# An extended variational formula for the Bismut–Cheeger eta form and its
applications

Man-Ho Ho<EMAIL_ADDRESS>Hong Kong

###### Abstract.

The purpose of this paper is to extend our previous work on the variational
formula for the Bismut–Cheeger eta form without the kernel bundle assumption
by allowing the spinc Dirac operators to be twisted by isomorphic vector
bundles, and to establish the $\mathbb{Z}_{2}$-graded additivity of the
Bismut–Cheeger eta form. Using these results, we give alternative proofs of
the fact that the analytic index in differential $K$-theory is a well defined
group homomorphism, and the Riemann–Roch–Grothendieck theorem in
$\mathbb{R}/\mathbb{Z}$ $K$-theory.

###### Key words and phrases:

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

###### 2020 Mathematics Subject Classification:

Primary 19K56, 19L50, 19L10

###### Contents

  1. 1 Introduction
  2. 2 Preliminaries
    1. 2.1 Notations and conventions
    2. 2.2 Some primary and secondary characteristic forms
  3. 3 An extended variational formula for ${\widehat{\eta}}$
    1. 3.1 Local index theory for twisted spinc Dirac operator
    2. 3.2 A proof of Theorem 1.1
  4. 4 Applications of Theorem 1.1
    1. 4.1 $\mathbb{Z}_{2}$-graded additivity of the Bismut–Cheeger eta form
    2. 4.2 The analytic index in differential and $\mathbb{R}/\mathbb{Z}$ $K$-theory
    3. 4.3 The RRG theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory for the twisted spinc Dirac operator

##  1\. Introduction

The Bismut–Cheeger eta form serves as a transgression form between the Chern
character of the index bundle and its Atiyah–Singer representative at the
differential form level [3, Theorems 4.35 and 4.95] and [4, Theorem 0.1]. In
this paper, we extend our previous work on the variational formula for the
Bismut–Cheeger eta form [7, Proposition 1] by allowing the spinc Dirac
operators to be twisted by isomorphic vector bundles, and prove the
$\mathbb{Z}_{2}$-graded additivity of the Bismut–Cheeger eta form. We then
present some applications of these results. All the Dirac operators in this
paper are not assumed to satisfy the kernel bundle assumption.

To put the paper into context, 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 denoted by
$(T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda})$. Suppose we have a
Hermitian bundle $(E,g^{E},\nabla^{E})$ over $X$ with a unitary connection,
and denote by $\mathsf{D}^{S\otimes E}$ the corresponding twisted spinc Dirac
operator. According to a result by Miščenko–Fomenko [14] (see also [5, Lemma
7.13]), there always exists a $\mathbb{Z}_{2}$-graded complex vector bundle
$L\to B$ such that it represents the analytic index of $\mathsf{D}^{S\otimes
E}$ in $K$-theory, i.e. $\operatorname{ind}^{a}([E])=[L^{+}]-[L^{-}]\in K(B)$.

The corresponding local family index theorem for $\mathsf{D}^{S\otimes E}$ by
Freed–Lott [5, (7.26)] states that

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

where $\nabla^{L}$ is a $\mathbb{Z}_{2}$-graded unitary connection on $L\to B$
with respect to a $\mathbb{Z}_{2}$-graded Hermitian metric $g^{L}$, and
${\widehat{\eta}}^{E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda},L)$
is the corresponding Bismut–Cheeger eta form.

The notation
${\widehat{\eta}}^{E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda},L)$
is to indicate the dependence on the geometric structures. If there exist two
sets of these geometric structures, denoted by subscripts 0 and 1, the
difference of the Bismut–Cheeger eta forms

${\widehat{\eta}}^{E}(g^{E}_{1},\nabla^{E}_{1},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{1})-{\widehat{\eta}}^{E}(g^{E}_{0},\nabla^{E}_{0},T^{H}_{0}X,g^{T^{V}X}_{0},g^{\lambda}_{0},\nabla^{\lambda}_{0},L_{0})$
(1.0.1)

is not zero in general. Variational formula for the Bismut–Cheeger eta form is
an explicit expression of the difference (1.0.1) in terms of the geometric
structures involved.

In recent years, variational formulas for the Bismut–Cheeger eta form have
been established in various settings, and have found numerous applications in
local index theory. For instance, Liu proves a variational formula for the
Bismut–Cheeger eta form in the equivariant setting [11, Theorem 1.4] (see also
[10, Theorem 1.2]), and uses it to prove the functoriality of the
Bismut–Cheeger eta form [11, Theorem 1.6] (see also [10, Theorem 1.3]). On the
other hand, we prove a variational formula for the Bismut–Cheeger eta form
without the kernel bundle assumption in the case $\dim(X/B)$ is even [7,
Proposition 1], and use it to prove the $\mathbb{Z}_{2}$-graded version of the
real part of the Riemann–Roch–Grothendieck (RRG) theorem for complex flat
vector bundles in the same case at the differential form level [7, Theorem 1].

The main result of this paper is an extension of [7, Proposition 1], in the
sense that the spinc Dirac operators are allowed to be twisted by isomorphic
vector bundles, i.e. an explicit expression for the difference

${\widehat{\eta}}^{F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{F})-{\widehat{\eta}}^{E}(g^{E},\nabla^{E},T^{H}_{0}X,g^{T^{V}X}_{0},g^{\lambda}_{0},\nabla^{\lambda}_{0},L_{E})$

in terms of the geometric structures involved, where the complex vector
bundles $E\to X$ and $F\to X$ are isomorphic. To establish the main result of
this paper, we first prove the following special case.

######  Proposition 1.1.

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
$(T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda})$. Let $(E,g^{E})$ be a
Hermitian bundle and $(F,g^{F},\nabla^{F})$ a Hermitian bundle with a unitary
connection. If there exists an isometric isomorphism
$\alpha:(E,g^{E})\to(F,g^{F})$ and $L_{E,\alpha}\to B$ is a
$\mathbb{Z}_{2}$-graded complex vector bundle representing the analytic index
of $\mathsf{D}^{S\otimes E,\alpha}$ (defined in terms of
$\alpha^{*}\nabla^{F}$) in $K$-theory, then there exist a unique
$\mathbb{Z}_{2}$-graded complex vector bundle $L_{F,\alpha}\to B$ 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 $L_{F,\alpha}\to B$ represents the analytic index of
$\mathsf{D}^{S\otimes F}$ (defined in terms of $\nabla^{F}$) in $K$-theory,
and

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

The main result of this paper is the following theorem, whose proof relies on
[7, Proposition 1] and Proposition 1.1.

######  Theorem 1.1.

Let $\pi:X\to B$ be a submersion with closed, oriented and spinc fibers of
even dimension, and $(E,g^{E},\nabla^{E})$ and $(F,g^{F},\nabla^{F})$ are
Hermitian bundles with unitary connections. Denote by $\mathsf{D}^{S\otimes
E}$ and $\mathsf{D}^{S\otimes F}$ the twisted spinc Dirac operators defined in
terms of the following Riemannian and differential spinc structures

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

on $\pi:X\to B$, respectively, where the underlying topological spinc
structures coincide. Let $L_{E}\to B$ and $L_{F}\to B$ be
$\mathbb{Z}_{2}$-graded complex vector bundles that represent the analytic
indices of $\mathsf{D}^{S\otimes E}$ and $\mathsf{D}^{S\otimes F}$ in
$K$-theory, respectively.

If there exists an isometric isomorphism $\alpha:(E,g^{E})\to(F,g^{F})$, then
there exist $\mathbb{Z}_{2}$-graded Hermitian bundles with
$\mathbb{Z}_{2}$-graded unitary connections $(W_{0},g^{W_{0}},\nabla^{W_{0}})$
and $(W_{1},g^{W_{1}},\nabla^{W_{1}})$, and a $\mathbb{Z}_{2}$-graded
isometric isomorphism $h:(L_{E}\oplus W_{0},g^{L_{E}}\oplus
g^{W_{0}})\to(L_{F}\oplus W_{1},g^{L_{F}}\oplus g^{W_{1}})$ such that

$\begin{split}&{\widehat{\eta}}^{F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{F})-{\widehat{\eta}}^{E}(g^{E},\nabla^{E},T^{H}_{0}X,g^{T^{V}X}_{0},g^{\lambda}_{0},\nabla^{\lambda}_{0},L_{E})\\\
&=\int_{X/B}\big{(}T{\widehat{A}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}c_{1}(\nabla^{\lambda}_{0})}+{\widehat{A}}(\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}Tc_{1}(\nabla^{\lambda}_{0},\nabla^{\lambda}_{1})}\big{)}\wedge\operatorname{ch}(\nabla^{E})\\\
&\quad+\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E},\alpha^{*}\nabla^{F})-\operatorname{CS}(\nabla^{L_{E}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}))\end{split}$

in $\displaystyle\frac{\Omega^{\operatorname{odd}}(B)}{\operatorname{Im}(d)}$.

Note that we do not impose any conditions on the relationship between the
unitary connections $\nabla^{E}$ and $\alpha^{*}\nabla^{F}$ on $E\to X$ in
Theorem 1.1.

We present some applications of Proposition 1.1 and Theorem 1.1 in this paper.
We use Proposition 1.1 to establish the $\mathbb{Z}_{2}$-graded additivity of
the Bismut–Cheeger eta form.

######  Theorem 1.2.

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
$(T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda})$. Let $(E,g^{E},\nabla^{E})$
be a $\mathbb{Z}_{2}$-graded Hermitian bundle with a $\mathbb{Z}_{2}$-graded
unitary connection. Denote by $\mathsf{D}^{S\otimes E}$, $\mathsf{D}^{S\otimes
E^{+}}$ and $\mathsf{D}^{S\otimes E^{-}}$ the spinc Dirac operators twisted by
$E\to X$, $E^{+}\to X$ and $E^{-}\to X$, respectively. If $L_{E^{+}}\to B$ and
$L_{E^{-}}\to B$ are $\mathbb{Z}_{2}$-graded complex vector bundles
representing the analytic indices of $\mathsf{D}^{S\otimes E^{+}}$ and
$\mathsf{D}^{S\otimes E^{-}}$ in $K$-theory, respectively, then
$L_{E^{+}}\oplus L_{E^{-}}^{\operatorname{op}}\to B$ represents the analytic
index of $\mathsf{D}^{S\otimes E}$ in $K$-theory, and

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

in $\displaystyle\frac{\Omega^{\operatorname{odd}}(B)}{\operatorname{Im}(d)}$,
where $L_{E^{-}}^{\operatorname{op}}\to B$ denotes the $\mathbb{Z}_{2}$-graded
complex vector bundle with the opposite grading of $L_{E^{-}}\to B$.

While the additivity of the Bismut–Cheeger eta form is well known for ungraded
direct sums (at least in the case when the Dirac operators satisfy the kernel
bundle assumption), its $\mathbb{Z}_{2}$-graded additivity is less well known.
We could not find a statement or proof of this result in the literature. We
also give a proof of the additivity of the Bismut–Cheeger eta form for
ungraded direct sums without the kernel bundle assumption (Proposition 4.1).

The first application of Theorem 1.1 concerns the analytic index in
differential $K$-theory. Given a submersion $\pi:X\to B$ with closed, oriented
and spinc fibers of even dimension that is equipped with a Riemannian and
differential spinc structure
$(T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda})$, the analytic index in
differential $K$-theory [5, Definition 7.27] is defined to be

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

where $\mathcal{E}$ is a ($\mathbb{Z}_{2}$-graded) generator of the Freed–Lott
differential $K$-group ${\widehat{K}}_{\operatorname{FL}}(X)$ and $L\to B$ is
a $\mathbb{Z}_{2}$-graded complex vector bundle representing the analytic
index of $\mathsf{D}^{S\otimes E}$ in $K$-theory. We use Theorem 1.1 and
Proposition 4.1 to prove the following result.

######  Proposition 1.2.

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.
The analytic index in differential $K$-theory

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

is a well defined group homomorphism.

Proposition 1.2 is first derived by Freed–Lott [5, (2) of Corollary 7.36] as a
consequence of the fact that the topological index in differential $K$-theory
$\operatorname{ind}^{t}_{{\widehat{K}}}:{\widehat{K}}_{\operatorname{FL}}(X)\to{\widehat{K}}_{\operatorname{FL}}(B)$
is a well defined group homomorphism [5, (2) of Proposition 4.18 and Lemma
5.30] and the family index theorem in differential $K$-theory [5, Theorem
7.35], i.e. for every generator $\mathcal{E}$ of
${\widehat{K}}_{\operatorname{FL}}(X)$,

$\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E})=\operatorname{ind}^{t}_{{\widehat{K}}}(\mathcal{E}).$

Our proof of Proposition 1.2 does not rely on these results. It is worth
noting that, as the construction of the topological index in differential
$K$-theory is complicated and the proofs that it is a well-defined group
homomorphism and the family index theorem in differential $K$-theory are
highly nontrivial, Proposition 1.2 is more nontrivial than it seems to be.

Since the analytic index in differential $K$-theory
$\operatorname{ind}^{a}_{{\widehat{K}}}:{\widehat{K}}_{\operatorname{FL}}(X)\to{\widehat{K}}_{\operatorname{FL}}(B)$
restricts to the analytic index in $\mathbb{R}/\mathbb{Z}$ $K$-theory [12,
Definition 14]

$\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}:K^{-1}_{\operatorname{L}}(X)\to
K^{-1}_{\operatorname{L}}(B),$ (1.0.2)

as an immediate consequence of Proposition 1.2, (1.0.2) is also a well defined
group homomorphism.

Finally, we use Theorems 1.1 and 1.2 to give an alternative proof of the RRG
theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory.

######  Theorem 1.3.

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.
The following diagram commutes.

${K^{-1}_{\operatorname{L}}(X)}$${H^{\operatorname{odd}}(X;\mathbb{R}/\mathbb{Q})}$${K^{-1}_{\operatorname{L}}(B)}$${H^{\operatorname{odd}}(B;\mathbb{R}/\mathbb{Q})}$$\scriptstyle{\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}}$$\scriptstyle{\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}}$$\scriptstyle{\int_{X/B}\operatorname{Todd}(T^{V}X)\cup(\cdot)}$$\scriptstyle{\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}}$
(1.0.3)

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

$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}(\mathcal{E}))=\int_{X/B}\operatorname{Todd}(T^{V}X)\cup\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E}).$
(1.0.4)

Here,
$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}:K^{-1}_{\operatorname{L}}(\cdot)\to
H^{\operatorname{odd}}(\cdot;\mathbb{R}/\mathbb{Q})$ is the Chern character in
$\mathbb{R}/\mathbb{Z}$ $K$-theory. We prove (1.0.4) at the differential form
level. Note that (1.0.4) implies diagram (1.0.3) commutes only if (1.0.2) is
well defined.

The paper is organized as follows. In §2 we set and fix the notations and
conventions used throughout the paper. In §3.1 we review the local family
index theorem for twisted spinc Dirac operators without the kernel bundle
assumption, and we prove the extended variational formula for the
Bismut–Cheeger eta form (Theorem 1.1) in §3.2. In §4.1, we establish some
intermediate results on the Bismut–Cheeger eta form, which, together with
Theorem 1.1, allow us to prove the $\mathbb{Z}_{2}$-graded additivity of the
Bismut–Cheeger eta form (Theorem 1.2). Furthermore, we prove the analytic
index in differential $K$-theory is a well defined group homomorphism
(Proposition 1.2) in §4.2 and the RRG theorem in $\mathbb{R}/\mathbb{Z}$
$K$-theory (Theorem 1.3) in §4.3.

## Acknowledgement

The author would like to thank Steve Rosenberg for his comments and
suggestions for this paper, and Jonathan Kin-Yue Lee, where the idea of
Proposition 4.2 is due to him.

##  2\. Preliminaries

###  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 $\Omega^{k}_{A}(X)$ the set of all real-valued closed $k$-forms on
$X$ with periods in $A$, where $A$ is a proper subgroup of $\mathbb{R}$. 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 [6, 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 $\nabla^{E,+}=\nabla^{E,-}$.

######  Remark 2.1.

Let $F\to X$ be another complex vector bundle, 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 [8, 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

$\alpha^{*}\nabla^{F}:=\alpha^{-1}\circ\nabla^{F}\circ\alpha.$ (2.1.2)

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 primary and 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

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

and the first Chern form of $\nabla^{E}$ is defined to be

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

Given two pairs of Hermitian metrics and unitary connections
$(g^{E}_{0},\nabla^{E}_{0})$ and $(g^{E}_{1},\nabla^{E}_{1})$ on $E\to X$, one
can define the Chern–Simons form
$\displaystyle\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})\in\frac{\Omega^{\operatorname{odd}}(X)}{\operatorname{Im}(d)}$
by

$\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1}):=-\int_{{\widetilde{X}}/X}\operatorname{ch}(\nabla^{\mathcal{E}})\mod\operatorname{Im}(d),$
(2.2.1)

where $\mathcal{E}\to{\widetilde{X}}$ is a complex vector bundle defined by
$\mathcal{E}=p_{X}^{*}E$ with a Hermitian metric $g^{\mathcal{E}}$, and
$\nabla^{\mathcal{E}}$ is the unitary connection on
$\mathcal{E}\to{\widetilde{X}}$ defined by [13, (B.5.21)], which satisfies
$i_{X,k}^{*}\nabla^{\mathcal{E}}=\nabla^{E}_{k}$ for $k\in\\{0,1\\}$. By [13,
(B.5.15)], $\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})$ satisfies the
following transgression formula

$d\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})=\operatorname{ch}(\nabla^{E}_{1})-\operatorname{ch}(\nabla^{E}_{0}).$
(2.2.2)

By [13, Theorem B.5.4], $\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})$ is
independent of the choice of $\nabla^{\mathcal{E}}$ satisfying
$i_{X,k}^{*}\nabla^{\mathcal{E}}=\nabla^{E}_{k}$ for $k\in\\{0,1\\}$.
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)

where $\nabla^{E}_{t}$ is a smooth path of unitary connections joining
$\nabla^{E}_{0}$ and $\nabla^{E}_{1}$, and $R^{E}_{t}$ is its curvature (see
[13, p.373] for the details). 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).

Similarly, one can define a differential form $\displaystyle
Tc_{1}(\nabla^{E}_{0},\nabla^{E}_{1})\in\frac{\Omega^{1}(X)}{\operatorname{Im}(d)}$
that satisfies the following transgression formula

$dTc_{1}(\nabla^{E}_{0},\nabla^{E}_{1})=c_{1}(\nabla^{E}_{1})-c_{1}(\nabla^{E}_{0}).$

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 with a Euclidean connection.
Denote by $R^{H}$ the curvature of $\nabla^{H}$. The ${\widehat{A}}$-genus
form of $\nabla^{H}$ is defined to be

${\widehat{A}}(\nabla^{H})=\sqrt{\det\bigg{(}\frac{-\frac{1}{4\pi
i}R^{H}}{\sinh(-\frac{1}{4\pi
i}R^{H})}\bigg{)}}\in\Omega^{4\bullet}_{\mathbb{Q}}(X).$

Similarly, one can define a differential form $\displaystyle
T{\widehat{A}}(\nabla^{H}_{0},\nabla^{H}_{1})\in\frac{\Omega^{4\bullet-1}(X)}{\operatorname{Im}(d)}$
that satisfies the following transgression formula

$dT{\widehat{A}}(\nabla^{H}_{0},\nabla^{H}_{1})={\widehat{A}}(\nabla^{H}_{1})-{\widehat{A}}(\nabla^{H}_{0}).$

##  3\. An extended variational formula for ${\widehat{\eta}}$

###  3.1. Local index theory for twisted spinc Dirac operator

In this subsection we review the statement of local family index theorem for
twisted spinc Dirac operators without the kernel bundle assumption. We refer
the readers to [1, Chapter 10] and [5, §7] for the details.

Let $\pi:X\to B$ be a submersion with closed, oriented and spinc fibers $Z$ of
even dimension $n$. Denote by $T^{V}X\to X$ the vertical tangent bundle.
Recall from [5, 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 $T^{V}X\to X$. 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 $\nabla^{T^{V}X}:=P^{T^{V}X}\nabla^{TX}$ 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

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

Then
$S:=\nabla^{TX}-{\widetilde{\nabla}}^{TX}\in\Omega^{1}(X,\operatorname{End}(TX))$.
By [2, Theorem 1.9] the $(3,0)$ tensor $g^{TX}(S(\cdot)\cdot,\cdot)$ depends
only on $(T^{H}X,g^{T^{V}X})$. 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 two $U,V\in\Gamma(B,TB)$,

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

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

Denote by $d\operatorname{vol}(Z)$ the Riemannian volume element of the fiber
$Z$, which is a section of $\Lambda^{n}(T^{V}X)^{*}\to X$.

Choose and fix a topological spinc structure on $T^{V}X\to X$. This fixes a
complex line bundle $\lambda\to X$ satisfying
$w_{2}(T^{V}X)=c_{1}(\lambda)\mod 2$ [9, p.397]. The spinor bundle
$S(T^{V}X)\to X$ associated to the chosen topological spinc structure of
$T^{V}X\to X$ is given by

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

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 $\lambda\to X$. Since $n$ is even, $S(T^{V}X)\to X$ is
$\mathbb{Z}_{2}$-graded. Recall from [5, p.918] that a differential spinc
structure on $\pi:X\to B$ consists of a topological spinc structure on
$T^{V}X\to X$, and a Hermitian metric $g^{\lambda}$ and a unitary connection
$\nabla^{\lambda}$ on $\lambda\to X$.

A Riemannian and differential spinc structure on $\pi:X\to B$ induce a
Hermitian metric $g^{S(T^{V}X)}$ and a unitary connection $\nabla^{S(T^{V}X)}$
on $S(T^{V}X)\to X$. Define the Todd form of $\nabla^{T^{V}X}$ by

$\operatorname{Todd}(\nabla^{T^{V}X})={\widehat{A}}(\nabla^{T^{V}X})\wedge
e^{\frac{1}{2}c_{1}(\nabla^{\lambda})}.$ (3.1.3)

Let $(E,g^{E},\nabla^{E})$ be a triple. Define the twisted spinc Dirac
operator $\mathsf{D}^{S\otimes E}:\Gamma(X,S(T^{V}X)\otimes
E)\to\Gamma(X,S(T^{V}X)\otimes E)$ by

$\mathsf{D}^{S\otimes
E}=\sum_{k=1}^{n}c(e_{k})\nabla_{e_{k}}^{S(T^{V}X)\otimes E},$ (3.1.4)

where $c$ is the Clifford multiplication, and $\nabla^{S(T^{V}X)\otimes E}$ is
the tensor product of $\nabla^{S(T^{V}X)}$ and $\nabla^{E}$. Note that
$\mathsf{D}^{S\otimes E}$ is odd self-adjoint.

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

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

The space of sections of $\pi_{*}E\to B$ is defined to be

$\Gamma(B,\pi_{*}E):=\Gamma(X,S(T^{V}X)\otimes E).$

Define an $L^{2}$-metric on $\pi_{*}E\to B$ by

$g^{\pi_{*}E}(s_{1},s_{2})(b)=\int_{Z_{b}}g^{S(T^{V}X)\otimes
E}(s_{1},s_{2})d\operatorname{vol}(Z).$ (3.1.5)

Define a connection on $\pi_{*}E\to B$ by

$\nabla^{\pi_{*}E}_{U}s:=\nabla^{S(T^{V}X)\otimes E}_{U^{H}}s,$ (3.1.6)

where $s\in\Gamma(B,\pi_{*}E)$ and $U\in\Gamma(B,TB)$. Then the connection on
$\pi_{*}E\to B$ defined by

$\nabla^{\pi_{*}E,u}:=\nabla^{\pi_{*}E}+\frac{1}{2}k,$ (3.1.7)

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

The Bismut superconnection on $\pi_{*}E\to B$ is defined to be

$\mathbb{B}^{E}=\mathsf{D}^{S\otimes E}+\nabla^{\pi_{*}E,u}-\frac{c(T)}{4},$
(3.1.8)

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

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

By [1, Theorem 10.23],

$\lim_{t\to
0}\operatorname{ch}(\mathbb{B}^{E}_{t})=\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{E}).$
(3.1.9)

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

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

Miščenko–Fomenko [14, p.96-97] (see also [5, Lemma 7.13]) prove that there
exist finite rank subbundles $L^{\pm}\to B$ and complementary closed
subbundles $K^{\pm}\to B$ of $(\pi_{*}E)^{\pm}\to B$ such that

$(\pi_{*}E)^{+}=K^{+}\oplus L^{+},\qquad(\pi_{*}E)^{-}=K^{-}\oplus L^{-},$
(3.1.10)

$\mathsf{D}^{S\otimes E}_{+}:(\pi_{*}E)^{+}\to(\pi_{*}E)^{-}$ is block
diagonal as a map with respect to (3.1.10), and $\mathsf{D}^{S\otimes
E}_{+}|_{K^{+}}:K^{+}\to K^{-}$ is an 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 $\mathsf{D}^{S\otimes
E}$. If $L\to B$ satisfies the MF property for $\mathsf{D}^{S\otimes E}$, then
the analytic index of $[E]\in K(X)$ is defined to be

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

It is proved in [14, p.96-97] that $\operatorname{ind}^{a}([E])$ does not
depend on the choice of $L\to B$ satisfying the MF property for
$\mathsf{D}^{S\otimes E}$.

Let $g^{L}$ be the $\mathbb{Z}_{2}$-graded Hermitian metric on $L\to B$
inherited from $g^{\pi_{*}E}$. Denote by $P:\pi_{*}E\to L$ be the
$\mathbb{Z}_{2}$-graded projection map with respect to (3.1.10). Define a
connection on $L\to B$ by

$\nabla^{L}:=P\circ\nabla^{\pi_{*}E,u}\circ P.$ (3.1.11)

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

Given an $L\to B$ satisfying the MF property for $\mathsf{D}^{S\otimes E}$,
consider the infinite rank $\mathbb{Z}_{2}$-graded complex vector bundle
$\pi_{*}E\oplus L^{\operatorname{op}}\to B$. Let
$i^{-}:L^{-}\to(\pi_{*}E)^{-}$ be the inclusion map and $z\in\mathbb{C}$.
Define a map ${\widetilde{\mathsf{D}}}^{S\otimes E}_{+}(z):(\pi_{*}E\oplus
L^{\operatorname{op}})^{+}\to(\pi_{*}E\oplus L^{\operatorname{op}})^{-}$ by

${\widetilde{\mathsf{D}}}^{S\otimes
E}_{+}(z)=\begin{pmatrix}\mathsf{D}^{S\otimes E}_{+}&zi^{-}\\\
zP^{+}&0\end{pmatrix}.$ (3.1.12)

Note that ${\widetilde{\mathsf{D}}}^{S\otimes E}_{+}(z)$ is invertible for all
$z\neq 0$ [5, Lemma 7.20]. Define a map ${\widetilde{\mathsf{D}}}^{S\otimes
E}(z):\pi_{*}E\oplus L^{\operatorname{op}}\to\pi_{*}E\oplus
L^{\operatorname{op}}$ by

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

Define a Bismut superconnection on $\pi_{*}E\oplus L^{\operatorname{op}}\to B$
by

${\widehat{\mathbb{B}}}^{E}={\widetilde{\mathsf{D}}}^{S\otimes
E}(1)+\nabla^{\pi_{*}E,u}\oplus\nabla^{L,\operatorname{op}}-\frac{c(T)}{4}.$
(3.1.13)

Choose and fix $a\in(0,1)$. Let $\alpha:[0,\infty)\to I$ be a smooth function
that satisfies $\alpha(t)=0$ for all $t\leq a$ and $\alpha(t)=1$ for all
$t\geq 1$. Define a rescaled Bismut superconnection by

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

Since ${\widetilde{\mathsf{D}}}^{S\otimes E}(\alpha(t))$ is invertible for
$t\geq 1$,

$\lim_{t\to\infty}\operatorname{ch}({\widehat{\mathbb{B}}}^{E}_{t})=0.$
(3.1.14)

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

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

By (3.1.9),

$\begin{split}\lim_{t\to
0}\operatorname{ch}({\widehat{\mathbb{B}}}^{E}_{t})&=\lim_{t\to
0}\operatorname{ch}(\mathbb{B}^{E}_{t})+\operatorname{ch}(\nabla^{L,\operatorname{op}})\\\
&=\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{E})-\operatorname{ch}(\nabla^{L}).\end{split}$
(3.1.15)

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

${\widehat{\eta}}^{E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda},L)=\frac{1}{\sqrt{\pi}}\int^{\infty}_{0}\operatorname{str}\bigg{(}\frac{d{\widehat{\mathbb{B}}}^{E}_{t}}{dt}e^{-\frac{1}{2\pi
i}({\widehat{\mathbb{B}}}^{E}_{t})^{2}}\bigg{)}dt.$ (3.1.16)

By (3.1.14) and (3.1.15) the local family index theorem for
$\mathsf{D}^{S\otimes E}$ is

$d{\widehat{\eta}}^{E}(g^{E},\nabla^{E},T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda},L)=\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{ch}(\nabla^{E})-\operatorname{ch}(\nabla^{L}).$
(3.1.17)

###  3.2. A proof of Theorem 1.1

In this subsection we prove an extended variational formula for the
Bismut–Cheeger eta form for spinc Dirac operators twisted by isomorphic
Hermitian bundles without the kernel bundle assumption (Theorem 1.1).

The following proposition is the spinc analog of [7, Proposition 1].

######  Proposition 3.1.

Let $\pi:X\to B$ be a submersion with closed, oriented and spinc fibers of
even dimension. Equip $\pi:X\to B$ with two sets of Riemannian and
differential spinc structures

$(T^{H}_{0}X,g^{T^{V}X}_{0},g^{\lambda}_{0},\nabla^{\lambda}_{0}),\qquad(T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1}),$

where the underlying topological spinc structures coincide. 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 $k\in\\{0,1\\}$, write $\mathsf{D}^{S\otimes
E}_{k}$ for the twisted spinc Dirac operator defined in terms of

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

If $(L_{k},g^{L_{k}},\nabla^{L_{k}})$ is a $\mathbb{Z}_{2}$-graded triple so
that $L_{k}\to B$ satisfies the MF property for $\mathsf{D}^{S\otimes E}_{k}$,
then 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

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

such that

$\begin{split}&{\widehat{\eta}}^{E}(g^{E}_{1},\nabla^{E}_{1},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{1})-{\widehat{\eta}}^{E}(g^{E}_{0},\nabla^{E}_{0},T^{H}_{0}X,g^{T^{V}X}_{0},g^{\lambda}_{0},\nabla^{\lambda}_{0},L_{0})\\\
&\equiv\int_{X/B}\big{(}T{\widehat{A}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}c_{1}(\nabla^{\lambda}_{0})}+{\widehat{A}}(\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}Tc_{1}(\nabla^{\lambda}_{0},\nabla^{\lambda}_{1})}\big{)}\wedge\operatorname{ch}(\nabla^{E}_{0})\\\
&\quad+\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E}_{0},\nabla^{E}_{1})-\operatorname{CS}(\nabla^{L_{0}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{1}}\oplus\nabla^{W_{1}})).\end{split}$
(3.2.2)

The proof of Proposition 3.1 is analogous to that of [7, Proposition 1],
except that for a given smooth path

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

where $t\in I$, we use the new path ${\widetilde{c}}(t)$ given by

${\widetilde{c}}(t)=\left\\{\begin{array}[]{ll}(T^{H}_{3t}X,g^{T^{V}X}_{3t},g^{\lambda}_{0},\nabla^{\lambda}_{0},g^{E}_{0},\nabla^{E}_{0}),&\displaystyle\textrm{
for }t\in\bigg{[}0,\frac{1}{3}\bigg{]}\\\ \\\
(T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{3t-1},\nabla^{\lambda}_{3t-1},g^{E}_{0},\nabla^{E}_{0}),&\displaystyle\textrm{
for }t\in\bigg{[}\frac{1}{3},\frac{2}{3}\bigg{]}\\\ \\\
(T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},g^{E}_{3t-2},\nabla^{E}_{3t-2}),&\displaystyle\textrm{
for }t\in\bigg{[}\frac{2}{3},1\bigg{]}\end{array}\right.$

to construct the Riemannian and differential spinc structure

$(T^{H}{\widetilde{X}},g^{T^{V}{\widetilde{X}}},g^{{\widetilde{\lambda}}},\nabla^{{\widetilde{\lambda}}})$

on ${\widetilde{\pi}}:{\widetilde{X}}\to{\widetilde{B}}$, where
${\widetilde{\lambda}}=p_{X}^{*}\lambda$, and the Hermitian metric
$g^{\mathcal{E}}$ and unitary connection $\nabla^{\mathcal{E}}$ on
$\mathcal{E}\to{\widetilde{X}}$, where $\mathcal{E}=p_{X}^{*}E$.

######  Remark 3.1.

The following remark is about a special case of Proposition 3.1, and we refer
to the proof of [7, Proposition 1] for the notations.

In Proposition 3.1, suppose there exists a $\mathbb{Z}_{2}$-graded isometric
isomorphism $\beta:(L_{0},g^{L_{0}})\to(L_{1},g^{L_{1}})$. Recall from [7,
(3.13)] that there exist complex vector bundles $G_{0}\to B$ and $H_{0}\to B$
such that

$L_{0}^{+}\oplus G_{0}\cong L^{+}\oplus H_{0},\qquad L_{0}^{-}\oplus
G_{0}\cong L^{-}\oplus H_{0}.$

Put Hermitian metrics $g^{G_{0}}$, $g^{H_{0}}$ and unitary connections
$\nabla^{G_{0}}$, $\nabla^{H_{0}}$ on $G_{0}\to X$ and $H_{0}\to X$,
respectively. Consider the balanced $\mathbb{Z}_{2}$-graded triples
$(W,g^{W},\nabla^{W})$ and $(H,g^{H},\nabla^{H})$, where $W\to B$ and $H\to B$
are defined by $W^{\pm}=G_{0}$ and $H^{\pm}=H_{0}$, and similarly for $g^{W}$,
$g^{H}$ and $\nabla^{W}$, $\nabla^{H}$. Let $f_{0}:(L\oplus H,g^{L}\oplus
g^{H})\to(L_{0}\oplus W,g^{L_{0}}\oplus g^{W})$ be a $\mathbb{Z}_{2}$-graded
isometric isomorphism. Take $h=\beta\oplus\operatorname{id}_{W}$ and
$f_{1}=h\circ f_{0}$. Then, as in [7, (3.15)], the following diagram commutes

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

Note that $f_{0}^{*}(\nabla^{L_{0}}\oplus\nabla^{W})$ and
$f_{1}^{*}(\nabla^{L_{1}}\oplus\nabla^{W})=f_{0}^{*}(h^{*}(\nabla^{L_{1}}\oplus\nabla^{W}))$
are $\mathbb{Z}_{2}$-graded unitary connections on $L\oplus H\to B$. By
following the proof of [7, Proposition 1], the last term of the right-hand
side of (3.2.2) is given by

$\begin{split}\operatorname{CS}(f_{0}^{*}(\nabla^{L_{0}}\oplus\nabla^{W}),f_{1}^{*}(\nabla^{L_{1}}\oplus\nabla^{W}))&=\operatorname{CS}(f_{0}^{*}(\nabla^{L_{0}}\oplus\nabla^{W}),f_{0}^{*}(h^{*}(\nabla^{L_{1}}\oplus\nabla^{W})))\\\
&=\operatorname{CS}(\nabla^{L_{0}}\oplus\nabla^{W},h^{*}(\nabla^{L_{1}}\oplus\nabla^{W}))\\\
&=\operatorname{CS}(\nabla^{L_{0}}\oplus\nabla^{W},\beta^{*}\nabla^{L_{1}}\oplus\operatorname{id}_{W}^{*}\nabla^{W})\\\
&\equiv\operatorname{CS}(\nabla^{L_{0}},\beta^{*}\nabla^{L_{1}}),\end{split}$

where the last equality follows from (2.2.6) and the fact that
$\operatorname{CS}(\nabla^{W},\operatorname{id}_{W}^{*}\nabla^{W})\equiv 0$.

Thus if $L_{0}\cong L_{1}$ as $\mathbb{Z}_{2}$-graded complex vector bundles
in Proposition 3.1, without loss of generality we can take $W_{0}\to B$ and
$W_{1}\to B$ to be the zero bundles.

######  Proposition 3.2.

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
$(T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda})$. 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 spinc Dirac operator $\mathsf{D}^{S\otimes
E,\alpha}$, which is 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 spinc Dirac
operator $\mathsf{D}^{S\otimes F}$, which is defined in terms of $\nabla^{F}$,
and

  2. (b)

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

One can think of Proposition 3.2 as the “variational” formula for the
Bismut–Cheeger eta form for the triples $(E,g^{E},\alpha^{*}\nabla^{F})$ and
$(F,g^{F},\nabla^{F})$.

###### Proof.

Note that $\alpha^{*}\nabla^{F}$ is a unitary connection on $E\to X$. By (a)
of Remark 2.1,

$\displaystyle\nabla^{S(T^{V}X)\otimes E}$
$\displaystyle=\nabla^{T^{V}X}\otimes\operatorname{id}+\operatorname{id}\otimes\alpha^{*}\nabla^{F}$
$\displaystyle=\alpha^{*}(\nabla^{T^{V}X}\otimes\operatorname{id}+\operatorname{id}\otimes\nabla^{F})$
$\displaystyle\Rightarrow\nabla^{S(T^{V}X)\otimes E}$
$\displaystyle=\alpha^{*}\nabla^{S(T^{V}X)\otimes F}.$ (3.2.3)

Thus

$\mathsf{D}^{S\otimes F}=\alpha\circ\mathsf{D}^{S\otimes
E,\alpha}\circ\alpha^{-1}.$ (3.2.4)

We write ${\widetilde{\alpha}}:\pi_{*}E\to\pi_{*}F$ for the
$\mathbb{Z}_{2}$-graded bundle isomorphism induced by $\alpha$. Since
$g^{E}=\alpha^{*}g^{F}$, it follows from (3.1.5) that

$g^{\pi_{*}E}={\widetilde{\alpha}}^{*}g^{\pi_{*}F}.$ (3.2.5)

That is, ${\widetilde{\alpha}}$ is isometric. By (3.1.6) and (3.2.3),
$\nabla^{\pi_{*}E}={\widetilde{\alpha}}^{*}\nabla^{\pi_{*}F}$, and therefore

$\nabla^{\pi_{*}E,u}={\widetilde{\alpha}}^{*}\nabla^{\pi_{*}F,u}.$ (3.2.6)

Let $K_{E,\alpha}\to B$ be a $\mathbb{Z}_{2}$-graded closed subbundle of
$\pi_{*}E\to B$ that is complementary to $L_{E,\alpha}\to B$, i.e.

$(\pi_{*}E)^{+}=K_{E,\alpha}^{+}\oplus
L_{E,\alpha}^{+},\qquad(\pi_{*}E)^{-}=K_{E,\alpha}^{-}\oplus
L_{E,\alpha}^{-},$ (3.2.7)

$\mathsf{D}^{S\otimes E,\alpha}_{+}:(\pi_{*}E)^{+}\to(\pi_{*}E)^{-}$ is block
diagonal as a map with respect to (3.2.7), and $\mathsf{D}^{S\otimes
E,\alpha}_{+}|_{K_{E,\alpha}^{+}}:K_{E,\alpha}^{+}\to K_{E,\alpha}^{-}$ is an
isomorphism. Write $\displaystyle\mathsf{D}^{S\otimes
E,\alpha}_{+}=\begin{pmatrix}a_{E,\alpha}&0\\\ 0&d_{E,\alpha}\end{pmatrix}$.
Define complex vector bundles $L_{F,\alpha}^{\pm}\to B$ and
$K_{F,\alpha}^{\pm}\to B$ by

$L_{F,\alpha}^{\pm}:={\widetilde{\alpha}}_{\pm}(L_{E,\alpha}^{\pm}),\qquad
K_{F,\alpha}^{\pm}:={\widetilde{\alpha}}_{\pm}(K_{E,\alpha}^{\pm}).$

Denote by ${\widetilde{\alpha}}_{\pm}^{L}:L_{E,\alpha}^{\pm}\to
L_{F,\alpha}^{\pm}$ and ${\widetilde{\alpha}}_{\pm}^{K}:K_{E,\alpha}^{\pm}\to
K_{F,\alpha}^{\pm}$ the corresponding restriction maps, which are
isomorphisms. Note that $L_{F,\alpha}^{\pm}\to B$ have finite ranks, and

$(\pi_{*}F)^{+}=K_{F,\alpha}^{+}\oplus
L_{F,\alpha}^{+},\qquad(\pi_{*}F)^{-}=K_{F,\alpha}^{-}\oplus
L_{F,\alpha}^{-}.$ (3.2.8)

With respect to (3.2.7) and (3.2.8), ${\widetilde{\alpha}}_{+}$ and
${\widetilde{\alpha}}_{-}$ are given by

${\widetilde{\alpha}}_{+}=\begin{pmatrix}{\widetilde{\alpha}}_{+}^{K}&0\\\
0&{\widetilde{\alpha}}_{+}^{L}\end{pmatrix},\qquad{\widetilde{\alpha}}_{-}=\begin{pmatrix}{\widetilde{\alpha}}_{-}^{K}&0\\\
0&{\widetilde{\alpha}}_{-}^{L}\end{pmatrix}.$ (3.2.9)

Define $\mathbb{Z}_{2}$-graded complex vector bundles $L_{F,\alpha}\to B$ and
$K_{F,\alpha}\to B$ by

$L_{F,\alpha}=L_{F,\alpha}^{+}\oplus L_{F,\alpha}^{-}\qquad\textrm{ and
}\qquad K_{F,\alpha}=K_{F,\alpha}^{+}\oplus K_{F,\alpha}^{-}.$

Then the map ${\widetilde{\alpha}}^{L}:L_{E,\alpha}\to L_{F,\alpha}$ given by

${\widetilde{\alpha}}^{L}={\widetilde{\alpha}}^{L}_{+}\oplus{\widetilde{\alpha}}^{L}_{-}$

is a $\mathbb{Z}_{2}$-graded smooth bundle isomorphism. Since the Hermitian
metric $g^{L_{E,\alpha}}$ on $L_{E,\alpha}\to B$ is inherited from
$g^{\pi_{*}E}$, it follows from (3.2.5) that the Hermitian metric
$g^{L_{F,\alpha}}$ on $L_{F,\alpha}\to B$ inherited from $g^{\pi_{*}F}$
satisfies $g^{L_{E,\alpha}}=({\widetilde{\alpha}}^{L})^{*}g^{L_{F,\alpha}}$.
Thus ${\widetilde{\alpha}}^{L}$ is isometric.

By (3.2.4) and (3.2.9),

$\begin{split}\mathsf{D}^{S\otimes
F}_{+}&={\widetilde{\alpha}}_{-}\circ\mathsf{D}^{S\otimes
E,\alpha}_{+}\circ{\widetilde{\alpha}}_{+}^{-1}\\\
&=\begin{pmatrix}{\widetilde{\alpha}}_{-}^{K}&0\\\
0&{\widetilde{\alpha}}_{-}^{L}\end{pmatrix}\begin{pmatrix}a_{E,\alpha}&0\\\
0&d_{E,\alpha}\end{pmatrix}\begin{pmatrix}({\widetilde{\alpha}}_{+}^{K})^{-1}&0\\\
0&({\widetilde{\alpha}}_{+}^{L})^{-1}\end{pmatrix}\\\
&=\begin{pmatrix}{\widetilde{\alpha}}_{-}^{K}\circ
a_{E,\alpha}\circ({\widetilde{\alpha}}_{+}^{K})^{-1}&0\\\
0&{\widetilde{\alpha}}_{-}^{L}\circ
d_{E,\alpha}\circ({\widetilde{\alpha}}_{+}^{L})^{-1}\end{pmatrix}.\end{split}$

Thus $\mathsf{D}^{S\otimes F}_{+}$ is block diagonal as a map with respect to
(3.2.8), and

$\mathsf{D}^{S\otimes F}|_{K_{F}^{+}}:{\widetilde{\alpha}}_{-}^{K}\circ
a_{E,\alpha}\circ({\widetilde{\alpha}}_{+}^{K})^{-1}:K_{F}^{+}\to K_{F}^{-}$

is an isomorphism. Therefore $L_{F,\alpha}\to B$ satisfies the MF property for
$\mathsf{D}^{S\otimes F}$.

Denote by $P_{E,\alpha}:\pi_{*}E\to L_{E,\alpha}$ and $P_{F}:\pi_{*}F\to
L_{F,\alpha}$ the $\mathbb{Z}_{2}$-graded projection maps with respect to
(3.2.7) and (3.2.8), respectively. First note that the following diagram
commutes.

${\pi_{*}E}$${\pi_{*}F}$${L_{E,\alpha}}$${L_{F,\alpha}}$$\scriptstyle{{\widetilde{\alpha}}}$$\scriptstyle{P_{E,\alpha}}$$\scriptstyle{P_{F}}$$\scriptstyle{{\widetilde{\alpha}}^{L}}$

That is,

${\widetilde{\alpha}}^{L}\circ P_{E,\alpha}=P_{F}\circ{\widetilde{\alpha}}.$
(3.2.10)

Let $s\in\Gamma(B,L_{E,\alpha})$. By (3.1.11), (3.2.6) and (3.2.10),

$\begin{split}\nabla^{L_{E,\alpha}}s&=(P_{E,\alpha}\circ\nabla^{\pi_{*}E,u}\circ
P_{E,\alpha})(s)\\\ &=(P_{E,\alpha}\circ\nabla^{\pi_{*}E,u})(s)\\\
&=(P_{E,\alpha}\circ{\widetilde{\alpha}}^{-1}\circ\nabla^{\pi_{*}F,u}\circ{\widetilde{\alpha}})(s)\\\
&=(({\widetilde{\alpha}}^{L})^{-1}\circ
P_{F}\circ\nabla^{\pi_{*}F,u})({\widetilde{\alpha}}^{L}(s))\\\
&=(({\widetilde{\alpha}}^{L})^{-1}\circ P_{F}\circ\nabla^{\pi_{*}F,u}\circ
P_{F})({\widetilde{\alpha}}^{L}(s))\\\
&=(({\widetilde{\alpha}}^{L})^{-1}\circ\nabla^{L_{F,\alpha}})({\widetilde{\alpha}}^{L}(s))\\\
&=(({\widetilde{\alpha}}^{L})^{-1}\circ\nabla^{L_{F,\alpha}}\circ{\widetilde{\alpha}}^{L})(s).\end{split}$

Thus

$\nabla^{L_{E,\alpha}}=({\widetilde{\alpha}}^{L})^{*}\nabla^{L_{F,\alpha}}.$
(3.2.11)

Since ${\widetilde{\alpha}}:\pi_{*}E\to\pi_{*}F$ and
${\widetilde{\alpha}}^{L}:L_{E,\alpha}\to L_{F,\alpha}$ are
$\mathbb{Z}_{2}$-graded isometric isomorphisms, the same is true for
${\widehat{\alpha}}:={\widetilde{\alpha}}\oplus({\widetilde{\alpha}}^{L})^{\operatorname{op}}:\pi_{*}E\oplus
L_{E,\alpha}^{\operatorname{op}}\to\pi_{*}F\oplus
L_{F,\alpha}^{\operatorname{op}}$. By the definition of
${\widetilde{\mathsf{D}}}^{S\otimes E,\alpha}_{+}(z)$, it follows from (3.2.4)
that

${\widetilde{\mathsf{D}}}^{S\otimes
E,\alpha}(z)={\widehat{\alpha}}^{-1}\circ{\widetilde{\mathsf{D}}}^{S\otimes
F}(z)\circ{\widehat{\alpha}}$ (3.2.12)

for every $z\in\mathbb{C}$. On the other hand, by (3.2.6) and (3.2.11),

$\nabla^{\pi_{*}E,u}\oplus\nabla^{L_{E,\alpha},\operatorname{op}}={\widehat{\alpha}}^{*}(\nabla^{\pi_{*}F,u}\oplus\nabla^{L_{F,\alpha},\operatorname{op}}).$
(3.2.13)

By the definition of ${\widehat{\mathbb{B}}}^{E,\alpha}$, it follows from
(3.2.12) and (3.2.13) that

${\widehat{\mathbb{B}}}^{E,\alpha}={\widehat{\alpha}}^{*}{\widehat{\mathbb{B}}}^{F}.$
(3.2.14)

Since

$\begin{split}({\widehat{\mathbb{B}}}^{E,\alpha}_{t})^{2k}&=({\widehat{\alpha}}^{-1}\circ{\widehat{\mathbb{B}}}^{F}_{t}\circ{\widehat{\alpha}})^{2k}={\widehat{\alpha}}^{-1}\circ({\widehat{\mathbb{B}}}^{F}_{t})^{2k}\circ{\widehat{\alpha}},\\\
\frac{d{\widehat{\mathbb{B}}}^{E,\alpha}_{t}}{dt}&=\frac{d({\widehat{\alpha}}^{*}{\widehat{\mathbb{B}}}^{F}_{t})}{dt}={\widehat{\alpha}}^{-1}\circ\frac{d{\widehat{\mathbb{B}}}^{F}_{t}}{dt}\circ{\widehat{\alpha}},\end{split}$

for any nonnegative integer $k$, it follows that

$\begin{split}\operatorname{str}\bigg{(}\frac{d{\widehat{\mathbb{B}}}^{E,\alpha}_{t}}{dt}e^{-\frac{1}{2\pi
i}({\widehat{\mathbb{B}}}^{E,\alpha}_{t})^{2}}\bigg{)}&=\operatorname{str}\bigg{(}{\widehat{\alpha}}^{-1}\circ\frac{d{\widehat{\mathbb{B}}}^{F}_{t}}{dt}e^{-\frac{1}{2\pi
i}({\widehat{\mathbb{B}}}^{F}_{t})^{2}}\circ{\widehat{\alpha}}\bigg{)}\\\
&=\operatorname{str}\bigg{(}\frac{d{\widehat{\mathbb{B}}}^{F}_{t}}{dt}e^{-\frac{1}{2\pi
i}({\widehat{\mathbb{B}}}^{F}_{t})^{2}}\bigg{)}.\end{split}$

It follows that

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

We now prove Theorem 1.1.

######  Theorem 3.1.

Let $\pi:X\to B$ be a submersion with closed, oriented and spinc fibers of
even dimension. Equip $\pi:X\to B$ with two sets of Riemannian and
differential spinc structures

$(T^{H}_{0}X,g^{T^{V}X}_{0},g^{\lambda}_{0},\nabla^{\lambda}_{0}),\qquad(T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1}),$

where the underlying topological spinc structures coincide. Let
$(E,g^{E},\nabla^{E})$ and $(F,g^{F},\nabla^{F})$ be triples, denote by
$\mathsf{D}^{S\otimes E}$ and $\mathsf{D}^{S\otimes F}$ the twisted spinc
Dirac operators defined in terms of

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

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 $\mathsf{D}^{S\otimes E}$ and
$\mathsf{D}^{S\otimes F}$, 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

$\begin{split}&{\widehat{\eta}}^{F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{F})-{\widehat{\eta}}^{E}(g^{E},\nabla^{E},T^{H}_{0}X,g^{T^{V}X}_{0},g^{\lambda}_{0},\nabla^{\lambda}_{0},L_{E})\\\
&\equiv\int_{X/B}\big{(}T{\widehat{A}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}c_{1}(\nabla^{\lambda}_{0})}+{\widehat{A}}(\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}Tc_{1}(\nabla^{\lambda}_{0},\nabla^{\lambda}_{1})}\big{)}\wedge\operatorname{ch}(\nabla^{E})\\\
&\quad+\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E},\alpha^{*}\nabla^{F})-\operatorname{CS}(\nabla^{L_{E}}\oplus\nabla^{W_{E}},{\widetilde{h}}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F}})).\end{split}$
(3.2.15)

###### Proof.

Write $\mathsf{D}^{S\otimes E,\alpha}$ for the twisted spinc Dirac operator
defined in terms of

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

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 $\mathsf{D}^{S\otimes E,\alpha}$. 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

$h:(L_{E}\oplus W_{0},g^{L_{E}}\oplus g^{W_{0}})\to(L_{E,\alpha}\oplus
W_{1},g^{L_{E,\alpha}}\oplus g^{W_{1}})$ (3.2.16)

such that

$\begin{split}&{\widehat{\eta}}^{E}(g^{E},\alpha^{*}\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{E,\alpha})-{\widehat{\eta}}^{E}(g^{E},\nabla^{E},T^{H}_{0}X,g^{T^{V}X}_{0},g^{\lambda}_{0},\nabla^{\lambda}_{0},L_{E})\\\
&\equiv\int_{X/B}\big{(}T{\widehat{A}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}c_{1}(\nabla^{\lambda}_{0})}+{\widehat{A}}(\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}Tc_{1}(\nabla^{\lambda}_{0},\nabla^{\lambda}_{1})}\big{)}\wedge\operatorname{ch}(\nabla^{E})\\\
&\quad+\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E},\alpha^{*}\nabla^{F})-\operatorname{CS}(\nabla^{L_{E}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}})).\end{split}$
(3.2.17)

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

${\widetilde{\alpha}}^{L}:(L_{E,\alpha},g^{L_{E,\alpha}})\to(L_{F,\alpha},g^{L_{F,\alpha}})$
(3.2.18)

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

${\widehat{\eta}}^{E}(g^{E},\alpha^{*}\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{E,\alpha})={\widehat{\eta}}^{F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{F,\alpha}).$
(3.2.19)

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 both $L_{F}\to B$ and $L_{F,\alpha}\to
B$ satisfy the MF property for $\mathsf{D}^{S\otimes F}$, it follows from [7,
Corollary 1] that 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

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

such that

$\begin{split}&{\widehat{\eta}}^{F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{F})-{\widehat{\eta}}^{F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{F,\alpha})\\\
&\equiv-\operatorname{CS}(\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{F,0}},h_{F}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F,1}})).\end{split}$
(3.2.21)

By (3.2.16), (3.2.18) and (3.2.20), the map

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

given by 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,
(3.2.19) and (3.2.21) imply

$\begin{split}&{\widehat{\eta}}^{E}(g^{E},\alpha^{*}\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{E,\alpha})\\\
&={\widehat{\eta}}^{F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{F,\alpha})\\\
&\equiv{\widehat{\eta}}^{F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{F})+\operatorname{CS}(\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{F,0}},h_{F}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F,1}})).\end{split}$
(3.2.22)

By (3.2.22), (3.2.17) is equal to

$\begin{split}&{\widehat{\eta}}^{F}(g^{F},\nabla^{F},T^{H}_{1}X,g^{T^{V}X}_{1},g^{\lambda}_{1},\nabla^{\lambda}_{1},L_{F})-{\widehat{\eta}}^{E}(g^{E},\nabla^{E},T^{H}_{0}X,g^{T^{V}X}_{0},g^{\lambda}_{0},\nabla^{\lambda}_{0},L_{E})\\\
&\equiv\int_{X/B}\big{(}T{\widehat{A}}(\nabla^{T^{V}X}_{0},\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}c_{1}(\nabla^{\lambda}_{0})}+{\widehat{A}}(\nabla^{T^{V}X}_{1})\wedge
e^{\frac{1}{2}Tc_{1}(\nabla^{\lambda}_{0},\nabla^{\lambda}_{1})}\big{)}\wedge\operatorname{ch}(\nabla^{E})\\\
&\quad+\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X}_{1})\wedge\operatorname{CS}(\nabla^{E},\alpha^{*}\nabla^{F})-\operatorname{CS}(\nabla^{L_{E}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}))\\\
&\quad-\operatorname{CS}(\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{F,0}},h_{F}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F,1}})).\end{split}$
(3.2.23)

Since
$\nabla^{L_{E,\alpha}}=({\widetilde{\alpha}}^{L})^{*}\nabla^{L_{F,\alpha}}$ by
(3.2.10), it follows that

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

Thus

$\begin{split}&\operatorname{CS}(\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}},(h_{F}\oplus\operatorname{id}_{W_{1}})^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}}))\\\
&\equiv\operatorname{CS}((h\oplus\operatorname{id}_{W_{F,0}})^{*}({\widetilde{\alpha}}^{L}\oplus\operatorname{id}_{W_{1}}\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}),\\\
&\qquad(h\oplus\operatorname{id}_{W_{F,0}})^{*}({\widetilde{\alpha}}^{L}\oplus\operatorname{id}_{W_{1}}\oplus\operatorname{id}_{W_{F,0}})^{*}(h_{F}\oplus\operatorname{id}_{W_{1}})^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}}))\\\
&\equiv\operatorname{CS}((h\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}),{\widetilde{h}}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}})).\end{split}$
(3.2.24)

Since
$\operatorname{CS}(\nabla^{W_{F,0}},\operatorname{id}_{W_{F,0}}^{*}\nabla^{W_{F,0}})\equiv
0$ and
$\operatorname{CS}(\nabla^{W_{1}},\operatorname{id}_{W_{1}}^{*}\nabla^{W_{1}})\equiv
0$, it follows from (3.2.24) and (2.2.5) that the last two terms of the right-
hand side of (3.2.23) is equal to

$\begin{split}&\operatorname{CS}(\nabla^{L_{E}}\oplus\nabla^{W_{0}},h^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}))+\operatorname{CS}(\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{F,0}},h_{F}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{F,1}}))\\\
&\equiv\operatorname{CS}(\nabla^{L_{E}}\oplus\nabla^{W_{0}}\oplus\nabla^{W_{F,0}},(h\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}))\\\
&\quad+\operatorname{CS}(\nabla^{L_{F,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}},(h_{F}\oplus\operatorname{id}_{W_{1}})^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}}))\\\
&\equiv\operatorname{CS}(\nabla^{L_{E}}\oplus\nabla^{W_{0}}\oplus\nabla^{W_{F,0}},(h\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}))\\\
&\quad+\operatorname{CS}((h\oplus\operatorname{id}_{W_{F,0}})^{*}(\nabla^{L_{E,\alpha}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,0}}),{\widetilde{h}}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}}))\\\
&\equiv\operatorname{CS}(\nabla^{L_{E}}\oplus\nabla^{W_{0}}\oplus\nabla^{W_{F,0}},{\widetilde{h}}^{*}(\nabla^{L_{F}}\oplus\nabla^{W_{1}}\oplus\nabla^{W_{F,1}})).\end{split}$
(3.2.25)

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 $\nabla^{W_{E}}$, $\nabla^{W_{F}}$,
(3.2.23) and (3.2.25) show that (3.2.15) holds. ∎

Note that Remark 3.1 applies to Theorem 3.1 as well.

##  4\. Applications of Theorem 1.1

In this section we present some applications of Proposition 1.1 and Theorem
1.1. All the results in this subsection are under the following setup. Let
$\pi:X\to B$ be a submersion with closed, oriented and spinc fibers of even
dimension, equipped with a fixed Riemannian and differential spinc structure
$(T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda})$. Henceforth, we suppress
the dependence of the Bismut–Cheeger eta form on
$(T^{H}X,g^{T^{V}X},g^{\lambda},\nabla^{\lambda})$.

###  4.1. $\mathbb{Z}_{2}$-graded additivity of the Bismut–Cheeger eta form

In this subsection, we establish some intermediate results for the
Bismut–Cheeger eta form that enable us, along with Proposition 3.2, to prove
its $\mathbb{Z}_{2}$-graded additivity (Theorem 1.2).

The following proposition says the Bismut–Cheeger eta form is additive with
respect to direct sums.

######  Proposition 4.1.

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 $\mathsf{D}^{S\otimes E}$ and $\mathsf{D}^{S\otimes F}$,
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
$\mathsf{D}^{S\otimes(E\oplus F)}$, and

${\widehat{\eta}}^{E\oplus F}(g^{E}\oplus
g^{F},\nabla^{E}\oplus\nabla^{F},L_{E\oplus
F})\equiv{\widehat{\eta}}^{E}(g^{E},\nabla^{E},L_{E})+{\widehat{\eta}}^{F}(g^{F},\nabla^{F},L_{F}).$

###### Proof.

Let $K_{E}\to B$ and $K_{F}\to B$ be $\mathbb{Z}_{2}$-graded closed subbundles
of $\pi_{*}E\to B$ and $\pi_{*}F\to B$ that are complementary to $L_{E}\to B$
and $L_{F}\to B$, respectively, i.e.

$\begin{split}(\pi_{*}E)^{+}=K_{E}^{+}\oplus
L_{E}^{+},\qquad(\pi_{*}E)^{-}=K_{E}^{-}\oplus L_{E}^{-},\\\
(\pi_{*}F)^{+}=K_{F}^{+}\oplus L_{F}^{+},\qquad(\pi_{*}F)^{-}=K_{F}^{-}\oplus
L_{F}^{-},\end{split}$ (4.1.1)

$\mathsf{D}^{S\otimes E}_{+}$ and $\mathsf{D}^{S\otimes F}_{+}$ are block
diagonal as maps with respect to (4.1.1), say

$\mathsf{D}^{S\otimes E}_{+}=\begin{pmatrix}a_{E}&0\\\
0&d_{E}\end{pmatrix},\qquad\mathsf{D}^{S\otimes
F}_{+}=\begin{pmatrix}a_{F}&0\\\ 0&d_{F}\end{pmatrix},$ (4.1.2)

and $a_{E}:K_{E}^{+}\to K_{E}^{-}$ and $a_{F}:K_{F}^{+}\to K_{F}^{-}$ are
isomorphisms, respectively.

Define a $\mathbb{Z}_{2}$-graded triple $(L_{E\oplus F},g^{L_{E\oplus
F}},\nabla^{L_{E\oplus F}})$ by

$L_{E\oplus F}=L_{E}\oplus L_{F},\qquad g^{L_{E\oplus F}}=g^{L_{E}}\oplus
g^{L_{F}},\qquad\nabla^{L_{E\oplus F}}=\nabla^{L_{E}}\oplus\nabla^{L_{F}}.$
(4.1.3)

By (4.1.1),

$\begin{split}(\pi_{*}(E\oplus
F))^{+}&=(\pi_{*}E)^{+}\oplus(\pi_{*}F)^{+}=(K_{E}^{+}\oplus
K_{F}^{+})\oplus(L_{E}^{+}\oplus L_{F}^{+}),\\\ (\pi_{*}(E\oplus
F))^{-}&=(\pi_{*}E)^{-}\oplus(\pi_{*}F)^{-}=(K_{E}^{-}\oplus
K_{F}^{-})\oplus(L_{E}^{-}\oplus L_{F}^{-}).\end{split}$ (4.1.4)

Thus $K_{E}\oplus K_{F}\to B$ is complementary to $L_{E\oplus F}\to B$. With
respect to (4.1.4), $\mathsf{D}^{S\otimes(E\oplus F)}_{+}$ is given by

$\mathsf{D}^{S\otimes(E\oplus F)}_{+}=\begin{pmatrix}a_{E}&0&0&0\\\
0&a_{F}&0&0\\\ 0&0&d_{E}&0\\\ 0&0&0&d_{F}\end{pmatrix},$

which is block diagonal as a map with respect to (4.1.4), and shows that
$\mathsf{D}^{S\otimes(E\oplus F)}_{+}|_{K_{E}^{+}\oplus K_{F}^{+}}=a_{E}\oplus
a_{F}$ is an isomorphism. Thus $L_{E\oplus F}\to B$ satisfies the MF property
for $\mathsf{D}^{S\otimes(E\oplus F)}$.

We show that the Bismut superconnection ${\widehat{\mathbb{B}}}^{E\oplus F}$
is additive in the following sense:

${\widehat{\mathbb{B}}}^{E\oplus
F}={\widehat{\mathbb{B}}}^{E}\oplus{\widehat{\mathbb{B}}}^{F}.$ (4.1.5)

With respect to the decomposition

$\pi_{*}(E\oplus F)\oplus L_{E\oplus F}^{\operatorname{op}}=(\pi_{*}E\oplus
L_{E}^{\operatorname{op}})\oplus(\pi_{*}F\oplus L_{F}^{\operatorname{op}}),$

we have

$\mathsf{D}^{S\otimes(E\oplus F)}_{+}=\mathsf{D}^{S\otimes
E}_{+}\oplus\mathsf{D}^{S\otimes F}_{+},\quad i_{E\oplus
F}^{-}=i_{E}^{-}\oplus i_{F}^{-},\quad P_{E\oplus F}^{+}=P_{E}^{+}\oplus
P_{F}^{+}.$

Thus ${\widetilde{\mathsf{D}}}^{S\otimes(E\oplus
F)}_{+}(z)={\widetilde{\mathsf{D}}}^{S\otimes
E}_{+}(z)\oplus{\widetilde{\mathsf{D}}}^{S\otimes F}_{+}(z)$ for any
$z\in\mathbb{C}$, and therefore

${\widetilde{\mathsf{D}}}^{S\otimes(E\oplus
F)}(z)={\widetilde{\mathsf{D}}}^{S\otimes
E}(z)\oplus{\widetilde{\mathsf{D}}}^{S\otimes F}(z).$ (4.1.6)

Since $\nabla^{\pi_{*}(E\oplus
F),u}=\nabla^{\pi_{*}E,u}\oplus\nabla^{\pi_{*}F,u}$, it follows that

$\nabla^{\pi_{*}(E\oplus F),u}\oplus\nabla^{L_{E\oplus
F},\operatorname{op}}=(\nabla^{\pi_{*}E,u}\oplus\nabla^{L_{E},\operatorname{op}})\oplus(\nabla^{\pi_{*}F,u}\oplus\nabla^{L_{F},\operatorname{op}}).$
(4.1.7)

By (4.1.6) and (4.1.7), (4.1.5) holds.

Let $t,T\in(0,\infty)$ satisfy $t<T$. By (4.1.5) and (2.2.6),

$\operatorname{CS}({\widehat{\mathbb{B}}}^{E\oplus
F}_{t},{\widehat{\mathbb{B}}}^{E\oplus
F}_{T})\equiv\operatorname{CS}({\widehat{\mathbb{B}}}^{E}_{t}\oplus{\widehat{\mathbb{B}}}^{F}_{t},{\widehat{\mathbb{B}}}^{E}_{T}\oplus{\widehat{\mathbb{B}}}^{F}_{T})\equiv\operatorname{CS}({\widehat{\mathbb{B}}}^{E}_{t},{\widehat{\mathbb{B}}}^{E}_{T})+\operatorname{CS}({\widehat{\mathbb{B}}}^{F}_{t},{\widehat{\mathbb{B}}}^{F}_{T}).$
(4.1.8)

By letting $t\to 0$ and $T\to\infty$ in (4.1.8), the result follows. ∎

Proposition 4.1 is also valid if not all of the triples $(E,g^{E},\nabla^{E})$
and $(F,g^{F},\nabla^{F})$ are $\mathbb{Z}_{2}$-graded.

The following lemma says for any given $\mathbb{Z}_{2}$-graded triple
$(E,g^{E},\nabla^{E})$, the $\mathbb{Z}_{2}$-graded complex vector bundle
satisfying the MF property for $\mathsf{D}^{S\otimes E}$ can be expressed in
terms of those for $\mathsf{D}^{S\otimes E^{+}}$ and $\mathsf{D}^{S\otimes
E^{-}}$.

######  Lemma 4.1.

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 $\mathsf{D}^{S\otimes E^{+}}$
and $\mathsf{D}^{S\otimes E^{-}}$, 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 $\mathsf{D}^{S\otimes E}$.

###### Proof.

Let $K_{E^{+}}\to B$ and $K_{E^{-}}\to B$ be $\mathbb{Z}_{2}$-graded closed
subbundles of $\pi_{*}E^{+}\to B$ and $\pi_{*}E^{-}\to B$ that are
complementary to $L_{E^{+}}\to B$ and $L_{E^{-}}\to B$, respectively, i.e.

$\begin{split}(\pi_{*}E^{+})^{+}&=K_{E^{+}}^{+}\oplus
L_{E^{+}}^{+},\qquad(\pi_{*}E^{+})^{-}=K_{E^{+}}^{-}\oplus L_{E^{+}}^{-},\\\
(\pi_{*}E^{-})^{+}&=K_{E^{-}}^{+}\oplus
L_{E^{-}}^{+},\qquad(\pi_{*}E^{-})^{-}=K_{E^{-}}^{-}\oplus
L_{E^{-}}^{-},\end{split}$ (4.1.9)

$\mathsf{D}^{S\otimes E^{+}}_{+}$ and $\mathsf{D}^{S\otimes E^{-}}_{+}$ are
block diagonal as maps with respect to (4.1.9), say

$\mathsf{D}^{S\otimes E^{+}}_{+}=\begin{pmatrix}a_{E^{+}}&0\\\
0&d_{E^{+}}\end{pmatrix},\qquad\mathsf{D}^{S\otimes
E^{-}}_{+}=\begin{pmatrix}a_{E^{-}}&0\\\ 0&d_{E^{-}}\end{pmatrix},$

and $a_{E^{+}}:K_{E^{+}}^{+}\to K_{E^{+}}^{-}$ and $a_{E^{-}}:K_{E^{-}}^{+}\to
K_{E^{-}}^{-}$ are isomorphisms.

By (4.1.9),

$\begin{split}(\pi_{*}E)^{+}&=(\pi_{*}E^{+})^{+}\oplus(\pi_{*}E^{-})^{-}\\\
&=K_{E^{+}}^{+}\oplus L_{E^{+}}^{+}\oplus K_{E^{-}}^{-}\oplus L_{E^{-}}^{-}\\\
&=(K_{E^{+}}\oplus K_{E^{-}}^{\operatorname{op}})^{+}\oplus(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})^{+}\end{split}$ (4.1.10)

and

$\begin{split}(\pi_{*}E)^{-}&=(\pi_{*}E^{+})^{-}\oplus(\pi_{*}E^{-})^{+}\\\
&=K_{E^{+}}^{-}\oplus L_{E^{+}}^{-}\oplus K_{E^{-}}^{+}\oplus L_{E^{-}}^{+}\\\
&=(K_{E^{+}}\oplus K_{E^{-}}^{\operatorname{op}})^{-}\oplus(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})^{-}.\end{split}$ (4.1.11)

By (4.1.10) and (4.1.11), $\pi_{*}E=(K_{E^{+}}\oplus
K_{E^{-}}^{\operatorname{op}})\oplus(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})$. With respect to (4.1.10) and (4.1.11),
$\mathsf{D}^{S\otimes E}_{+}$ is given by

$\mathsf{D}^{S\otimes E}_{+}=\begin{pmatrix}a_{E^{+}}&0&0&0\\\
0&a_{E^{-}}^{*}&0&0\\\ 0&0&d_{E^{+}}&0\\\ 0&0&0&d_{E^{-}}^{*}\end{pmatrix},$

where $\mathsf{D}^{S\otimes E^{-}}_{-}=(\mathsf{D}^{S\otimes
E^{-}}_{+})^{*}=\begin{pmatrix}a_{E^{-}}^{*}&0\\\
0&d_{E^{-}}^{*}\end{pmatrix}$. Note that $\mathsf{D}^{S\otimes
E}_{+}|_{(K_{E^{+}}\oplus K_{E^{-}}^{\operatorname{op}})^{+}}=a_{E^{+}}\oplus
a_{E^{-}}^{*}$, which is an isomorphism. Thus $L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}}\to B$ satisfies the MF property for
$\mathsf{D}^{S\otimes E}$. ∎

Let $E\to X$ be a $\mathbb{Z}_{2}$-graded complex vector bundle. Since
$\operatorname{End}(E)^{+}=\operatorname{End}(E^{+})\oplus\operatorname{End}(E^{-})$,
it follows that

$\operatorname{End}(E^{\operatorname{op}})^{+}=\operatorname{End}(E^{\operatorname{op},+})\oplus\operatorname{End}(E^{\operatorname{op},-})=\operatorname{End}(E^{-})\oplus\operatorname{End}(E^{+})=\operatorname{End}(E)^{+}.$

Thus $T\in\operatorname{End}(E^{\operatorname{op}})^{+}$ if and only if
$T\in\operatorname{End}(E)^{+}$. Moreover,
$\operatorname{str}_{E^{\operatorname{op}}}(T)$ exists if and only if
$\operatorname{str}_{E}(T)$ exists. If either one of them exists, then

$\operatorname{str}_{E^{\operatorname{op}}}(T)=\operatorname{tr}(T|_{E^{\operatorname{op},+}})-\operatorname{tr}(T|_{E^{\operatorname{op},-}})=\operatorname{tr}(T|_{E^{-}})-\operatorname{tr}(T|_{E^{+}})=-\operatorname{str}_{E}(T).$
(4.1.12)

The following lemma says switching the $\mathbb{Z}_{2}$-grading of a
$\mathbb{Z}_{2}$-graded triple induces a minus sign in the corresponding
Bismut–Cheeger eta form.

######  Lemma 4.2.

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 $\mathsf{D}^{S\otimes E}$, 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 $\mathsf{D}^{S\otimes E^{\operatorname{op}}}$,
and

${\widehat{\eta}}^{E^{\operatorname{op}}}(g^{E,\operatorname{op}},\nabla^{E,\operatorname{op}},L^{\operatorname{op}})\equiv-{\widehat{\eta}}^{E}(g^{E},\nabla^{E},L).$
(4.1.13)

###### Proof.

Let $K\to B$ be a $\mathbb{Z}_{2}$-graded closed subbundle of $\pi_{*}E\to B$
that is complementary to $L\to B$, i.e.

$(\pi_{*}E)^{+}=K^{+}\oplus L^{+},\qquad(\pi_{*}E)^{-}=K^{-}\oplus L^{-},$
(4.1.14)

$\mathsf{D}^{S\otimes E}_{+}$ is block diagonal as a map with respect to
(4.1.14) and $\mathsf{D}^{S\otimes E}|_{K^{+}}$ is an isomorphism. Since

$\begin{split}(S(T^{V}X)\otimes
E^{\operatorname{op}})^{+}&=(S(T^{V}X)^{+}\otimes
E^{-})\oplus(S(T^{V}X)^{-}\otimes E^{+})=(S(T^{V}X)\otimes E)^{-},\\\
(S(T^{V}X)\otimes E^{\operatorname{op}})^{-}&=(S(T^{V}X)^{+}\otimes
E^{+})\oplus(S(T^{V}X)^{-}\otimes E^{-})=(S(T^{V}X)\otimes E)^{+},\end{split}$

it follows that

$\begin{split}(\pi_{*}(E^{\operatorname{op}}))^{+}&=(\pi_{*}E)^{-}=K^{-}\oplus
L^{-},\\\ (\pi_{*}(E^{\operatorname{op}}))^{-}&=(\pi_{*}E)^{+}=K^{+}\oplus
L^{+}.\end{split}$ (4.1.15)

That is, $\pi_{*}(E^{\operatorname{op}})=K^{\operatorname{op}}\oplus
L^{\operatorname{op}}$. Moreover, $\mathsf{D}^{S\otimes
E^{\operatorname{op}}}_{+}=\mathsf{D}^{S\otimes E}_{-}=(\mathsf{D}^{S\otimes
E}_{+})^{*}$. Thus $\mathsf{D}^{S\otimes E^{\operatorname{op}}}_{+}$ is also
block diagonal as a map with respect to (4.1.15), and $\mathsf{D}^{S\otimes
E^{\operatorname{op}}}_{+}|_{K^{\operatorname{op},+}}$ is the inverse of
$\mathsf{D}^{S\otimes E}_{+}|_{K^{+}}$, which is an isomorphism. Thus
$L^{\operatorname{op}}\to B$ satisfies the MF property for
$\mathsf{D}^{S\otimes E^{\operatorname{op}}}$.

Note that $g^{L,\operatorname{op}}$ is inherited from
$g^{\pi_{*}(E^{\operatorname{op}})}$ and
$\nabla^{L,\operatorname{op}}=P^{\operatorname{op}}\circ\nabla^{\pi_{*}(E^{\operatorname{op}}),u}\circ
P^{\operatorname{op}}$, where
$P^{\operatorname{op}}:\pi_{*}(E^{\operatorname{op}})\to
L^{\operatorname{op}}$ is the obvious $\mathbb{Z}_{2}$-graded projection map.
By applying (4.1.12) to
$\displaystyle\frac{d{\widehat{\mathbb{B}}}^{E^{\operatorname{op}}}_{t}}{dt}e^{-\frac{1}{2\pi
i}({\widehat{\mathbb{B}}}^{E^{\operatorname{op}}}_{t})^{2}}$, (4.1.13) holds.
∎

######  Proposition 4.2.

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 $\mathsf{D}^{S\otimes
V^{+}}$, 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
$\mathsf{D}^{S\otimes V}$ and

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

###### Proof.

Since the $\mathbb{Z}_{2}$-graded triple $(V,g^{V},\nabla^{V})$ is balanced,
$(V^{+},g^{V,+},\nabla^{V,+})=(V^{-},g^{V,-},\nabla^{V,-})$. Thus $L_{+}\to B$
also satisfies the MF property for $\mathsf{D}^{S\otimes V^{-}}$. Define a
$\mathbb{Z}_{2}$-graded triple $(L,g^{L},\nabla^{L})$ by

$L=L_{+}\oplus L_{+}^{\operatorname{op}},\qquad g^{L}=g^{L_{+}}\oplus
g^{L_{+},\operatorname{op}},\qquad\nabla^{L}=\nabla^{L_{+}}\oplus\nabla^{L_{+},\operatorname{op}}.$

By Lemma 4.1, $L\to B$ satisfies the MF property for $\mathsf{D}^{S\otimes
V}$. Since

$L^{+}=L_{+}^{+}\oplus L_{+}^{-}\qquad\textrm{ and }\qquad
L^{-}=L_{+}^{-}\oplus L_{+}^{+},$

and similarly for $g^{L}$ and $\nabla^{L}$, the $\mathbb{Z}_{2}$-graded triple
$(L,g^{L},\nabla^{L})$ is balanced. Since
$(V^{\operatorname{op}},g^{V,\operatorname{op}},\nabla^{V,\operatorname{op}})=(V,g^{V},\nabla^{V})$,
it follows from Lemma 4.2 and the fact
$(L^{\operatorname{op}},g^{L,\operatorname{op}},\nabla^{L,\operatorname{op}})=(L,g^{L},\nabla^{L})$
that

${\widehat{\eta}}^{V}(g^{V},\nabla^{V},L)\equiv-{\widehat{\eta}}^{V^{\operatorname{op}}}(g^{V,\operatorname{op}},\nabla^{V,\operatorname{op}},L^{\operatorname{op}})\equiv-{\widehat{\eta}}^{V}(g^{V},\nabla^{V},L).$

Thus ${\widehat{\eta}}^{V}(g^{V},\nabla^{V},L)\equiv 0$. ∎

We now prove Theorem 1.2.

######  Theorem 4.1.

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 $\mathsf{D}^{S\otimes E^{+}}$ and $\mathsf{D}^{S\otimes E^{-}}$,
respectively, then

${\widehat{\eta}}^{E}(g^{E},\nabla^{E},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\equiv{\widehat{\eta}}^{E^{+}}(g^{E,+},\nabla^{E,+},L_{E^{+}})-{\widehat{\eta}}^{E^{-}}(g^{E,-},\nabla^{E,-},L_{E^{-}}).$
(4.1.16)

###### Proof.

By Lemma 4.1, $L_{E^{+}}\oplus L_{E^{-}}^{\operatorname{op}}\to B$ satisfies
the MF property for $\mathsf{D}^{S\otimes E}$. Thus the left-hand side of
(4.1.16) makes sense.

Consider the (ungraded) triple $(E\oplus E^{-},g^{E}\oplus
g^{E,-},\nabla^{E}\oplus\nabla^{E,-})$. By Proposition 4.1,

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

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 $\mathsf{D}^{S\otimes(E\oplus E^{-})}$, 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

$\alpha^{*}(\nabla^{E,+}\oplus\nabla^{H})=\nabla^{E}\oplus\nabla^{E,-}.$
(4.1.17)

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 $\mathsf{D}^{S\otimes(E\oplus E^{-})}$, which is
defined in terms of
$\nabla^{E}\oplus\nabla^{E,-}=\alpha^{*}(\nabla^{E,+}\oplus\nabla^{H})$ by
(4.1.17), 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
$\mathsf{D}^{S\otimes(E^{+}\oplus H)}$, which is defined in terms of
$\nabla^{E,+}\oplus\nabla^{H}$, and

$\begin{split}&{\widehat{\eta}}^{E\oplus E^{-}}(g^{E}\oplus
g^{E,-},\alpha^{*}(\nabla^{E,+}\oplus\nabla^{H}),(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{E^{-}})\\\
&={\widehat{\eta}}^{E^{+}\oplus H}(g^{E,+}\oplus
g^{H},\nabla^{E,+}\oplus\nabla^{H},L_{\alpha}).\end{split}$ (4.1.18)

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

$\big{(}L_{E^{+}}\oplus(L_{E^{-}}\oplus
L_{E^{-}}^{\operatorname{op}}),g^{L_{E^{+}}}\oplus(g^{L_{E^{-}}}\oplus
g^{L_{E^{-}},\operatorname{op}}),\nabla^{L_{E^{+}}}\oplus(\nabla^{L_{E^{-}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})\big{)}.$
(4.1.19)

By (4.1.17) and (4.1.19), (4.1.18) becomes

$\begin{split}&{\widehat{\eta}}^{E\oplus E^{-}}(g^{E}\oplus
g^{E,-},\nabla^{E}\oplus\nabla^{E,-},(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{E^{-}})\\\
&={\widehat{\eta}}^{E^{+}\oplus H}(g^{E,+}\oplus
g^{H},\nabla^{E,+}\oplus\nabla^{H},L_{E^{+}}\oplus(L_{E^{-}}\oplus
L_{E^{-}}^{\operatorname{op}})).\end{split}$ (4.1.20)

By Propositions 4.1 and 4.2, (4.1.20) becomes

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

Thus (4.1.16) holds. ∎

###  4.2. The analytic index in differential and $\mathbb{R}/\mathbb{Z}$
$K$-theory

In this subsection we prove that the analytic index in differential $K$-theory
$\operatorname{ind}^{a}_{{\widehat{K}}}:{\widehat{K}}_{\operatorname{FL}}(X)\to{\widehat{K}}_{\operatorname{FL}}(B)$
is a well defined group homomorphism (Proposition 1.2).

The Freed–Lott differential $K$-group ${\widehat{K}}_{\operatorname{FL}}(X)$
[5, 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 ${\widehat{K}}_{\operatorname{FL}}(X)$ 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

$\omega_{E}-\omega_{F}\equiv\operatorname{CS}(\nabla^{E}\oplus\nabla^{V_{E}},\alpha^{*}(\nabla^{F}\oplus\nabla^{V_{F}})).$
(4.2.1)

The analytic index of a $\mathbb{Z}_{2}$-graded generator $\mathcal{E}$ of
${\widehat{K}}_{\operatorname{FL}}(X)$ is defined to be

$\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E};L)=\bigg{(}L,g^{L},\nabla^{L},\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\omega+{\widehat{\eta}}^{E}(g^{E},\nabla^{E},L)\bigg{)},$
(4.2.2)

where $(L,g^{L},\nabla^{L})$ is a $\mathbb{Z}_{2}$-graded triple so that $L\to
B$ satisfies the MF property for $\mathsf{D}^{S\otimes E}$. By [7, Corollary
1], $\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E};L)$ does not depend on
the choice of $(L,g^{L},\nabla^{L})$ for which $L\to B$ satisfies the MF
property for $\mathsf{D}^{S\otimes E}$. Henceforth we write
$\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E})$ for
$\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E};L)$.

Let $\mathcal{E}$ and $\mathcal{F}$ be $\mathbb{Z}_{2}$-graded generators of
${\widehat{K}}_{\operatorname{FL}}(X)$. By [7, Corollary 1] and Proposition
4.1, the analytic index of
$\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E}+\mathcal{F})$ is given by

$\begin{split}\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E}+\mathcal{F})&=\bigg{(}L_{E\oplus
F},g^{L_{E\oplus F}},\nabla^{L_{E\oplus
F}},\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge(\omega_{E}+\omega_{F}),\\\
&\qquad{\widehat{\eta}}^{E\oplus F}(g^{E}\oplus
g^{F},\nabla^{E}\oplus\nabla^{F},L_{E\oplus F})\bigg{)},\end{split}$ (4.2.3)

where $(L_{E\oplus F},g^{L_{E\oplus F}},\nabla^{L_{E\oplus F}})$ is the
$\mathbb{Z}_{2}$-graded triple given by (4.1.3). Thus the analytic index in
differential $K$-theory is additive, i.e.

$\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E}+\mathcal{F})=\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E})+\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{F}).$
(4.2.4)

We now prove Proposition 1.2.

###### Proof of Proposition 1.2.

We first show that $\operatorname{ind}^{a}_{{\widehat{K}}}$ passes to a well
defined map
${\widehat{K}}_{\operatorname{FL}}(X)\to{\widehat{K}}_{\operatorname{FL}}(B)$,
i.e. if $\mathcal{E}$ and $\mathcal{F}$ are $\mathbb{Z}_{2}$-graded generators
of ${\widehat{K}}_{\operatorname{FL}}(X)$ satisfying
$\mathcal{E}=\mathcal{F}$, then

$\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{E})=\operatorname{ind}^{a}_{{\widehat{K}}}(\mathcal{F}).$
(4.2.5)

Once (4.2.5) is established, (4.2.4) immediately implies that
$\operatorname{ind}^{a}_{{\widehat{K}}}:{\widehat{K}}_{\operatorname{FL}}(X)\to{\widehat{K}}_{\operatorname{FL}}(B)$
is a group homomorphism.

Since $\mathcal{E}=\mathcal{F}$, 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 (4.2.1) holds. By Proposition 4.2 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 $\mathsf{D}^{S\otimes V_{E}}$
and $\mathsf{D}^{S\otimes V_{F}}$, respectively, and

${\widehat{\eta}}^{V_{E}}(g^{V_{E}},\nabla^{V_{E}},L_{V_{E}})\equiv
0\equiv{\widehat{\eta}}^{V_{F}}(g^{V_{F}},\nabla^{V_{F}},L_{V_{F}}).$ (4.2.6)

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 $\mathsf{D}^{S\otimes E^{+}}$ and $\mathsf{D}^{S\otimes E^{-}}$,
respectively. By Lemma 4.1, $(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}},g^{L_{E^{+}}}\oplus
g^{L_{E^{-}},\operatorname{op}},\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})$
is a $\mathbb{Z}_{2}$-graded triple so that $L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}}\to B$ satisfies the MF property for
$\mathsf{D}^{S\otimes E}$. By Proposition 4.1,

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

is a $\mathbb{Z}_{2}$-graded triple so that $(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{V_{E}}\to B$ satisfies the MF property
for $\mathsf{D}^{S\otimes(E\oplus V_{E})}$.

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 $\mathsf{D}^{S\otimes F^{+}}$ and $\mathsf{D}^{S\otimes F^{-}}$,
respectively. Similarly,

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

is a $\mathbb{Z}_{2}$-graded triple so that $(L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})\oplus L_{V_{F}}\to B$ satisfies the MF property
for $\mathsf{D}^{S\otimes(F\oplus V_{F})}$.

By applying Theorem 3.1 to the $\mathbb{Z}_{2}$-graded isometric isomorphism
$\alpha:(E\oplus V_{E},g^{E}\oplus g^{V_{E}})\to(F\oplus V_{F},g^{F}\oplus
g^{V_{F}})$, 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{)}}$${\big{(}(L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})\oplus L_{V_{F}}\oplus
W_{F},(g^{L_{F^{+}}}\oplus g^{L_{F^{-}},\operatorname{op}})\oplus
g^{L_{V_{F}}}\oplus g^{V_{F}})\big{)}}$$\scriptstyle{h}$ (4.2.7)

such that

$\begin{split}&{\widehat{\eta}}^{F\oplus V_{F}}(g^{F}\oplus
g^{V_{F}},\nabla^{F}\oplus\nabla^{V_{F}},(L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})\oplus L_{V_{F}})\\\
&\quad-{\widehat{\eta}}^{E\oplus V_{E}}(g^{E}\oplus
g^{V_{E}},\nabla^{E}\oplus\nabla^{V_{E}},(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus L_{V_{E}})\\\
&\equiv\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{CS}(\nabla^{E}\oplus\nabla^{V_{E}},\alpha^{*}(\nabla^{F}\oplus\nabla^{V_{F}}))\\\
&\quad-\operatorname{CS}\big{(}(\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{E}}}\oplus\nabla^{W_{E}},h^{*}((\nabla^{L_{F^{+}}}\oplus\nabla^{L_{F^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{F}}}\oplus\nabla^{W_{F}})\big{)}.\end{split}$
(4.2.8)

By Proposition 4.1 and (4.2.6),

${\widehat{\eta}}^{E\oplus V_{E}}(g^{E}\oplus
g^{V_{E}},\nabla^{E}\oplus\nabla^{V_{E}},(L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\oplus
L_{V_{E}})\equiv{\widehat{\eta}}^{E}(g^{E},\nabla^{E},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}}),$ (4.2.9)

and similarly,

${\widehat{\eta}}^{F\oplus V_{F}}(g^{F}\oplus
g^{V_{F}},\nabla^{F}\oplus\nabla^{V_{F}},(L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})\oplus
L_{V_{F}})\equiv{\widehat{\eta}}^{F}(g^{F},\nabla^{F},L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}}).$ (4.2.10)

By (4.2.1), (4.2.9) and (4.2.10), (4.2.8) becomes

$\begin{split}&{\widehat{\eta}}^{F}(g^{F},\nabla^{F},L_{F^{+}}\oplus
L_{F^{-}}^{\operatorname{op}})-{\widehat{\eta}}^{E}(g^{E},\nabla^{E},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\equiv\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge(\omega_{E}-\omega_{F})\\\
&\quad-\operatorname{CS}\big{(}(\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{E}}}\oplus\nabla^{W_{E}},h^{*}((\nabla^{L_{F^{+}}}\oplus\nabla^{L_{F^{-}},\operatorname{op}})\oplus\nabla^{L_{V_{F}}}\oplus\nabla^{W_{F}})\big{)}.\end{split}$
(4.2.11)

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 (4.2.7) and (4.2.11) that (4.2.5) holds. ∎

###  4.3. The RRG theorem in $\mathbb{R}/\mathbb{Z}$ $K$-theory for the
twisted spinc Dirac operator

In this subsection we give an alternative proof of the RRG theorem in
$\mathbb{R}/\mathbb{Z}$ $K$-theory for the twisted spinc Dirac operators
without the kernel bundle assumption (Theorem 1.3).

Define a map
$\operatorname{ch}_{{\widehat{K}}}:{\widehat{K}}_{\operatorname{FL}}(X)\to\Omega^{\operatorname{even}}_{\mathbb{Q}}(X)$
by
$\operatorname{ch}_{{\widehat{K}}}(\mathcal{E})=\operatorname{ch}(\nabla^{E})+d\omega$.
The $\mathbb{R}/\mathbb{Z}$ $K$-group $K^{-1}_{\operatorname{L}}(X)$ [12,
Definition 7] (cf. [5, (2.20)]) can be defined as
$K^{-1}_{\operatorname{L}}(X)=\ker(\operatorname{ch}_{{\widehat{K}}})$. Thus a
$\mathbb{Z}_{2}$-graded generator $\mathcal{E}$ of
$K^{-1}_{\operatorname{L}}(X)$ is a $\mathbb{Z}_{2}$-graded generator of
${\widehat{K}}_{\operatorname{FL}}(X)$ satisfying

$\operatorname{ch}(\nabla^{E,+})-\operatorname{ch}(\nabla^{E,-})=-d\omega.$
(4.3.1)

Note that (4.3.1) implies that
$\operatorname{rank}(E^{+})=\operatorname{rank}(E^{-})$.

The $\mathbb{R}/\mathbb{Q}$ Chern character
$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}:K^{-1}_{\operatorname{L}}(X)\to
H^{\operatorname{odd}}(X;\mathbb{R}/\mathbb{Q})$ is defined as follows. Let
$\mathcal{E}$ be a $\mathbb{Z}_{2}$-graded generator of
$K^{-1}_{\operatorname{L}}(X)$. By (4.3.1), there exist a $k\in\mathbb{N}$ and
an isometric isomorphism $\alpha:(kE^{+},kg^{E,+})\to(kE^{-},kg^{E,-})$ (see
[7, Remark 1] for a proof). Define
$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E})$ by

$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E})=\bigg{[}\frac{1}{k}\operatorname{CS}(\alpha^{*}(k\nabla^{E,-}),k\nabla^{E,+})+\omega\bigg{]}\mod\mathbb{Q}.$
(4.3.2)

It is easy to check that the odd form of the right-hand side of (4.3.2) is
closed. Note that $\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E})$ is
independent of the choices of $k$ and $\alpha$ [12, p.289].

The analytic index $\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}$ in
$\mathbb{R}/\mathbb{Z}$ $K$-theory of a $\mathbb{Z}_{2}$-graded generator
$\mathcal{E}$ of $K^{-1}_{\operatorname{L}}(X)$ is defined by the same formula
(4.2.2). It is easy to check that
$\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}(\mathcal{E})\in
K^{-1}_{\operatorname{L}}(B)$. As an immediate consequence of Proposition 1.2,
the analytic index in $\mathbb{R}/\mathbb{Z}$ $K$-theory

$\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}:K^{-1}_{\operatorname{L}}(X)\to
K^{-1}_{\operatorname{L}}(B)$

is a well-defined group homomorphism.

We now prove Theorem 1.3.

###### Proof of Theorem 1.3.

As mentioned in §1, we prove (1.0.4) at the differential form level. Let
$\mathcal{E}$ be a $\mathbb{Z}_{2}$-graded generator of
$K^{-1}_{\operatorname{L}}(X)$. By (4.3.1), 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
$(L_{E^{-}},g^{L_{E^{-}}},\nabla^{L_{E^{-}}})$ be $\mathbb{Z}_{2}$-graded
triples so that $L_{E^{+}}\to B$ and $L_{E^{-}}\to B$ satisfy the MF property
for $\mathsf{D}^{S\otimes E^{+}}$ and $\mathsf{D}^{S\otimes E^{-}}$,
respectively. By Lemma 4.1,

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

is a the $\mathbb{Z}_{2}$-graded triple so that $L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}}\to B$ satisfies the MF property for
$\mathsf{D}^{S\otimes E}$. By [7, Corollary 1],
$\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}(\mathcal{E})$ is given by

$\begin{split}\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}(\mathcal{E})&=\bigg{(}L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}},g^{L_{E^{+}}}\oplus
g^{L_{E^{-}},\operatorname{op}},\nabla^{L_{E^{+}}}\oplus\nabla^{L_{E^{-}},\operatorname{op}},\\\
&\qquad\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\omega+{\widehat{\eta}}^{E}(g^{E},\nabla^{E},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\bigg{)}.\end{split}$ (4.3.3)

On the other hand, by (4.3.1) and Theorem 4.1,

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

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

Let $k$ be the least common multiple of $k_{1}$ and $k_{2}$, and let
$\alpha:(kE^{+},kg^{E,+})\to(kE^{-},kg^{E,-})$ be an 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

$\begin{split}&{\widehat{\eta}}^{kE^{-}}(kg^{E,-},k\nabla^{E,-},L_{kE^{-}})-{\widehat{\eta}}^{kE^{+}}(kg^{E,+},k\nabla^{E,+},L_{kE^{+}})\\\
&\equiv\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{CS}(k\nabla^{E,+},\alpha^{*}(k\nabla^{E,-}))-\operatorname{CS}(\nabla^{L_{kE^{+}}}\oplus\nabla^{W_{+}},h^{*}(\nabla^{L_{kE^{-}}}\oplus\nabla^{W_{-}})).\end{split}$
(4.3.4)

Recall from (4.1.3) that

$L_{kE^{\pm}}=kL_{E^{\pm}},\qquad
g^{L_{kE^{\pm}}}=kg^{L_{E^{\pm}}},\qquad\nabla^{L_{kE^{\pm}}}=k\nabla^{L_{E^{\pm}}}.$
(4.3.5)

By applying Remark 3.1 to $kL_{E^{+}}\cong kL_{E^{-}}$, we take $W_{+}\to B$
and $W_{-}\to B$ to be the zero bundles. By (4.3.5), (4.3.4) becomes

$\begin{split}&{\widehat{\eta}}^{kE^{-}}(kg^{E,-},k\nabla^{E,-},kL_{E^{-}})-{\widehat{\eta}}^{kE^{+}}(kg^{E,+},k\nabla^{E,+},kL_{E^{+}})\\\
&\equiv\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{CS}(k\nabla^{E,+},\alpha^{*}(k\nabla^{E,-}))-\operatorname{CS}(k\nabla^{L_{E^{+}}},h^{*}(k\nabla^{L_{E^{-}}})).\end{split}$
(4.3.6)

By Proposition 4.1 and Theorem 4.1,

$\begin{split}&{\widehat{\eta}}^{kE^{-}}(kg^{E,-},k\nabla^{E,-},kL_{E^{-}})-{\widehat{\eta}}^{kE^{+}}(kg^{E,+},k\nabla^{E,+},kL_{E^{+}})\\\
&\equiv
k\big{(}{\widehat{\eta}}^{E^{-}}(g^{E,-},\nabla^{E,-},L_{E^{-}})-{\widehat{\eta}}^{E^{+}}(g^{E,+},\nabla^{E,+},L_{E^{+}})\big{)}\\\
&\equiv-k{\widehat{\eta}}^{E}(g^{E},\nabla^{E},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}}).\end{split}$ (4.3.7)

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),

$\displaystyle\operatorname{CS}\big{(}k\nabla^{L_{E^{+}}},h^{*}(k\nabla^{L_{E^{-}}})\big{)}$
$\displaystyle=\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+}\oplus
k\nabla^{L_{E^{+}},-},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\oplus
h_{-}^{*}(k\nabla^{L_{E^{-}},-})\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\big{)}-\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},-},h_{-}^{*}(k\nabla^{L_{E^{-}},-})\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\big{)}-\operatorname{CS}\big{(}(h_{-}^{-1})^{*}(k\nabla^{L_{E^{+}},-}),k\nabla^{L_{E^{-}},-}\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\big{)}+\operatorname{CS}\big{(}k\nabla^{L_{E^{-}},-},(h_{-}^{-1})^{*}(k\nabla^{L_{E^{+}},-})\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k\nabla^{L_{E^{+}},+}\oplus
k\nabla^{L_{E^{-}},-},h_{+}^{*}(k\nabla^{L_{E^{-}},+})\oplus(h_{-}^{-1})^{*}(k\nabla^{L_{E^{+}},-})\big{)}$
$\displaystyle\equiv\operatorname{CS}\big{(}k(\nabla^{L_{E^{+}},+}\oplus\nabla^{L_{E^{-}},-}),(h_{+}\oplus
h_{-}^{-1})^{*}(k(\nabla^{L_{E^{-}},+}\oplus\nabla^{L_{E^{+}},-}))\big{)}.$
(4.3.8)

By (4.3.7) and (4.3.8), (4.3.6) becomes

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

By (2.2.4) it becomes

$\begin{split}&k{\widehat{\eta}}^{E}(g^{E},\nabla^{E},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})+\operatorname{CS}\big{(}(h_{+}\oplus
h_{-}^{-1})^{*}(k(\nabla^{L_{E^{-}},+}\oplus\nabla^{L_{E^{+}},-})),k(\nabla^{L_{E^{+}},+}\oplus\nabla^{L_{E^{-}},-})\big{)}\\\
&\equiv\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\operatorname{CS}(\alpha^{*}(k\nabla^{E,-}),k\nabla^{E,+}).\end{split}$
(4.3.9)

By first dividing both sides of (4.3.9) by $k$, and adding
$\displaystyle\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\omega$ to
both sides of (4.3.9), (4.3.9) becomes

$\begin{split}&\frac{1}{k}\operatorname{CS}\big{(}(h_{+}\oplus
h_{-}^{-1})^{*}(k(\nabla^{L_{E^{-}},+}\oplus\nabla^{L_{E^{+}},-})),k(\nabla^{L_{E^{+}},+}\oplus\nabla^{L_{E^{-}},-})\big{)}\\\
&\qquad+\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\omega+{\widehat{\eta}}^{E}(g^{E},\nabla^{E},L_{E^{+}}\oplus
L_{E^{-}}^{\operatorname{op}})\\\
&\equiv\int_{X/B}\operatorname{Todd}(\nabla^{T^{V}X})\wedge\bigg{(}\frac{1}{k}\operatorname{CS}(\alpha^{*}(k\nabla^{E,-}),k\nabla^{E,+})+\omega\bigg{)}.\end{split}$
(4.3.10)

Since the left-hand side of (4.3.10) is a differential form representative of
$\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\operatorname{ind}^{a}_{\mathbb{R}/\mathbb{Z}}(\mathcal{E}))$
and the right-hand side of (4.3.10) is that of
$\displaystyle\int_{X/B}\operatorname{Todd}(T^{V}X)\cup\operatorname{ch}_{\mathbb{R}/\mathbb{Q}}(\mathcal{E})$,
(1.0.4) holds. ∎

Note that (4.3.10) is a refinement of (1.0.4) at the differential form level.

By following the argument given in the paragraph before [5, Theorem 7.42] and
[5, Theorem A.2, Lemma A.7], all the results in this paper extend to
submersions with noncompact total and base spaces.

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