$\mathrm{pr}_{X}^{*}\mathcal{A}$-module bundle $\mathrm{pr}_{X}^{*}\not{S}$
over $I\times X$ compatible with $\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}}$, we
define its Chern-Simons form by

(3.66)
$\displaystyle\mathrm{CS}_{\mathrm{self}}(\nabla\\!\\!\\!\\!\nabla^{\not{S},I};\nabla^{\mathcal{A}}):=\int_{I}\mathrm{Ph}_{\mathrm{self}}\left(\nabla\\!\\!\\!\\!\nabla^{\not{S},I};\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}}\right)\in\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(X;\mathrm{Ori}(\mathcal{A})).$

  2. (2)

In the same settings, if $\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ is skew-adjoint,
we define its Pontryagin character form by

(3.67)
$\displaystyle\mathrm{Ph}_{\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\nabla^{\mathcal{A}}):=\mathrm{Tr}_{\mathcal{A}}(e^{F(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\nabla^{\mathcal{A}})})\in\Omega_{\mathrm{clo}}^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})-1}(X;\mathrm{Ori}(\mathcal{A})).$

If $\nabla\\!\\!\\!\\!\nabla^{\not{S},I}$ is skew-adjoint, we define its
Chern-Simons form by

(3.68)
$\displaystyle\mathrm{CS}_{\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S},I};\nabla^{\mathcal{A}}):=\int_{I}\mathrm{Ph}_{\mathrm{skew}}\left(\nabla\\!\\!\\!\\!\nabla^{\not{S},I};\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}}\right)\in\Omega^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})-2}(X;\mathrm{Ori}(\mathcal{A})).$

By Proposition 3.56, setting
$\nabla\\!\\!\\!\\!\nabla_{i}^{\not{S}}:=\nabla\\!\\!\\!\\!\nabla^{\not{S},I}|_{\\{i\\}\times
X}$ we have

(3.69)
$\displaystyle\mathrm{Ph}_{\mathrm{self}/\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla_{1}^{\not{S}};\nabla^{\mathcal{A}})-\mathrm{Ph}_{\mathrm{self}/\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla_{0}^{\not{S}};\nabla^{\mathcal{A}})=d\mathrm{CS}_{\mathrm{self}/\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S},I};\nabla^{\mathcal{A}}).$

By Lemma 3.59, we get the following.

######  Lemma 3.70.

For any $C\in\Omega^{1}(X;\mathcal{A}^{0}_{\mathrm{skew}})$, we have

$\displaystyle\mathrm{Ph}_{\mathrm{self}/\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\nabla^{\mathcal{A}})=\mathrm{Ph}_{\mathrm{self}/\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S}}+C;\nabla^{\mathcal{A}}+\\{C,-\\}).$

Moreover, if we have an $\mathcal{A}$-superconnection
$\nabla\\!\\!\\!\\!\nabla^{\not{S},I}$ as in Definition 3.64, we have

$\displaystyle\mathrm{CS}_{\mathrm{self}/\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S},I};\nabla^{\mathcal{A}})=\mathrm{CS}_{\mathrm{self}/\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S},I}+\mathrm{pr}_{X}^{*}C;\nabla^{\mathcal{A}}+\\{C,-\\}).$

####  3.3.1. The relation with the superconnection formalism by Quillen
[Qui85]

Here we explain how to recover the Quillen’s superconnection formalism [Qui85]
as a special case of the formalisms developed in this subsection.

First of all, Quillen works in the $\mathbb{C}$-linear setting. Thus,
precisely speaking we recover the $\mathbb{R}$-linear version of Quillen’s
superconnection formalism (by the obvious restriction of the coefficients
$\mathbb{R}\subset\mathbb{C}$) as a special case of the formalisms in this
subsection. As we will explain in Subsection 6.1, the formalism in this
subsection can be modified to the $\mathbb{C}$-linear setting in a
straightforward way. Based on the modified formalism, we can generalize the
following discussion in a straightforward manner to recover the formalism in
[Qui85].

Recall that in [Qui85] the superconnections are considered for
$\mathbb{Z}_{2}$-graded vector bundles (which we call the even superconnection
formalism) and ungraded vector bundles (which we call the odd superconnection
formalism).

For the former, the relation with our formulation is very simple. The even
superconnection formalism is our superconnection formalism in the case where
$\mathcal{A}$ is the trivial bundle with fiber $Cl_{0,1}$. Indeed, a
$\mathbb{Z}_{2}$-grading is equivalent to a $Cl_{0,1}$-module structure, and
the trace $\mathrm{Tr}_{u}$ for a volume element $u\in Cl_{0,1}$ recovers the
usual supertrace.

Now we explain the latter. The odd superconnection formalism can be regarded
as our superconnection formalism in the case where $\mathcal{A}$ is the
trivial bundle with fiber $Cl_{1,1}$, as follows. In [Qui85, Section 5], we
start with an ungraded vector bundle $\not{S}$, form a vector bundle
$\not{S}^{\prime}:=Cl_{0,1}\otimes\not{S}$, regard $\not{S}^{\prime}$ as a
right $Cl_{0,1}$-module, and consider the bundle
$\mathrm{End}_{\sigma}(\not{S}^{\prime})$ of endomorphisms on
$\not{S}^{\prime}$ commuting with the right $Cl_{0,1}$-action, where
$\sigma\in Cl_{0,1}$ denotes a generator. We have

$\displaystyle\mathrm{End}_{\sigma}(\not{S}^{\prime})=\\{\mathrm{id}\otimes
a+(\sigma\cdot)\otimes b\ |\ a,b\in\mathrm{End}({\not{S}})\\}.$

Then the odd supertrace is defined as

(3.71)
$\displaystyle\mathrm{tr}_{\sigma}\colon\Omega^{*}(X;\mathrm{End}_{\sigma}(\not{S}^{\prime}))$
$\displaystyle\to\Omega^{*}(X)$ $\displaystyle\mathrm{id}\otimes
a+(\sigma\cdot)\otimes b$ $\displaystyle\mapsto\mathrm{Tr}_{\not{S}}(b).$

Now we turn to our formalism. Recall that in the generalized Clifford
superconnection formalism of this subsection, we required that $\mathcal{A}$
is a bundle of nondegenerete algebras. Thus, for an ungraded vector bundle
${\not{S}}$ regarded as an $\mathbb{R}=Cl_{0,0}$-module bundle, we cannot
apply the formalism directly. In such a case, we tensor a negligible module to
produce a nondegenerate algebra, use the isomorphism in Lemma 2.12 (3), and
apply the superconnection formalism to the latter. Actually, the above process
of producing $\mathrm{tr}_{\sigma}$ on
$\mathrm{End}_{\sigma}(\not{S}^{\prime})$ can be regarded as performing this
procedure. Indeed, consider the $\mathbb{Z}_{2}$-graded vector space
$V=V^{0}\oplus V^{1}=\mathbb{R}\oplus\mathbb{R}$ and the associated negligible
module $\mathrm{End}(V)\simeq Cl_{1,1}$. The grading operator $\gamma_{V}$ on
$V$ is a volume element for $\mathrm{End}(V)$. Then, given an ungraded vector
bundle ${\not{S}}$, as in Lemma 2.12 (3) we consider the
$\mathrm{End}(V)$-module bundle $V\otimes{\not{S}}$. The bundle
$V\otimes{\not{S}}$ corresponds to the above
$\not{S}^{\prime}=Cl_{0,1}\otimes\not{S}$, and the action by
$\gamma_{V}\otimes\mathrm{id}_{\not{S}}$ corresponds to
$(\sigma\cdot)\otimes\mathrm{id}_{\not{S}}$777 Be careful that in this
correspondence the $\mathbb{Z}_{2}$-grading on $V$ does not correspond to the
$\mathbb{Z}_{2}$-grading on $Cl_{0,1}$. In our generalized Clifford
superconnection formalism, as an $\mathrm{End}(V)$-module we just regard
$V\otimes{\not{S}}$ as an ungraded module, so this difference is irrelevant. .
We have

(3.72) $\displaystyle\mathrm{End}^{0}_{\mathrm{End}(V)}(V\otimes\not{S})$
$\displaystyle=\\{\mathrm{id}_{V}\otimes a\ |\ a\in\mathrm{End}(\not{S})\\},$
(3.73) $\displaystyle\mathrm{End}^{1}_{\mathrm{End}(V)}(V\otimes\not{S})$
$\displaystyle=\\{\gamma_{V}\otimes b\ |\ b\in\mathrm{End}(\not{S})\\},$

and the $\mathrm{Tr}_{\gamma_{V}}$ in Definition 3.1 is given by

(3.74)
$\displaystyle\mathrm{Tr}_{\gamma_{V}}\colon\Omega^{*}(X;\mathrm{End}^{1}_{\mathrm{End}(V)}(V\otimes\not{S}))$
$\displaystyle\to\Omega^{*}(X)$ $\displaystyle\mathrm{id}_{V}\otimes
a+\gamma_{V}\otimes b$ $\displaystyle\mapsto\mathrm{Tr}_{\not{S}}(b).$

Thus we see that we recover the odd supertrace $\mathrm{tr}_{\sigma}$ in
(3.71), as desired.

###  3.4. The Pontryagin character forms for gradations

Let $X$, $\mathcal{A}$, $\nabla^{\mathcal{A}}$ and $\not{S}$ with an inner
product be as in the last subsection. Note that we are assuming that the
fibers of $\mathcal{A}$ are nondegenerate. Let $\nabla^{\not{S}}$ be a self-
adjoint $\mathcal{A}$-connection (i.e., one which increases the form degree by
one) on $\not{S}$ compatible with $\nabla^{\mathcal{A}}$. In this setting,
given a smooth invertible section $h\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$, we are going to
define its Pontryagin character form
$\mathrm{Ph}_{\mathrm{self}}(h;\nabla^{\mathcal{A}},\nabla^{\not{S}})$
(Definition 3.80), generalizing Definition 3.15.

Consider the manifold $(0,\infty)_{t}\times X$, and the bundles
$\mathrm{pr}_{X}^{*}\mathcal{A}$ and $\mathrm{pr}_{X}^{*}\not{S}$ on it. We
consider the following $\mathrm{pr}_{X}^{*}\mathcal{A}$-superconnection on
$\mathrm{pr}_{X}^{*}\not{S}$ compatible with
$\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}}$,

(3.75) $\displaystyle\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th,$

where $t$ denotes the corrdinate on $(0,\infty)$. Then (3.75) is self-adjoint
in the sense of Definition 3.60, so we have

(3.76)
$\displaystyle\mathrm{Ph}_{\mathrm{self}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th;\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})\in\Omega_{\mathrm{clo}}^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})+1}((0,\infty)\times
X;\mathrm{pr}_{X}^{*}\mathrm{Ori}(\mathcal{A})).$

######  Lemma 3.77.

In the above settings, the following integration converges pointwise,

(3.78)
$\displaystyle\int_{(0,\infty)}\mathrm{Ph}_{\mathrm{self}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th;\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})\in\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(X;\mathrm{Ori}(\mathcal{A})).$

We have

(3.79) $\displaystyle
d\int_{(0,\infty)}\mathrm{Ph}_{\mathrm{self}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th;\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})=-\mathrm{Ph}_{\mathrm{self}}(\nabla^{\not{S}};\nabla^{\mathcal{A}}).$

###### Proof.

Working locally, we may trivialize $\mathcal{A}$ and $\not{S}$. By Lemma 3.70
and the fact that under the action (3.45) the superconnection (3.75)
transforms as
$\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th\mapsto\mathrm{pr}_{X}^{*}(\nabla^{\not{S}}+C)+th$,
we may assume $\nabla^{\mathcal{A}}=d$. In this case we have
$(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th)^{2}\in\Omega^{*}((0,\infty)\times
X;\mathrm{End}_{\mathcal{A}}(\not{S}))$, and
$\mathrm{Ph}_{\mathrm{self}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th)=\mathrm{Tr}_{\mathcal{A}}(e^{-(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th)^{2}})$.
Fix any Riemannian metric on $X$. It is enough to show that, for each $x\in
X$, there exists $c>0$ and $t_{0}\in(0,\infty)$ such that

$\displaystyle\|e^{-(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th)^{2}}\|<e^{-t^{2}c}$

on $[t_{0},\infty)\times\\{x\\}$. We have

$\displaystyle(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th)^{2}=h+(\nabla^{\not{S}})^{2}+t\\{\nabla^{\not{S}},h\\}+t^{2}h^{2}.$

Since $h(x)$ is self-adjoint and invertible, there exists a constant $a>0$
such that $h(x)^{2}>a\cdot\mathrm{Id}$. From this we get the convergence of
(3.78). (3.79) follows from (3.69) and the convergence
$\lim_{t\to\infty}\mathrm{Ph}_{\mathrm{self}}(\nabla^{\not{S}}+th;\nabla^{\mathcal{A}})=0$
shown similarly. ∎

By Lemma 3.77, we get

$\displaystyle-\pi^{-1/2}\mathcal{R}\circ\int_{(0,\infty)}\mathrm{Ph}_{\mathrm{self}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th;\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})\in\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(X;\mathrm{Ori}(\mathcal{A})).$

Here $\mathcal{R}$ is the endomorphism on differential forms defined in
(2.57).

######  Definition 3.80
($\mathrm{Ph}_{\mathrm{self}}(h;\nabla^{\mathcal{A}},\nabla^{\not{S}})$,
$\mathrm{CS}_{\mathrm{self}}(h_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})$).

Let $X$, $\mathcal{A}$, $\not{S}$, $\nabla^{\mathcal{A}}$ and
$\nabla^{\not{S}}$ be as above.

  1. (1)

For any smooth invertible section $h\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$, we define

(3.81)
$\mathrm{Ph}_{\mathrm{self}}(h;\nabla^{\mathcal{A}},\nabla^{\not{S}}):=-\pi^{-1/2}\mathcal{R}\circ\int_{(0,\infty)}\mathrm{Ph}_{\mathrm{self}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th;\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})\\\
\in\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(X;\mathrm{Ori}(\mathcal{A})).$

The convergence of the integration follows from Lemma 3.77. Note that the form
above is not a closed one (see (3.79)).

  2. (2)

For any smooth invertible section $h_{I}\in C^{\infty}(I\times
X;\mathrm{pr}_{X}^{*}\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$, we define

(3.82)
$\mathrm{CS}_{\mathrm{self}}(h_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}}):=\int_{I}\mathrm{Ph}_{\mathrm{self}}\left(h_{I};\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}},\mathrm{pr}_{X}^{*}\nabla^{\not{S}}\right)\\\
\in\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})-1}(X;\mathrm{Ori}(\mathcal{A})).$

######  Lemma 3.83.

We have

(3.84)
$d\mathrm{CS}_{\mathrm{self}}\left(h_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}}\right)\\\
=\mathrm{Ph}_{\mathrm{self}}\left(h_{I}|_{\\{1\\}\times
X};\nabla^{\mathcal{A}},\nabla^{\not{S}}\right)-\mathrm{Ph}_{\mathrm{self}}\left(h_{I}|_{\\{0\\}\times
X};\nabla^{\mathcal{A}},\nabla^{\not{S}}\right).$

###### Proof.

Based on (3.79), the difference of the left and the right hand side of (3.84)
is computed as

$\displaystyle\int_{I}d\mathrm{Ph}_{\mathrm{self}}\left(h_{I};\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}},\mathrm{pr}_{X}^{*}\nabla^{\not{S}}\right)$
$\displaystyle=(\mbox{const})\cdot\mathcal{R}\circ\int_{I}\mathrm{Ph}_{\mathrm{self}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}};\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})$
$\displaystyle=(\mbox{const})\cdot\mathcal{R}\circ\int_{I}\mathrm{pr}_{X}^{*}\mathrm{Ph}_{\mathrm{self}}(\nabla^{\not{S}};\nabla^{\mathcal{A}})=0.$

∎

Note that, in Definition 3.80 (1), if we have a submanifold $Y\subset X$ and
two smooth sections $h_{0},h_{1}\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$ such that
$h_{0}|_{Y}=h_{1}|_{Y}$, we have

(3.85)
$\displaystyle\mathrm{Ph}_{\mathrm{self}}(h_{1};\nabla^{\mathcal{A}},\nabla^{\not{S}})-\mathrm{Ph}_{\mathrm{self}}(h_{0};\nabla^{\mathcal{A}},\nabla^{\not{S}})\in\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}_{\mathrm{clo}}(X,Y;\mathrm{Ori}(\mathcal{A})).$

The closedness follows from (3.79). Similarly, in Definition 3.80 (2), if
$h_{I}|_{I\times Y}$ is constant in the $I$-direction, we have
$\mathrm{CS}_{\mathrm{self}}(h_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})|_{Y}=0$,
so that we have

(3.86)
$\displaystyle\mathrm{CS}_{\mathrm{self}}(h_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})\in\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})-1}(X,Y;\mathrm{Ori}(\mathcal{A})).$

Moreover, we can also easily show that the above Chern-Simons form modulo
exact forms only depends on the homotopy class of the homotopy $h_{I}$, that
is, if we have two choices of homotopies $h_{I}$ and $h^{\prime}_{I}$ which
are homotopic in the obvious sense, then we have

(3.87)
$\displaystyle\mathrm{CS}_{\mathrm{self}}(h_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})-\mathrm{CS}_{\mathrm{self}}(h^{\prime}_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})\in\mathrm{Im}(d).$

Now we investigate into properties of
$\mathrm{Ph}_{\mathrm{self}}(h;\nabla^{\mathcal{A}},\nabla^{\not{S}})$. Our
goal is Theorem 3.96, which says that
$\mathrm{Ph}_{\mathrm{self}}(h;\nabla^{\mathcal{A}},\nabla^{\not{S}})$
realizes the twisted topological Pontryagin character homomorphism in
Subsection 2.6. It generalizes Corollary 3.31.

First, by Lemma 3.70 we get the following.

######  Lemma 3.88.

For any $C\in\Omega^{1}(X;\mathcal{A}^{0}_{\mathrm{skew}})$, we have

$\displaystyle\mathrm{Ph}_{\mathrm{self}}(h;\nabla^{\mathcal{A}},\nabla^{\not{S}})$
$\displaystyle=\mathrm{Ph}_{\mathrm{self}}(h;\nabla^{\mathcal{A}}+C,\nabla^{\not{S}}+C),$
$\displaystyle\mathrm{CS}_{\mathrm{self}}(h_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})$
$\displaystyle=\mathrm{CS}_{\mathrm{self}}(h_{I};\nabla^{\mathcal{A}}+C,\nabla^{\not{S}}+C).$

Next, we show that the cohomology class of (3.85) does not depend on the
connections, as follows.

######  Lemma 3.89.

Let $X$, $\mathcal{A}$ and $\not{S}$ be as above. Assume we have a submanifold
$Y\subset X$ and two smooth sections $h_{0},h_{1}\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$ such that
$h_{0}|_{Y}=h_{1}|_{Y}$. Then the cohomology class of the difference element
(3.85),

(3.90)
$\mathrm{Rham}\left(\mathrm{Ph}_{\mathrm{self}}(h_{1};\nabla^{\mathcal{A}},\nabla^{\not{S}})-\mathrm{Ph}_{\mathrm{self}}(h_{0};\nabla^{\mathcal{A}},\nabla^{\not{S}})\right)\\\
\in
H^{4\mathbb{Z}+{\mathrm{type}}(\mathcal{A})}(X,Y;\mathrm{Ori}(\mathcal{A})),$

does not depend on the compatible pair
$(\nabla^{\mathcal{A}},\nabla^{\not{S}})$.

###### Proof.

Any two compatible pairs $(\nabla_{0}^{V},\nabla_{0}^{\not{S}})$ and
$(\nabla_{1}^{V},\nabla_{1}^{\not{S}})$ are connected by a homotopy. As
always, we can transgress the difference of the forms (3.85) corresponding to
those connections, by integrating the form (3.85) constructed by using the
homotopy in the $I$-direction. ∎

Recall that twists in $KO_{+}$ by negligible bundles are trivial (Fact 2.33).
Here we show the corresponding statement for the Pontryagin character forms.
Let $E=E^{0}\oplus E^{1}$ be a smooth $\mathbb{Z}_{2}$-graded vector bundle
over $X$ equipped with a fiberwise positive definite inner product on each
$E^{i}$, associating the negligible bundle $\mathrm{End}(E)$. Recall Lemma
2.12. Given any other $\mathcal{A}$, for an $\mathcal{A}$-module bundle
$\not{S}$ with an inner product we get an
$\mathrm{End}(E)\widehat{\otimes}\mathcal{A}$-module bundle $E\otimes\not{S}$,
and we have a canonical isomorphism
$\psi^{E}\colon\mathrm{Self}^{*}_{\mathcal{A}}(\not{S})\simeq\mathrm{Self}^{*}_{\mathrm{End}(E)\widehat{\otimes}\mathcal{A}}(E\otimes\not{S})$.

######  Lemma 3.91.

In the above setting, let $\nabla^{\mathcal{A}}$ be a connection on
$\mathcal{A}$ and $\nabla^{\not{S}}$ be an $\mathcal{A}$-connection on
$\not{S}$ compatible with $\nabla^{\mathcal{A}}$. Let $\nabla^{E}$ be any
orthogonal connection on $E$, and we denote by $\nabla^{\mathrm{End}(E)}$ the
induced connection on $\mathrm{End}(E)$. Then $\nabla^{E}\otimes
1+1\otimes\nabla^{\not{S}}$ is an
$\mathrm{End}(E)\widehat{\otimes}\mathcal{A}$-connection on
$\mathrm{End}(E)\otimes\not{S}$ compatible with
$\nabla^{\mathrm{End}(E)}\otimes 1+1\otimes\nabla^{\mathcal{A}}$. Furthermore,
for any $h\in C^{\infty}(X;\mathrm{Self}^{*}_{\mathcal{A}}(\not{S}))$, we have

(3.92)
$\mathrm{Ph}_{\mathrm{self}}(h;\nabla^{\mathcal{A}},\nabla^{\not{S}})\\\
=\mathrm{Ph}_{\mathrm{self}}(\psi^{E}(h);\nabla^{\mathrm{End}(E)}\otimes
1+1\otimes\nabla^{\mathcal{A}},\nabla^{E}\otimes 1+1\otimes\nabla^{\not{S}}).$

Here we are using
$\mathrm{Ori}(\mathcal{A})\simeq\mathrm{Ori}(\mathrm{End}(E)\widehat{\otimes}\mathcal{A})$
given by $u\mapsto\gamma_{E}\widehat{\otimes}u$.

###### Proof.

Locally we can trivialize $E$ and write $\nabla^{E}=d+C^{E}$ for some
$\mathrm{End}(E)^{0}$-valued one-form $C^{E}$. Then we have
$\nabla^{\mathrm{End}(E)}=d+\\{C^{E},-\\}$, so the result follows from Lemma
3.88 and Lemma 3.3. ∎

Now we turn to the relation with the twisted topological Pontryagin character
homomorphism in Subsection 2.6. For this, we use the explanation of the
twisted $KO$-theory in Subsection 2.5 in terms of
$\mathrm{Self}_{A}^{*}(-)$-bundles.

Recall that in the untwisted case, Pontryagin character forms can be regarded
as the pullbacks of the universal Pontryagin form (3.21). Here in the twisted
case, we also have the corresponding universal form. Let $X$, $\mathcal{A}$
and $\not{S}$ be as before, and consider the fiber bundle
$\pi\colon\mathrm{Self}_{\mathcal{A}}^{*}(\not{S})\to X$. We consider the
pullback of $\mathcal{A}$ and $\not{S}$ on
$\mathrm{Self}_{\mathcal{A}}^{*}(\not{S})$. Then we have the universal
gradation, which is given by the tautological section,

$\displaystyle h^{\mathcal{A},\not{S}}_{\mathrm{univ}}\in
C^{\infty}(\mathrm{Self}_{\mathcal{A}}^{*}(\not{S});\mathrm{Self}_{\pi^{*}\mathcal{A}}^{*}(\pi^{*}\not{S})).$

Given a pair $(\nabla^{\mathcal{A}},\nabla^{\not{S}})$ as before, we pull them
back by $\pi$, and get the universal Pontryagin form,

(3.93)
$\displaystyle\mathrm{Ph}_{\mathrm{self}}\left(h^{\mathcal{A},\not{S}}_{\mathrm{univ}};\pi^{*}\nabla^{\mathcal{A}},\pi^{*}\nabla^{\not{S}}\right)\in\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(\mathrm{Self}_{\mathcal{A}}^{*}(\not{S});\mathrm{Ori}(\mathcal{A})),$

where we abuse the notation to write the pullback of
$\mathrm{Ori}(\mathcal{A})$ by the same symbol. Note that it is not closed
(3.79). It is universal in the following sense. Any section $h\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$ is in particular a
smooth map from $X$ to $\mathrm{Self}_{\mathcal{A}}^{*}(\not{S})$, and the
gradation $h$ is the pullback of the universal gradation
$h^{\mathcal{A},\not{S}}_{\mathrm{univ}}$ by this map. By the obvious
naturality of the Pontryagin character forms, we see that

$\displaystyle\mathrm{Ph}_{\mathrm{self}}(h;\nabla^{\mathcal{A}},\nabla^{\not{S}})=h^{*}\mathrm{Ph}_{\mathrm{self}}\left(h^{\mathcal{A},\not{S}}_{\mathrm{univ}};\pi^{*}\nabla^{\mathcal{A}},\pi^{*}\nabla^{\not{S}}\right).$

Recall that, as explained in Subsection 2.5, twisted $KO$-theory groups are
given in terms of sections of bundles of based spaces. As explained around
(2.66), it corresponds to considering pairs $(\not{S},h_{0})$ of an
$\mathcal{A}$-module bundle and a section $h_{0}\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$. For such a pair, we
get a closed form,

(3.94)
$\mathrm{Ph}_{\mathrm{self}}\left(h^{\mathcal{A},\not{S}}_{\mathrm{univ}};\pi^{*}\nabla^{\mathcal{A}},\pi^{*}\nabla^{\not{S}}\right)-\mathrm{Ph}_{\mathrm{self}}\left(\pi^{*}h_{0};\pi^{*}\nabla^{\mathcal{A}},\pi^{*}\nabla^{\not{S}}\right)\\\
\in\Omega_{\mathrm{clo}}^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}),\\{h_{0}\\};\mathrm{Ori}(\mathcal{A})).$

The cohomology class of the form (3.94) is independent of
$(\nabla^{\mathcal{A}},\nabla^{\not{S}})$ by Lemma 3.89, which we denote by

(3.95) $\displaystyle\mathrm{Ph}_{\mathrm{self}}(\mathcal{A},\not{S},h_{0})\in
H^{4\mathbb{Z}+{\mathrm{type}}(\mathcal{A})}(\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}),\\{h_{0}\\};\mathrm{Ori}(\mathcal{A})).$

######  Theorem 3.96.

Let $X$ be connected. The class (3.95) is the pullback of the universal
topological Pontryagin character class

$\displaystyle[\mathrm{Ph}_{\mathrm{top}}]\in
H^{4\mathbb{Z}+{\mathrm{type}}(\mathcal{A})}((\mathcal{KO}_{\mathrm{type}(\mathcal{A})})_{\iota},\\{\beta\\};E\mathbb{Z}_{2}\times_{\mathbb{Z}_{2}}\mathbb{R})$

in (2.74) under the map given as the composition

$\displaystyle(\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}),\\{h_{0}\\})\hookrightarrow(\mathcal{KO}_{\mathcal{A}},\\{\beta\\})\to((\mathcal{KO}_{\mathrm{type}(\mathcal{A})})_{\iota},\\{\beta\\}),$

where the first inclusion map is (2.67) and the second map comes from the
composition of the classifying map $X\to B\mathrm{Aut}_{*,\mathbb{Z}_{2}}(A)$
of $\mathcal{A}$ and the map $(w_{1},w_{2})\colon
B\mathrm{Aut}_{*,\mathbb{Z}_{2}}(A)\to B\mathrm{O}\langle 0,1,2\rangle$.

###### Proof.

We use the notations in Subsection 2.6. By the naturality of the Pontryagin
character forms, the class (3.95) gives an element

(3.97)
$\displaystyle\\{\mathrm{Ph}_{\mathrm{self}}(V,\not{S},h_{0})\\}_{(S,h_{0})}$
$\displaystyle\in\varprojlim_{(\not{S},h_{0})}H^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}\left(\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}),\\{h_{0}\\};\mathrm{Ori}(\mathcal{A})\right)$
$\displaystyle\simeq
H^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(\mathcal{KO}_{\mathcal{A}},\\{\beta\\};\mathrm{Ori}(\mathcal{A})).$

Here the inverse limit runs over the pairs $(\not{S},h_{0})$ of
$\mathcal{A}$-module bundles with inner product and sections $h_{0}\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{\dagger}(\not{S}))$ with respect to
their inclusions. The isomorphism in (3.97) is given by Lemma 2.68. Moreover,
by the same naturality, we see that the class (3.97) is natural in
$(X,\mathcal{A})$. Since $B\mathrm{Aut}_{*,\mathbb{Z}_{2}}(A)$ can be realized
as a direct limit of closed manifolds, we get the element

(3.98) $\displaystyle\\{\mathrm{Ph}(V,\not{S},h_{0})\\}_{(X,V,S,h_{0})}\in
H^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(\mathcal{KO}_{E\mathrm{Aut}_{*,\mathbb{Z}_{2}}(A)},\\{\beta\\};\mathrm{Ori}(\mathcal{A})).$

Here we abuse the notation to denote by
$\mathcal{KO}_{E\mathrm{Aut}_{*,\mathbb{Z}_{2}}(A)}$ the bundle over
$B\mathrm{Aut}_{*,\mathbb{Z}_{2}}(A)$ defined by applying (2.64) to
$P=E\mathrm{Aut}_{*,\mathbb{Z}_{2}}(A)$. In turn, by Lemma 3.91 this element
is the pullback under
$(w_{1},w_{2})\colon\mathcal{KO}_{E\mathrm{Aut}_{*,\mathbb{Z}_{2}}(A)}\to(\mathcal{KO}_{\mathrm{type}(A)})_{\iota}$
of an element on $(\mathcal{KO_{\mathrm{type}(\mathcal{A})}})_{\iota}$ denoted
temporarily as

(3.99) $\displaystyle[\mathrm{Ph}^{\prime}]\in
H^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}((\mathcal{KO}_{\mathrm{type}(A))})_{\iota},\\{\beta\\};E\mathbb{Z}_{2}\times_{\mathbb{Z}_{2}}\mathbb{R}).$

Moreover, the element (3.99) restricts to the universal class for the
untwisted topological Pontryagin character

$[\mathrm{Ph}_{\mathrm{top}}]\in
H^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(KO_{\mathrm{type}(A)},\\{*\\};\mathrm{Ori}(\mathcal{A}))$

on each fiber. To see this, it is enough to show the corresponding statement
for any of $\mathcal{KO}_{\mathcal{A}}$. When $X=\mathrm{pt}$, the element
(3.95) coincides with the untwisted universal Pontryagin form in (3.21). Using
Theorem 3.29 and taking the inverse limit, we get the desired claim.

By Lemma 2.75, this property establishes

$\displaystyle[\mathrm{Ph}^{\prime}]=[\mathrm{Ph}_{\mathrm{top}}],$

which leads to the theorem. ∎

Recall that, for a CW-pair $(X,Y)$, given $\mathcal{A}$ over $X$, we have
(2.73)

$\displaystyle\mathrm{Ph}_{\mathrm{top}}\colon KO^{\mathcal{A}}_{+}(X,Y)\to
H^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}(X,Y;\mathrm{Ori}(\mathcal{A})).$

Theorem 3.96 and the definition of the isomorphism (2.66) imply the following.

######  Corollary 3.100.

In the setting of Definition 3.80, let $(X,Y)$ be an object of
$\mathrm{MfdPair}_{f}$. Suppose we have two sections $h_{0},h_{1}\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$ such that
$h_{0}|_{Y}=h_{1}|_{Y}$. Then we have

(3.101)
$\displaystyle\mathrm{Ph}_{\mathrm{top}}([\not{S},h_{0},h_{1}])=\mathrm{Rham}\left(\mathrm{Ph}_{\mathrm{self}}(h_{1};\nabla^{\mathcal{A}},\nabla^{\not{S}})-\mathrm{Ph}_{\mathrm{self}}(h_{0};\nabla^{\mathcal{A}},\nabla^{\not{S}})\right).$

Here we note that the right hand side is well-defined by (3.85).

##  4\. The definition of differential $KO$-theory $\widehat{KO}_{+}$

In this section we define our model of differential $KO$ groups. We deal with
the untwisted case in Subsection 4.1, and the twisted case in Subsection 4.2.
In this section we work over $\mathbb{R}$.

###  4.1. The untwisted groups $\widehat{KO}^{A}_{+}$

In this subsection we give a model $\widehat{KO}^{A}_{+}$ for the untwisted
differential $KO$-theory groups, by refining Definition 2.21. In this
subsection we use the notations in Subsection 3.2.

######  Definition 4.1.

Let $A$ be a nondegenerate simple central graded $*$-algebra. Let $(X,Y)$ be
an object in $\mathrm{MfdPair}_{f}$.

  * •

A $\widehat{KO}_{+}$-cocycle $(S,h_{0},h_{1},\eta)$ on $(X,Y)$ consists of an
$A$-module $S$ with an inner product, two smooth maps $h_{0},h_{1}\in
C^{\infty}(X,\mathrm{Self}_{A}^{*}(S))$ such that $h_{0}|_{Y}=h_{1}|_{Y}$, and
an element

$\eta\in\Omega^{4\mathbb{Z}+\mathrm{type}(A)-1}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d).$

  * •

Two $\widehat{KO}_{+}$-cocycles $(S,h_{0},h_{1},\eta)$ and
$(S^{\prime},h^{\prime}_{0},h^{\prime}_{1},\eta^{\prime})$ are isomorphic if
there exists an isometric isomorphism of $A$-modules $f\colon S\simeq
S^{\prime}$ such that $f\circ h_{i}=h^{\prime}_{i}\circ f$ for $i=0,1$, and we
have $\eta=\eta^{\prime}$.

######  Definition 4.2 ($\widehat{KO}_{+}^{A}(X,Y)$).

Let $A$ and $(X,Y)$ be as in Definition 4.1

  * •

We introduce an abelian monoid structure on the set $\widehat{M}^{A}_{+}(X,Y)$
of isomorphism classes of $\widehat{KO}_{+}$-cocycles $(S,h_{0},h_{1},\eta)$
on $(X,Y)$ by

$\displaystyle[S,h_{0},h_{1},\eta]+[S^{\prime},h^{\prime}_{0},h^{\prime}_{1},\eta^{\prime}]=[S\oplus
S^{\prime},h_{0}\oplus h^{\prime}_{0},h_{1}\oplus
h^{\prime}_{1},\eta+\eta^{\prime}].$

  * •

We define $\widehat{Z}^{A}_{+}(X,Y)$ to be the submonoid of
$\widehat{M}^{A}_{+}(X,Y)$ consisting of elements of the form

$\displaystyle[S,h_{0},h_{1},\mathrm{CS}_{\mathrm{self}}(h_{I})],$

where $h_{I}$ is a smooth homotopy between $h_{0}$ and $h_{1}$ which is
constant on $Y$, i.e., a smooth map $h_{I}\in C^{\infty}(I\times
X,\mathrm{Self}^{*}_{A}(S))$ with $h_{I}|_{\\{i\\}\times X}=h_{i}$ for $i=0,1$
and $h_{I}|_{\\{t\\}\times Y}=h_{0}|_{Y}$ for all $t\in I$.

  * •

We define
$\widehat{KO}^{A}_{+}(X,Y):=\widehat{M}^{A}_{+}(X,Y)/\widehat{Z}^{A}_{+}(X,Y)$.

######  Lemma 4.3.

$\widehat{KO}_{+}^{A}(X,Y)$ is an abelian group.

###### Proof.

The additive inverse of an element
$[S,h_{0},h_{1},\eta]\in\widehat{KO}_{+}^{A}(X,Y)$ is given as follows.
Consider the $A$-module bundle $S\oplus S$, and note that $h_{0}\oplus h_{1}$
and $h_{1}\oplus h_{0}$ are homotopic relative to $Y$ in
$C^{\infty}(X;\mathrm{Self}_{A}^{*}(S\oplus S))$. Take any smooth homotopy
$h_{I}$ between them, which is constant on $Y$. Then we have the following
equality in $\widehat{KO}_{+}^{A}(X,Y)$.

$\displaystyle\left[S\oplus S,h_{0}\oplus h_{1},h_{1}\oplus
h_{0},\mathrm{CS}_{\mathrm{self}}\left(h_{I}\right)\right]=0.$

This means that we have

(4.4)
$\displaystyle-[S,h_{0},h_{1},\eta]=\left[S,h_{1},h_{0},-\eta+\mathrm{CS}_{\mathrm{self}}\left(h_{I}\right)\right].$

∎

The abelian group $\widehat{KO}_{+}^{A}(X,Y)$ has the obvious naturality in
$(X,Y)$, so we regard $\widehat{KO}_{+}^{A}$ as a contravariant functor,

(4.5)
$\displaystyle\widehat{KO}_{+}^{A}\colon\mathrm{MfdPair}^{\mathrm{op}}_{f}\to\mathrm{Ab}.$

The remainder of this subsection is devoted to showing that
$\widehat{KO}_{+}^{A}$ satisfies the axioms of differential extension as
developed by Bunke and Schick [BS09, BS12]. First of all, we define the
structure homomorphisms for $\widehat{KO}^{A}_{+}$ as follows.

######  Definition 4.6 (Structure homomorphisms for $\widehat{KO}^{A}_{+}$).

We define the following structure homomorphisms, which are natural in $(X,Y)$.

$\displaystyle R$
$\displaystyle\colon\widehat{KO}_{+}^{A}(X,Y)\to\Omega_{\mathrm{clo}}^{4\mathbb{Z}+\mathrm{type}(A)}(X,Y;\mathrm{Ori}(A))$
$\displaystyle[S,h_{0},h_{1},\eta]\mapsto\mathrm{Ph}_{\mathrm{self}}(h_{1})-\mathrm{Ph}_{\mathrm{self}}(h_{0})+d\eta.$
$\displaystyle I$ $\displaystyle\colon\widehat{KO}_{+}^{A}(X,Y)\to
KO_{+}^{A}(X,Y)$ $\displaystyle[S,h_{0},h_{1},\eta]\mapsto[S,h_{0},h_{1}].$
$\displaystyle a$
$\displaystyle\colon\Omega^{4\mathbb{Z}+\mathrm{type}(A)-1}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d)\to\widehat{KO}_{+}^{A}(X,Y)$
$\displaystyle\eta\mapsto[0,0,0,\eta].$

The well-definedness of $R$ follows from (3.19).

Now we check that the functor $\widehat{KO}_{+}^{A}$ satisfies the axioms for
the differential $KO^{\mathrm{type}(A)}$.

######  Theorem 4.7.

In the notations of Definition 4.6, we have the following.

  1. (1)

We have $R\circ a=d$.

  2. (2)

The following diagram commutes.

  3. (3)

The following sequence is exact.

(4.12)

###### Proof.

(1) is obvious. (2) follows from Corollary 3.31. We prove (3). The exactness
of (4.12) at ${KO}_{+}^{A}(X,Y)$, i.e., the surjectivity of $I$, is obvious.
We then check the exactness at
$\Omega^{4\mathbb{Z}+\mathrm{type}(A)-1}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d)$.
We have the following commutative diagram.

(4.17)

We would like to apply (2) to the top row of the above diagram, but a
technical point is that the subset $\partial I\times X\cup I\times Y$ is not a
manifold. So we choose smooth neighborhood $U$ of $\partial I\times X\cup
I\times Y$ in $I\times X$ which has a deformation retraction to $\partial
I\times X\cup I\times Y$. Then by (2) and the surjectivity of $I$, we have the
following commutative diagram.

(4.24)

Thus, noting that the suspension in the de Rham cohomology is given by the
integration $\int_{I}$ on closed forms, the desired exactness is equivalent to
the exactness of the following sequence.

(4.25) $\displaystyle\widehat{KO}_{+}^{A}(I\times
X,\overline{U})\xrightarrow{\mathrm{Rham}\circ\int_{I}\circ
R}\Omega^{4\mathbb{Z}+\mathrm{type}(A)-1}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d)\xrightarrow{a}\widehat{KO}_{+}^{A}(X,Y).$

Now we prove that the composition $a\circ\mathrm{Rham}\circ\int_{I}\circ R$ in
(4.25) is zero. By the commutativity of the upper square of (4.24), the image
under $\mathrm{Rham}\circ\int_{I}\circ R$ of any element in
$\widehat{KO}_{+}^{A}(I\times X,\overline{U})$ only depends on its image in
$KO_{+}^{A}(I\times X,\overline{U})$ under $I$. Moreover, by the excision, we
know that the group $KO_{+}^{A}(I\times X,\overline{U})$ is generated by
triples of the form

(4.26) $\displaystyle(S,h_{0},h_{1}),$

where $h_{0},h_{1}$ are smooth and $h_{0}|_{I\times Y}=h_{1}|_{I\times Y}$ is
constant in the $I$-direction, i.e., there exists a smooth map $h_{Y}\in
C^{\infty}(Y,\mathrm{Self}_{A}^{*}(S))$ such that $h_{i}|_{\\{t\\}\times
Y}=h_{Y}$ for all $t\in[0,1]$, $i=0,1$. It is enough to prove that the element

$\displaystyle[S,h_{0},h_{1},0]\in\widehat{KO}_{+}^{A}(I\times
X,\overline{U})$

maps to zero under the composition (4.25). We have

$\displaystyle\int_{I}\circ
R[S,h_{0},h_{1},0]=\int_{I}\mathrm{Ph}_{\mathrm{self}}(h_{1})-\int_{I}\mathrm{Ph}_{\mathrm{self}}(h_{0}).$

So we have

$\displaystyle a\circ\mathrm{Rham}\int_{I}\circ
R[S,h_{0},h_{1},0]=\left[0,0,0,\int_{I}\mathrm{Ph}_{\mathrm{self}}(h_{1})-\int_{I}\mathrm{Ph}_{\mathrm{self}}(h_{0})\right].$

To see that this element is zero, notice that, for each $i=0,1$ the following
$\widehat{KO}^{A}_{+}$-cocycle is an element in $\widehat{Z}^{A}_{+}(X,Y)$,

$\displaystyle[S,h_{i}|_{\\{0\\}\times X},h_{i}|_{\\{1\\}\times
X},\mathrm{CS}_{\mathrm{self}}(h_{i})]\in\widehat{Z}^{A}_{+}(X,Y),$

where $h_{i}\in C^{\infty}(I\times X,\mathrm{Self}^{*}_{A}(S))$ is regarded as
a homotopy from $h_{i}|_{\\{0\\}\times X}\in
C^{\infty}(X,\mathrm{Self}^{*}_{A}(S))$ to $h_{i}|_{\\{1\\}\times X}\in
C^{\infty}(X,\mathrm{Self}^{*}_{A}(S))$. For each $i=0,1$ we have

$\displaystyle\mathrm{CS}_{\mathrm{self}}(h_{i})=\int_{I}\mathrm{Ph}_{\mathrm{self}}(h_{i}).$

Also notice that for $i=0,1$ we have

$\displaystyle h_{0}|_{\\{i\\}\times X}=h_{1}|_{\\{i\\}\times X},$

since $\partial I\times X\subset\overline{U}$. Thus we get

$\displaystyle\left[0,0,0,\int_{I}\mathrm{Ph}_{\mathrm{self}}(h_{1})-\int_{I}\mathrm{Ph}_{\mathrm{self}}(h_{0})\right]$
$\displaystyle=[S,h_{1}|_{\\{0\\}\times X},h_{1}|_{\\{1\\}\times
X},\mathrm{CS}_{\mathrm{self}}(h_{1})]-[S,h_{0}|_{\\{0\\}\times
X},h_{0}|_{\\{1\\}\times X},\mathrm{CS}_{\mathrm{self}}(h_{0})]$
$\displaystyle=0,$

as desired. This proves that the composition in (4.25) is zero.

To complete the proof of exactness of (4.25), take any element
$\eta\in\Omega^{4\mathbb{Z}+\mathrm{type}(A)-1}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d)$
such that $a(\eta)=[0,0,0,\eta]=0$ in the group $\widehat{KO}_{+}^{A}(X,Y)$.
This means that there exists an element $x\in\widehat{Z}_{+}^{A}(X,Y)$ such
that $x+a(\eta)$ is also an element in $\widehat{Z}_{+}^{A}(X,Y)$. This
implies that, under the description
$x=[S,h_{0},h_{1},\mathrm{CS}_{\mathrm{self}}(h_{I})]$ where $h_{I}$ is a
homotopy between $h_{0}$ and $h_{1}$, there exists another homotopy
$h^{\prime}_{I}$ between $h_{0}$ and $h_{1}$ such that

(4.27)
$\displaystyle\mathrm{CS}_{\mathrm{self}}(h^{\prime}_{I})=\mathrm{CS}_{\mathrm{self}}(h_{I})+\eta\mod(\mathrm{Im}(d)).$

Notice that we have

$\displaystyle h_{I}|_{\partial I\times X\cup I\times
Y}=h^{\prime}_{I}|_{\partial I\times X\cup I\times Y}.$

We may replace each of $h_{I}$ and $h^{\prime}_{I}$ by a homotopic one so that
they satisfy

$\displaystyle h_{I}|_{\overline{U}}=h^{\prime}_{I}|_{\overline{U}}.$

By (3.87), the equality (4.27) is preserved by this replacement. Now we have
the element

(4.28) $\displaystyle[S,h^{\prime}_{I},h_{I},0]\in\widehat{KO}_{+}^{A}(I\times
X,\overline{U}).$

We have

(4.29) $\displaystyle
R[S,h^{\prime}_{I},h_{I},0]=\mathrm{Ph}_{\mathrm{self}}(h^{\prime}_{I})-\mathrm{Ph}_{\mathrm{self}}(h_{I}).$

By (4.27), (4.29) and the definition of $\mathrm{CS}_{\mathrm{self}}$, we see
that the element (4.28) maps to $\eta$ under the left arrow in (4.25). This
completes the proof of the exactness of (4.12) at
$\Omega^{4\mathbb{Z}+\mathrm{type}(A)-1}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d)$.

Finally we prove the exactness of (4.12) at $\widehat{KO}_{+}^{A}(X,Y)$. The
equality $I\circ a=0$ is obvious. Take any element
$[S,h_{0},h_{1},\eta]\in\widehat{KO}_{+}^{A}(X,Y)$ such that
$I[S,h_{0},h_{1},\eta]=[S,h_{0},h_{1}]=0$ in ${KO}_{+}^{A}(X,Y)$. It is enough
to consider the case $(S,h_{0},h_{1})\in Z_{+}^{A}(X,Y)$. Then there exists a
homotopy $h_{I}$ between $h_{0}$ and $h_{1}$ which is constant on $Y$.
Moreover, we can choose $h_{I}$ so that it is smooth. Then we have

$\displaystyle[S,h_{0},h_{1},\eta]=[S,h_{0},h_{1},\mathrm{CS}_{\mathrm{self}}(h_{I})]+a(\eta-\mathrm{CS}_{\mathrm{self}}(h_{I}))=a(\eta-\mathrm{CS}_{\mathrm{self}}(h_{I})).$

This completes the proof of the exactness of (4.12) at
$\widehat{KO}_{+}^{A}(X,Y)$ and finishes the proof of the theorem. ∎

Thus, together with Fact 2.24 we get the following.

######  Theorem 4.30.

The quadruple $\left(\widehat{KO}^{A}_{+},R,I,a\right)$ is a differential
extension of the $KO$-theory $KO^{A}_{+}\simeq KO^{\mathrm{type}(A)}$ on
$\mathrm{MfdPair}_{f}$.

Let $V:=V^{0}\oplus V^{1}$ be a $\mathbb{Z}_{2}$-graded real vector space with
a positive definite inner product on each $V^{i}$. We denote the
$\mathbb{Z}_{2}$-grading operator on $V$ by $\gamma_{V}$. Since the algebra
$\mathrm{End}(V)$ is negligible, Theorem 4.30 in particular means that
$\widehat{KO}_{+}^{A}$ and
$\widehat{KO}_{+}^{\mathrm{End}(V)\widehat{\otimes}A}$ are both differential
extensions of $KO^{\mathrm{type}(A)}$. Actually, we have a canonical
isomorphism
$\widehat{KO}_{+}^{A}\simeq\widehat{KO}_{+}^{\mathrm{End}(V)\widehat{\otimes}A}$
as follows.

######  Proposition 4.31.

In the above settings, we have a natural isomorphism

(4.32)
$\displaystyle\widehat{KO}^{A}_{+}\simeq\widehat{KO}^{\mathrm{End}(V)\widehat{\otimes}A}_{+}$

which is compatible with the corresponding isomorphisms of other functors
appearing in Definition 4.6 via the structure homomorphisms (under the
isomorphism
$\mathrm{Ori}(A)\simeq\mathrm{Ori}(\mathrm{End}(V)\widehat{\otimes}A)$ given
by $u\mapsto\gamma_{V}\widehat{\otimes}u$).

###### Proof.

The natural isomorphism (4.32) on an object $(X,Y)$ in $\mathrm{MfdPair}_{f}$
is given by the following refinement of the untwisted version of the map
(2.35),

(4.33) $\displaystyle[S,h_{0},h_{1},\eta]\mapsto[E\otimes
S,\psi^{E}(h_{0}),\psi^{E}(h_{1}),\eta].$

The well-definedness and the compatibility with the structure maps follow from
Lemma 3.91. By the exactness of (4.12) and the five lemma, we see that (4.33)
defines an isomorphism, as desired. ∎

In particular we get

######  Corollary 4.34.

The isomorphism class of the differential extension

$\left(\widehat{KO}_{+}^{A},R,I,a\right)$

of $KO^{\mathrm{type}(A)}$ only depends on the type of $A$.

###  4.2. The twisted groups $\widehat{KO}^{\mathcal{A}}_{+}$

In this subsection we give a model $\widehat{KO}^{\mathcal{A}}_{+}$ for the
twisted differential $KO$-theory groups, by refining Definition 2.28. First we
introduce the category $\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}$ of twists on
$\widehat{KO}_{+}$.

######  Definition 4.35 ($\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}$).

  1. (1)

For each object $X\in\mathrm{Mfd}_{f}$, we define
$\mathfrak{Twist}_{\widehat{KO}_{+},X}$ to be the groupoid of smooth bundles
of nondegenerate simple central graded $*$-algebras $\mathcal{A}$ over $X$,
where morphisms are ismorphisms between such bundles.

  2. (2)

For each morphism $f\colon X\to X^{\prime}$ in $\mathrm{Mfd}_{f}$, we define a
functor
$f^{*}\colon\mathfrak{Twist}_{\widehat{KO}_{+},X^{\prime}}\to\mathfrak{Twist}_{\widehat{KO}_{+},X}$
by the pullback.

  3. (3)

We define the category $\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}$ such that an
object $(X,Y,\mathcal{A})$ consists of $(X,Y)\in\mathrm{MfdPair}_{f}$ and
$\mathcal{A}\in\mathfrak{Twist}_{\widehat{KO}_{+},X}$ and a morphism from
$(X,Y,\mathcal{A})$ to $(X^{\prime},Y^{\prime},\mathcal{A}^{\prime})$ consists
of a morphism $f\colon(X,Y)\to(X^{\prime},Y^{\prime})$ in
$\mathrm{MfdPair}_{f}$ and an isomorphism $\mathcal{A}\simeq
f^{*}\mathcal{A}^{\prime}$.

We have the forgetful functor

$\displaystyle\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}\to\mathfrak{Twist}_{KO_{+}}^{2}.$

######  Definition 4.36.

Let $(X,Y,\mathcal{A})\in\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}$.

  * •

A $\widehat{KO}_{+}$-cocycle
$(\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta)$ on
$(X,Y,\mathcal{A})$ consists of an $\mathcal{A}$-module bundle $\not{S}$ with
an inner product, a connection $\nabla^{\mathcal{A}}$ on $\mathcal{A}$, a
self-adjoint $\mathcal{A}$-connnection $\nabla^{\not{S}}$ on $\not{S}$
compatible with $\nabla^{\mathcal{A}}$, two smooth sections $h_{0},h_{1}\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$ such that
$h_{0}|_{Y}=h_{1}|_{Y}$, and an element
$\eta\in\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})-1}(X,Y;\mathrm{Ori}(\mathcal{A}))/\mathrm{Im}(d)$.

  * •

Two $\widehat{KO}_{+}$-cocycle
$(\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta)$ and
$(\not{S}^{\prime},(\nabla^{\mathcal{A}})^{\prime},\nabla^{\not{S}^{\prime}},h^{\prime}_{0},h^{\prime}_{1},\eta^{\prime})$
are isomorphic if we have
$\nabla^{\mathcal{A}}=(\nabla^{\mathcal{A}})^{\prime}$ and
$\eta=\eta^{\prime}$, and there exists an isometric isomorphism of smooth
$\mathcal{A}$-module bundles $f\colon\not{S}\simeq\not{S}^{\prime}$ such that
$f^{*}\nabla^{\not{S}^{\prime}}=\nabla^{\not{S}}$ and $f\circ
h_{i}=h^{\prime}_{i}\circ f$ for $i=0,1$.

######  Definition 4.37 ($\widehat{KO}_{+}^{\mathcal{A}}(X,Y)$).

Let $(X,Y,\mathcal{A})\in\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}$.

  * •

We introduce an abelian monoid structure on the set
$\widehat{M}^{\mathcal{A}}_{+}(X,Y)$ of isomorphism classes of
$\widehat{KO}_{+}$-cocycles
$(\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta)$ by

$\displaystyle[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta]+[\not{S}^{\prime},(\nabla^{\mathcal{A}})^{\prime},\nabla^{\not{S}^{\prime}},h^{\prime}_{0},h^{\prime}_{1},\eta^{\prime}]$
$\displaystyle\quad=[\not{S}\oplus\not{S}^{\prime},\nabla^{\mathcal{A}}\oplus(\nabla^{\mathcal{A}})^{\prime},\nabla^{\not{S}}\oplus\nabla^{\not{S}^{\prime}},h_{0}\oplus
h^{\prime}_{0},h_{1}\oplus h^{\prime}_{1},\eta+\eta^{\prime}].$

  * •

We define $\widehat{Z}^{\mathcal{A}}_{+}(X,Y)$ to be the submonoid of
$\widehat{M}^{\mathcal{A}}_{+}(X,Y)$ consisting of elements of the form

$\displaystyle[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\mathrm{CS}_{\mathrm{self}}(h_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})],$

where $h_{I}$ is a smooth homotopy between $h_{0}$ and $h_{1}$ which is
constant on $Y$, i.e., a smooth section $h_{I}\in C^{\infty}(I\times
X;\mathrm{pr}_{X}^{*}\mathrm{Self}^{*}_{\mathcal{A}}(\not{S}))$ with
$h_{I}|_{\\{i\\}\times X}=h_{i}$ for $i=0,1$ and $h_{I}|_{\\{t\\}\times
Y}=h_{0}|_{Y}$ for all $t\in I$ (see (3.86)).

  * •

We define
$\widehat{KO}_{+}^{\mathcal{A}}(X,Y):=\widehat{M}^{\mathcal{A}}_{+}(X,Y)/\widehat{Z}^{\mathcal{A}}_{+}(X,Y)$.

######  Lemma 4.38.

$\widehat{KO}_{+}^{\mathcal{A}}(X,Y)$ is an abelian group.

###### Proof.

The proof is similar to the untwisted case (Lemma 4.3). The additive inverse
of an element
$[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta]\in\widehat{KO}_{+}^{\mathcal{A}}(X,Y)$
is given by

(4.39) $-[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta]\\\
=\left[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{1},h_{0},-\eta+\mathrm{CS}_{\mathrm{self}}\left(h_{I};\nabla^{\mathcal{A}}\oplus\nabla^{\mathcal{A}},\nabla^{\not{S}}\oplus\nabla^{\not{S}}\right)\right].$

Here $h_{I}$ is a homotopy between $h_{0}\oplus h_{1}$ and $h_{1}\oplus h_{0}$
on $\not{S}\oplus\not{S}$. ∎

Thus we get the functor

(4.40)
$\displaystyle\widehat{KO}_{+}\colon\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}\to\mathrm{Ab}.$

The structure homomorphisms in the twisted case are given as follows.

######  Definition 4.41 (Structure homomorphisms for
$\widehat{KO}^{\mathcal{A}}_{+}$).

We define the following structure homomorphisms, which are natural in
$(X,Y,\mathcal{A})\in\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}$.

$\displaystyle R$
$\displaystyle\colon\widehat{KO}_{+}^{\mathcal{A}}(X,Y)\to\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})}_{\mathrm{clo}}(X,Y;\mathrm{Ori}(\mathcal{A}))$
$\displaystyle[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta]\mapsto\mathrm{Ph}_{\mathrm{self}}(h_{1};\nabla^{\mathcal{A}},\nabla^{\not{S}})-\mathrm{Ph}_{\mathrm{self}}(h_{0};\nabla^{\mathcal{A}},\nabla^{\not{S}})+d\eta.$
$\displaystyle I$ $\displaystyle\colon\widehat{KO}_{+}^{\mathcal{A}}(X,Y)\to
KO_{+}^{\mathcal{A}}(X,Y)$
$\displaystyle[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta]\mapsto[\not{S},h_{0},h_{1}].$
$\displaystyle a$
$\displaystyle\colon\Omega^{4\mathbb{Z}+\mathrm{type}(\mathcal{A})-1}(X,Y;\mathrm{Ori}(\mathcal{A}))/\mathrm{Im}(d)\to\widehat{KO}_{+}^{\mathcal{A}}(X,Y)$
$\displaystyle\eta\mapsto[0,0,0,0,0,\eta].$

The well-definedness of $R$ follows from (3.84) and (3.85). The well-
definedness of $a$ and $I$ is obvious.

Recall that, by (3.45), $\Omega^{1}(X;\mathcal{A}^{0}_{\mathrm{skew}})$ acts
on the set of pairs $(\nabla^{\mathcal{A}},\nabla^{\not{S}})$.

######  Lemma 4.42.

For any $KO^{\mathcal{A}}_{+}$-cycle
$(\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta)$ and any
element $C\in\Omega^{1}(X;\mathcal{A}^{0}_{\mathrm{skew}})$, we have the
following equality in $\widehat{KO}_{+}^{\mathcal{A}}(X,Y)$.

$\displaystyle[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta]=[\not{S},\nabla^{\mathcal{A}}+C,\nabla^{\not{S}}+C,h_{0},h_{1},\eta].$

In particular, if we choose and fix a connection $\nabla^{\mathcal{A}}$ on
$\mathcal{A}$, then any element in $\widehat{KO}^{\mathcal{A}}_{+}$ is
represented by a $\widehat{KO}_{+}$-cocycle of the form
$(\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta)$.

###### Proof.

By (4.39), it is enough to prove that the following $\widehat{KO}_{+}$-cocycle
is in $\widehat{Z}^{\mathcal{A}}_{+}(X,Y)$.

$\displaystyle[\not{S},\nabla^{\mathcal{A}}+C,\nabla^{\not{S}}+C,h_{0},h_{1},\eta]$
$\displaystyle\quad+\left[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{1},h_{0},-\eta+\mathrm{CS}_{\mathrm{self}}\left(h_{I};\nabla^{\mathcal{A}}\oplus\nabla^{\mathcal{A}},\nabla^{\not{S}}\oplus\nabla^{\not{S}}\right)\right]$
$\displaystyle=\left[\not{S}\oplus\not{S},(\nabla^{\mathcal{A}}+C)\oplus\nabla^{\mathcal{A}},(\nabla^{\not{S}}+C)\oplus\nabla^{\not{S}},h_{0}\oplus
h_{1},h_{1}\oplus h_{0},\right.$ $\displaystyle\hskip
170.71652pt\left.\mathrm{CS}_{\mathrm{self}}\left(h_{I};\nabla^{\mathcal{A}}\oplus\nabla^{\mathcal{A}},\nabla^{\not{S}}\oplus\nabla^{\not{S}}\right)\right].$

Here $h_{I}$ is any homotopy between $h_{0}\oplus h_{1}$ and $h_{1}\oplus
h_{0}$ on $\not{S}\oplus\not{S}$. Indeed, this claim follows from

$\mathrm{CS}_{\mathrm{self}}\left(h_{I};\nabla^{\mathcal{A}}\oplus\nabla^{\mathcal{A}},\nabla^{\not{S}}\oplus\nabla^{\not{S}}\right)\\\
=\mathrm{CS}_{\mathrm{self}}\left(h_{I};(\nabla^{\mathcal{A}}+C)\oplus\nabla^{\mathcal{A}},(\nabla^{\not{S}}+C)\oplus\nabla^{\not{S}}\right)$

as a result of Lemma 3.88. ∎

######  Theorem 4.43.

Let $(X,Y,\mathcal{A})\in\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}$.

  1. (1)

We have $R\circ a=d$.

  2. (2)

The following diagram commutes.

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  3. (3)

The following sequence is exact.

(4.48)

###### Proof.

(1) is obvious. (2) follows from Theorem 3.96. The proof of (3) is analogous
to the untwisted case in Theorem 4.7 (3), so we leave the details to the
reader. ∎

We use the axiom of differential extensions of twisted generalized cohomology
theories given in [BS12]. By Theorem 4.43, we get the following.

######  Theorem 4.49.

The quadruple $\left(\widehat{KO}_{+},R,I,a\right)$ is a differential
extension of the twisted $KO$-theory ${KO}_{+}$ on
$\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}$.

Now we investigate into the dependence of $\widehat{KO}^{\mathcal{A}}_{+}$ on
the bundle $\mathcal{A}$. Recall that the twist of topological $KO$-theory is
classified by
$(\mathrm{type}(\mathcal{A}),w(\mathcal{A}))\in\mathrm{GBrO}(X)=\mathbb{Z}_{8}\times
HO(X)$ (Corollary 2.36). Actually the same classification holds for our
differential refinement. We start with the following Proposition corresponding
to Fact 2.33.

######  Proposition 4.50.

Let $(X,Y)$ be an object of $\mathrm{MfdPair}_{f}$. Let $E=E^{0}\oplus E^{1}$
be a $\mathbb{Z}_{2}$-graded vector bundle over $X$ with fiberwise positive
definite inner product on each $E^{i}$. For any smooth bundle of nondegenerate
simple central graded $*$-algebras $\mathcal{A}$, we have a canonical
isomorphism

(4.51)
$\displaystyle\widehat{KO}^{\mathcal{A}}_{+}(X,Y)\simeq\widehat{KO}^{\mathrm{End}(E)\widehat{\otimes}\mathcal{A}}_{+}(X,Y)$

which is compatible with the corresponding isomorphisms of other functors
appearing in Definition 4.41 via the structure homomorphisms (under the
isomorphism
$\mathrm{Ori}(\mathcal{A})\simeq\mathrm{Ori}(\mathrm{End}(E)\widehat{\otimes}\mathcal{A})$
given by $u\mapsto\gamma_{E}\widehat{\otimes}u$).

###### Proof.

Fix an orthogonal connection $\nabla^{E}$ on $E$, and denote by
$\nabla^{\mathrm{End}(E)}$ the induced connection on $\mathrm{End}(E)$. The
isomorphism (4.51) is given by the following refinement of the map (2.35),

(4.52) $[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta]\\\
\mapsto[E\otimes\not{S},\nabla^{\mathrm{End}(E)}\otimes
1+1\otimes\nabla^{\mathcal{A}},\nabla^{E}\otimes
1+1\otimes\nabla^{\not{S}},\psi^{E}(h_{0}),\psi^{E}(h_{1}),\eta].$

The well-definedness and the compatibility with the structure maps follow from
Lemma 3.91. By the exactness of (4.48) and the five lemma, we see that (4.52)
defines an isomorphism. Actually, the right hand side of (4.52) is independent
of the choice of $\nabla_{E}$ by Lemma 4.42. Thus we see that (4.42) defines a
canonical isomorphism (4.51) independent of any additional choice, as desired.
∎

Thus we get the following differential version of Corollary 2.36.

######  Corollary 4.53.

Given an object $(X,Y,\mathcal{A})\in\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}$,
the isomorphism class of the twisted differential $KO$-theory group
$\widehat{KO}^{\mathcal{A}}_{+}(X,Y)$ only depends on the class of
$\mathcal{A}$ in $\mathrm{GBrO}(X)$.

##  5\. The model in terms of skew-adjoint operators

In this section, we introduce another model of (twisted) differential
$KO$-theory, $\widehat{KO}_{-}^{A}$ ($\widehat{KO}_{-}^{\mathcal{A}}$), in
terms of skew-adjoint operators. In this section we work over $\mathbb{R}$,
unless otherwise stated (e.g., in Remarks 5.7 and 5.19).

###  5.1. Algebraic preparations

This subsection is devoted to algebraic preparations.

######  Definition 5.1 ($\widetilde{\Sigma}^{0,1}A$).

Let $A$ be a simple central graded $*$-algebra. We define
$\widetilde{\Sigma}^{0,1}A$ to be $\Sigma^{0,1}A=A\widehat{\otimes}Cl_{0,1}$
as a $*$-algebra, and introduce a $\mathbb{Z}_{2}$ grading on
$\widetilde{\Sigma}^{0,1}A$ by $1\otimes\gamma$, where $\gamma$ is the
$\mathbb{Z}_{2}$-grading operator on $Cl_{0,1}$.

Thus we have

$\displaystyle(\widetilde{\Sigma}^{0,1}A)^{0}$
$\displaystyle=\\{a\widehat{\otimes}1\ |\ a\in A\\},$
$\displaystyle(\widetilde{\Sigma}^{0,1}A)^{1}$
$\displaystyle=\\{a\widehat{\otimes}\beta\ |\ a\in A\\}.$

######  Example 5.2.

In the case where $A=Cl_{p,q}$ for nonnegative integers $p$ and $q$, we have
$\Sigma^{0,1}A=Cl_{p,q+1}$ and $\widetilde{\Sigma}^{0,1}A\simeq Cl_{q,p+1}$.
An isomorphism of $*$-algebras $\Sigma^{0,1}A\simeq\widetilde{\Sigma}^{0,1}A$
does not preserve the $\mathbb{Z}_{2}$-grading:

$\displaystyle Cl_{p,q+1}$ $\displaystyle\simeq Cl_{q,p+1}$
$\displaystyle\beta_{0}$ $\displaystyle\mapsto\beta^{\prime}_{0}$
$\displaystyle\alpha_{i}$
$\displaystyle\mapsto\beta^{\prime}_{0}\beta^{\prime}_{i}\ (1\leq i\leq p)$
$\displaystyle\beta_{j}$
$\displaystyle\mapsto\beta^{\prime}_{0}\alpha^{\prime}_{j}\ (1\leq j\leq q)$

where we denoted by $\alpha_{*}$ ($\alpha^{\prime}_{*}$) and $\beta_{*}$
($\beta^{\prime}_{*}$) the negative and positive Clifford generators of
$Cl_{p,q+1}$ ($Cl_{q,p+1}$), respectively.

######  Lemma 5.3.

For any simple central graded $*$-algebra $A$, the $\mathbb{Z}_{2}$-graded
$*$-algebra $\widetilde{\Sigma}^{0,1}A$ is a simple central graded $*$-algebra
with

$\displaystyle\mathrm{type}(\widetilde{\Sigma}^{0,1}A)=-\mathrm{type}(A)-1.$

If $u\in A$ is a volume element of $A$, the element
$u\otimes\beta^{\mathrm{type}(A)+1}\in\widetilde{\Sigma}^{0,1}A$ is a volume
element of $\widetilde{\Sigma}^{0,1}A$.

###### Proof.

The proof can be given by a case-by-case check of Table 1, and we leave the
details to the reader. One possible simplification is to use the
$\mathbb{Z}_{2}$-grading-preserving isomorphism
$\widetilde{\Sigma}^{0,1}\Sigma^{0,1}A\simeq\Sigma^{1,0}\widetilde{\Sigma}^{0,1}A$
to reduce the problem to the case $\mathrm{type}(A)=0$. ∎

The following lemma can also be proved easily.

######  Lemma 5.4.

Let $A$ be a simple central graded $*$-algebra and $S$ be a
$\Sigma^{0,1}A$-module with an inner product. We also regard $S$ as a
$\widetilde{\Sigma}^{0,1}A$-module. Then we have a bijection

(5.5) $\displaystyle\psi_{\beta}\colon\mathrm{End}_{\Sigma^{0,1}A}(S)$
$\displaystyle\simeq\mathrm{End}_{\widetilde{\Sigma}^{0,1}A}(S)$
$\displaystyle f$ $\displaystyle\mapsto\beta^{|f|_{\Sigma^{0,1}A}}f$

which preserves the $\mathbb{Z}_{2}$ grading. Here
$|f|_{\Sigma^{0,1}A}\in\mathbb{Z}_{2}$ in the right hand side of (5.5) is the
$\mathbb{Z}_{2}$-grading of $f$ as an element in
$\mathrm{End}_{\Sigma^{0,1}A}(S)$. Moreover, the bijection restricts to the
following bijections relating $\mathrm{Self}$ with $\mathrm{Skew}$,

$\displaystyle\mathrm{Self}_{\Sigma^{0,1}A}(S)\simeq\mathrm{Skew}_{\widetilde{\Sigma}^{0,1}A}(S),\quad$
$\displaystyle\mathrm{Self}^{*}_{\Sigma^{0,1}A}(S)\simeq\mathrm{Skew}^{*}_{\widetilde{\Sigma}^{0,1}A}(S),$
$\displaystyle\mathrm{Skew}_{\Sigma^{0,1}A}(S)\simeq\mathrm{Self}_{\widetilde{\Sigma}^{0,1}A}(S),\quad$
$\displaystyle\mathrm{Skew}^{*}_{\Sigma^{0,1}A}(S)\simeq\mathrm{Self}^{*}_{\widetilde{\Sigma}^{0,1}A}(S).$

Applying Lemma 5.4 fiberwise, we also get the bijection in the bundle case,

(5.6)
$\displaystyle\psi_{\beta}\colon\mathrm{End}_{\Sigma^{0,1}\mathcal{A}}(\not{S})$
$\displaystyle\simeq\mathrm{End}_{\widetilde{\Sigma}^{0,1}\mathcal{A}}(\not{S}).$

######  Remark 5.7.

In the $\mathbb{C}$-linear setting, we simply have the bijection

(5.8) $\displaystyle\mathrm{End}^{1}_{A}(S)\simeq\mathrm{End}^{1}_{A}(S),\
f\mapsto\sqrt{-1}f,$

which transforms $\mathrm{Skew}$ to $\mathrm{Self}$. We use this bijection
when we relate $K_{+}$ with $K_{-}$ in Section 6.

###  5.2. The topological model $KO_{-}$

In this subsection we define another model of the (twisted) $KO$-theory in
terms of skew-adjoint operators. The functor $KO_{-}^{A}$ is shown to be a
model of $KO^{-\mathrm{type}(A)-2}$.

We start with the untwisted groups. The definition of $KO_{-}$ is given by
simply replacing $\mathrm{Self}$ with $\mathrm{Skew}$ in Definitions 2.20 and
2.21 of untwisted $KO_{+}$, as follows.

######  Definition 5.9.

Let $(X,Y)$ be a finite CW-pair and $A$ be a simple central graded
$*$-algebra.

  * •

A triple $(S,m_{0},m_{1})$ on $(X,Y)$ consists of an $A$-module $S$ with an
inner product and two continuous maps
$m_{0},m_{1}\in\mathrm{Map}(X,\mathrm{Skew}_{A}^{*}(S))$ with
$m_{0}|_{Y}=m_{1}|_{Y}$. Such $m_{i}$’s are called mass terms on $S$ (c.f.
Subsubsection 1.1.1).

  * •

Triples $(S,m_{0},m_{1})$ and $(S^{\prime},m^{\prime}_{0},m^{\prime}_{1})$ on
$(X,Y)$ are isomorphic if there exists an isometric isomorphism of $A$-modules
$f\colon S\simeq S^{\prime}$ such that $f\circ m_{i}=m^{\prime}_{i}\circ f$
for $i=0,1$.

######  Definition 5.10 ($KO_{-}^{A}(X,Y)$).

Let $(X,Y)$ and $A$ be as above.

  * •

On the set $M^{p,q}_{-}(X,Y)$ of isomorphism classes of triples
$(S,m_{0},m_{1})$ on $(X,Y)$, we introduce an abelian monoid structure by the
direct sum.

  * •

We define $Z^{A}_{-}(X,Y)$ to be the submonoid of $M^{A}_{-}(X,Y)$ consisting
of isomorphism classes of triples $(S,m_{0},m_{1})$ on $(X,Y)$ such that there
exists a homotopy between $m_{0}$ and $m_{1}$ which is constant on $Y$, i.e.,
a map $m_{I}\in\mathrm{Map}(I\times X,\mathrm{Skew}^{*}_{A}(S))$ with
$m_{I}|_{\\{i\\}\times X}=m_{i}$ for $i=0,1$ and $m_{I}|_{\\{t\\}\times
Y}=m_{0}|_{Y}$ for all $t\in I$.

  * •

We define $KO^{A}_{-}(X,Y)=M^{A}_{-}(X,Y)/Z^{A}_{-}(X,Y)$ to be the quotient
monoid.

Next we define twisted groups. The category of twists
$\mathfrak{Twist}^{2}_{{KO}_{-}}$ in ${KO}_{-}$ is just the same as that for
${KO}_{+}$ in Definition 2.26,

(5.11)
$\displaystyle\mathfrak{Twist}^{2}_{{KO}_{-}}:=\mathfrak{Twist}^{2}_{{KO}_{+}}.$

The definition of ${KO}_{-}$ is given by modifiying Definition 2.28 for
twisted $KO_{+}$ in the same way, so we go briefly.

######  Definition 5.12 ($KO^{\mathcal{A}}_{-}(X,Y)$).

Let $(X,Y,\mathcal{A})\in\mathfrak{Twist}^{2}_{{KO}_{-}}$. By replacing
$\mathrm{Self}$ to $\mathrm{Skew}$ in Definition 2.28, we define the abelian
group $KO^{\mathcal{A}}_{-}(X,Y)$.

Thus elements in $KO^{\mathcal{A}}_{-}(X,Y)$ is of the form
$[\not{S},m_{0},m_{1}]$ with
$m_{0},m_{1}\in\Gamma(X,\mathrm{Skew}_{\mathcal{A}}^{*}(S))$ such that
$m_{0}|_{Y}=m_{1}|_{Y}$. We have $[\not{S},m_{0},m_{1}]=0$ if $m_{0}$ and
$m_{1}$ are homotopic relative to $Y$.

The result corresponding to Fact 2.33 also holds for $KO_{-}$. For
$E=E^{0}\oplus E^{1}$ and any $\mathcal{A}$ as in the setting there, we have a
canonical isomorphism using Lemma 2.12,

(5.13) $\displaystyle KO^{\mathcal{A}}_{-}(X,Y)\simeq
KO^{\mathrm{End}(E)\widehat{\otimes}\mathcal{A}}_{-}(X,Y).$

We can relate $KO_{-}$ with $KO_{+}$ using the results in Subsection 5.1.

######  Proposition 5.14.

Let $(X,Y,\mathcal{A})\in\mathfrak{Twist}^{2}_{{KO}_{-}}$. We have an
isomorphism

(5.15) $\displaystyle KO_{-}^{\Sigma^{0,1}\mathcal{A}}(X,Y)\simeq
KO_{+}^{\widetilde{\Sigma}^{0,1}\mathcal{A}}(X,Y),$

which is natural in $(X,Y)$ and $\mathcal{A}$.

###### Proof.

The map (5.15) is given by

$\displaystyle[\not{S},m_{0},m_{1}]\mapsto[\not{S},\psi_{\beta}(m_{0}),\psi_{\beta}(m_{1})],$

where
$\psi_{\beta}\colon\mathrm{Skew}^{*}_{\Sigma^{0,1}\mathcal{A}}(\not{S})\simeq\mathrm{Self}^{*}_{\widetilde{\Sigma}^{0,1}\mathcal{A}}(\not{S})$
is the isomorphism in (5.6). The well-definedness and the fact that this map
gives a natural isomorphism are obvious. ∎

Since we have
$\Sigma^{0,1}\Sigma^{1,0}\mathcal{A}=\mathcal{A}\hat{\otimes}Cl_{1,1}$ and
$Cl_{1,1}$ is negligible, by Proposition 5.14 and (5.13), we have a natural
isomorphism

(5.16) $\displaystyle KO_{-}^{\mathcal{A}}(X,Y)\simeq
KO_{+}^{\widetilde{\Sigma}^{0,1}\Sigma^{1,0}\mathcal{A}}(X,Y).$

Thus we see that the two models $KO_{+}$ and $KO_{-}$ are equivalent under
explicit algebraic operations. In particular, in the case of untwisted groups,
also using Lemma 5.3 we have the following result corresponding to Fact 2.24.

######  Proposition 5.17.

Let $A$ be a simple central graded $*$-algebra. We have a natural isomorphism
on the category of finite CW-pairs,

(5.18) $\displaystyle KO_{-}^{A}\simeq KO^{-\mathrm{type}(A)-2}.$

######  Remark 5.19.

In the $\mathbb{C}$-linear setting, just replacing the coefficients
$\mathbb{R}$ by $\mathbb{C}$ in Definitions 5.10 and 5.12 as we did in Remark
2.37, we define groups $K_{-}^{A}(X,Y)$ and $K_{-}^{\mathcal{A}}(X,Y)$. In
this case, we have an isomorphism

(5.20) $\displaystyle K_{-}^{\mathcal{A}}\simeq K_{+}^{\mathcal{A}}$

by $[\not{S},m_{0},m_{1}]\mapsto[\not{S},\sqrt{-1}m_{0},\sqrt{-1}m_{1}]$ (see
Remark 5.7). By (2.38) and (5.20), we have a natural isomorphism on the
category of finite CW-pairs,

(5.21) $\displaystyle K_{-}^{A}\simeq K^{\mathrm{type}(A)}.$

###  5.3. The Pontryagin character forms for mass terms

Recall that the definition of the differential model $\widehat{KO}_{+}^{A}$
used the Pontryagin character forms $\mathrm{Ph}_{\mathrm{self}}(h)$ for
invertible elements $h\in C^{\infty}(X,\mathrm{Self}^{*}_{A}(S))$. In this
subsection, we define their skew-adjoint variants.

####  5.3.1. The case of trivial bundles

Here we explain the skew-adjoint version of Subsubsection 3.2.2. Let $X$ be a
manifold and $A$ be nondegenerate. We define the Pontryagin character form
$\mathrm{Ph}_{\mathrm{skew}}(m)$ for an invertible element (a mass term) $m\in
C^{\infty}(X,\mathrm{Skew}^{*}_{A}(S))$.

For such $m$, consider the section $tm\in C^{\infty}((0,\infty)\times
X,\mathrm{Skew}^{*}_{A}(S))$, where $t$ is the coordinate in $(0,\infty)$. We
have

(5.22) $\mathrm{Ph}_{\mathrm{skew}}(d_{(0,\infty)\times
X}+tm;d)=dt\wedge\mathrm{Tr}_{A}\left(me^{td_{X}m+t^{2}m^{2}}\right)\\\
\in\Omega_{\mathrm{clo}}^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})-1}(X;\mathrm{Ori}(\mathcal{A})).$

Here the first term is defined in (3.67), in which $d$ is the trivial
connection on $\underline{A}$. Since $m^{2}$ is strictly negative, the
convergence result analogous to Lemma 3.13 holds and allows us to define the
following.

######  Definition 5.23 ($\mathrm{Ph}_{\mathrm{skew}}(m)$).

For $m\in C^{\infty}(X,\mathrm{Skew}^{*}_{A}(S))$, its Pontryagin character
form

$\displaystyle\mathrm{Ph}_{\mathrm{skew}}(m)\in\Omega^{4\mathbb{Z}-\mathrm{type}(A)-2}_{\mathrm{clo}}(X;\mathrm{Ori}(A))$

is defined by

(5.24) $\displaystyle\mathrm{Ph}_{\mathrm{skew}}(m)$
$\displaystyle:=-\pi^{-1/2}\mathcal{R}\circ\int_{(0,\infty)}\mathrm{Ph}_{\mathrm{skew}}(d_{(0,\infty)\times
X}+tm)$
$\displaystyle=-\pi^{-1/2}\mathcal{R}\circ\int_{(0,\infty)}dt\wedge\mathrm{Tr}_{A}\left(me^{td_{X}m+t^{2}m^{2}}\right).$

We also define

######  Definition 5.25 ($\mathrm{CS}_{\mathrm{skew}}(m_{I})$).

For $m_{I}\in C^{\infty}(I\times X,\mathrm{Skew}^{*}_{A}(S))$, we define its
Chern-Simons form by

(5.26)
$\displaystyle\mathrm{CS}_{\mathrm{skew}}(m_{I}):=\int_{I}\mathrm{Ph}_{\mathrm{skew}}\left(m_{I}\right)\in\Omega^{4\mathbb{Z}-\mathrm{type}(A)-3}(X;\mathrm{Ori}(A)).$

####  5.3.2. The general case

Now we explain the skew-adjoint version of Subsection 3.4. Let $X$,
$\mathcal{A}$, $\nabla^{\mathcal{A}}$ and $\not{S}$ with an inner product be
as in that subsection, where we always assume that the fibers of $\mathcal{A}$
are nondegenerate. Let $\nabla^{\not{S}}$ be a skew-adjoint
$\mathcal{A}$-connection on $\not{S}$ compatible with $\nabla^{\mathcal{A}}$.
In this setting, given a smooth invertible section $m\in
C^{\infty}(X;\mathrm{Skew}_{\mathcal{A}}^{*}(\not{S}))$, we are going to
define its Pontryagin character form
$\mathrm{Ph}_{\mathrm{skew}}(m;\nabla^{\mathcal{A}},\nabla^{\not{S}})$.

On the manifold $(0,\infty)\times X$, We consider the skew-adjoint
$\mathrm{pr}_{X}^{*}\mathcal{A}$-superconnection
$\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+tm$ on $\mathrm{pr}_{X}^{*}\not{S}$
compatible with $\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}}$. We have

$\displaystyle\mathrm{Ph}_{\mathrm{skew}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+tm;\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})\in\Omega_{\mathrm{clo}}^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})-1}((0,\infty)\times
X;\mathrm{pr}_{X}^{*}\mathrm{Ori}(\mathcal{A})).$

Using the convergence result analogous to Lemma 3.77, we define the following.

######  Definition 5.27
($\mathrm{Ph}(m;\nabla^{\mathcal{A}},\nabla^{\not{S}})$,
$\mathrm{CS}(m_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})$).

Let $X$, $\mathcal{A}$, $\nabla^{\mathcal{A}}$ and $\not{S}$ be as above.

  1. (1)

For any smooth invertible section $m\in
C^{\infty}(X;\mathrm{Skew}_{\mathcal{A}}^{*}(\not{S}))$, we define

(5.28)
$\mathrm{Ph}_{\mathrm{skew}}(m;\nabla^{\mathcal{A}},\nabla^{\not{S}}):=-\pi^{-1/2}\mathcal{R}\circ\int_{(0,\infty)}\mathrm{Ph}_{\mathrm{skew}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+tm;\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})\\\
\in\Omega^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})-2}(X;\mathrm{Ori}(\mathcal{A}))$

  2. (2)

For any smooth invertible section $m_{I}\in C^{\infty}(I\times
X;\mathrm{pr}_{X}^{*}\mathrm{Skew}_{\mathcal{A}}^{*}(\not{S}))$, we define

(5.29)
$\mathrm{CS}_{\mathrm{skew}}(m_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}}):=\int_{I}\mathrm{Ph}_{\mathrm{skew}}\left(m_{I};\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}},\mathrm{pr}_{X}^{*}\nabla^{\not{S}}\right)\\\
\in\Omega^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})-3}(X;\mathrm{Ori}(\mathcal{A})).$

####  5.3.3. The relation between $\mathrm{Ph}_{\mathrm{skew}}$ and
$\mathrm{Ph}_{\mathrm{self}}$

In Subsection 5.1, we saw that we have an isomorphism
$\psi_{\beta}\colon\mathrm{Skew}^{*}_{\Sigma^{0,1}\mathcal{A}}(\not{S})\simeq\mathrm{Self}^{*}_{\widetilde{\Sigma}^{0,1}\mathcal{A}}(\not{S})$.
In this subsection, we show that the Pontryagin character forms
$\mathrm{Ph}_{\mathrm{skew}}$ and $\mathrm{Ph}_{\mathrm{self}}$ are also
related under this isomorphism.

Let $(\mathcal{A},\nabla^{\mathcal{A}})$ be a bundle of simple central graded
$*$-algebras equipped with a connection. The induced connection on
$\Sigma^{0,1}\mathcal{A}$ and $\widetilde{\Sigma}^{0,1}\mathcal{A}$ are
denoted by $\Sigma^{0,1}\nabla^{\mathcal{A}}$ and
$\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}}$, respectively. Assume that a
$\Sigma^{0,1}\mathcal{A}$-module bundle $\not{S}$ is equipped with an inner
product and let $\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ be a
$\Sigma^{0,1}\mathcal{A}$-superconnection compatible with
$\Sigma^{0,1}\nabla^{\mathcal{A}}$. Then decompose
$\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ as in (3.40),

$\displaystyle\nabla\\!\\!\\!\\!\nabla^{\not{S}}=\nabla+\sum_{j}\omega_{j}\widehat{\otimes}\xi_{j},$

where $\nabla$ is an $\mathcal{A}$-connection, $\omega_{j}\in\Omega^{j}(X)$
and $\xi_{j}\in C^{\infty}(X;\mathrm{End}_{\Sigma^{0,1}\mathcal{A}}(\not{S}))$
with $|\omega_{j}|+|\xi_{j}|=1\in\mathbb{Z}_{2}$. Then consider the following
operator on $\Omega^{*}(X;\not{S})$

(5.30)
$\displaystyle\psi_{\beta}(\nabla\\!\\!\\!\\!\nabla^{\not{S}}):=\nabla+\sum_{j}(-1)^{\nu(j)}\omega_{j}\widehat{\otimes}\psi_{\beta}(\xi_{j}),$

where

(5.31) $\displaystyle\nu(j):=\begin{cases}0&j\equiv 0,1\pmod{4}\\\ 1&j\equiv
2,3\pmod{4}.\end{cases}$

The expression (5.30) does not depend on the decomposition (3.40).

######  Lemma 5.32.

The operator $\psi_{\beta}(\nabla\\!\\!\\!\\!\nabla^{\not{S}})$ in (5.30) is a
$\widetilde{\Sigma}^{0,1}\mathcal{A}$-superconnection on $\not{S}$ compatible
with $\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}}$. Moreover, if
$\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ is a skew-adjoint superconnection, then
$\psi_{\beta}(\nabla\\!\\!\\!\\!\nabla^{\not{S}})$ is a self-adjoint
superconnection.

###### Proof.

The first statement follows from the fact that $\psi_{\beta}$ preserves the
$\mathbb{Z}_{2}$-grading. The second statement follows from a degree-wise
check by using the definition of self/skew-adjointness of superconnections
(Definition 3.60). ∎

######  Proposition 5.33.

In the above setting, assume that $\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ is
skew-adjoint. For any element $f\in\mathbb{R}[[z]]$, we have

(5.34)
$\displaystyle\mathrm{Tr}_{\Sigma^{0,1}\mathcal{A}}(f(F(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\Sigma^{0,1}\nabla^{\mathcal{A}})))=\mathrm{Tr}_{\widetilde{\Sigma}^{0,1}\mathcal{A}}(f(-F(\psi_{\beta}(\nabla\\!\\!\\!\\!\nabla^{\not{S}});\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}})))$
$\displaystyle\in\Omega_{\mathrm{clo}}^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})}(X;\mathrm{Ori}(\mathcal{A})),$

under the following identifications of orientation bundles,

$\displaystyle\mathrm{Ori}(\mathcal{A})$
$\displaystyle\simeq\mathrm{Ori}(\Sigma^{0,1}\mathcal{A}),$ $\displaystyle u$
$\displaystyle\mapsto u\widehat{\otimes}\beta,$
$\displaystyle\mathrm{Ori}(\mathcal{A})$
$\displaystyle\simeq\mathrm{Ori}(\widetilde{\Sigma}^{0,1}\mathcal{A}),$
$\displaystyle u$
$\displaystyle\mapsto(-1)^{\nu(\mathrm{type}(\mathcal{A}))}u\widehat{\otimes}\beta^{\mathrm{type}(\mathcal{A})+1},$

where $\nu$ is given by (5.31).

###### Proof.

The both sides of (5.34) are closed forms of degree
$-\mathrm{type}(\mathcal{A})\pmod{4}$ by Proposition 3.63 and Lemma 5.3.

In this proof, we denote

$\displaystyle F$
$\displaystyle:=F(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\Sigma^{0,1}\nabla^{\mathcal{A}}),$
$\displaystyle\widetilde{F}$
$\displaystyle:=F(\psi_{\beta}(\nabla\\!\\!\\!\\!\nabla^{\not{S}});\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}}).$

Also, for elements $\Xi\in\Omega^{*}(X;\mathrm{End}(\not{S}))$, we denote by
$\Xi=\sum_{j\in\mathbb{Z}_{4}}\Xi_{j}$ the decomposition with respect to the
form-degree modulo $4$, in which
$\Xi_{j}\in\Omega^{4\mathbb{Z}+j}(U;\mathrm{End}(\not{S}))$.

It is enough to prove in the case that $f(z)=z^{n}$ for each $n$. We may work
locally, and use the local expression as in (3.43). As explained there, we may
write $\nabla^{\mathcal{A}}=d+C$ for some
$C\in\Omega^{1}(U;A^{0}_{\mathrm{skew}})$, and
$\nabla\\!\\!\\!\\!\nabla^{\not{S}}=d+C+B$ for some
$B\in\Omega^{*}(U;\mathrm{End}_{\Sigma^{0,1}\mathcal{A}}(\not{S}))^{1}$. Then
we have

$\displaystyle\psi_{\beta}(\nabla\\!\\!\\!\\!\nabla^{\not{S}})=d+C+\beta\cdot
B_{0}+B_{1}-\beta\cdot B_{2}-B_{3}.$

Here we denote
$\beta\cdot(\omega\widehat{\otimes}\xi):=\omega\widehat{\otimes}(\beta\xi)$
for
$\omega\widehat{\otimes}\xi\in\Omega^{*}(U;\mathrm{End}_{\Sigma^{0,1}\mathcal{A}}(\not{S}))$.
Let us denote $\widetilde{B}:=\beta\cdot B_{0}+B_{1}-\beta\cdot B_{2}-B_{3}$.
Then we have (3.48)

$\displaystyle F$ $\displaystyle=dB+B^{2},$ $\displaystyle\widetilde{F}$
$\displaystyle=d\widetilde{B}+\widetilde{B}^{2}.$

Using $\beta\cdot B_{j}\cdot=(-1)^{|j|+1}B_{j}\cdot\beta\cdot$ as operators on
$\Omega^{*}(U;\mathrm{End}_{\Sigma^{0,1}\mathcal{A}}(\not{S}))$, we can easily
check that

$\displaystyle\widetilde{F}_{0}$ $\displaystyle=-F_{0},$
$\displaystyle\widetilde{F}_{1}$ $\displaystyle=\beta\cdot F_{1},$
$\displaystyle\widetilde{F}_{2}$ $\displaystyle=F_{2},$
$\displaystyle\widetilde{F}_{3}$ $\displaystyle=-\beta\cdot F_{3}.$

It holds that $\beta\cdot F_{j}\cdot=(-1)^{|j|}F_{j}\cdot\beta\cdot$ since
$F\in\Omega^{*}(U;\mathrm{End}_{\Sigma^{0,1}\mathcal{A}}(\not{S}))^{0}$. Using
this formula, we can check that

$\displaystyle((-\widetilde{F})^{n})_{0}$ $\displaystyle=(F^{n})_{0},$
$\displaystyle((-\widetilde{F})^{n})_{1}$
$\displaystyle=-\beta\cdot(F^{n})_{1},$
$\displaystyle((-\widetilde{F})^{n})_{2}$ $\displaystyle=-(F^{n})_{2},$
$\displaystyle((-\widetilde{F})^{n})_{3}$
$\displaystyle=\beta\cdot(F^{n})_{3},$

by an induction on $n$. As we have checked at the beginning of the proof, the
both sides of (5.34) are forms of degree
$-\mathrm{type}(\mathcal{A})\pmod{4}$. Using this fact, we see that

$\displaystyle\mathrm{Tr}_{u\widehat{\otimes}\beta}(F^{n})$
$\displaystyle=\mathrm{Tr}_{u\widehat{\otimes}\beta}((F^{n})_{-\mathrm{type}(\mathcal{A})})$
$\displaystyle=\mathrm{Tr}_{(-1)^{\nu(\mathrm{type}(\mathcal{A}))}u\widehat{\otimes}\beta^{\mathrm{type}(\mathcal{A})+1}}(((-\widetilde{F})^{n})_{-\mathrm{type}(\mathcal{A})})$
$\displaystyle=\mathrm{Tr}_{(-1)^{\nu(\mathrm{type}(\mathcal{A}))}u\widehat{\otimes}\beta^{\mathrm{type}(\mathcal{A})+1}}((-\widetilde{F})^{n}),$

as desired. ∎

By Proposition 5.33, we get the following.

######  Corollary 5.35.

Let $X$ and $(\mathcal{A},\nabla^{\mathcal{A}})$ be as above and $\not{S}$ be
a $\Sigma^{0,1}\mathcal{A}$-module bundle with inner product. Then for any
skew-adjoint $\Sigma^{0,1}\mathcal{A}$-superconnection
$\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ compatible with
$\Sigma^{0,1}\nabla^{\mathcal{A}}$, we have

(5.36)
$\displaystyle\mathrm{Ph}_{\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\Sigma^{0,1}\nabla^{\mathcal{A}})=\mathrm{Ph}_{\mathrm{self}}(\psi_{\beta}(\nabla\\!\\!\\!\\!\nabla^{\not{S}});\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}}).$

On the manifold $I\times X$, for any skew-adjoint
$\mathrm{pr}_{X}^{*}(\Sigma^{0,1}\mathcal{A})$-superconnection
$\nabla\\!\\!\\!\\!\nabla^{\not{S},I}$ compatible with
$\mathrm{pr}_{X}^{*}(\Sigma^{0,1}\nabla^{\mathcal{A}})$, we have

(5.37)
$\displaystyle\mathrm{CS}_{\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S},I};\Sigma^{0,1}\nabla^{\mathcal{A}})=\mathrm{CS}_{\mathrm{self}}(\psi_{\beta}(\nabla\\!\\!\\!\\!\nabla^{\not{S},I});\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}}).$

Here the orientation bundles are identified as in Proposition 5.33.

Using this corollary, we also get the correspondence of the Pontryagin
character forms for mass terms and gradations.

######  Corollary 5.38.

Let $X$ and $(\mathcal{A},\nabla^{\mathcal{A}})$ be as above and $\not{S}$ be
a $\Sigma^{0,1}\mathcal{A}$-module bundle with inner product, equipped with a
skew-adjoint $\Sigma^{0,1}\mathcal{A}$-connection $\nabla^{\not{S}}$. Then for
any element $m\in
C^{\infty}(X;\mathrm{Skew}^{*}_{\Sigma^{0,1}\mathcal{A}}(\not{S}))$, we have

(5.39)
$\displaystyle\mathrm{Ph}_{\mathrm{skew}}(m;\Sigma^{0,1}\nabla^{\mathcal{A}},\nabla^{\not{S}})=\mathrm{Ph}_{\mathrm{self}}(\psi_{\beta}(m);\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}},\nabla^{\not{S}}),$

in which the identity $\psi_{\beta}(\nabla^{\not{S}})=\nabla^{\not{S}}$ is
used. For any element $m_{I}\in C^{\infty}(I\times
X;\mathrm{pr}_{X}^{*}\mathrm{Skew}_{\Sigma^{0,1}\mathcal{A}}^{*}(\not{S}))$,
we also have

(5.40)
$\displaystyle\mathrm{CS}_{\mathrm{skew}}(m_{I};\Sigma^{0,1}\nabla^{\mathcal{A}},\nabla^{\not{S}})=\mathrm{CS}_{\mathrm{self}}(\psi_{\beta}(m_{I});\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}},\nabla^{\not{S}}).$

Here the orientation bundles are identified as in Proposition 5.33.

###  5.4. The differential model

####  5.4.1. The untwisted groups $\widehat{KO}_{-}^{A}(X,Y)$

Now we define the untwisted differential $KO$-theory $\widehat{KO}_{-}$. As in
the case for topological $KO_{-}$, the definition is given by just replacing
$\mathrm{Self}$ to $\mathrm{Skew}$ and changing the form degree from
$(4\mathbb{Z}+\mathrm{type}(A)-1)$ to $(4\mathbb{Z}-\mathrm{type}(A)-3)$ in
Definitions 4.1 and 4.2 of $\widehat{KO}_{+}$, as follows.

######  Definition 5.41.

Let $A$ be a nondegenerate simple central graded $*$-algebra. Let $(X,Y)$ be
an object in $\mathrm{MfdPair}_{f}$.

  * •

A $\widehat{KO}_{-}$-cocycle $(S,m_{0},m_{1},\eta)$ on $(X,Y)$ consists of an
$A$-module $S$ with an inner product, two smooth maps $m_{0},m_{1}\in
C^{\infty}(X,\mathrm{Skew}_{A}^{*}(S))$ such that $m_{0}|_{Y}=m_{1}|_{Y}$, and
an element

$\eta\in\Omega^{4\mathbb{Z}-\mathrm{type}(A)-3}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d).$

  * •

Two $\widehat{KO}_{-}$-cocycles $(S,m_{0},m_{1},\eta)$ and
$(S^{\prime},m^{\prime}_{0},m^{\prime}_{1},\eta^{\prime})$ are isomorphic if
there exists an isometric isomorphism of $A$-modules $f\colon S\simeq
S^{\prime}$ such that $f\circ m_{i}=m^{\prime}_{i}\circ f$ for $i=0,1$, and we
have $\eta=\eta^{\prime}$.

######  Definition 5.42 ($\widehat{KO}_{-}^{A}(X,Y)$).

Let $A$ and $(X,Y)$ be as in Definition 5.41

  * •

We introduce an abelian monoid structure on the set $\widehat{M}^{A}_{-}(X,Y)$
of isomorphism classes of $\widehat{KO}_{-}$-cocycles $(S,m_{0},m_{1},\eta)$
on $(X,Y)$ by

$\displaystyle[S,m_{0},m_{1},\eta]+[S^{\prime},m^{\prime}_{0},m^{\prime}_{1},\eta^{\prime}]=[S\oplus
S^{\prime},m_{0}\oplus m^{\prime}_{0},m_{1}\oplus
m^{\prime}_{1},\eta+\eta^{\prime}].$

  * •

We define $\widehat{Z}^{A}_{-}(X,Y)$ to be the submonoid of
$\widehat{M}^{A}_{-}(X,Y)$ consisting of elements of the form

$\displaystyle[S,m_{0},m_{1},\mathrm{CS}_{\mathrm{skew}}(m_{I})],$

where $m_{I}$ is a smooth homotopy between $m_{0}$ and $m_{1}$ which is
constant on $Y$, i.e., a smooth map $m_{I}\in C^{\infty}(I\times
X,\mathrm{Skew}^{*}_{A}(S))$ with $m_{I}|_{\\{i\\}\times X}=m_{i}$ for $i=0,1$
and $m_{I}|_{\\{t\\}\times Y}=m_{0}|_{Y}$ for all $t\in I$.

  * •

We define
$\widehat{KO}^{A}_{-}(X,Y):=\widehat{M}^{A}_{-}(X,Y)/\widehat{Z}^{A}_{-}(X,Y)$.

In a way similar to the proof of Lemma 4.3, we can show that
$\widehat{KO}^{A}_{-}(X,Y)$ is an abelian group, and thus
$\widehat{KO}^{\mathcal{A}}$ defines a functor
$\mathrm{MfdPair}^{\mathrm{op}}_{f}\to\mathrm{Ab}$.

######  Definition 5.43 (Structure homomorphisms for $\widehat{KO}^{A}_{-}$).

We define the following structure homomorphisms, which are natural in $(X,Y)$.

$\displaystyle R$
$\displaystyle\colon\widehat{KO}_{-}^{A}(X,Y)\to\Omega^{4\mathbb{Z}-\mathrm{type}(A)-2}_{\mathrm{clo}}(X,Y;\mathrm{Ori}(A))$
$\displaystyle[S,m_{0},m_{1},\eta]\mapsto\mathrm{Ph}_{\mathrm{skew}}(m_{1})-\mathrm{Ph}_{\mathrm{skew}}(m_{0})+d\eta.$
$\displaystyle I$ $\displaystyle\colon\widehat{KO}_{-}^{A}(X,Y)\to
KO_{-}^{A}(X,Y)$ $\displaystyle[S,m_{0},m_{1},\eta]\mapsto[S,m_{0},m_{1}].$
$\displaystyle a$
$\displaystyle\colon\Omega^{4\mathbb{Z}-\mathrm{type}(A)-3}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d)\to\widehat{KO}_{-}^{A}(X,Y)$
$\displaystyle\eta\mapsto[0,0,0,\eta].$

We have the natural isomorphism corresponding to Proposition 4.31,

(5.44)
$\displaystyle\widehat{KO}^{A}_{-}\simeq\widehat{KO}^{\mathrm{End}(V)\widehat{\otimes}A}_{-},$

by sending $[S,m_{0},m_{1},\eta]$ to $[E\otimes
S,\psi^{E}(m_{0}),\psi^{E}(m_{1}),\eta]$.

Moreover, we have the following refinement of the untwisted version of
Proposition 5.14.

######  Proposition 5.45.

We have a natural isomorphism

(5.46)
$\displaystyle\widehat{KO}_{-}^{\Sigma^{0,1}A}\simeq\widehat{KO}_{+}^{\widetilde{\Sigma}^{0,1}A}$

which is compatible with the corresponding isomorphisms of other functors
appearing in Definitions 5.43 and 4.6 via the structure homomorphisms. Here we
use Proposition 5.14 and the identification
$\mathrm{Ori}(\Sigma^{0,1}A)\simeq\mathrm{Ori}(\widetilde{\Sigma}^{0,1}A)$ in
Proposition 5.33.

###### Proof.

The transformation (5.46) on an object $(X,Y)$ in $\mathrm{MfdPair}_{f}$ is
given by

$\displaystyle[S,m_{0},m_{1},\eta]\mapsto[S,\psi_{\beta}(m_{0}),\psi_{\beta}(m_{1}),\eta],$

where
$\psi_{\beta}\colon\mathrm{Skew}^{*}_{\Sigma^{0,1}A}(S)\simeq\mathrm{Self}^{*}_{\widetilde{\Sigma}^{0,1}{A}}(S)$
is the isomorphism in (5.6). The well-definedness follows from Corollary 5.38.
Since $\psi_{\beta}$ is an isomorphism, we easily see that this map gives a
natural isomorphism. ∎

Since we have $\Sigma^{0,1}\Sigma^{1,0}A=A\hat{\otimes}Cl_{1,1}$ and
$Cl_{1,1}$ is negligible, by Proposition 5.14 and (5.13), we have a natural
isomorphism

(5.47)
$\displaystyle\widehat{KO}_{-}^{A}\simeq\widehat{KO}_{+}^{\widetilde{\Sigma}^{0,1}\Sigma^{1,0}A},$

which is compatible with the structure homomorphisms. Recall that by Theorem
4.30 we know that
$(\widehat{KO}^{\widetilde{\Sigma}^{0,1}\Sigma^{1,0}A}_{+},R,I,a)$ is a
differential extension of
$KO^{\widetilde{\Sigma}^{0,1}\Sigma^{1,0}A}_{+}\simeq
KO^{-\mathrm{type}(A)-2}$. Thus we get the following.

######  Theorem 5.48.

The quadruple $\left(\widehat{KO}^{A}_{-},R,I,a\right)$ is a differential
extension of the $KO$-theory $KO^{A}_{-}\simeq KO^{-\mathrm{type}(A)-2}$ on
$\mathrm{MfdPair}_{f}$.

The natural isomorphism (5.47), being an explicit algebraic manipulation,
allows us to translate results on $\widehat{KO}_{+}$ into corresponding ones
on $\widehat{KO}_{-}$.

####  5.4.2. The twisted groups $\widehat{KO}_{-}^{\mathcal{A}}(X,Y)$

Now we define the twisted groups $\widehat{KO}_{-}^{\mathcal{A}}(X,Y)$. Again
the category $\mathfrak{Twist}^{2}_{\widehat{KO}_{-}}$ of twists for
$\widehat{KO}_{-}$ is just the same as that for $\widehat{KO}_{+}$,

(5.49)
$\displaystyle\mathfrak{Twist}^{2}_{\widehat{KO}_{-}}:=\mathfrak{Twist}^{2}_{\widehat{KO}_{+}}.$

The definition of the twisted $\widehat{KO}_{-}$ is also a simple modification
of Definition 4.37, so we go briefly.

######  Definition 5.50 ($\widehat{KO}_{-}^{\mathcal{A}}(X,Y)$).

Let $(X,Y,\mathcal{A})\in\mathfrak{Twist}^{2}_{\widehat{KO}_{-}}$. Replacing
$\mathrm{Self}$ by $\mathrm{Skew}$ and changing the form degree from
$(4\mathbb{Z}+\mathrm{type}(A)-1)$ to $(4\mathbb{Z}-\mathrm{type}(A)-3)$ in
Definition 4.37, we define the abelian group
$\widehat{KO}_{-}^{\mathcal{A}}(X,Y)$.

An element of $\widehat{KO}_{-}^{\mathcal{A}}(X,Y)$ is of the form
$[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},m_{0},m_{1},\eta]$, where
$m_{i}\in C^{\infty}(X;\mathrm{Skew}_{\mathcal{A}}^{*}(\not{S}))$ are such
that $m_{0}|_{Y}=m_{1}|_{Y}$, and

$\eta\in\Omega^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})-3}(X,Y;\mathrm{Ori}(\mathcal{A}))/\mathrm{Im}(d).$

We have
$[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},m_{0},m_{1},\mathrm{CS}_{\mathrm{skew}}(m_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})]=0$
for a homotopy $m_{I}$ from $m_{0}$ to $m_{1}$ relative to $Y$. We clearly
have the functor

$\displaystyle\widehat{KO}_{-}\colon\mathfrak{Twist}^{2}_{\widehat{KO}_{-}}\to\mathrm{Ab}.$

######  Definition 5.51 (Structure homomorphisms for
$\widehat{KO}^{\mathcal{A}}_{-}$).

We define the structure homomorphisms

$\displaystyle R$
$\displaystyle\colon\widehat{KO}_{-}^{\mathcal{A}}(X,Y)\to\Omega^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})-2}_{\mathrm{clo}}(X,Y;\mathrm{Ori}(\mathcal{A})),$
$\displaystyle I$ $\displaystyle\colon\widehat{KO}_{-}^{\mathcal{A}}(X,Y)\to
KO_{-}^{\mathcal{A}}(X,Y),$ $\displaystyle a$
$\displaystyle\colon\Omega^{4\mathbb{Z}-\mathrm{type}(\mathcal{A})-3}(X,Y;\mathrm{Ori}(\mathcal{A}))/\mathrm{Im}(d)\to\widehat{KO}_{-}^{\mathcal{A}}(X,Y)$

applying the same modification as in Definition 5.50 to Definition 4.41.

As a counterprt of Proposition 4.50, we have an isomorphism

(5.52)
$\displaystyle\widehat{KO}^{\mathcal{A}}_{-}(X,Y)\simeq\widehat{KO}^{\mathrm{End}(E)\widehat{\otimes}\mathcal{A}}_{-}(X,Y),$

which are compatible with the structure homomorphisms. Moreover, as in the
untwisted case (Proposition 5.45), we get the refinement of Proposition 5.14
as follows.

######  Proposition 5.53.

We have an isomorphism

(5.54)
$\displaystyle\widehat{KO}_{-}^{\Sigma^{0,1}\mathcal{A}}(X,Y)\simeq\widehat{KO}_{+}^{\widetilde{\Sigma}^{0,1}\mathcal{A}}(X,Y)$

which is compatible with the corresponding isomorphisms of other functors
appearing in Definitions 5.51 and 4.41 via the structure homomorphisms. Here
we use Proposition 5.14 and the identification
$\mathrm{Ori}(\Sigma^{0,1}\mathcal{A})\simeq\mathrm{Ori}(\widetilde{\Sigma}^{0,1}\mathcal{A})$
in Proposition 5.33.

###### Proof.

Take a connection $\nabla^{\mathcal{A}}$ on $\mathcal{A}$. Recall that, by
Lemma 4.42, any element in
$\widehat{KO}_{+}^{\widetilde{\Sigma}^{0,1}\mathcal{A}}(X,Y)$ is represented
by a $\widehat{KO}_{+}$-cocycle of the form
$(\not{S},\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}},\nabla^{\not{S}},h_{0},h_{1},\eta)$.
The $\widehat{KO}_{-}$-version of the lemma is shown by exactly the same
proof, so we can represent any element in
$\widehat{KO}_{-}^{\Sigma^{0,1}\mathcal{A}}(X,Y)$ by a
$\widehat{KO}_{-}$-cocycle of the form
$(\not{S},{\Sigma}^{0,1}\nabla^{\mathcal{A}},\nabla^{\not{S}},m_{0},m_{1},\eta)$.

Then, the homomorphism (5.54) is given by

$\displaystyle[\not{S},{\Sigma}^{0,1}\nabla^{\mathcal{A}},\nabla^{\not{S}},m_{0},m_{1},\eta]\mapsto[\not{S},\widetilde{\Sigma}^{0,1}\nabla^{\mathcal{A}},\nabla^{\not{S}},\psi_{\beta}(m_{0}),\psi_{\beta}(m_{1}),\eta],$

where
$\psi_{\beta}\colon\mathrm{Skew}^{*}_{\Sigma^{0,1}\mathcal{A}}(\not{S})\simeq\mathrm{Self}^{*}_{\widetilde{\Sigma}^{0,1}\mathcal{A}}(\not{S})$
is the isomorphism in (5.6). The well-definedness follows from Corollary 5.38,
and the resulting map is independent on the choice of $\nabla^{\mathcal{A}}$
by Lemma 4.42 and its $\widehat{KO}_{-}$-version. Since $\psi_{\beta}$ is an
isomorphism, we easily see that this map gives an isomorphism. ∎

The above results give a twisted generalization of (5.47)

(5.55)
$\displaystyle\widehat{KO}_{-}^{\mathcal{A}}(X,Y)\simeq\widehat{KO}_{+}^{\widetilde{\Sigma}^{0,1}\Sigma^{1,0}\mathcal{A}}(X,Y),$

which is compatible with the structure homomorphisms. Recall that by Theorem
4.49 we know that $(\widehat{KO}_{+},R,I,a)$ is a differential extension of
the twisted $KO$-theory $KO_{+}$. Thus we get the following.

######  Theorem 5.56.

The quadruple $\left(\widehat{KO}_{-},R,I,a\right)$ is a differential
extension of the twisted $KO$-theory $KO_{-}$ on
$\mathfrak{Twist}^{2}_{\widehat{KO}_{-}}$.

Again, the natural isomorphism (5.55) allows us to translate results on
twisted $\widehat{KO}_{+}$ into corresponding ones on twisted
$\widehat{KO}_{-}$.

##  6\. The complex case : $\widehat{K}_{+}$ and $\widehat{K}_{-}$

So far we have worked in the $\mathbb{R}$-linear setting. In this section, we
explain that the $\mathbb{C}$-linear version of the above story works by
straightforward modifications, producing differential extensions of the
Karoubi’s $K$-theory $K_{+}$ (Remark 2.37).

###  6.1. The complex superconnections

In this subsection we explain the $\mathbb{C}$-linear version of Section 3. In
the $\mathbb{C}$-linear setting, we work with simple central graded
$*$-algebras over $\mathbb{C}$. The traces $\mathrm{Tr}_{u}$ and
$\mathrm{Tr}_{A}$ are defined exactly as in Subsection 3.1, by replacing
$\dim_{\mathbb{R}}$ with $\dim_{\mathbb{C}}$.

We explain the complex superconnection formalism by modifying Subsection 3.3.
We start with a bundle of nondegenerate simple central graded $*$-algebras
$\mathcal{A}$ over $\mathbb{C}$ and an $\mathcal{A}$-module bundle $\not{S}$
with an inner product on a manifold. Then we define
$\mathcal{A}$-superconnections $\nabla\\!\\!\\!\\!\nabla^{\not{S}}$, the
curvature $F(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\nabla^{\mathcal{A}})$, the
self/skew-adjointness of superconnections exactly in the same way as in that
subsection. We easily see that the obvious $\mathbb{C}$-linear version of the
results until Remark 3.61 hold. In the $\mathbb{C}$-linear case, we have
(recall $\mathrm{type}(\mathcal{A})\in H^{0}(X;\mathbb{Z}_{2})$ in the complex
settings)

(6.1)
$\displaystyle\mathrm{Tr}_{\mathcal{A}}(f(F(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\nabla^{\mathcal{A}})))\in\Omega_{\mathrm{clo}}^{2\mathbb{Z}+\mathrm{type}(\mathcal{A})+1}(X;\mathrm{Ori}(\mathcal{A})\otimes_{\mathbb{R}}\mathbb{C}).$

for a $\mathcal{A}$-superconnection $\nabla\\!\\!\\!\\!\nabla^{\not{S}}$, and
any $f\in\mathbb{C}[[z]]$. For $\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ and
$\nabla\\!\\!\\!\\!\nabla^{\not{S},I}$ self/skew-adjoint, we define

$\displaystyle\mathrm{Ch}_{\mathrm{self}/\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\nabla^{\mathcal{A}})$
$\displaystyle:=\mathrm{Tr}_{\mathcal{A}}(e^{\mp
F(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\nabla^{\mathcal{A}})})\in\Omega_{\mathrm{clo}}^{2\mathbb{Z}+\mathrm{type}(\mathcal{A})+1}(X;\mathrm{Ori}(\mathcal{A})\otimes_{\mathbb{R}}\mathbb{C}),$
$\displaystyle\mathrm{CS}_{\mathrm{self}/\mathrm{skew}}(\nabla\\!\\!\\!\\!\nabla^{\not{S},I};\nabla^{\mathcal{A}})$
$\displaystyle:=\int_{I}\mathrm{Ch}_{\mathrm{self}/\mathrm{skew}}\left(\nabla\\!\\!\\!\\!\nabla^{\not{S},I};\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}}\right)\in\Omega^{2\mathbb{Z}+\mathrm{type}(\mathcal{A})}(X;\mathrm{Ori}(\mathcal{A})\otimes_{\mathbb{R}}\mathbb{C}).$

Now we move on to the constructions corresponding to Subsections 3.4 and 5.3.
Recall the endomorphism $\mathcal{R}_{\mathbb{C}}$ on $\mathbb{C}$-valued
differential forms defined in (2.58). Given $\nabla^{\mathcal{A}}$ and a self-
adjoint (=skew-adjoint) $\mathcal{A}$-connection $\nabla^{\not{S}}$, for an
invertible section $h\in
C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$, we consider the
manifold $(0,\infty)_{t}\times X$ with the self-adjoint superconnection
$\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th$, and define its Chern character form
by

(6.2)
$\mathrm{Ch}_{\mathrm{self}}(h;\nabla^{\mathcal{A}},\nabla^{\not{S}}):=-\pi^{-1/2}\mathcal{R}_{\mathbb{C}}\circ\int_{(0,\infty)}\mathrm{Ch}_{\mathrm{self}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+th;\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})\\\
\in\Omega^{2\mathbb{Z}+\mathrm{type}(\mathcal{A})}(X;\mathrm{Ori}(\mathcal{A})).$

Note that $\mathrm{Ori}(\mathcal{A})$ is a real line bundle. Thus it can be
nontrivial, but can be checked directly, that the formula (6.2) defines an
element in
$\Omega^{2\mathbb{Z}+\mathrm{type}(\mathcal{A})}(X;\mathrm{Ori}(\mathcal{A}))$.

Similarly for $m\in C^{\infty}(X;\mathrm{Skew}_{\mathcal{A}}^{*}(\not{S}))$,
we define

(6.3)
$\mathrm{Ch}_{\mathrm{skew}}(m;\nabla^{\mathcal{A}},\nabla^{\not{S}}):=-\pi^{-1/2}(-\sqrt{-1})^{\deg}\mathcal{R}_{\mathbb{C}}\circ\int_{(0,\infty)}\mathrm{Ch}_{\mathrm{skew}}(\mathrm{pr}_{X}^{*}\nabla^{\not{S}}+tm;\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}})\\\
\in\Omega^{2\mathbb{Z}+\mathrm{type}(\mathcal{A})}(X;\mathrm{Ori}(\mathcal{A})).$

Here $(-\sqrt{-1})^{\deg}$ is an operator on $\mathbb{C}$-valued differential
forms which multiplies $d$-forms by $(-\sqrt{-1})^{d}$. For a homotopy
$h_{I}\in C^{\infty}(I\times
X;\mathrm{pr}_{X}^{*}\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$, we define the
Chern-Simons form by

(6.4)
$\mathrm{CS}_{\mathrm{self}}(h_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}}):=\int_{I}\mathrm{Ch}_{\mathrm{self}}\left(h_{I};\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}},\mathrm{pr}_{X}^{*}\nabla^{\not{S}}\right)\\\
\in\Omega^{2\mathbb{Z}+\mathrm{type}(\mathcal{A})-1}(X;\mathrm{Ori}(\mathcal{A})).$

Similarly for $m_{I}\in C^{\infty}(I\times
X;\mathrm{pr}_{X}^{*}\mathrm{Skew}_{\mathcal{A}}^{*}(\not{S}))$, we define

(6.5)
$\mathrm{CS}_{\mathrm{skew}}(m_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}}):=\int_{I}\mathrm{Ch}_{\mathrm{skew}}\left(m_{I};\mathrm{pr}_{X}^{*}\nabla^{\mathcal{A}},\mathrm{pr}_{X}^{*}\nabla^{\not{S}}\right)\\\
\in\Omega^{2\mathbb{Z}+\mathrm{type}(\mathcal{A})-1}(X;\mathrm{Ori}(\mathcal{A})).$

As a special case of the above constructions applied to trivial bundles
$\underline{A}$ and $\underline{S}$, we get the $\mathbb{C}$-linear version of
Subsection 3.2 and Subsubsection 5.3.1. We use the corresponding notations
such as $\mathrm{Ch}_{\mathrm{self}}(h)$ for $h\in\mathrm{Skew}^{*}_{A}(S)$.

We can show that the analogous properties of Chern character forms
$\mathrm{Ch}_{\mathrm{self}}(h)$ corresponding to those listed in Subsection
3.2.3. The invariance under tensoring negligible modules (Lemma 3.23) can be
shown in the same way. The compatibility with the suspension (Proposition
3.27) is also basically the same, but we need to take care of the
multiplications by $(\sqrt{-1})^{\bullet}$ appearing in the definition (2.58)
of $\mathcal{R}_{\mathbb{C}}$. The statement becomes the following.

######  Proposition 6.6.

For any element $h\in C^{\infty}(X,\mathrm{Self}_{\Sigma A}^{\dagger}(S))$, we
define $\widetilde{h}\in C^{\infty}(I\times X,\mathrm{Self}_{A}^{\dagger}(S))$
by the same formula as (3.25). We have

$\displaystyle\mathrm{Ch}_{\mathrm{self}}(h)=\int_{I}\mathrm{Ch}_{\mathrm{self}}(\widetilde{h})$

under the identification of orientation bundles (cf. (2.2))

$\displaystyle\mathrm{Ori}(A)$ $\displaystyle\simeq\mathrm{Ori}(\Sigma A),$
$\displaystyle u$
$\displaystyle\mapsto\left\\{\begin{array}[]{ll}u\widehat{\otimes}\beta,&(\mathrm{type}(A)=0)\\\
\sqrt{-1}u\widehat{\otimes}\beta.&(\mathrm{type}(A)=1)\end{array}\right.$

To show that the Chern character forms represent the correct cohomology
classes, we need the result corresponding to Theorem 3.29 about the Chern
character forms for the universal (tautological) gradation
$h_{\mathrm{univ}}\in
C^{\infty}(\mathrm{Self}^{*}_{A}(S),\mathrm{Self}^{*}_{A}(S))$. The
topological Chern character homomorphism (2.53) can be regarded as an element

$\displaystyle\mathrm{Ch}_{\mathrm{top}}\in{H}^{2\mathbb{Z}+{\mathrm{type}}(A)}\left(K_{\mathrm{type}(A)},\\{*\\};\mathbb{R}\right).$

We claim that the equality

(6.7)
$\displaystyle\mathrm{Rham}\left(\mathrm{Ch}_{\mathrm{self}}(h_{\mathrm{univ}})\right)=\iota_{S,u}^{*}\mathrm{Ch}_{\mathrm{top}}$

holds in
$H^{2\mathbb{Z}+{\mathrm{type}}(A)}\left(\mathrm{Self}_{A}^{*}(S),\\{\beta\\};\mathbb{R}\right)$.
Indeed, the proof of Theorem 3.29 can be modified to the $K$-theory. In that
proof, we used the fact that the degree-zero part of the topological
Pontryagin character is realized as the Chern-Weil construction on the real
Grassmannians, and used the diffeomorphism
$\mathrm{Self}^{\dagger}_{A}(S_{n})\simeq\mathrm{Gr}(\mathbb{R}^{2n})$ in
(2.49) to compare $\mathrm{Ph}_{\mathrm{self}}(h_{\mathrm{univ}})$ with
$\mathcal{R}\circ\mathrm{Tr}(e^{-\nabla_{\mathrm{Gr}}^{2}})$. The complex
analogue of this argument works since the topological Chern character is
realized as the Chern-Weil construction on the complex Grassmannians
$\mathrm{Gr}(\mathbb{C}^{2n})$, and we have a diffeomorphism among models of
the $K$-theory spectrum corresponding to that detailed in the study of the
$KO$-spectrum in Subsubsection 2.3.2. These results imply the following one,
corresponding to Corollary 3.31.

######  Corollary 6.8.

Let $A$ be a simple central graded $*$-algebra over $\mathbb{C}$ and $(X,Y)$
be an object of $\mathrm{MfdPair}_{f}$. Represent classes of $K_{+}^{A}(X,Y)$
by triples $(S,h_{0},h_{1})$ with $h_{i}\in
C^{\infty}(X;\mathrm{Self}_{A}^{*}(S))$ as in Remark 2.37. Then the
topological Chern character homomorphism

$\displaystyle\mathrm{Ch}_{\mathrm{top}}\colon K_{+}^{A}(X,Y)\to
H^{2\mathbb{Z}+{\mathrm{type}}(A)}(X,Y;\mathrm{Ori}(A))$

is given by

$\displaystyle[S,h_{0},h_{1}]\mapsto\mathrm{Rham}\left(\mathrm{Ch}_{\mathrm{self}}(h_{1})-\mathrm{Ch}_{\mathrm{self}}(h_{0})\right).$

Using this result on untwisted Chern character forms, we also get the
corresponding statement about the twisted case. Recall that, for a CW-pair
$(X,Y)$, we have the twisted Chern character homomorphism,

$\displaystyle\mathrm{Ch}_{\mathrm{top}}\colon K^{\mathcal{A}}_{+}(X,Y)\to
H^{2\mathbb{Z}+{\mathrm{type}}(\mathcal{A})}(X,Y;\mathrm{Ori}(\mathcal{A})).$

We then have the following result corresponding to Corollary 3.100.

######  Corollary 6.9.

let $(X,Y)$ be an object of $\mathrm{MfdPair}_{f}$, and $\mathcal{A}$ and
$\not{S}$ over $\mathbb{C}$ as above. Suppose we have two smooth sections
$h_{0},h_{1}\in C^{\infty}(X;\mathrm{Self}_{\mathcal{A}}^{*}(\not{S}))$ such
that $h_{0}|_{Y}=h_{1}|_{Y}$. Then we have

(6.10)
$\displaystyle\mathrm{Ch}_{\mathrm{top}}([\not{S},h_{0},h_{1}])=\mathrm{Rham}\left(\mathrm{Ch}_{\mathrm{self}}(h_{1};\nabla^{\mathcal{A}},\nabla^{\not{S}})-\mathrm{Ch}_{\mathrm{self}}(h_{0};\nabla^{\mathcal{A}},\nabla^{\not{S}})\right).$

Now we turn to the discussion corresponding to Subsubsection 5.3.3, relating
$\mathrm{Ch}_{\mathrm{self}}$ and $\mathrm{Ch}_{\mathrm{skew}}$. In the
$\mathbb{C}$-linear setting here, the argument is simpler. Recall that we have
a bijection $\mathrm{Skew}^{*}_{A}(S)\simeq\mathrm{Self}_{A}^{*}(S)$ by
$m\mapsto\sqrt{-1}m$ (Remarks 5.7 and 5.19). Let us decompose a skew-adjoint
$\mathcal{A}$-superconnection $\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ compatible
with $\nabla^{\mathcal{A}}$ as (3.43),

$\displaystyle\nabla\\!\\!\\!\\!\nabla^{\not{S}}=\nabla+\sum_{j}\omega_{j}\widehat{\otimes}\xi_{j},$

where $\omega_{j}\in\Omega^{j}(X)$ and $\xi_{j}\in
C^{\infty}(X;\mathrm{End}_{\mathcal{A}}(\not{S}))$ with
$|\omega_{j}|+|\xi_{j}|=1\in\mathbb{Z}_{2}$. Then consider the following
operator on $\Omega^{*}(X;\not{S})$

(6.11)
$\displaystyle\psi_{\sqrt{-1}}(\nabla\\!\\!\\!\\!\nabla^{\not{S}}):=\nabla+\sum_{j}(-\sqrt{-1})^{j-1}\omega_{j}\widehat{\otimes}\xi_{j}.$

We easily see that the expression (6.11) is independent of the decomposition
and defines a self-adjoint $\mathcal{A}$-superconnection compatible with
$\nabla^{\mathcal{A}}$. We have the following result corresponding to
Proposition 5.33, whose proof is an easy computation checking the case
$f(z)=z^{n}$ and so omitted.

######  Proposition 6.12.

For a skew-adjoint $\mathcal{A}$-superconnection
$\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ compatible with $\nabla^{\mathcal{A}}$
and for any $f\in\mathbb{C}[[z]]$, we have the equality

(6.13) $\displaystyle
f(F(\nabla\\!\\!\\!\\!\nabla^{\not{S}};\nabla^{\mathcal{A}}))=(-\sqrt{-1})^{\deg}\circ
f(-F(\psi_{\sqrt{-1}}(\nabla\\!\\!\\!\\!\nabla^{\not{S}});\nabla^{\mathcal{A}}))$

in
$\Omega^{2\mathbb{Z}+\mathrm{type}(\mathcal{A})+1}(X;\mathrm{Ori}(A)\otimes_{\mathbb{R}}\mathbb{C})$.
Here $(-\sqrt{-1})^{\deg}$ is the operator on differential forms which
multiplies $d$-forms by $(-\sqrt{-1})^{d}$.

Thus we conclude:

######  Corollary 6.14.

For any element $m\in C^{\infty}(X;\mathrm{Skew}^{*}_{\mathcal{A}}(\not{S}))$,
we have

(6.15)
$\displaystyle\mathrm{Ch}_{\mathrm{skew}}(m;\nabla^{\mathcal{A}},\nabla^{\not{S}})=\mathrm{Ch}_{\mathrm{self}}(\sqrt{-1}m;\nabla^{\mathcal{A}},\nabla^{\not{S}}).$

For any element $m_{I}\in C^{\infty}(I\times
X;\mathrm{pr}_{X}^{*}\mathrm{Skew}_{\mathcal{A}}^{*}(\not{S}))$, we have

(6.16)
$\displaystyle\mathrm{CS}_{\mathrm{skew}}(m_{I};\nabla^{\mathcal{A}},\nabla^{\not{S}})=\mathrm{CS}_{\mathrm{self}}(\sqrt{-1}(m_{I});\nabla^{\mathcal{A}},\nabla^{\not{S}}).$

###  6.2. The definitions of differential $K$-theories $\widehat{K}_{+}$ and
$\widehat{K}_{-}$

Now it should be obvious to the reader that the definitions of the
differential extensions of $K_{+}$ and $K_{-}$ are given by just replacing the
coefficient from $\mathbb{R}$ to $\mathbb{C}$ in the definitions of
$\widehat{KO}_{+}$ and $\widehat{KO}_{-}$, and by using the
$\mathbb{C}$-linear version of the characteristic forms defined in the last
subsection.

######  Definition 6.17 ($\widehat{K}_{+}^{A}(X,Y)$ and
$\widehat{K}_{-}^{A}(X,Y)$).

Let $A$ be a nondegenerate simple central graded $*$-algebra over $\mathbb{C}$
and let $(X,Y)$ be an object in $\mathrm{MfdPair}_{f}$. Replacing the
coefficient $\mathbb{R}$ by $\mathbb{C}$ and the form degree $4\mathbb{Z}$ by
$2\mathbb{Z}$ and using characteristic forms in Subsection 6.1, we define
$\widehat{K}_{+}^{A}(X,Y)$ and $\widehat{K}_{-}^{A}(X,Y)$, respectively.

Thus, elements of the group $\widehat{K}_{+}^{A}(X,Y)$ are of the form
$[S,h_{0},h_{1},\eta]$, where $h_{i}\in
C^{\infty}(X,\mathrm{Self}_{A}^{*}(S))$ are such that $h_{0}|_{Y}=h_{1}|_{Y}$
and

$\eta\in\Omega^{2\mathbb{Z}+\mathrm{type}(A)-1}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d).$

We have $[S,h_{0},h_{1},\mathrm{CS}_{\mathrm{self}}(h_{I})]=0$ when $h_{I}$ is
a homotopy relative to $Y$ between $h_{0}$ and $h_{1}$. Similarly, elements of
the group $\widehat{K}_{-}^{A}(X,Y)$ are of the form $[S,m_{0},m_{1},\eta]$,
where $m_{i}\in C^{\infty}(X,\mathrm{Skew}_{A}^{*}(S))$ are such that
$m_{0}|_{Y}=m_{1}|_{Y}$ and

$\eta\in\Omega^{2\mathbb{Z}+\mathrm{type}(A)-1}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d).$

We have $[S,m_{0},m_{1},\mathrm{CS}_{\mathrm{skew}}(m_{I})]=0$ when $m_{I}$ is
a homotopy relative to $Y$ between $m_{0}$ and $m_{1}$.

For twisted groups, we define the (same) categories
$\mathfrak{Twist}^{2}_{\widehat{K}_{+}}=\mathfrak{Twist}^{2}_{\widehat{K}_{-}}$
by the $\mathbb{C}$-linear analogue of Definition 4.35. We define the
following.

######  Definition 6.18 ($\widehat{K}_{+}^{\mathcal{A}}(X,Y)$ and
$\widehat{K}_{-}^{\mathcal{A}}(X,Y)$).

Let
$(X,Y,\mathcal{A})\in\mathfrak{Twist}^{2}_{\widehat{K}_{+}}=\mathfrak{Twist}^{2}_{\widehat{K}_{-}}$.
By the same replacement as in Definition 6.17 of Definitions 4.37 and 5.50, we
define $\widehat{K}_{+}^{\mathcal{A}}(X,Y)$ and
$\widehat{K}_{-}^{\mathcal{A}}(X,Y)$, respectively.

######  Definition 6.19 (Structure homomorphisms for $\widehat{K}_{+}$ and
$\widehat{K}_{-}$).

We define the structure homomorphisms $R$, $I$, $a$ for $\widehat{K}^{A}_{+}$,
$\widehat{K}^{\mathcal{A}}_{+}$, $\widehat{K}^{A}_{-}$ and
$\widehat{K}^{\mathcal{A}}_{-}$ by the same replacement as in Definition 6.17
of Definitions 4.6, 4.41, 5.43 and 5.51, respectively.

For example in the case $\widehat{K}^{A}_{+}$ we define

$\displaystyle R$
$\displaystyle\colon\widehat{K}_{+}^{A}(X,Y)\to\Omega_{\mathrm{clo}}^{2\mathbb{Z}+\mathrm{type}(A)}(X,Y;\mathrm{Ori}(A))$
$\displaystyle[S,h_{0},h_{1},\eta]\mapsto\mathrm{Ch}_{\mathrm{self}}(h_{1})-\mathrm{Ch}_{\mathrm{self}}(h_{0})+d\eta.$
$\displaystyle I$ $\displaystyle\colon\widehat{K}_{+}^{A}(X,Y)\to
K_{+}^{A}(X,Y)$ $\displaystyle[S,h_{0},h_{1},\eta]\mapsto[S,h_{0},h_{1}].$
$\displaystyle a$
$\displaystyle\colon\Omega^{2\mathbb{Z}+\mathrm{type}(A)-1}(X,Y;\mathrm{Ori}(A))/\mathrm{Im}(d)\to\widehat{K}_{+}^{A}(X,Y)$
$\displaystyle\eta\mapsto[0,0,0,\eta].$

We have natural isomorphisms for both untwisted and twisted cases
corresponding to Propositions 5.45 and 5.53

(6.20) $\displaystyle\widehat{K}_{-}\simeq\widehat{K}_{+},$

which are compatible with the structure homomorphisms. The isomorphisms (6.20)
are given by
$[S,m_{0},m_{1},\eta]\mapsto[S,\sqrt{-1}m_{0},\sqrt{-1}m_{1},\eta]$ in the
untwisted case and
$[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},m_{0},m_{1},\eta]\mapsto[\not{S},\nabla^{\mathcal{A}},\nabla^{\not{S}},\sqrt{-1}m_{0},\sqrt{-1}m_{1},\eta]$
in the twisted case. The well-definedness follows from Corollary 6.14.

Moreover, by the cohomological properties of the Chern character forms
$\mathrm{Ch}_{\mathrm{self}}$ in Corollaries 6.8 and 6.9, we can check the
axioms of differential cohomology theories as in Theorems 4.7 and 4.43. Thus
we conclude the following.

######  Theorem 6.21.

The quadruples $\left(\widehat{K}^{A}_{+},R,I,a\right)$ and
$\left(\widehat{K}^{A}_{-},R,I,a\right)$ are differential extensions of the
$K$-theories $K^{A}_{+}\simeq K^{A}_{-}\simeq KO^{\mathrm{type}(A)}$ on
$\mathrm{MfdPair}_{f}$.

The quadruples $\left(\widehat{K}_{+},R,I,a\right)$ and
$\left(\widehat{K}_{-},R,I,a\right)$ are differential extensions of the
twisted $K$-theories ${K}_{+}\simeq K_{-}$ on
$\mathfrak{Twist}^{2}_{\widehat{K}_{+}}=\mathfrak{Twist}^{2}_{\widehat{K}_{-}}$.

##  7\. The proofs

###  7.1. The proof of Proposition 3.27

Here we prove Proposition 3.27. Let $S$ be a $\Sigma^{0,1}A$-module. For $h\in
C^{\infty}(X,\mathrm{Self}_{\Sigma^{0,1}A}^{\dagger}(S))$, we have the
corresponding $\widetilde{h}\in C^{\infty}(I\times
X,\mathrm{Self}^{\dagger}_{A}(S))$. Choose a volume element $u$ of $A$ to
trivialize $\mathrm{Ori}(A)$.

By Definition 3.15, using $h^{2}=1$ and $\widetilde{h}^{2}=1$ we have

$\displaystyle\mathrm{Ph}_{\mathrm{self}}(h)$
$\displaystyle=\pi^{-1/2}\mathcal{R}\circ\int_{(0,\infty)}dte^{-t^{2}}\mathrm{Tr}_{u\widehat{\otimes}\beta}(h\exp(-tdh)),$
$\displaystyle\mathrm{Ph}_{\mathrm{self}}(\widetilde{h})$
$\displaystyle=\pi^{-1/2}\mathcal{R}\circ\int_{(0,\infty)}dte^{-t^{2}}\mathrm{Tr}_{u}(\widetilde{h}\exp(-td\widetilde{h})).$

We compute (note that we are working in the algebra $\Omega^{*}(I\times
X,\mathrm{End}_{A}(S))$ with the multiplication (3.33)),

$\displaystyle\exp(-td\widetilde{h})$
$\displaystyle=\exp(-t(d\theta\cdot(\partial_{\theta}\widetilde{h})+(\sin\pi\theta)dh))$
$\displaystyle=\sum_{n=0}^{\infty}\frac{(-t)^{n}}{n!}(d\theta\cdot(\partial_{\theta}\widetilde{h})+(\sin\pi\theta)dh)^{n}$
$\displaystyle=d\theta\cdot(\partial_{\theta}\widetilde{h})\left(\sum_{n=0}^{\infty}\frac{(-t)^{n+1}}{n!}\cdot(\sin^{n}\pi\theta)(dh)^{n}\right)+\exp(-t(\sin\pi\theta)dh).$

Here the last equality used the fact that $dh$ commutes with
$d\theta\cdot(\partial_{\theta}\widetilde{h})$, which can be checked easily by
$h\in C^{\infty}(X,\mathrm{Self}_{\Sigma^{0,1}A}^{\dagger}(S))$ and the
formula for $\widetilde{h}$. We also note

$\displaystyle\widetilde{h}(\partial_{\theta}\widetilde{h})=\pi(\beta\cos\pi\theta+h\sin\pi\theta)(-\beta\sin\pi\theta+h\cos\pi\theta)=\pi\beta
h.$

Thus we have (note that $\widetilde{h}d\theta=-d\theta\widetilde{h}$)

(7.1) $\int_{I}\mathrm{Ph}_{\mathrm{self}}(\widetilde{h})\\\
=\pi^{1/2}\sum_{n=0}^{\infty}\left(\frac{1}{n!}\int_{0}^{\infty}(-t)^{n+1}e^{-t^{2}}dt\int_{I}\sin^{n}(\pi\theta)\cdot\mathcal{R}\left(d\theta\mathrm{Tr}_{u}(-\beta{h}(d{h})^{n})\right)\right).$

Now assume $\mathrm{type}(A)$ is even. In this case
$\mathrm{Ph}_{\mathrm{self}}(\widetilde{h})$ is an even form and
$\mathrm{Ph}_{\mathrm{self}}(h)$ is an odd form, so that (7.1) becomes

(7.2)
$\displaystyle\pi^{1/2}\sum_{l=0}^{\infty}\left(\frac{1}{(2l+1)!}\int_{0}^{\infty}t^{2l+2}e^{-t^{2}}dt\int_{I}\sin^{2l+1}(\pi\theta)\cdot\mathcal{R}\left(d\theta\mathrm{Tr}_{u}(-\beta
h(dh)^{2l+1})\right)\right),$

and $\mathrm{Ph}_{\mathrm{self}}(h)$ is expanded as

(7.3)
$\displaystyle\mathrm{Ph}_{\mathrm{self}}(h)=-\pi^{-1/2}\sum_{l=0}^{\infty}\frac{1}{(2l+1)!}\int_{0}^{\infty}t^{2l+1}e^{-t^{2}}dt\cdot\mathcal{R}\left(\mathrm{Tr}_{u\widehat{\otimes}\beta}(h(dh)^{2l+1})\right).$

We now check that (7.2) and (7.3) coincide. We have:

(7.4) $\displaystyle\mathrm{Tr}_{u}(-\beta
h(dh)^{2l+1})=-\mathrm{Tr}_{u\widehat{\otimes}\beta}(h(dh)^{2l+1}),$

(7.5)
$\displaystyle\int_{0}^{\infty}t^{2l+2}e^{-t^{2}}dt\int_{I}d\theta\sin^{2l+1}(\pi\theta)=\frac{(2l+1)!!\sqrt{\pi}}{2^{l+2}}\cdot\frac{(2l)!!\cdot
2}{(2l+1)!!\pi}=\frac{l!}{2\sqrt{\pi}},$ (7.6)
$\displaystyle\int_{0}^{\infty}t^{2l+1}e^{-t^{2}}dt=\frac{l!}{2}.$

Using the above formulas and (2.57), we see that (7.2) and (7.3) coincide.
This finishes the proof in the case where $\mathrm{type}(A)$ is even.

Now assume $\mathrm{type}(A)$ is odd. In this case
$\mathrm{Ph}_{\mathrm{self}}(\widetilde{h})$ is an odd form and
$\mathrm{Ph}_{\mathrm{self}}(h)$ is an even form, so that (7.1) becomes

(7.7)
$\displaystyle-\pi^{1/2}\sum_{l=0}^{\infty}\left(\frac{1}{(2l)!}\int_{0}^{\infty}t^{2l+1}e^{-t^{2}}dt\int_{I}\sin^{2l}(\pi\theta)\mathcal{R}\left(d\theta\mathrm{Tr}_{u}(-\beta
h(dh)^{2l})\right)\right),$

and $\mathrm{Ph}_{\mathrm{self}}(h)$ is expanded as

(7.8)
$\displaystyle\mathrm{Ph}_{\mathrm{self}}(h)=\pi^{-1/2}\sum_{l=0}^{\infty}\frac{1}{(2l)!}\int_{0}^{\infty}t^{2l}e^{-t^{2}}dt\cdot\mathcal{R}\left(\mathrm{Tr}_{u\widehat{\otimes}\beta}(h(dh)^{2l})\right).$

We now check that (7.7) and (7.8) coincide. In this case we have

(7.9) $\displaystyle-\mathrm{Tr}_{u}(-\beta
h(dh)^{2l+1})=2\mathrm{Tr}_{u\widehat{\otimes}\beta}(h(dh)^{2l+1}),$ (7.10)
$\displaystyle\int_{0}^{\infty}t^{2l+1}e^{-t^{2}}dt\int_{I}\sin^{2l}(\pi\theta)d\theta=\frac{l!}{2}\cdot\frac{(2l-1)!!}{(2l)!!}=\frac{(2l-1)!!}{2^{l+1}},$
(7.11)
$\displaystyle\int_{0}^{\infty}t^{2l}e^{-t^{2}}dt=\frac{(2l-1)!!\sqrt{\pi}}{2^{l+1}}.$

Using above formulas and (2.57), we see that (7.7) and (7.8) coincide. This
finishes the proof in the case where $\mathrm{type}(A)$ is odd and completes
the proof of Proposition 3.27.

###  7.2. Proof of Theorem 3.29

Here we prove Theorem 3.29. The first step is to show the following.

######  Lemma 7.12.

Theorem 3.29 holds if $\mathrm{type}(A)=0$.

###### Proof.

Choose and fix a volume element $u\in A$ to trivialize $\mathrm{Ori}(A)$.
Recall that, in Subsubsection 2.3.2, we explained the model of the
$KO$-spectrum realized as a direct limit of $\mathrm{Skew}^{\dagger}_{A}(-)$,
and in the case that $\mathrm{type}(A)=0$ we gave an explicit homeomorphism
(2.52) with the model of $KO_{0}$ realized by using Grassmannians. Also, as
explained in Subsection 2.4, the degree-zero part of the topological
Pontryagin character is realized as the Chern-Weil construction on vector
bundles. For each $n$, set
$S_{n}:=(S_{+}\otimes\mathbb{R}^{n})\oplus(S_{-}\otimes\mathbb{R}^{n})$ in the
notation there. Recall the diffeomorphism (2.49)

$\displaystyle\mathrm{Self}^{\dagger}_{A}(S_{n})=\mathrm{Self}^{\dagger}_{A}((S_{+}\otimes\mathbb{R}^{n})\oplus(S_{-}\otimes\mathbb{R}^{n}))$
$\displaystyle\simeq\mathrm{Gr}(\mathbb{R}^{2n})$ $\displaystyle u\otimes a$
$\displaystyle\mapsto\mathrm{Im}((1-a)/2)=\mathrm{Im}(P),$

where $P:=(1-a)/2$. The canonical connection on the tautological real vector
bundle $\theta_{2n}$ over $\mathrm{Gr}(\mathbb{R}^{2n})$ is given by

$\displaystyle\nabla_{\mathrm{Gr}}=PdP,$

where the inclusions
$\Omega^{*}(\mathrm{Gr}(\mathbb{R}^{2n});\theta_{2n})\subset\Omega^{*}(\mathrm{Gr}(\mathbb{R}^{2n});\underline{\mathbb{R}}^{2n})$
are understood. We know that the topological Pontryagin character is the
direct limit with respect to $n$ of the Pontryagin character form
$\mathcal{R}\circ\mathrm{Tr}(e^{\nabla_{\mathrm{Gr}}^{2}})-n$ for the
connection $\nabla_{\mathrm{Gr}}$. Thus it is enough to show that (see (2.47))

(7.13)
$\displaystyle\mathcal{R}\circ\mathrm{Tr}(e^{\nabla_{\mathrm{Gr}}^{2}})-n-\mathrm{Ph}_{\mathrm{self}}(h_{\mathrm{univ}})\in\mathrm{Im}(d)$

on $\mathrm{Self}^{\dagger}_{A}(S_{n})$ for each $n$.

First we compute $\mathrm{Tr}(e^{\nabla_{\mathrm{Gr}}^{2}})$. We have

$\displaystyle\nabla_{\mathrm{Gr}}^{2}=PdP\wedge dP=\frac{1}{4}Pda\wedge da.$

Using the relation $ada+da\cdot a=0$ and $P^{2}=P$, we have

(7.14) $\displaystyle\mathrm{Tr}(e^{\nabla_{\mathrm{Gr}}^{2}})$
$\displaystyle=\sum_{k=0}^{\infty}\frac{1}{2^{2k+1}k!}\left(\mathrm{Tr}(-a(da)^{2k})+\mathrm{Tr}((da)^{2k})\right).$

Next we compute $\mathrm{Ph}_{\mathrm{self}}(h_{\mathrm{univ}})$. Applying
Definition 3.15 to $h_{\mathrm{univ}}=u\otimes a$ and using
$h_{\mathrm{univ}}^{2}=1$, we have

$\displaystyle\mathrm{Ph}_{\mathrm{self}}(h_{\mathrm{univ}})=\mathcal{R}\circ\pi^{-1/2}\int_{(0,\infty)}e^{-t^{2}}dt\wedge\mathrm{Tr}_{u}\left((u\otimes
a)\cdot e^{-tu\widehat{\otimes}da}\right).$

We have

$\displaystyle
e^{-tu\widehat{\otimes}da}=\sum_{k=0}^{\infty}\frac{t^{2k}}{(2k)!}\cdot
1\widehat{\otimes}(da)^{2k}-\sum_{k=0}^{\infty}\frac{t^{2k+1}}{(2k+1)!}\cdot
u\widehat{\otimes}(da)^{2k+1},$

so that, using $\mathrm{Tr}_{u}(u\otimes B)=-\mathrm{Tr}(B)$ and
$\mathrm{Tr}_{u}(1\otimes B)=0$ for any $B\in\mathrm{End}(\mathbb{R}^{2n})$,

$\displaystyle\mathrm{Tr}_{u}\left((u\otimes
a)e^{-tu\widehat{\otimes}da}\right)=\sum_{k=0}^{\infty}\frac{t^{2k}}{(2k)!}\cdot\mathrm{Tr}\left(-a(da)^{2k}\right).$

Using the formula

$\displaystyle\int_{0}^{\infty}e^{-t^{2}}t^{2k}dt=\sqrt{\pi}\cdot\frac{(2k-1)!!}{2^{k+1}},$

we get

(7.15)
$\displaystyle\mathrm{Ph}_{\mathrm{self}}(h_{\mathrm{univ}})=\mathcal{R}\circ\sum_{k=0}^{\infty}\frac{1}{2^{2k+1}k!}\mathrm{Tr}\left(-a(da)^{2k}\right).$

Now we can compute the element (7.13). By (7.14) and (7.15), we have

$\displaystyle\mathcal{R}\circ\mathrm{Tr}(e^{\nabla_{\mathrm{Gr}}^{2}})-n-\mathrm{Ph}_{\mathrm{self}}(h_{\mathrm{univ}})$
$\displaystyle=\mathcal{R}\circ\left(-n+\sum_{k=0}^{\infty}\frac{1}{2^{2k+1}k!}\mathrm{Tr}((da)^{2k})\right)$
$\displaystyle=\mathcal{R}\circ\frac{1}{2}\mathrm{Tr}(e^{\frac{1}{4}da\wedge
da}-1).$

This is exact, so we have proved (7.13). This completes the proof of Lemma
7.12. ∎

###### Proof of Theorem 3.29.

By Proposition 3.27, we know that $\mathrm{Ph}(m)$ is compatible with the
suspension. Thus we can reduce to the case $\mathrm{type}(A)=0$, and the
result follows from Lemma 7.12. ∎

###  7.3. The proof of Proposition 3.63

Here we prove Proposition 3.63. We only give the proof for (1), whose
straightforward modification leads to (2).

It is enough to consider the case that $f(z)=z^{n}$ for $n\in\mathbb{Z}_{\geq
0}$. Moreover, working locally, we may assume that $V=\underline{A}$ for some
$A$ and $\not{S}=\underline{S}$ for an $A$-module $S$ with an inner product.
Furthermore, by Lemma 3.59 it is enough to consider the case that
$\nabla^{\mathcal{A}}=d$, since the action (3.45) preserves the self-
adjointness. Fix a volume element $u$ for $A$ to trivialize $\mathrm{Ori}(A)$.
Then we can decompose the self-adjoint $\mathcal{A}$-superconnection
$\nabla\\!\\!\\!\\!\nabla^{\not{S}}$ as

$\displaystyle\nabla\\!\\!\\!\\!\nabla^{\not{S}}=d+\sum_{j\in
J}\omega_{j}\widehat{\otimes}\xi_{j},$

for $\omega_{j}\in\Omega^{*}(X)$ and $\xi_{j}\in\mathrm{End}_{A}(S)$ with
$|\omega_{j}|+|\xi_{j}|\equiv 1\pmod{2}$. We have

$\displaystyle
F({\nabla\\!\\!\\!\\!\nabla^{\not{S}}};d)=(\nabla\\!\\!\\!\\!\nabla^{\not{S}})^{2}=\sum_{j}d\omega_{j}\widehat{\otimes}\xi_{j}+\sum_{j,k}(\omega_{j}\widehat{\otimes}\xi_{j})\cdot(\omega_{k}\widehat{\otimes}\xi_{k}).$

Thus
$(F({\nabla\\!\\!\\!\\!\nabla^{\not{S}}};d))^{n}=(\nabla\\!\\!\\!\\!\nabla^{\not{S}})^{2n}$
is the sum of the all possible terms of the form

(7.16)
$\displaystyle(\alpha_{j_{1}}\widehat{\otimes}\xi_{j_{1}})\cdot(\alpha_{j_{2}}\widehat{\otimes}\xi_{j_{2}})\cdot\cdots\cdot(\alpha_{j_{l}}\widehat{\otimes}\xi_{j_{l}}),$

where $j_{t}\in J$ and $\alpha_{j_{t}}$ is either of $d\omega_{j_{t}}$ or
$\omega_{j_{t}}$ for each $t$, and if the number of $t$ such that
$\alpha_{j_{t}}=d\omega_{j_{t}}$ is $m$, we have $l+m=2n$.

We show that, for each term (7.16) as above, the sum

(7.17)
$\mathrm{Tr}_{u}\left((\alpha_{j_{1}}\widehat{\otimes}\xi_{j_{1}})\cdot(\alpha_{j_{2}}\widehat{\otimes}\xi_{j_{2}})\cdot\cdots\cdot(\alpha_{j_{l}}\widehat{\otimes}\xi_{j_{l}})\right)\\\
+\mathrm{Tr}_{u}\left((\alpha_{j_{l}}\widehat{\otimes}\xi_{j_{l}})\cdot(\alpha_{j_{l-1}}\widehat{\otimes}\xi_{j_{l-1}})\cdot\cdots\cdot(\alpha_{j_{1}}\widehat{\otimes}\xi_{j_{1}})\right)$

can be nonzero only if we have
$\sum_{t}|\alpha_{j_{t}}|\equiv\mathrm{type}(A)+1\pmod{4}$. This implies the
desired result.

Fix a term (7.16). We define $I\subset\\{1,\cdots,l\\}$ so that
$\alpha_{j_{t}}=d\omega_{j_{t}}$ for $t\in I$ and
$\alpha_{j_{t}}=\omega_{j_{t}}$ for $t\notin I$. We have $|I|=m$. We also
define $\sigma\colon\mathbb{Z}\to\mathbb{Z}_{2}$ by

$\displaystyle\sigma(x):=\begin{cases}1&\mbox{if }x\equiv 1,2\pmod{4},\\\
0&\mbox{if }x\equiv 0,3\pmod{4}.\end{cases}$

Then the self-adjointness of $\nabla\\!\\!\\!\\!\nabla$ (Definition 3.60)
implies that

(7.18) $\displaystyle\xi_{j}^{*}=(-1)^{\sigma(|\omega_{j}|)}\xi_{j}$

for all $j\in J$. Also, a straightforward computation shows that

(7.19) $\displaystyle u^{*}=(-1)^{\sigma(\mathrm{type}(A))}u.$

Now we compute (7.16). We define $A,B,C,D\in\mathbb{Z}_{2}$ by the following.

(7.20)
$\displaystyle(\alpha_{j_{1}}\widehat{\otimes}\xi_{j_{1}})\cdot(\alpha_{j_{2}}\widehat{\otimes}\xi_{j_{2}})\cdot\cdots\cdot(\alpha_{j_{l}}\widehat{\otimes}\xi_{j_{l}})$
$\displaystyle=(-1)^{A}\alpha_{j_{1}}\wedge\alpha_{j_{2}}\wedge\cdots\wedge\alpha_{j_{l}}\widehat{\otimes}\xi_{j_{1}}\xi_{j_{2}}\cdots\xi_{j_{l}},$
$\displaystyle(\alpha_{j_{l}}\widehat{\otimes}\xi_{j_{l}})\cdot(\alpha_{j_{l-1}}\widehat{\otimes}\xi_{j_{l-1}})\cdot\cdots\cdot(\alpha_{j_{1}}\widehat{\otimes}\xi_{j_{1}})$
$\displaystyle=(-1)^{B}\alpha_{j_{l}}\wedge\alpha_{j_{l-1}}\wedge\cdots\wedge\alpha_{j_{1}}\widehat{\otimes}\xi_{j_{l}}\xi_{j_{l-1}}\cdots\xi_{j_{1}},$
$\displaystyle\alpha_{j_{1}}\wedge\alpha_{j_{2}}\wedge\cdots\wedge\alpha_{j_{l}}$
$\displaystyle=(-1)^{C}\alpha_{j_{l}}\wedge\alpha_{j_{l-1}}\wedge\cdots\wedge\alpha_{j_{1}},$
$\displaystyle(u\xi_{j_{1}}\xi_{j_{2}}\cdots\xi_{j_{l}})^{*}$
$\displaystyle=(-1)^{D}\xi_{j_{l}}\xi_{j_{l-1}}\cdots\xi_{j_{1}}u.$

Then we have

$\displaystyle\mathrm{Tr}\left(u\xi_{j_{1}}\xi_{j_{2}}\cdots\xi_{j_{l}}\right)$
$\displaystyle=\mathrm{Tr}\left((u\xi_{j_{1}}\xi_{j_{2}}\cdots\xi_{j_{l}})^{*}\right)$
$\displaystyle=(-1)^{D}\mathrm{Tr}\left(\xi_{j_{l}}\xi_{j_{l-1}}\cdots\xi_{j_{1}}u\right)$
$\displaystyle=(-1)^{D}\mathrm{Tr}\left(u\xi_{j_{l}}\xi_{j_{l-1}}\cdots\xi_{j_{1}}\right),$

so that

$\displaystyle\mathrm{Tr}_{u}\left(\xi_{j_{1}}\xi_{j_{2}}\cdots\xi_{j_{l}}\right)=(-1)^{D}\mathrm{Tr}_{u}\left(\xi_{j_{l}}\xi_{j_{l-1}}\cdots\xi_{j_{1}}\right).$

Using this, we get

(7.21)
$\displaystyle\mathrm{Tr}_{u}\left((\alpha_{j_{1}}\widehat{\otimes}\xi_{j_{1}})\cdot(\alpha_{j_{2}}\widehat{\otimes}\xi_{j_{2}})\cdot\cdots\cdot(\alpha_{j_{l}}\widehat{\otimes}\xi_{j_{l}})\right)$
$\displaystyle=(-1)^{A}\mathrm{Tr}_{u}\left(\xi_{j_{1}}\xi_{j_{2}}\cdots\xi_{j_{l}}\right)\alpha_{j_{1}}\wedge\alpha_{j_{2}}\wedge\cdots\wedge\alpha_{j_{l}}$
$\displaystyle=(-1)^{A+C+D}\mathrm{Tr}_{u}\left(\xi_{j_{l}}\xi_{j_{l-1}}\cdots\xi_{j_{1}}\right)\alpha_{j_{l}}\wedge\alpha_{j_{l-1}}\wedge\cdots\wedge\alpha_{j_{1}}$
$\displaystyle=(-1)^{A+B+C+D}\mathrm{Tr}_{u}\left((\alpha_{j_{l}}\widehat{\otimes}\xi_{j_{l}})\cdot(\alpha_{j_{l-1}}\widehat{\otimes}\xi_{j_{l-1}})\cdot\cdots\cdot(\alpha_{j_{1}}\widehat{\otimes}\xi_{j_{1}})\right).$

Now we compute $A+B+C+D$. We have

(7.22) $\displaystyle A$
$\displaystyle=\sum_{s<t}|\alpha_{j_{s}}|\cdot|\xi_{j_{t}}|,$ $\displaystyle
B$ $\displaystyle=\sum_{t<s}|\alpha_{j_{s}}|\cdot|\xi_{j_{t}}|,$
$\displaystyle C$
$\displaystyle=\sum_{s<t}|\alpha_{j_{s}}|\cdot|\alpha_{j_{t}}|,$
$\displaystyle D$
$\displaystyle=\sum_{t}\sigma(|\omega_{j_{t}}|)+\sigma(\mathrm{type}(A)),$

where the last equality used (7.18) and (7.19). Recall that
$|\xi_{j_{t}}|\equiv|\alpha_{j_{t}}|\pmod{2}$ if $t\in I$ and
$|\xi_{j_{t}}|\equiv|\alpha_{j_{t}}|-1\pmod{2}$ if $t\notin I$. We get

(7.23) $\displaystyle A+B$ $\displaystyle=\sum_{t\neq
s}|\alpha_{j_{s}}|\cdot|\xi_{j_{t}}|$ $\displaystyle=\sum_{t\neq
s}|\alpha_{j_{s}}|\cdot|\alpha_{j_{t}}|-(l-m)\sum_{s\in
I}|\alpha_{j_{s}}|-(l-m-1)\sum_{s\in J\setminus I}|\alpha_{j_{s}}|$
$\displaystyle=\sum_{s\in J\setminus I}|\alpha_{j_{s}}|,$

where the last equality follows from the fact that the first term in the
second line is even, and that $l-m$ is even since $l+m=2n$. Using
$\sigma(x-1)=\sigma(x)+x$ and the fact that
$|\omega_{j_{t}}|=|\alpha_{j_{t}}|-1$ if $t\in I$ and
$|\omega_{j_{t}}|=|\alpha_{j_{t}}|$ if $t\notin I$, we have

(7.24) $\displaystyle D$ $\displaystyle=\sum_{t\in
J}\sigma(|\alpha_{j_{t}}|)+\sum_{t\in
I}|\alpha_{j_{t}}|+\sigma(\mathrm{type}(A)).$

Moreover, using $\sigma(x)+\sigma(y)+xy=\sigma(x+y)$, we get

(7.25) $\displaystyle C+D$ $\displaystyle=\sigma\left(\sum_{t\in
J}|\alpha_{j_{t}}|\right)+\sum_{t\in
I}|\alpha_{j_{t}}|+\sigma(\mathrm{type}(A)).$

By (7.23) and (7.25), we conclude

(7.26) $\displaystyle A+B+C+D=\sigma\left(\sum_{t\in
J}|\alpha_{j_{t}}|\right)+\sum_{t\in
J}|\alpha_{j_{t}}|+\sigma(\mathrm{type}(A)).$

Recall that we already know that the map (3.51) preserves the
$\mathbb{Z}_{2}$-grading if $\mathrm{type}(A)$ is odd, and reverses the
$\mathbb{Z}_{2}$-grading if $\mathrm{type}(A)$ is even. In particular, by
Lemma 3.46, we know that (7.17) can be nonzero only when

(7.27) $\displaystyle\sum_{t\in
J}|\alpha_{j_{t}}|\equiv\mathrm{type}(A)+1\pmod{2}.$

In this case (7.26) becomes

$\displaystyle A+B+C+D$ $\displaystyle=\sigma\left(\sum_{t\in
J}|\alpha_{j_{t}}|\right)+\mathrm{type}(A)+1+\sigma(\mathrm{type}(A))$
$\displaystyle=\sigma\left(\sum_{t\in
J}|\alpha_{j_{t}}|\right)+\sigma(\mathrm{type}(A)+1)$

where the last equality used $\sigma(x+1)=\sigma(x)+x+1$. Using (7.21), the
sum (7.17) can be nonzero only when both (7.27) and

$\displaystyle\sigma\left(\sum_{t}|\alpha_{j_{t}}|\right)=\sigma(\mathrm{type}(A)+1)$

are satisfied. But $\sigma(x)=\sigma(y)$ and $x\equiv y\pmod{2}$ imply
$x\equiv y\pmod{4}$, so these conditions imply

$\displaystyle\sum_{t}|\alpha_{j_{t}}|\equiv\mathrm{type}(A)+1\pmod{4}.$

Thus we get the desired result.

## Acknowledgment

The authors are grateful to Mikio Furuta, Yuji Tachikawa and Kazuya Yonekura
for helpful discussion and comments. KG is supported by JSPS KAKENHI Grant
Numbers 20K03606 and JP17H06461. MY is supported by JSPS KAKENHI Grant Number
20K14307 and JST CREST program JPMJCR18T6.

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