},L_{1})\\\
&\equiv-\operatorname{CS}\big{(}\nabla^{L_{1}}\oplus\nabla^{W_{0}},h_{2}^{*}(\nabla^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}\oplus\nabla^{W_{1}})\big{)}.\end{split}$
(4.2.13)

Since the map $h:(L_{0}\oplus W_{0},g^{L_{0}}\oplus
g^{W_{0}})\to(\ker(\mathsf{D}^{Z,\operatorname{dR}})\oplus
W_{1},g^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}\oplus g^{W_{1}})$ defined to
be the composition

${(L_{0}\oplus W_{0},g^{L_{0}}\oplus g^{W_{0}})}$${(L_{1}\oplus
W_{0},g^{L_{1}}\oplus
g^{W_{0}})}$${(\ker(\mathsf{D}^{Z,\operatorname{dR}})\oplus
W_{1},g^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}\oplus
g^{W_{1}})}$${(H(Z,F|_{Z})\oplus W_{1},g^{H(Z,F|_{Z})}\oplus
g^{W_{1}})}$$\scriptstyle{h_{1}\oplus\operatorname{id}_{W_{0}}}$$\scriptstyle{h_{2}}$$\scriptstyle{h_{0}^{-1}\oplus\operatorname{id}_{W_{1}}}$

where $h_{0}$ is given by (4.1.5), is a $\mathbb{Z}_{2}$-graded isometric
isomorphism, (4.2.10) holds. On the other hand, by adding (4.2.12) to (4.2.13)
and noting that
$\nabla^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}=(h_{0}^{-1})^{*}\nabla^{H(Z,F|_{Z}),u}$
by (4.1.5), it follows from (2.2.6) and (2.2.5) that

$\begin{split}&{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},\ker(\mathsf{D}^{Z,\operatorname{dR}}))-{\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},L_{0})\\\
&\equiv-\operatorname{CS}(\nabla^{L_{0}},h_{1}^{*}\nabla^{L_{1}})-\operatorname{CS}\big{(}\nabla^{L_{1}}\oplus\nabla^{W_{0}},h_{2}^{*}(\nabla^{\ker(\mathsf{D}^{Z,\operatorname{dR}})}\oplus\nabla^{W_{1}})\big{)}\\\
&\equiv-\operatorname{CS}\big{(}\nabla^{L_{0}}\oplus\nabla^{W_{0}},(h_{1}\oplus\operatorname{id}_{W_{0}})^{*}\nabla^{L_{1}}\oplus\nabla^{W_{0}}\big{)}\\\
&\qquad-\operatorname{CS}\big{(}(h_{1}\oplus\operatorname{id}_{W_{0}})^{*}(\nabla^{L_{1}}\oplus\nabla^{W_{0}}),(h_{1}\oplus\operatorname{id}_{W_{0}})^{*}h_{2}^{*}(h_{0}^{-1}\oplus\operatorname{id}_{W_{1}})^{*}(\nabla^{H(Z,F|_{Z}),u}\oplus\nabla^{W_{1}})\big{)}\\\
&\equiv-\operatorname{CS}\big{(}\nabla^{L_{0}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{H(Z,F|_{Z}),u}\oplus\nabla^{W_{1}})\big{)}.\end{split}$

Thus, to prove (4.2.11) it suffices to show that

${\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},\ker(\mathsf{D}^{Z,\operatorname{dR}}))\equiv
0,$ (4.2.14)

where the Bismut–Cheeger eta form in (4.2.14) is associated to the rescaled
Bismut superconnection

${\widehat{\mathbb{B}}}^{\Lambda,F}_{t}=\sqrt{t}\begin{pmatrix}\mathsf{D}^{Z,\operatorname{dR}}_{+}&\alpha(t)i^{-}\\\
\alpha(t)P^{+}&0\end{pmatrix}+\big{(}\nabla^{\pi^{\Lambda}_{*}F,u}\oplus(\nabla^{H(Z,F|_{Z}),u})^{\operatorname{op}}\big{)}-\frac{1}{\sqrt{t}}\bigg{(}\frac{c(T)}{4}\oplus
0\bigg{)}.$

By an argument similar to the proof of (3.4.4), we have

${\widehat{\eta}}^{\Lambda,F}(g^{F},\nabla^{F,u},T^{H}X,g^{T^{V}X},\ker(\mathsf{D}^{Z,\operatorname{dR}}))\equiv{\widetilde{\eta}}^{\operatorname{dR}}.$
(4.2.15)

It follows from (4.1.9) that (4.2.14) holds. ∎

Let $F\to X$ be a complex flat vector bundle of rank $\ell$ with flat
connection $\nabla^{F}$. Since $\operatorname{ch}(\nabla^{F})=\ell$, there
exists a $k\in\mathbb{N}$ such that $kF\cong k\mathbb{C}^{\ell}$. Put a
Hermitian metric $g^{F}$ on $F\to X$. Let
$\alpha:(kF,kg^{F})\to(\mathbb{C}^{k\ell},g^{k\ell})$ be an isometric
isomorphism. The real part of the Cheeger–Chern–Simons class of
$(F,\nabla^{F})$ is defined to be

$\operatorname{Re}(\operatorname{CCS}(F,\nabla^{F}))=\bigg{[}\frac{1}{k}\operatorname{CS}(\alpha^{*}d^{k\ell},k\nabla^{F,u})\bigg{]}\mod\mathbb{Q}.$
(4.2.16)

If $F\to X$ and $\nabla^{F}$ are $\mathbb{Z}_{2}$-graded and
$\operatorname{rank}(F^{+})=\operatorname{rank}(F^{-})$, then
$\operatorname{Re}(\operatorname{CCS}(F,\nabla^{F}))$ is defined to be

$\operatorname{Re}(\operatorname{CCS}(F,\nabla^{F}))=\bigg{[}\frac{1}{k}\operatorname{CS}(\alpha^{*}(k(\nabla^{F,u})^{-}),k(\nabla^{F,u})^{+})\bigg{]}\mod\mathbb{Q},$
(4.2.17)

where $\alpha:(kF^{+},kg^{F,+})\to(kF^{-},kg^{F,-})$ is an isometric
isomorphism.