# The families Clifford index and differential KO-theory

Daniel Berwick-Evans

###### Abstract.

Extending ideas of Atiyah–Bott–Shapiro and Quillen, we construct a model for
differential ${\rm KO}$-theory whose cocycles are families of Clifford modules
with superconnection. The model is built to accommodate an analytic
pushforward for bundles of spin manifolds, affording a differential refinement
of Atiyah and Singer’s families index.

###### Contents

  1. 1 Introduction
  2. 2 The main idea from the families index
  3. 3 Clifford modules
  4. 4 The Atiyah–Bott–Shapiro construction and the spectrum ${\rm KO}$
  5. 5 ${\rm KO}$-classes from ${\rm Cl}_{n}$-bundles
  6. 6 The Pontryagin class of a ${\rm Cl}_{n}$-bundle as a Čech–de Rham cocycle
  7. 7 Differential cocycles
  8. 8 The families differential index

##  1\. Introduction

This paper gives a model for differential ${\rm KO}$-theory in degree
$n\in{\mathbb{Z}}$ that generalizes a generators and relations description of
$\widehat{\rm K}{}^{0}$, differential complex K-theory in degree 0.

For a smooth manifold $M$, generators for $\widehat{\rm K}{}^{0}(M)$ are
triples $(V,{\mathbb{A}},\phi)$ where $V\to M$ is a ${\mathbb{C}}$-vector
bundle with hermitian metric, ${\mathbb{A}}$ is a hermitian superconnection,
and $\phi$ is an odd differential form on $M$. These data are subject to an
equivalence relation involving a Chern–Simons form; see §7.1 for details. At
first pass, generators for $\widehat{\rm KO}{}^{n}(M)$ replace ${\mathbb{C}}$
by the real Clifford algebra ${\rm Cl}_{n}$: consider $(V,{\mathbb{A}},\phi)$
where $V$ is a real, metrized vector bundle with self-adjoint ${\rm
Cl}_{n}$-action, ${\mathbb{A}}$ is a ${\rm Cl}_{n}$-linear metric-preserving
connection on $V$, and $\phi$ is a differential form in degrees $n-1$ mod 4.
The appropriate equivalence relation on these data generalizes the
Atiyah–Bott–Shapiro equivalence relation on Clifford modules [ABS64], see also
Donovan–Karoubi [DK70]. Roughly, triples $(V,{\mathbb{A}},\phi)$ and
$(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})$ are equivalent when there
is an inclusion $g\colon V\hookrightarrow V^{\prime}$ of metrized vector
bundles together with the data of a ${\rm Cl}_{n}\otimes{\rm Cl}_{-1}$-action
on the orthogonal complement $V^{\perp}\subset V^{\prime}$, plus a condition
on a Chern–Simons form. The main subtlety is that this equivalence relation
needs to be implemented locally on $M$, meaning an element of $\widehat{\rm
KO}{}^{n}(M)$ is given by an open cover $\\{U_{\alpha}\\}$ with
$(V_{\alpha},{\mathbb{A}}_{\alpha},\phi_{\alpha})$ over each $U_{\alpha}$ and
equivalences $g_{\alpha\beta}$ for each $U_{\alpha}\bigcap U_{\beta}$
satisfying a cocycle condition that refines the one in the Čech–de Rham
complex; §6 and §7 for details.

######  Definition 1.1 (Sketch of Definition 7.6).

Let $M$ be a smooth manifold. Consider the free abelian group on equivalence
classes of tuples $\\{V_{\alpha}\to
U_{\alpha},{\mathbb{A}}_{\alpha},\phi_{\alpha},g_{\alpha\beta}\\}$, modulo
concordances with exact Chern–Simons form. Define $\widehat{\rm KO}{}^{n}(M)$
as the quotient gotten from identifying the direct sum of bundles with
addition in $\widehat{\rm KO}{}^{n}(M)$; see (118).

######  Theorem 1.2.

The graded ring $\widehat{\rm KO}{}^{\bullet}(M)$ is naturally isomorphic to
the differential KO-theory of $M$. When $M={\rm pt}$, this description refines
the Atiyah–Bott–Shapiro isomorphism that sends a Clifford module to a class in
${\rm KO}^{\bullet}({\rm pt})$.

We emphasize that the real vector bundles $V_{\alpha}\to U_{\alpha}$ in
Definition 1.1 are finite rank. By contrast, Atiyah and Singer’s construction
of (non-differential) KO-theory comes from families of ${\rm Cl}_{n}$-linear
Fredholm operators acting on infinite-rank bundles [AS69]. As we explain
below, an index bundle construction permits a comparison between Atiyah and
Singer’s description of ${\rm KO}^{n}(M)$ and the one from Theorem 1.2; see
§4.2-§4.5 and Proposition 5.12. As outlined in §2, Definition 1.1 emerges
naturally when considering a differential refinement of the index bundle for a
family of Clifford linear Dirac operators. The kernel of such a family is
usually not a vector bundle (it need not have locally constant rank), which
leads to the subtle gluing data for the index bundle and its differential
refinement. Definition 1.1 is purpose-built to accommodate this information,
providing a natural home for the analytic index in differential ${\rm
KO}$-theory.

######  Theorem 1.3.

Let $\pi\colon X\to B$ be a proper family of $n$-dimensional Riemannian spin
manifolds and $V\to X$ be a metrized real vector bundle with compatible
connection $\nabla$. The families Clifford index determines a degree $-n$
differential cocycle $\widehat{\pi}_{!}(V,\nabla)$ on $B$ whose associated
class $[\widehat{\pi}_{!}(V,\nabla)]\in\widehat{\rm KO}{}^{-n}(B)$ refines the
analytic pushforward in ${\rm KO}$-theory.

###  1.1. Comparison with other work

In contrast to differential K-theory for complex bundles, the literature on
differential KO-theory is quite sparse. The most thorough treatment to date is
by Grady and Sati [GS21]. They work with Hopkins and Singer’s model for
differential cohomology [HS05] phrased in terms of sheaves of spectra [BNM16].
When applied to KO, this general construction elucidates structural aspects
like differential twists and communicates with powerful homotopical tools,
e.g., the differential Atiyah–Hirzebruch spectral sequence.

The Hopkins–Singer differential refinement of complex K-theory is frequently
augmented by geometrically-motivated cocycle models. These more tailored
presentations of $\widehat{\rm K}{}$ are crucial in certain constructions such
as the differential analytic index in [GL15] and differential index theorem in
[FL10]. In turn, these constructions allow one to refine and enhance classical
results, e.g., [Lot94, Fre01, Bun09, Bun11]. The analog of these geometric
models and their ensuing applications remain under-developed in differential
KO-theory. Freed sketches one candidate construction [Fre00, page 34], and
further suggestions are given by Bunke and Schick [BS12, §4.9]; however, these
ideas were never fleshed out. More recently, Gomi and Yamashita [GY22]
developed a model as an enhancement of Karoubi’s KO-theory [Kar78], with the
goal of studying differential refinements of invariants of condensed matter
systems [FH21].

The cocycle model for differential KO constructed in this paper is motivated
by the geometry of index bundles with intended applications in real
differential index theory. At the heart of this model is the definition of a
${\rm Cl}_{n}$-bundle (see Definition 5.2), which arises as a finite-
dimensional families index of ${\rm Cl}_{n}$-linear Dirac operators, see §2.2.
This geometry resonates with Furuta’s vectorial bundles [Fur02, Gom10] and
Henrique’s quasi-bundles [Hen09], though below we emphasize the connection
with the Atiyah–Bott–Shapiro construction [ABS64]. In future work we plan to
construct a differential topological index via a realization of the
Atiyah–Bott–Shapiro Thom class in the framework of Theorem 1.2.

Replacing the real Clifford algebras ${\rm Cl}_{n}$ with complex Clifford
algebras $\mathbb{C}{\rm l}_{n}$ leads to analogs of Theorems 1.2 and 1.3 for
differential complex K-theory. This makes contact with previous results. For
example, when $n=0$ and $M$ is compact, Theorem 1.2 generalizes the generators
and relations presentation of $\widehat{{\rm K}}^{0}(M)$ in terms of finite-
rank vector bundles with connection and an odd degree differential form; see
Remark 7.7. One also finds overlap with models of a more analytic flavor:
applying the index bundle construction (see §2) to the Hilbert bundle model
for differential K-theory from [GL15] leads to a version of Definition 1.1
with ${\mathbb{C}}$-coefficients, where the odd degree groups in [GL15, §6]
can be understood in terms of $\mathbb{C}{\rm l}_{1}$-linear superconnections.
Finally, Theorem 1.3 (both in the statement and proof) generalizes Freed and
Lott’s construction of the differential analytic index for families of spinc-
manifolds [FL10].

Definition 1.1 is also designed to receive a cocycle map from a class of
$1|1$-dimensional Euclidean field theories in the style of Stolz and Teichner
[ST04, ST11]. A fundamental assertion of the Stolz–Teichner program is that
quantization of such field theories should correspond to the analytic
pushforward in KO-theory. It is conjectured that a $2|1$-dimensional
generalization supplies an analytic pushforward in the cohomology theory of
topological modular forms. Theorems 1.2 and 1.3 give a context in which one
can start to make some of these ideas precise, see [BE23a, BE23b, BE23c].
Owing to its independent interest in geometry, the present paper is written to
be independent from this discussion of field theories.

###  1.2. Acknowledgements

I thank Theo Johnson-Freyd for patiently (and repeatedly) entertaining my
questions about superalgebra signs rules. I also thank Yigal Kamel his careful
reading and feedback on previous drafts. This work was supported by the
National Science Foundation under grant DMS-2205835.

##  2\. The main idea from the families index

Definition 1.1 emerges naturally when generalizing Freed and Lott’s families
differential index [FL10, §3] to families of Clifford linear Dirac operators.
In this section, we review versions of the families index and describe the
relationship with our definitions and results.

###  2.1. The families index in ${\rm K}$-theory

Let $X$ be a compact even-dimensional spinc manifold. The index of the Dirac
operator $\not{D}$ on $X$ is

(1) $\displaystyle{\rm Ind}(\not{D})=[{\rm ker}(\not{D})_{\rm ev}]-[{\rm
ker}(\not{D})_{\rm odd}]\in{\rm K}({\rm pt})\simeq{\mathbb{Z}},$

where we consider ${\rm ker}(\not{D})$ as a ${\mathbb{Z}}/2$-graded
${\mathbb{C}}$-vector space with even and odd subspaces ${\rm
ker}(\not{D})_{\rm ev}$ and ${\rm ker}(\not{D})_{\rm odd}$, respectively. A
family of Riemannian spinc manifolds $\pi\colon X\to B$ with compact even-
dimensional fibers determines a family of operators $\not{D}$ on $B$. The
fiber at $b\in B$ is a Dirac operator $\not{D}_{b}$ on the spinc manifold
$\pi^{-1}(\\{b\\})=X_{b}$. One would like to construct a class in ${\rm K}(B)$
from this data. Taking the fiberwise index yields a ${\mathbb{Z}}/2$-graded
vector space ${\rm ker}(\not{D}_{b})$ for each $b\in B$. However, these
typically fail to assemble into a ${\mathbb{Z}}/2$-graded vector bundle, e.g.,
the dimension of the kernel need not be locally constant. Hence, this naive
attempt at a families index does not determine a class in ${\rm K}(B)$
generalizing (1).

There are two well-known ways to remedy this. The first is analytic: the space
of Fredholm operators represents the functor ${\rm K}(-)$, e.g., see [LM94,
§III.8]. A family of spinc manifolds $\pi\colon X\to B$ determines a
$B$-family of Dirac operators, which in turn provides a map from $B$ into the
space of Fredholm operators. The homotopy class of this map defines the
_families analytic index_ ${\rm Ind}(\not{D})\in{\rm K}(B)$. The second
approach is geometric: consider the open cover
$\\{U_{\lambda}\\}_{\lambda\in{\mathbb{R}}_{>0}}$ of $B$ defined as

(2) $\displaystyle U_{\lambda}:=\\{b\in B\mid\lambda\notin{\rm
Spec}(\not{D}_{b}^{2})\\},$

where ${\rm Spec}(\not{D}_{b}^{2})$ denotes the spectrum of the self-adjoint,
nonnegative Dirac Laplacian $\not{D}_{b}^{2}$. For $b\in U_{\lambda}$, let
$\mathcal{H}^{<\lambda}_{b}$ denote the sum of eigenspaces of
$\not{D}_{b}^{2}$ with eigenvalues less than $\lambda$. Ellipticity of
$\not{D}_{b}^{2}$ implies that the $\mathcal{H}^{<\lambda}_{b}$ fit together
into a finite rank, smooth, ${\mathbb{Z}}/2$-graded vector bundle
$\mathcal{H}^{<\lambda}\to U_{\lambda}$. Hence we obtain a ${\rm K}$-theory
class $[\mathcal{H}^{<\lambda}]=[\mathcal{H}^{<\lambda}_{\rm
ev}]-[\mathcal{H}^{<\lambda}_{\rm odd}]\in{\rm K}(U_{\lambda})$. On overlaps
$U_{\lambda}\bigcap U_{\mu}$ with $\lambda<\mu$, there is an evident inclusion
of ${\mathbb{Z}}/2$-graded vector bundles,

(3) $\displaystyle
g_{\lambda\mu}\colon\mathcal{H}^{<\lambda}\hookrightarrow\mathcal{H}^{<\mu}.$

The restriction of $\not{D}$ to the orthogonal complement of the inclusion (3)
determines an invertible odd endomorphism $e_{\lambda\mu}$ that witnesses an
equality of ${\rm K}$-theory classes
$[\mathcal{H}^{<\lambda}]=[\mathcal{H}^{<\mu}]\in{\rm K}(U_{\lambda}\bigcap
U_{\mu})$, see (142). In fact, the inclusions (3) and odd isomorphisms
$e_{\lambda\mu}$ satisfy a cocycle condition on $U_{\lambda}\bigcap
U_{\mu}\bigcap U_{\mu}$. This constructs a well-defined class

(4) $\displaystyle{\rm Ind}(\not{D}):=[\\{\mathcal{H}^{<\lambda}\to
U_{\lambda},g_{\lambda\mu},e_{\lambda\mu}\\}]\in{\rm K}(B)$

equal to the families analytic index of $\pi\colon X\to B$; see [Fre87, §7.2],
[BGV92, §9.5].

The geometric description (4) of the index also leads to a natural refinement
in the differential ${\rm K}$-theory of $B$ that uses the _Bismut
superconnection_ , denoted ${\mathbb{B}}$ and recalled in (143) below. The
restriction of the degree 1 part of ${\mathbb{B}}$ to $\mathcal{H}^{<\lambda}$
determines ordinary connections $\nabla^{<\lambda}$ on each finite-rank bundle
$\mathcal{H}^{<\lambda}\subset\mathcal{H}|_{U_{\lambda}}$ with Chern forms

(5) $\displaystyle{\rm Ch}(\nabla^{<\lambda})\in\Omega^{\rm
even}(U_{\lambda};{\mathbb{R}}[u^{\pm 1}]),\qquad|u|=-2.$

Properties of the Bismut superconnection further determine a unique extension
of the data (5) to a cocycle in the Čech–de Rham complex of $M$ relative to
the open cover (2) (see Proposition 6.15),

(6) $\displaystyle{\rm
Ch}({\mathbb{B}},\\{U_{\lambda}\\}_{\lambda\in{\mathbb{R}}_{>0}})\in\Omega^{\bullet}(U_{*};{\mathbb{R}}[u^{\pm
1}]),\qquad(d+\delta){\rm
Ch}({\mathbb{B}},\\{U_{\lambda}\\}_{\lambda\in{\mathbb{R}}_{>0}})=0.$

By construction, this Čech–de Rham cocycle represents the Chern character of
the index bundle. By the Riemann–Roch theorem, another representative is the
integral along the fibers

(7) $\displaystyle\int_{X/B}{\rm Ch}(\nabla^{V})\wedge{\rm
Todd}(\nabla^{X/B})\in\Omega^{\bullet}_{\rm
cl}(B;{\mathbb{R}}[\alpha,\alpha^{-1}])$

of the Chern form of $V$ modified by the ${\rm Todd}(\nabla^{X/B})$, the Todd
form of the spinc-family.

The families differential index comes from specifying a coboundary in the
Čech–de Rham complex that mediates between the cocycles (6) and (7). Adapting
methods of Bismut [Bis86], Bismut–Cheeger [BC89] and Freed–Lott [FL10, §3],
there are forms $\eta_{\lambda}\in\Omega^{\rm
odd}(U_{\lambda},{\mathbb{R}}[u,u^{-1}])$ satisfying

(8) $\displaystyle\int_{X/B}{\rm Ch}(\nabla^{V})\wedge{\rm Todd}(X/B)-{\rm
Ch}({\mathbb{B}},\\{U_{\lambda}\\})=(d+\delta)\\{\eta_{\lambda}\\}\in\Omega^{\bullet}(U_{*};{\mathbb{R}}[u,u^{-1}]),$

where $d$ is the de Rham differential and $\delta$ is the Čech differential,
i.e., the $\eta_{\lambda}$ determine the desired coboundary in the Čech–de
Rham complex. This determines a differential ${\rm K}$-cocycle

(9) $\displaystyle\widehat{{\rm Ind}}(\not{D})=[\\{(\mathcal{H}^{<\lambda}\to
U_{\lambda},\nabla^{<\lambda},\eta_{\lambda}),(g_{\lambda\mu},e_{\lambda\mu})\\}]\in\widehat{{\rm
K}}(B)$

representing the differential index.

###  2.2. The families index in ${\rm KO}$-theory

Now consider the Clifford linear Dirac operator $\not{D}$ on an
$n$-dimensional Riemannian spin manifold. The kernel of $\not{D}$ is a finite-
rank Clifford module [LM94, II.7]. The equivalence class of this module under
the Atiyah–Bott–Shapiro map gives a class in ${\rm KO}^{-n}({\rm pt})$ [ABS64]

(10) $\displaystyle{\rm Ind}(\not{D}):=[{\rm ker}(\not{D})]\in{\rm
KO}^{-n}({\rm pt}).$

This is the _Clifford index_ of $\not{D}$. We refer to §8.1 for further
details on the Clifford linear Dirac operator and §4.1 for a review of the
Atiyah–Bott–Shapiro construction.

When generalizing (10) to families of $n$-dimensional spin manifolds
$\pi\colon X\to B$, the kernels of the fiberwise Clifford linear Dirac
operators fail to determine a smooth bundle of Clifford-modules for the same
reasons as in the spinc case. One again has an analytic remedy: the space
${\rm Fred}_{n}$ of Clifford linear Fredholm operators represents the functor
${\rm KO}^{-n}(-)$ [AS69], [LM94, III.10]. A family of Clifford linear Dirac
operators associated to a bundle of spin manifolds $\pi\colon X\to B$
determines a map $B\to{\rm Fred}_{n}$ whose homotopy class defines the
_families analytic index_

(11) $\displaystyle{\rm Ind}(\not{D})\in[B,{\rm Fred}_{n}]\simeq{\rm
KO}^{-n}(B).$

The main goal of this paper is to develop the complementary geometric approach
to the families Clifford index, in parallel to the index bundle for the spinc-
Dirac operator. In the remainder of this section we sketch the main ideas.

######  Remark 2.1.

There are several sign conventions to fix when working with Clifford linear
Dirac operators. Our conventions (explained in Remarks 3.7 and 4.1) lead to
the kernel of the Clifford linear Dirac operator being a (left) ${\rm
Cl}_{-n}$-module.

For a family $\not{D}$ of ${\rm Cl}_{-n}$-linear Dirac operators, define an
open cover (2) as before: consider the subspace where
$\lambda\in{\mathbb{R}}_{>0}$ is not an eigenvalue of $\not{D}^{2}$. Taking
the direct sum of eigenspaces with eigenvalues $<\lambda$ yields finite-rank
bundles of Clifford modules $\mathcal{H}^{<\lambda}\to U_{\lambda}$ over each
component. Smoothness of this bundle follows from ellipticity of the Dirac
Laplacian $\not{D}^{2}$. On an overlap $U_{\lambda}\bigcap U_{\mu}$, the
restriction of $\not{D}$ to the orthogonal complement to the inclusion of
bundles (3) determines an invertible odd endomorphism that commutes with the
${\rm Cl}_{-n}$-action, see (142). This odd endomorphism generates a ${\rm
Cl}_{-1}$-action, leading to a ${\rm Cl}_{-n}\otimes{\rm Cl}_{-1}\simeq{\rm
Cl}_{-n-1}$-action on the orthogonal complement extending the prior ${\rm
Cl}_{-n}$-action. The existence of such an extension implies that the bundles
of Clifford modules over $U_{\lambda}\bigcap U_{\mu}$ represent the same class
in ${\rm KO}$-theory. In fact, the analogous information (4) in the Clifford
linear case determines a class in ${\rm KO}^{-n}(B)$.

To summarize, the index bundle for a family of ${\rm Cl}_{-n}$-linear Dirac
operators is data:

  1. (i)

an open cover with finite-rank ${\rm Cl}_{-n}$-module bundles
$\mathcal{H}^{<\lambda}\to U_{\lambda}$;

  2. (ii)

for each intersection $U_{\lambda}\bigcap U_{\mu}$, inclusions (3) of ${\rm
Cl}_{-n}$-modules together with the data of a ${\rm Cl}_{-n-1}$-action on the
orthogonal complement extending the ${\rm Cl}_{-n}$-action.

These data satisfy a cocycle condition. This essentially recovers Definition
5.2 of a ${\rm Cl}_{-n}$-bundle. In turn, ${\rm Cl}_{-n}$-bundles can be used
to construct ${\rm KO}^{-n}(M)$ (Proposition 5.12), yielding a version of
Theorem 1.2 in (non-differential) ${\rm KO}$-theory.

A differential refinement comes from generalizing the differential index (9)
in complex K-theory. To start, the Bismut superconnection ${\mathbb{B}}$
defined in (143) for a family of Clifford linear Dirac operators is
automatically ${\rm Cl}_{-n}$-linear. It restricts to an ordinary ${\rm
Cl}_{-n}$-linear connection $\nabla^{<\lambda}$ on each
$\mathcal{H}^{<\lambda}$ with Chern forms analogous to (5) comprising part of
the data of a Čech–de Rham cocycle (6) representing the Pontryagin character
of the index bundle. Replacing the Todd form in (7) by the $\hat{A}$-form, the
remaining data of the differential index comes from $\eta_{\lambda}$-forms
determining a Čech–de Rham coboundary as in (8). These are constructed from
Clifford linear generalizations of the standard constructions. With these
definitions in hand, the differential ${\rm KO}$-index is defined essentially
the same as in (9),

(12) $\displaystyle\widehat{\rm Ind}(\not{D})=[\\{(\mathcal{H}^{<\lambda}\to
U_{\lambda},\nabla^{<\lambda},\eta_{\lambda}),(g_{\lambda\mu},e_{\lambda\mu})\\}]\in\widehat{\rm
KO}{}^{-n}(B)$

where $\mathcal{H}^{<\lambda}\to U_{\lambda}$ is a bundle of ${\rm
Cl}_{-n}$-modules, $g_{\lambda\mu}$ is an inclusion of Clifford module
bundles, $e_{\lambda\mu}$ extends a ${\rm Cl}_{-n}$-action on the orthogonal
complement of the inclusion to a ${\rm Cl}_{-n-1}$-action, $\nabla^{<\lambda}$
is a Clifford linear connection on $\mathcal{H}^{<\lambda}$, and the forms
$\\{\eta_{\lambda}\\}$ determine a Čech–de Rham coboundary.

Definition 7.3 of a differential cocycle is designed so that the differential
index (12) is an immediate example. Constructing a map from data like (12) to
differential ${\rm KO}$-theory basically follows the same route as in complex
K-theory. The main work in Theorem 1.2 is to show that this map is surjective;
this comes down to a computation (Proposition 7.23). Theorem 1.3 then follows
from the differential index construction sketched above.

##  3\. Clifford modules

In this section we review the theory of Clifford modules. Our focus is on real
Clifford algebras and real Clifford modules, with a few pieces of the theory
over ${\mathbb{C}}$. For definitions and constructions that apply to either
real or complex coefficients, we use ${\mathbb{F}}$ to denote either
${\mathbb{R}}$ or ${\mathbb{C}}$.

###  3.1. ${\mathbb{Z}}/2$-graded $*$-algebras

The category of _real_ , respectively _complex_ , _super vector spaces_ is the
category whose objects are ${\mathbb{Z}}/2$-graded vector spaces over
${\mathbb{R}}$, respectively ${\mathbb{C}}$, and morphisms are linear maps
that preserve the grading. We promote this to a symmetric monoidal category
via the graded tensor product, where the adjective “super” refers to the sign
in the braiding isomorphism

$V\otimes_{\mathbb{F}}W\stackrel{{\scriptstyle\sim}}{{\to}}W\otimes_{\mathbb{F}}V\qquad
v\otimes w\mapsto(-1)^{|v||w|}w\otimes v\quad v\in V,\ w\in W$

where $v$ and $w$ are homogeneous elements of degree
$|v|,|w|\in{\mathbb{Z}}/2$. For a super vector space $V$, let $V_{\rm
ev}\oplus V_{\rm odd}$ denote the direct sum decomposition of $V$ into its
even and odd subspaces. The _grading involution_ is the linear map $(-1)^{\sf
F}\colon V\to V$ that acts by $+1$ on $V_{\rm ev}$ and $-1$ on $V_{\rm
odd}$.111The notation $(-1)^{\sf F}$ is from physics where ${\sf F}$ stands
for “fermion”, e.g., see [Wit82, §1]. Let $\Pi V$ denote the parity reversal
of $V$, i.e., $(\Pi V)_{\pm}=V_{\mp}$. Throughout, ${\mathbb{F}}^{n|m}$ will
denote the super vector space with $({\mathbb{F}}^{n|m})_{\rm
ev}={\mathbb{F}}^{n}$ and $({\mathbb{F}}^{n|m})_{\rm odd}={\mathbb{F}}^{m}$.

A _superalgebra_ over ${\mathbb{F}}$ is an algebra object in super vector
spaces over ${\mathbb{F}}$. The _super commutator_ of elements in a
superalgebra is defined as

(13) $\displaystyle[a,b]=ab-(-1)^{|a||b|}ba\qquad a,b\in A,$

for homogenous elements $a$ and $b$. The super commutator satisfies a graded
version of the Jacobi identity.

######  Example 3.1.

For a super vector space $V$, let ${\rm End}(V)$ denote the set of all linear
maps $V\to V$ (not necessarily grading-preserving), where grading-preserving
maps are regarded as even and grading-reversing maps are odd. Composition of
linear maps endows ${\rm End}(V)$ with the structure of a superalgebra.

######  Remark 3.2.

Generalizing the previous example, for super vector spaces $V$ and $W$ one can
consider the super vector space $\underline{{\sf Hom}}(V,W)$ of all linear
maps $V\to W$, graded according to whether a linear map is grading preserving
versus grading reversing. This endows the category of super vector spaces with
a closed monoidal structure, i.e., $-\otimes V$ is left adjoint to
$\underline{{\sf Hom}}(V,-)$.

A _super trace_ on a superalgebra is a map of super vector spaces

(14) $\displaystyle{\sf sTr}_{A}\colon A\to{\mathbb{F}}\qquad{\sf
sTr}_{A}([a,b])=0$

that vanishes on super commutators, and hence is determined by a linear map
$A/[A,A]\to{\mathbb{F}}$. The _opposite_ of a superalgebra $A$ is a
superalgebra $A^{\sf{op}}$ with the same underlying super vector space and
multiplication

$a\cdot_{{\sf{op}}}b:=(-1)^{|a||b|}b\cdot a$

where $b\cdot a$ is the multiplication in $A$. For superalgebras $A$ and $B$,
let $A\otimes B$ denote the superalgebra with multiplication

$(a\otimes b)\cdot(a^{\prime}\otimes
b^{\prime})=(-1)^{|b||a^{\prime}|}aa^{\prime}\otimes bb^{\prime}.$

######  Example 3.3.

The _super trace_ of a linear endomorphism $T\colon V\to V$ is defined as

(15) $\displaystyle{\sf sTr}(T):={\sf Tr}((-1)^{\sf F}\circ T)$

where ${\sf Tr}$ is the ordinary trace. The super trace determines a trace on
the superalgebra ${\rm End}(V)$ from Example 3.1.

A _$*$ -structure_ on a superalgebra is a $*$-structure on its underlying
(ungraded) algebra, i.e., an anti-involution of the underlying algebra (that
is conjugate-linear when ${\mathbb{F}}={\mathbb{C}}$).

######  Example 3.4.

The usual transpose determines a $*$-structure on ${\rm End}(V)$.

######  Remark 3.5.

A closely related notion is that of a _$*$ -superalgebra_, which is a
superalgebra together with an involutive _graded_ anti-involution, meaning a
homomorphism to the opposite algebra in the graded sense. There is a
dictionary relating superalgebras with $*$-structures to $*$-superalgebras
[DEF+99, pages 89-92]. The signs associated with $*$-superalgebras (while
perhaps more categorically pleasing) compare awkwardly with the standard
conventions in index theory. This is why we have chosen to work with ordinary
$*$-structures.

###  3.2. Modules over superalgebras

Given a superalgebra $A$, let ${}_{A}{\sf Mod}$ and ${\sf Mod}_{A}$ denote the
categories of left and right $A$-modules, respectively, whose objects are
super vector spaces with a (graded) $A$-action. These categories have
symmetric monoidal structures from the direct sum of $A$-modules. Similarly,
let ${}_{A}{\sf Mod}_{B}$ denote the category of $A-B$-bimodules. There are
canonical equivalences of categories

(16) $\displaystyle{}_{A\otimes B}{\sf Mod}\simeq{}_{A}{\sf
Mod}_{B^{\sf{op}}}\simeq{\sf Mod}_{A^{\sf{op}}\otimes B^{\sf{op}}}$

where a left $A\otimes B$-bimodule $W$ is sent to the $A-B^{\sf{op}}$-bimodule
with the same underlying super vector space and action

$a\cdot w\cdot b:=(-1)^{|w||b|}(a\otimes b)\cdot w$

with similar formulas defining a right $A^{\sf{op}}\otimes
B^{\sf{op}}$-module. We often use notation like ${}_{A}W$ for an object of
${}_{A}{\sf Mod}$, and ${}_{A}W_{B}$ for an object of ${}_{A}{\sf Mod}_{B}$.

A super trace on a superalgebra $A$ induces a super trace on the category of
(finite-dimensional) $A$-modules. For right $A$-modules, this super trace
comes from the composition

(17) $\displaystyle{\rm
End}_{A}(W)\stackrel{{\scriptstyle\sim}}{{\leftarrow}}W\otimes_{A}W^{\vee}\to
A/[A,A]\stackrel{{\scriptstyle{\sf sTr}_{A}}}{{\to}}{\mathbb{F}}$

where $W^{\vee}:={\rm Hom}_{A}(W,A)$ is the dual (left) $A$-module, and the
map to $A/[A,A]$ is the Hattori–Stallings trace [Hat65, Sta65] induced by the
evaluation pairing,

${\rm eval}\colon W\otimes_{\mathbb{F}}W^{\vee}\to A.$

One obtains traces for bimodules (and left modules) from traces on right
modules using (16).

Super algebras $A$ and $B$ are _Morita equivalent_ if there exist bimodules
${}_{A}W_{B}$ and ${}_{B}V_{A}$ such that
${}_{A}W\otimes_{B}V_{A}\simeq{}_{A}A_{A}$ and
${}_{B}V\otimes_{A}W_{B}\simeq{}_{B}B_{B}$, where $A$ and $B$ are regarded as
bimodules over themselves. For Morita equivalent algebras, there are
equivalences of categories

$({}_{B}V\otimes_{A}-)\colon{}_{A}{\sf Mod}\leftrightarrows{}_{B}{\sf
Mod}\colon({}_{A}W\otimes_{B}-).$

######  Remark 3.6.

Super algebras, bimodules, and bimodule maps are the objects, 1-morphisms and
2-morphisms of the (super) Morita bicategory, e.g., see [Fre12, §1.27].
Objects in this bicategory are 1-isomorphic when they are Morita equivalent.
We do not explicitly use this bicategorical structure below, but we do employ
some of the terminology. In particular, a Morita equivalence will be referred
to as an _invertible_ bimodule in reference to its role as an invertible
1-morphism in the Morita bicategory.

###  3.3. Clifford algebras

For a real (ungraded) vector space $V$ with quadratic form $q$, the _Clifford
algebra_ is the quotient of the tensor algebra of $V$

(18) $\displaystyle{\rm Cl}(V,q):=T(V)/\langle v\otimes v+q(v)\rangle.$

This is a superalgebra with grading involution determined by $v\mapsto-v$ for
$v\in V\subset{\rm Cl}(V,q)$.

######  Remark 3.7.

Sign conventions in (18) vary; ours are consistent with [ABS64, LM94, ST04]
but differ from [Bou88, IX.9].

Let ${\rm Cl}_{n,m}$ denote the Clifford algebra associated to the real vector
space $V={\mathbb{R}}^{n+m}={\mathbb{R}}^{n}\oplus{\mathbb{R}}^{m}$ for
quadratic form given by the direct sum of the standard positive definite inner
product on ${\mathbb{R}}^{n}$ and the standard negative definite inner product
on ${\mathbb{R}}^{m}$. Hence, we have the generators and relations
presentation,

(19) $\displaystyle{\rm Cl}_{n,m}=\langle
f_{1},\dots,f_{n},e_{1},\dots,e_{m}\mid[f_{i},f_{j}]=-2\delta_{ij},\
[e_{i},e_{j}]=2\delta_{ij},\ [f_{i},e_{j}]=0\rangle$

where the $f_{i}\in{\mathbb{R}}^{n}\subset{\rm Cl}_{n,m}$ and
$e_{i}\in{\mathbb{R}}^{m}\subset{\rm Cl}_{n,m}$ are odd, and the relations all
involve the super commutator (13). Since quadratic forms over ${\mathbb{R}}$
are classified by their signature, there exists an isomorphism ${\rm
Cl}(V,q)\simeq{\rm Cl}_{n,m}$ for any $(V,q)$ where $q$ has signature $(n,m)$.

We equip ${\rm Cl}_{n,m}$ with a $*$-structure determined by the values on
generators,

(20) $\displaystyle e_{i}^{*}=e_{i},\qquad f_{i}^{*}=-f_{i}.$

We also adopt the notation

(23) $\displaystyle{\rm Cl}_{n}:=\left\\{\begin{array}[]{ll}{\rm
Cl}_{n,0}&\phantom{BB}n\geq 0\\\ {\rm
Cl}_{0,n}&\phantom{BB}n<0\end{array}\right.$

and

(26) $\displaystyle\Gamma_{n,m}=2^{-(n+m)/2}f_{1}f_{2}\cdots
f_{n}e_{1}e_{2}\cdots
e_{m},\qquad\Gamma_{n}:=\left\\{\begin{array}[]{ll}\Gamma_{n,0}&n\geq 0\\\
\Gamma_{0,n}&n<0.\end{array}\right.$

The element $\Gamma_{n,m}$ depends on a choice of orientation of
${\mathbb{R}}^{n}\oplus{\mathbb{R}}^{m}$, and changing the orientation
introduces a sign. There are canonical isomorphisms

(27) $\displaystyle{\rm Cl}_{n}^{\sf{op}}\simeq{\rm Cl}_{-n},\quad{\rm
Cl}_{n}\simeq{\rm Cl}_{1}^{\otimes n},\quad{\rm Cl}_{n}\otimes{\rm
Cl}_{m}\simeq{\rm Cl}_{n+m},\quad{\rm Cl}_{n}\otimes{\rm Cl}_{-m}\simeq{\rm
Cl}_{n,m},$

for $n,m$ both nonnegative or both nonpositive. The isomorphism ${\rm
Cl}_{n}^{\sf{op}}\simeq{\rm Cl}_{-n}$ is determined by the identity map on
underlying super vector spaces, so below we will write ${\rm
Cl}_{n}^{\sf{op}}={\rm Cl}_{-n}$. The other isomorphisms critically use that
$\otimes$ is the graded tensor product of superalgebras.

Define the complex Clifford algebras by $\mathbb{C}{\rm l}_{n,m}:={\rm
Cl}_{n,m}\otimes{\mathbb{C}}$. These have $*$-structures by taking the
${\mathbb{C}}$-antilinear extension of the assignments (20). There are
isomorphisms $\mathbb{C}{\rm l}_{n,m}\simeq\mathbb{C}{\rm
l}_{n+m}\simeq\mathbb{C}{\rm l}_{-n-m}$ implemented by multiplying the
appropriate generators by $i=\sqrt{-1}$. These isomorphisms do not preserve
the elements $\Gamma_{n,m}\in{\rm Cl}_{n,m}\subset\mathbb{C}{\rm l}_{n,m}$ in
(26).

###  3.4. Clifford modules

A (real) _Clifford module_ is a graded left module over ${\rm Cl}(V,q)$. Right
Clifford modules and Clifford bimodules are defined analogously. The first
isomorphism in (27) determines equivalences of categories

(28) $\displaystyle{}_{{\rm Cl}_{n}\otimes{\rm Cl}_{-m}}{\sf
Mod}\simeq{}_{{\rm Cl}_{n}}{\sf Mod}_{{\rm Cl}_{m}}\simeq{\sf Mod}_{{\rm
Cl}_{-n}\otimes{\rm Cl}_{m}}$

which will allow us to translate any Clifford (bi)module into a left Clifford
module.

An _inner product_ on a real super vector space $W$ is an inner product on the
ungraded vector space underlying $W$ such that the even and odd subspaces of
$W$ are orthogonal. We take a similar definition for a hermitian inner product
on a complex super vector space.

######  Remark 3.8.

Sign conventions in differential geometry often use a mix of the graded and
ungraded tensor product for ${\mathbb{Z}}/2$-graded Hilbert spaces, Clifford
algebras, and differential forms valued in sections of a super vector bundle,
e.g., conventions in [BGV92]. Making these conventions consistent with each
other uses an equivalence between inner products on super vector spaces
defined via the graded and ungraded tensor product. This equivalence involves
signs and factors of $\sqrt{-1}$ (see [DEF+99, pages 90-91]). Such factors
appear when the two conventions collide. For example, self-adjoint
superconnections inherit signs from this translation, see Definition 6.2.

A left Clifford module $\rho\colon{\rm Cl}(V,q)\to{\rm End}(W)$ is _self-
adjoint_ [BGV92, Definition 3.3] if $W$ is equipped with an inner product and

(29) $\displaystyle\rho(v^{*})=\rho(v)^{\dagger}$

for all $v\in V\subset{\rm Cl}(V,q)$, using the $*$-structure (20) and where
$(-)^{\dagger}$ is the adjoint on ${\rm End}(W)$. Self-adjoint right modules
and self-adjoint bimodules are defined using the equivalences (28). Given a
self-adjoint left ${\rm Cl}(V,q)$-module $W$, a _Clifford linear endomorphism_
is a linear map $T\colon W\to W$ (not necessarily grading preserving) that
graded commutes with the Clifford action, i.e., for all $v\in V\subset{\rm
Cl}(V,q)$,

$T(\rho(v)\cdot w)=(-1)^{|T|}\rho(v)\cdot T(w),\quad\iff\quad[T,\rho(v)]=0$

where $[-,-]$ denotes the super commutator (13) on ${\rm End}(W)$. A Clifford
linear odd endomorphism is _self-adjoint_ when it is self adjoint with respect
to the inner product on $W$. Note that the grading involution on $W$ is always
self-adjoint.

Complex Clifford modules are defined analogously, where self-adjoint modules
are defined relative to a hermitian inner product.

######  Example 3.9.

Define the left ${\rm Cl}_{1,1}$-module whose underlying super vector space is
${\mathbb{R}}^{1|1}$ with action determined by

(34) $\displaystyle f\mapsto\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right],\qquad e\mapsto\left[\begin{array}[]{cc}0&1\\\
1&0\end{array}\right]$

using the notation (19) for generators. For the standard inner product this
${\rm Cl}_{1,1}$-action is self-adjoint. The assignment (34) induces an
isomorphism of superalgebras, ${\rm Cl}_{1,1}\simeq{\rm
End}({\mathbb{R}}^{1|1})$. This implies that ${\rm Cl}_{1,1}$ is Morita
equivalent to ${\mathbb{R}}$ with Morita bimodule ${\mathbb{R}}^{1|1}$. Using
(27), we find that ${\rm Cl}_{n}\otimes{\rm Cl}_{m}$ is Morita equivalent to
${\rm Cl}_{n+m}$ for any $n,m\in{\mathbb{Z}}$. From (28), we also see that
(34) determines an invertible ${\rm Cl}_{1}-{\rm Cl}_{1}$-bimodule. The
isomorphism ${\rm Cl}_{1,1}\simeq{\rm End}({\mathbb{R}}^{1|1})$ shows that
there are two isomorphism classes of invertible ${\rm Cl}_{1}-{\rm
Cl}_{1}$-bimodules, corresponding under Morita equivalence to the line
${\mathbb{R}}$ with its two possible gradings. The left ${\rm
Cl}_{1,1}$-module from (34) corresponds to ${\rm Cl}_{1}$ as a bimodule over
itself. Finally, the isometric automorphisms of ${\rm Cl}_{1}$ as a bimodule
is the group ${\mathbb{Z}}/2={\rm Spin}(1)$, acting by $\\{\pm 1\\}$ on
${\mathbb{R}}^{1|1}\simeq{\rm Cl}_{1}$.

######  Example 3.10.

Taking tensor powers of the previous example yields a Morita equivalence
between ${\rm Cl}_{n,n}$ and ${\mathbb{R}}$, determined by ${\rm Cl}_{n}$ as
an invertible ${\rm Cl}_{n}$-bimodule for $n\in{\mathbb{N}}$. The group ${\rm
SO}(n)$ acts on ${\rm Cl}_{\pm n}$ through algebra automorphisms via its
standard action on generators ${\mathbb{R}}^{n}\subset{\rm Cl}_{\pm n}$. This
gives a categorical action of ${\rm SO}(n)$ on the groupoid of invertible
${\rm Cl}_{\pm n}$-bimodules, given by twisting the left ${\rm Cl}_{\pm
n}$-action by an algebra automorphism of ${\rm Cl}_{\pm n}$. Categorical
actions are additional data, which in this case amounts to the ${\rm
Spin}(n)$-extension of ${\rm SO}(n)$. To see this, we first observe that by
Schur’s lemma there exists an isomorphism between ${\rm Cl}_{\pm n}$ as a
bimodule over itself for the standard action, and the bimodule that twists the
left action by an element of ${\rm SO}(n)$. This isomorphism is given by a
(metric-preserving) scalar, i.e., $\\{\pm 1\\}$. Hence, the group of
automorphisms of ${\rm Cl}_{\pm n}$ as a bimodule is a
${\mathbb{Z}}/2$-extension of ${\rm SO}(n)$, which turns out to be the spin
double cover, e.g., see [ST04, Definition 2.3.1].

######  Example 3.11.

Let $\mathbb{H}$ denote the quaternions, and fix the isomorphism of vector
spaces $\mathbb{H}\simeq{\mathbb{R}}^{4}$ sending $\\{1,i,j,k\\}$ to the
standard basis of ${\mathbb{R}}^{4}$. Define a ${\rm Cl}_{4}$-module whose
underlying super vector space is ${\mathbb{R}}^{4|4}$ with action determined
by the action of generators ${\mathbb{R}}^{4}\subset{\rm Cl}_{4}$

(37) $\displaystyle{\mathbb{R}}^{4}\to{\rm End}(\mathbb{R}^{4|4})\qquad
q\mapsto\left[\begin{array}[]{cc}0&q\\\ -\bar{q}&0\end{array}\right],\quad
q\in\mathbb{H}$

where $q\in\mathbb{H}$ is an element of the quaternions acting on
${\mathbb{R}}^{4}$ via the left action of the quaternions on themselves, and
the block diagonal matrix is for the decomposition
${\mathbb{R}}^{4|4}\simeq{\mathbb{R}}^{4|0}\oplus{\mathbb{R}}^{0|4}$, i.e.,
into even and odd subspaces. Checking the Clifford relations, the assignment
(37) extends to an injective homomorphism of superalgebras ${\rm
Cl}_{4}\hookrightarrow{\rm End}({\mathbb{R}}^{4|4})$ with image the
$\mathbb{H}$-linear endomorphisms of
${\mathbb{R}}^{4|4}\simeq\mathbb{H}\oplus\mathbb{H}$.

######  Example 3.12.

Let $\mathbb{O}$ denote the octonians, and fix an isomorphism of vector spaces
$\mathbb{O}\simeq{\mathbb{R}}^{8}$ by sending the generators of the octonians
to the standard basis of ${\mathbb{R}}^{8}$. Define a ${\rm Cl}_{8}$-module
whose underlying super vector space is ${\mathbb{R}}^{8|8}$ with an action
determined by the action of generators ${\mathbb{R}}^{8}\subset{\rm Cl}_{8}$

(40) $\displaystyle{\mathbb{R}}^{8}\to{\rm End}(\mathbb{R}^{8|8})\qquad
x\mapsto\left[\begin{array}[]{cc}0&x\\\ -\bar{x}&0\end{array}\right],\quad
x\in\mathbb{O}$

where $x\in\mathbb{O}$ acts on ${\mathbb{R}}^{8}$ (on the left) via the
identification with left multiplication in the octonions, and the block
diagonal matrix is for the decomposition
${\mathbb{R}}^{8|8}\simeq{\mathbb{R}}^{8|0}\oplus{\mathbb{R}}^{0|8}$. The
assignment (40) leads to an injective map ${\rm Cl}_{8}\hookrightarrow{\rm
End}({\mathbb{R}}^{8|8})$; since $\dim({\rm Cl}_{8})=2^{8}=16\cdot
16=\dim({\rm End}({\mathbb{R}}^{8|8}))$, this map is an isomorphism. Together
with Example 3.9, this induces the famous Morita equivalences between the
Clifford algebras ${\rm Cl}_{n+8}$ and ${\rm Cl}_{n}$ for all $n$. We also
note the isomorphism of ${\rm Cl}_{8}\simeq{\rm Cl}_{4}\otimes{\rm
Cl}_{4}$-modules [LM94, page 70]

(41) $\displaystyle({}_{{\rm
Cl}_{4}}{\mathbb{R}}^{4|4})\otimes_{\mathbb{R}}({}_{{\rm
Cl}_{4}}{\mathbb{R}}^{4|4})\simeq({}_{{\rm Cl}_{8}}{\mathbb{R}}^{8|8})^{\oplus
4}$

where ${}_{{\rm Cl}_{4}}{\mathbb{R}}^{4|4}$ is from the previous example.

######  Example 3.13.

The complexification of the module from Example 3.9 gives a ${\rm
Cl}_{1,1}\otimes{\mathbb{C}}\simeq\mathbb{C}{\rm l}_{2}$-module that is an
invertible ${\mathbb{C}}-\mathbb{C}{\rm l}_{2}$-bimodule. Explicitly, the
action of $\mathbb{C}{\rm l}_{2}$ on ${\mathbb{C}}^{1|1}$ is determined by the
action of the generators

(46) $\displaystyle f_{1}\mapsto\left[\begin{array}[]{cc}0&1\\\
-1&0\end{array}\right],\qquad f_{2}\mapsto\left[\begin{array}[]{cc}0&i\\\
i&0\end{array}\right],$

inducing an isomorphism $\mathbb{C}{\rm
l}_{2}\stackrel{{\scriptstyle\sim}}{{\to}}{\rm End}({\mathbb{C}}^{1|1})$ and
hence a Morita equivalence between $\mathbb{C}{\rm l}_{2}$ and ${\mathbb{C}}$.

###  3.5. Super traces for Clifford modules

There is a canonical isomorphism [BGV92, Proposition 3.21]

${\rm Cl}_{n}/[{\rm Cl}_{n},{\rm Cl}_{n}]\simeq{\rm Cl}_{n}/{\rm
Cl}_{n-1}\simeq\Lambda^{n}{\mathbb{R}}^{n}={\rm Det}({\mathbb{R}}^{n})$

with the determinant line bundle of ${\mathbb{R}}^{n}$. Hence, a (nontrivial)
super trace (14) on ${\rm Cl}_{n}$ is equivalent to the data of a
trivialization of the determinant line on ${\mathbb{R}}^{n}$, i.e., a choice
of volume form. Any such volume form is equivalent to an element
$c\cdot\Gamma_{n}$ for $c\in{\mathbb{R}}$ and $\Gamma_{n}$ from (26). This
gives a formula for the associated super trace,

(47) $\displaystyle{\sf sTr}_{{\rm Cl}_{n}}(a):=c\cdot{\sf
sTr}(\Gamma_{n}\circ a)\qquad c\in{\mathbb{R}}$

where the super trace on the right is taken for the action of $\Gamma_{n}\circ
a$ on the vector space $\Lambda^{\bullet}{\mathbb{R}}^{n}\simeq{\rm Cl}_{n}$
coming from the left action of the Clifford algebra on itself.

After choosing a scalar $c$, equations (17) and (47) determine a super trace
on free ${\rm Cl}_{n}$-modules. We extend this to a super trace on arbitrary
Clifford modules by writing a finite-rank module as a summand of a free
module. The resulting super trace

${\rm End}_{{\rm Cl}_{n}}(W)\to{\rm
Det}({\mathbb{R}}^{n})\stackrel{{\scriptstyle\sim}}{{\to}}{\mathbb{R}}$

inherits all the properties of the Hattori–Stallings trace (17), e.g., it
vanishes on super commutators of Clifford linear endomorphisms. When comparing
with the Pontryagin character in KO-theory, it turns out to be convenient to
modify the above to include a formal parameter; compare [FL10, §2.5].

######  Definition 3.14.

Given a left ${\rm Cl}_{n,m}$-module $W$, the _Clifford super trace_ of a
${\rm Cl}_{n,m}$-linear map $T$ is

(48) $\displaystyle{\rm sTr}_{{\rm Cl}_{n,m}}(T)={\rm
sTr}(u^{(m-n)/2}\Gamma_{n,m}\circ T)\in{\mathbb{R}}[u^{1/2},u^{-1/2}],$

where $u^{1/2}$ is an invertible formal variable of degree $-1$ with inverse
$u^{-1/2}$.

From properties of the ordinary super trace and the definition of
$\Gamma_{n,m}$ in (26), we observe that the Clifford super trace satisfies

(49) $\displaystyle{\rm sTr}_{{\rm Cl}_{n,m}}(T_{1}\oplus T_{2})={\rm
sTr}_{{\rm Cl}_{n,m}}(T_{1})+{\rm sTr}_{{\rm Cl}_{n,m}}(T_{2})$

for $T_{i}$ an endomorphism of a ${\rm Cl}_{n,m}$-module $W_{i}$ for $i=1,2$,
and

(50) $\displaystyle{\rm sTr}_{{\rm Cl}_{n,m}\otimes{\rm
Cl}_{n^{\prime},m^{\prime}}}(T_{1}\otimes T_{2})={\rm sTr}_{{\rm
Cl}_{n,m}}(T_{1})\cdot{\rm sTr}_{{\rm Cl}_{n^{\prime},m^{\prime}}}(T_{2})$

for $T_{1}$ an endomorphism of a ${\rm Cl}_{n,m}$-module $W_{1}$ and for
$T_{2}$ an endomorphism of a ${\rm Cl}_{n^{\prime},m^{\prime}}$-module
$W_{2}$.

There is a completely analogous trace theory for modules over complex Clifford
algebras, enjoying the same formal properties as the real Clifford super
trace. The choice of normalization in (48) determines a normalization of the
complex Clifford supertrace, using that ${\rm Det}({\mathbb{C}}^{n})\simeq{\rm
Det}({\mathbb{R}}^{n})\otimes{\mathbb{C}}$, i.e., that a real trivialization
of the determinant line also provides a trivialization of its
complexification.

######  Example 3.15.

For the ${\rm Cl}_{1,1}$-module in Example 3.9,

${\rm sTr}_{{\rm Cl}_{1,1}}({{\rm id}}_{{\mathbb{R}}^{1|1}})={\rm
sTr}(\Gamma_{1,1})=2/2=1.$

From this we see that the Clifford super trace of a ${\rm Cl}_{n,m}$-linear
operator $T$ is compatible with the Morita equivalences from Example 3.9

${\rm sTr}_{{\rm Cl}_{n,m}\otimes{\rm Cl}_{1,1}}(T\otimes{{\rm
id}}_{{\mathbb{R}}^{1|1}})={\rm sTr}_{{\rm Cl}_{n,m}}(T)\cdot{\rm sTr}_{{\rm
Cl}_{1,1}}({{\rm id}}_{{\mathbb{R}}^{1|1}})={\rm sTr}_{{\rm Cl}_{n,m}}(T).$

For the ${\rm Cl}_{4}$-module in Example 3.11 we find

(51) $\displaystyle{\rm sTr}_{{\rm Cl}_{4}}({{\rm
id}}_{{\mathbb{R}}^{4|4}})={\rm sTr}(u^{-4/2}\Gamma_{4})=2^{-4/2}\cdot
8u^{-2}=2u^{-2},$

and for the ${\rm Cl}_{8}$-module in Example 3.12,

(52) $\displaystyle{\rm sTr}_{{\rm Cl}_{8}}({{\rm
id}}_{{\mathbb{R}}^{8|8}})={\rm sTr}(u^{-8/2}\Gamma_{8})=u^{-4}2^{-8/2}\cdot
16=u^{-4}.$

We verify that (51) and (52) are compatible with the isomorphism of ${\rm
Cl}_{8}$-modules in (41),

$2u^{-2}\cdot 2u^{-2}={\rm sTr}_{{\rm Cl}_{4}\otimes{\rm Cl}_{4}}({{\rm
id}}_{{\mathbb{R}}^{4|4}}\otimes{{\rm id}}_{{\mathbb{R}}^{4|4}})={\rm
sTr}_{{\rm Cl}_{8}}(({{\rm id}}_{{\mathbb{R}}^{8|8}})^{\oplus 4})=4u^{-4}.$

From (52) we also find that for a ${\rm Cl}_{n}$-linear operator $T$,

${\rm sTr}_{{\rm Cl}_{n}\otimes{\rm Cl}_{8k}}(T\otimes({{\rm
id}}_{{\mathbb{R}}^{8|8}})^{\otimes k})={\rm sTr}_{{\rm Cl}_{n}}(T)\cdot({\rm
sTr}_{{\rm Cl}_{8}}({{\rm id}}_{{\mathbb{R}}^{8|8}}))^{k}={\rm sTr}_{{\rm
Cl}_{n}}(T)\cdot u^{-4k},$

so that the $8$-periodicity of Clifford modules is reflected by the Clifford
super trace, where the periodicity generator is $u^{\pm
4}\in{\mathbb{R}}[u^{1/2},u^{-1/2}]$.

######  Example 3.16.

The complex Clifford module from Example 3.13 has

${\rm sTr}_{\mathbb{C}{\rm l}_{2}}({{\rm id}}_{{\mathbb{C}}^{1|1}})={\rm
sTr}(\Gamma_{2})=2iu^{-1}/2=iu^{-1}.$

Hence, analogous to the 8-periodicity in the real case

(53) $\displaystyle{\rm sTr}_{\mathbb{C}{\rm l}_{n}\otimes\mathbb{C}{\rm
l}_{2k}}(T\otimes({{\rm id}}_{{\mathbb{C}}^{1|1}})^{\otimes k})={\rm
sTr}_{\mathbb{C}{\rm l}_{n}}(T)\cdot({\rm sTr}_{\mathbb{C}{\rm l}_{2}}({{\rm
id}}_{{\mathbb{C}}^{1|1}}))^{k}={\rm sTr}_{\mathbb{C}{\rm l}_{n}}(T)\cdot
i^{k}u^{-k}.$

######  Remark 3.17.

When working with complex Clifford modules, some authors incorporate a factor
of $i^{n/2}$ in the definition (26) of $\Gamma_{n}$ so that ${\rm
sTr}_{\mathbb{C}{\rm l}_{2}}({{\rm id}}_{{\mathbb{C}}^{1|1}})=u^{-1}$, e.g.,
see [BGV92, Proposition 3.21] and [ST04, Definition 3.2.16]. However, for real
Clifford modules this is a bit less natural as the resulting $\Gamma_{n}$ is
not an element of the real Clifford algebra ${\rm Cl}_{n}$. One reason to
incorporate the formal variable $u$ in (48) is that any two choices of
normalization can be related by rescaling $u$ by the appropriate scalar.

##  4\. The Atiyah–Bott–Shapiro construction and the spectrum ${\rm KO}$

In this section we review a Clifford module construction of the spaces
representing ${\rm KO}^{n}(-)$ from [HST10, §7] and [Che07, §1.3-1.4]. Our
presentation emphasizes the connection with the Atiyah–Bott–Shapiro
construction and the families index.

###  4.1. The Atiyah–Bott–Shapiro construction

Let $M$ be a manifold, not necessarily compact. A class in the compactly
supported real K-theory group ${\rm KO}^{0}_{c}(M)$ can be specified by a pair
$(V,Q)$ for a ${\mathbb{Z}}/2$-graded real vector bundle $V\to M$ and
$Q\in{\rm End}(V)_{\rm odd}$ an odd endomorphism that is invertible outside of
a compact subset of $M$. Roughly,

(54) $\displaystyle[V]=[V_{\rm ev}]-[V_{\rm odd}]\in{\rm KO}^{0}(M)$

determines a class, and on the subspace of $M$ where $Q$ is invertible we
obtain an isomorphism $V_{\rm ev}\stackrel{{\scriptstyle\sim}}{{\to}}V_{\rm
odd}$, so that the formal difference (54) vanishes on this subspace; we
explain this in greater detail in Remark 4.20.

Given a right ${\rm Cl}_{n}$-module $W$, consider the trivial bundle
$\underline{W}=W\times{\mathbb{R}}^{n}\to{\mathbb{R}}^{n}$. Viewing this as a
${\mathbb{Z}}/2$-graded vector bundle, equip it with the odd operator ${\rm
cl}_{x}\in{\rm End}(W)_{\rm odd}$ that at a point $x\in{\mathbb{R}}^{n}$ is
Clifford multiplication by $x\in{\mathbb{R}}^{n}\subset{\rm Cl}_{n}$ on $W$.
This constructs a map

(55) $\displaystyle{\sf Mod}_{{\rm Cl}_{n}}\to{\rm
KO}^{0}_{c}({\mathbb{R}}^{n})\simeq{\rm KO}^{-n}({\rm pt}),\qquad
W\mapsto(\underline{W},{\rm cl}_{x})$

where the class has compact support because the operator ${\rm cl}_{x}$ is
invertible away from $0\in{\mathbb{R}}^{n}$ as $({\rm cl}_{x})^{2}=-|x|^{2}$
is multiplication by a scalar. The isomorphism in (55) is the suspension
isomorphism, identifying the 1-point compactification of ${\mathbb{R}}^{n}$
with the $n$-sphere. Direct sums of ${\rm Cl}_{n}$-modules are sent to
addition of classes in ${\rm KO}^{-n}({\rm pt})$ under (55).

Let ${\mathfrak{M}}_{n}$ denote the Grothendieck group of ${\sf Mod}_{{\rm
Cl}_{n}}$. The standard inclusion ${\mathbb{R}}^{n}\subset{\mathbb{R}}^{n+1}$
yields a homomorphism of superalgebras $i\colon{\rm Cl}_{n}\hookrightarrow{\rm
Cl}_{n+1}$ determined by the map on generators (19). This induces restrictions

${\sf Mod}_{{\rm Cl}_{n+1}}\to{\sf Mod}_{{\rm Cl}_{n}},\quad
i^{*}\colon{\mathfrak{M}}_{n+1}\to{\mathfrak{M}}_{n},\qquad n\geq 0.$

Atiyah–Bott–Shapiro [ABS64, Theorem 11.5] show that (55) factors through the
quotient ${\mathfrak{M}}_{n}/i^{*}{\mathfrak{M}}_{n+1}$, and in fact
determines isomorphisms of abelian groups

(56)
$\displaystyle{\mathfrak{M}}_{n}/i^{*}{\mathfrak{M}}_{n+1}\stackrel{{\scriptstyle\sim}}{{\to}}{\rm
KO}^{-n}({\rm pt})\qquad n\geq 0.$

Furthermore, the tensor product of Clifford modules is compatible with the
graded ring structure on $\bigoplus_{n\in{\mathbb{N}}}{\rm KO}^{-n}({\rm
pt})$. Together with the 8-fold periodicity of Clifford algebras and ${\rm
KO}$, the maps (56) give a completely algebraic description of the graded ring
${\rm KO}^{\bullet}({\rm pt})$.

######  Remark 4.1.

We compare sign conventions in the Atiyah–Bott–Shapiro isomorphism (56).
First, we recall the equivalence of categories ${\sf Mod}_{{\rm
Cl}_{n}}\simeq{}_{{\rm Cl}_{-n}}{\sf Mod}$ using the canonical isomorphism
${\rm Cl}_{\pm n}^{\sf{op}}={\rm Cl}_{\mp n}$. This allows one to rephrase
(56) in terms of left Clifford modules,

(57)
$\displaystyle{}_{-n}{\mathfrak{M}}/i^{*}{}_{-n-1}{\mathfrak{M}}\simeq{\rm
KO}^{-n}({\rm pt})\qquad n\geq 0,\quad i\colon{\rm Cl}_{-n}\hookrightarrow{\rm
Cl}_{-n-1}$

but with the indicated sign change. One can instead use the ungraded anti-
automorphism of ${\rm Cl}_{n}$ (sometimes called the _transpose_) that gives a
left-module version of (56) but without a sign change. This second option is a
bit awkward when considering tensor products of modules and the corresponding
products in ${\rm KO}^{\bullet}({\rm pt})$: the ungraded tensor product of
${\rm Cl}_{m}$ and ${\rm Cl}_{n}$ is not a Clifford algebra.

Our choice of sign convention below is fixed by our preference for left
modules and the example of the Clifford linear Dirac operator on an
$n$-dimensional Riemannian spin manifold. If one demands that the Dirac
Laplacian is a nonnegative operator, then it must be defined on the associated
bundle for the ${\rm Spin}(n)$ action on ${\rm Cl}_{n}$ as in (134) (the
associated bundle for ${\rm Cl}_{-n}$ leads to a nonpositive Dirac Laplacian).
This convention agrees with [LM94, II.7]. Using the canonical isomorphism
${\rm Cl}_{n}^{\sf{op}}={\rm Cl}_{-n}$, the kernel of the Clifford linear
Dirac operator is then a left ${\rm Cl}_{-n}$-module, which determines an
element of ${\rm KO}^{-n}({\rm pt})$ using the convention (57).

######  Remark 4.2.

The Atiyah–Bott–Shapiro isomorphism (57) can be verified completely explicitly
using the generators and relations description [LM94, I.9]

(58) $\displaystyle\pi_{*}{\rm KO}={\mathbb{Z}}[\eta,\alpha,\beta^{\pm
1}]/\langle
2\eta,\eta^{3},\eta\alpha,\alpha^{2}-4\beta\rangle,\quad|\eta|=-1,\
|\alpha|=-4,\ |\beta|=-8.$

We sketch surjectivity of (57). The Clifford modules constructed in the
previous section correspond to classes

$[_{{\rm Cl}_{-1}}{\mathbb{R}}^{1|1}]=\eta\in{\rm KO}^{-1}({\rm
pt}),\quad[_{{\rm Cl}_{4}}{\mathbb{R}}^{4|4}]=\beta^{-1}\alpha\in{\rm
KO}^{4}({\rm pt}),\quad[_{{\rm Cl}_{8}}{\mathbb{R}}^{8|8}]=\beta^{-1}\in{\rm
KO}^{8}({\rm pt})$

where the ${\rm Cl}_{-1}$-action on ${\mathbb{R}}^{1|1}$ is defined in Example
3.9 (by forgetting the ${\rm Cl}_{1}$-action), the ${\rm Cl}_{4}$-action on
${\mathbb{R}}^{4|4}$ in Example 3.11 and the ${\rm Cl}_{8}$-action on
${\mathbb{R}}^{8|8}$ in Example 3.12. To see that $2\eta=\eta^{3}=0$, one must
show that ${\mathbb{R}}^{1|1}\oplus{\mathbb{R}}^{1|1}\simeq{\mathbb{R}}^{2|2}$
extends to a ${\rm Cl}_{-1}\otimes{\rm Cl}_{-1}\simeq{\rm Cl}_{-2}$-module and
the ${\rm Cl}_{-3}$-module $({\mathbb{R}}^{1|1})^{\otimes
3}\simeq{\mathbb{R}}^{4|4}$ extends to a ${\rm Cl}_{-4}$-module; for the
former, the second Clifford generator can be taken to be the isomorphism that
exchanges the copies of ${\mathbb{R}}^{1|1}$ in the direct sum, and for the
latter one uses a module structure closely related to (37). To see
$\eta\alpha=0$, it suffices to check $\eta\alpha\beta^{-1}=0$; starting with
the ${\rm Cl}_{-1}\otimes{\rm Cl}_{4}$-module
${\mathbb{R}}^{1|1}\otimes{\mathbb{R}}^{4|4}$, a Morita equivalence provides a
${\rm Cl}_{3}$-module with underlying vector space ${\mathbb{R}}^{4|4}$, and
this similarly has an extension to a ${\rm Cl}_{4}$-module. Invertibility of
$\beta$ follows from the Morita equivalence implemented by (40), and the
relation involving $\alpha$ and $\beta$ is equivalent to the isomorphism (41).

###  4.2. Unbounded operators that represent ${\rm KO}$

Next we review a space of unbounded operators that is known to represent ${\rm
KO}$, following the presentation in [HST10, §3]. Fix a real, infinite-
dimensional, separable Hilbert space $\mathcal{H}$. An _unbounded operator_
$D$ on $\mathcal{H}$ is the data of a subspace ${\rm
dom}(D)\subset\mathcal{H}$ (the domain of $D$) and a linear map $D\colon{\rm
dom}(D)\to\mathcal{H}$. An unbounded operator is _self-adjoint_ if it is self-
adjoint on the closure of its domain. An unbounded, self-adjoint operator $D$
has _compact resolvent_ if the spectrum of $D$ consists of eigenvalues of
finite multiplicity that do not have an accumulation point in ${\mathbb{R}}$.

######  Definition 4.3.

Let ${\rm Inf}(\mathcal{H})$ denote the set of unbounded, self-adjoint
operators on $\mathcal{H}$ with compact resolvent.

######  Remark 4.4.

The notation ${\rm Inf}$ comes from the terminology in [HST10]: $D\in{\rm
Inf}(\mathcal{H})$ is an _infinitesimal generator_ of a super semigroup
representation. Although we do not use super semigroup representations, we
keep the same notation for ease of comparison with [HST10]. We caution that
our sign conventions differ slightly from [HST10]; see Remark 4.5.

To simplify the exposition, hereafter we will typically omit the words
“compact resolvent” and “self-adjoint,” referring to an element of ${\rm
Inf}(\mathcal{H})$ simply as an unbounded operator on $\mathcal{H}$. The
Cayley transform determines an injective map [HST10, Proposition 3.7]

(59) $\displaystyle{\rm Inf}(\mathcal{H})\hookrightarrow O(\mathcal{H})$

from unbounded operators to the infinite orthogonal group, sending eigenspaces
of an operator $D$ with (real) eigenvalue $\lambda$ to an orthogonal operator
with (complex) eigenvalue $\frac{\lambda+i}{\lambda-i}$. Equip the set of
unbounded operators with the subspace topology inherited from
$O(\mathcal{H})$, where $O(\mathcal{H})$ is given the norm topology.

Next we consider variations on the space ${\rm Inf}(\mathcal{H})$ of unbounded
operators that involve gradings and Clifford actions. Define a
_${\mathbb{Z}}/2$ -graded Hilbert space_ as a separable Hilbert space whose
underlying vector space has a ${\mathbb{Z}}/2$-grading such that the even and
odd subspaces are infinite-dimensional and orthogonal to each other. A _stable
${\rm Cl}_{n}$-module_ is a ${\mathbb{Z}}/2$-graded Hilbert space with a self-
adjoint left ${\rm Cl}_{n}$-module structure such that every isomorphism class
of irreducible ${\rm Cl}_{n}$-module appears with infinite multiplicity; for
example, starting with a ${\mathbb{Z}}/2$-graded Hilbert space $\mathcal{H}$,
then $\mathcal{H}_{n}:={\rm Cl}_{n}\otimes\mathcal{H}$ is a stable ${\rm
Cl}_{n}$-module. Any pair of stable ${\rm Cl}_{n}$-modules are isomorphic, and
the tensor product of a stable ${\rm Cl}_{n}$-module with a stable ${\rm
Cl}_{m}$-module determines a stable ${\rm Cl}_{n+m}$-module (using the Morita
equivalence from Example 3.9 when $n$ and $m$ have different signs). We refer
to [Joa01, §3] for further discussion of this tensor product of stable ${\rm
Cl}_{n}$-modules.

A self-adjoint unbounded operator $D$ on $\mathcal{H}_{n}$ is _odd_ if its
domain is a graded subspace and $D\colon{\rm dom}(D)\to\mathcal{H}$ is an odd
linear map. An odd self-adjoint unbounded operator $D$ on $\mathcal{H}_{n}$ is
_Clifford linear_ if ${\rm dom}(D)$ is a Clifford submodule and $D$ graded
commutes with the ${\rm Cl}_{n}$-action. Let ${\rm Inf}_{-n}$ denote the set
of odd, Clifford linear unbounded operators on $\mathcal{H}_{n}$. We equip
this set with the subspace topology for the evident inclusion

(60) $\displaystyle{\rm Inf}_{-n}\subset{\rm Inf}(\mathcal{H}_{n}).$

######  Remark 4.5.

In [HST10], the authors use stable _right_ ${\rm Cl}_{n}$-modules rather than
the stable left ${\rm Cl}_{n}$-modules above, compare Remark 4.1. The spaces
${\rm Inf}_{-n}$ in the present paper are homeomorphic to the ones in [HST10]
(with the sign unchanged) by trading a left ${\rm Cl}_{n}$-action on
$\mathcal{H}_{n}={\rm Cl}_{n}\otimes\mathcal{H}$ for a right ${\rm
Cl}_{-n}$-action.

It is important in applications that the domain of an unbounded operator
$D\in{\rm Inf}_{-n}$ is _not_ required to be dense in $\mathcal{H}_{n}$; this
permits the following example.

######  Example 4.6.

Consider a graded, ${\rm Cl}_{n}$-invariant subspace
$W\hookrightarrow\mathcal{H}_{n}$. Define the unbounded operator $P_{W}$ whose
domain is $W$ and $P_{W}\colon W\to\mathcal{H}_{n}$ is the inclusion. If $W$
is finite-dimensional, then $P_{W}$ has compact resolvent and $P_{W}\in{\rm
Inf}_{-n}$.

Perhaps the most well-known analytic model for the ${\rm KO}$-spectrum comes
from the Atiyah–Singer spaces ${\rm Fred}_{n}$ of ${\rm Cl}_{n}$-linear
Fredholm operators that for $n\geq 0$ represent ${\rm KO}^{-n}$ [AS69], [LM94,
III.10]. The spaces ${\rm Inf}_{n}$ give an unbounded version of these spaces
in the following sense (we continue to use the sign convention from Remark
4.1).

######  Theorem 4.7 ([HST10] Proposition 3.11 and Theorem 8.2).

For $n\geq 0$, there are homotopy equivalences ${\rm
Inf}_{n}\stackrel{{\scriptstyle\sim}}{{\to}}{\rm Fred}_{n}$. Hence, ${\rm
Inf}_{n}$ represents the functor ${\rm KO}^{-n}(-)$.

The 8-periodicity of Clifford modules gives homeomorphisms ${\rm
Inf}_{n}\simeq{\rm Inf}_{n-8}$, and so (by Bott periodicity) one concludes
that ${\rm Inf}_{n}$ represents ${\rm KO}^{-n}(-)$ for all $n\in{\mathbb{Z}}$.
Alternatively, one can extend the construction of the spaces ${\rm Fred}_{n}$
to arbitrary $n\in{\mathbb{Z}}$, and prove a version of Theorem 4.7 for all
$n$, e.g., see [HST10, Remark 0.1].

###  4.3. Topological categories and their classifying spaces

A _topological category_ is a category in which the objects and morphisms are
topological spaces for which the structure maps (source, target, unit,
composition) are continuous. For a topological category $\mathcal{C}$, let
$\mathcal{C}_{0}$ denote the space of objects and $\mathcal{C}_{1}$ the space
of morphisms.

Given an open cover of a topological space, the following two examples
construct topological categories.

######  Example 4.8.

Let $Z$ be a topological space, and $\\{U_{i}\\}_{i\in I}$ an open cover of
$Z$. For $\sigma\subset I$ a nonempty finite subset, let
$U_{\sigma}=\bigcap_{i\in\sigma}U_{i}$ denote the intersection. Define the
_Čech category_ of the cover, denoted $\check{{\rm C}}(U_{i})$, as the
topological category whose objects are $\check{{\rm
C}}(U_{i})_{0}=\coprod_{\sigma}U_{\sigma}$ and whose morphisms are the space
$\check{{\rm C}}(U_{i})_{1}=\coprod_{\sigma\subseteq\tau}U_{\tau}$. The source
and target maps are determined by the projection and inclusion
$U_{\tau}\subseteq U_{\sigma}$, the unit is induced by the identity map
$U_{\sigma}\to U_{\sigma}$, and composition comes from nested inclusions.

######  Example 4.9.

Let $Z$ be a topological space, and $\\{U_{i}\\}_{i\in I}$ an open cover of
$Z$ with the data of an ordering on the indexing set $I$. Then define a
topological category $\check{{\rm C}}^{\rm ord}(U_{i})$ with objects and
morphisms

$\check{{\rm C}}^{\rm ord}(U_{i})_{0}=\coprod_{i\in I}U_{i},\qquad\check{{\rm
C}}^{\rm ord}(U_{i})_{1}=\coprod_{i\leq j}U_{ij},\qquad U_{ij}=U_{i}\bigcap
U_{j}.$

The source and target maps are the inclusions $U_{i}\bigcap U_{j}\subset
U_{i},U_{j}$, the unit map is the identity on $U_{i}=U_{i}\bigcap U_{i}$, and
composition comes from inclusions of triple intersections into double
intersections.

######  Definition 4.10.

The _nerve_ of a topological category $\mathcal{C}$ is the simplicial space
whose $k$-simplices are $N_{k}\mathcal{C}=\mathcal{C}_{k}$ for $k=0,1$, and
for $k>1$ are length $k$ chains of composable morphisms

$\displaystyle N_{k}\mathcal{C}$ $\displaystyle=$
$\displaystyle\\{x_{0}\stackrel{{\scriptstyle
f_{1}}}{{\to}}x_{1}\stackrel{{\scriptstyle
f_{2}}}{{\to}}x_{2}\stackrel{{\scriptstyle
f_{3}}}{{\to}}\cdots\stackrel{{\scriptstyle f_{k}}}{{\to}}x_{k}\mid
x_{i}\in\mathcal{C}_{0},f_{i}\in\mathcal{C}_{1}\\}$ $\displaystyle=$
$\displaystyle\mathcal{C}_{1}\times_{\mathcal{C}_{0}}\cdots\times_{\mathcal{C}_{0}}\mathcal{C}_{1},$

topologized as the fibered product. The face maps in $N_{\bullet}\mathcal{C}$
are determined by composing morphisms, and degeneracies are determined by the
unit map in $\mathcal{C}$.

The _geometric realization_ of a simplicial space $Z_{\bullet}$ is

(61)
$\displaystyle|Z_{\bullet}|:=(\coprod_{n}Z_{n}\times\Delta^{n})/{\sim}\qquad(f^{*}x,t)\sim(x,f_{*}t)\quad\forall
f\colon[l]\to[k]$

where $\Delta^{n}$ is the standard $n$-simplex, and $f\colon[l]\to[k]$ is an
order-preserving map. Geometric realization is natural: a map of simplicial
spaces $F\colon Z_{\bullet}\to Y_{\bullet}$ determines a map between
realizations, $|F|\colon|Z_{\bullet}|\to|Y_{\bullet}|$.

######  Definition 4.11.

Given a topological category $\mathcal{C}$, define the _classifying space of
$\mathcal{C}$_ as the geometric realization of its nerve,
${\rm{B}}\mathcal{C}:=|N_{\bullet}\mathcal{C}|.$

######  Example 4.12.

The nerve of the topological category $\check{C}^{\rm ord}(U_{i})$ from
Example 4.9 has as $k$-simplices the intersections

(62) $\displaystyle N_{k}\check{C}(U_{i})=\coprod_{i_{0}\leq
i_{1}\leq\cdots\leq i_{k}}U_{i_{0}i_{1}\cdots i_{k}},\qquad
U_{i_{0}i_{1}\cdots i_{k}}=U_{i_{0}}\bigcap U_{i_{1}}\bigcap\cdots\bigcap
U_{i_{k}}.$

whereas the nerve of the topological category $\check{C}(U_{i})$ from Example
4.8 has as $k$-simplices

$N_{k}\check{C}(U_{i})=\coprod_{\sigma_{0}\subseteq\sigma_{1}\subseteq\cdots\subseteq\sigma_{k}}U_{\sigma_{k}}.$

Results of Segal [Seg68, Proposition 4.12], and Dugger–Isaksen [DI04, Theorem
2.1 and Proposition 2.6] show that the canonical maps

(63) $\displaystyle{\rm{B}}\check{C}^{\rm
ord}(U_{i})\stackrel{{\scriptstyle\sim}}{{\to}}{\rm{B}}\check{C}(U_{i}),\qquad{\rm{B}}\check{C}^{\rm
ord}(U_{i})\stackrel{{\scriptstyle\sim}}{{\to}}Z,\qquad{\rm{B}}\check{C}(U_{i})\stackrel{{\scriptstyle\sim}}{{\to}}Z,$

are homotopy equivalences.

There is a homotopy equivalence
${\rm{B}}\mathcal{C}^{\sf{op}}\simeq{\rm{B}}\mathcal{C}$ between the
realization of a category and the realization of its opposite. Below we will
often identify ${\rm{B}}\check{C}(U_{i})$ with
${\rm{B}}\check{C}(U_{i})^{\sf{op}}$.

###  4.4. The Atiyah–Bott–Shapiro category

######  Definition 4.13.

Let ${\mathcal{M}}_{n}^{\delta}$ be the category whose objects are finite-
dimensional ${\rm Cl}_{n}$-submodules $W\subset\mathcal{H}_{n}$ and whose
morphisms are

(69) $\displaystyle{\rm
Mor}(W_{0},W_{1}):=\left\\{\begin{array}[]{cl}\begin{array}[]{c}{\rm
Cl}_{n}\otimes{\rm Cl}_{-1}-{\rm action\ on\ }W_{0}^{\perp}\\\ {\rm extending\
the}\ {\rm Cl}_{n}{\rm-action}\end{array}&{\rm if}\ W_{0}\subseteq
W_{1}\subset\mathcal{H}_{n}\\\ \hbox{}\hfil\\\ \emptyset&{\rm
else}\end{array}\right.$

where $W_{0}^{\perp}\subseteq W_{1}$ denotes the orthogonal complement of the
subspace $W_{0}\subseteq W_{1}$. Composition for a nested inclusion
$W_{0}\subseteq W_{1}\subseteq W_{2}$ uses the direct sum of Clifford modules
to obtain a ${\rm Cl}_{n}\otimes{\rm Cl}_{-1}$-module structure on the
orthogonal complement of $W_{0}\subseteq W_{2}$.

######  Remark 4.14.

An extension to a ${\rm Cl}_{n}\otimes{\rm Cl}_{-1}$-action is the data of a
self-adjoint odd endomorphism $e$ on $W_{0}^{\perp}$ that graded commutes with
the ${\rm Cl}_{n}$-action and satisfies $e^{2}=1$.

######  Remark 4.15.

For $n\geq 0$, a ${\rm Cl}_{-n}\otimes{\rm Cl}_{-1}$-module is the same data
as a ${\rm Cl}_{-n-1}$-module. Hence, for $n\leq 0$,
${\mathcal{M}}_{n}^{\delta}$ is a category whose morphisms parameterize how
the Atiyah–Bott–Shapiro map (57) identifies (left) Clifford modules in ${\rm
KO}^{n}({\rm pt})$.

Let $\mathcal{B}(\mathcal{H}_{n})$ denote the space of bounded ${\rm
Cl}_{n}$-linear operators on $\mathcal{H}_{n}$. There are injections

(70) $\displaystyle{\rm
Obj}({\mathcal{M}}_{n}^{\delta})\hookrightarrow\mathcal{B}(\mathcal{H}_{n}),\qquad{\rm
Mor}({\mathcal{M}}_{n}^{\delta})\hookrightarrow\mathcal{B}(\mathcal{H}_{n})^{\times
3}.$

The first map comes from identifying a submodule $W_{0}\subset\mathcal{H}_{n}$
with a finite rank projection operator. Similarly, $(W_{0},W_{1},e)\in{\rm
Mor}({\mathcal{M}}_{n}^{\delta})$ determines a pair of finite rank projection
operators and a bounded operator gotten by extending $e$ (in the notation of
Remark 4.14) to an operator on $\mathcal{H}_{n}$ via the zero operator on the
orthogonal complement of $W_{0}^{\perp}\subset\mathcal{H}_{n}$.

######  Definition 4.16.

Define the _$n$ th Atiyah–Bott–Shapiro category ${\mathcal{M}}_{n}$_ as the
topological category with underlying category ${\mathcal{M}}_{n}^{\delta}$
whose objects and morphisms are endowed with the subspace topology for the
inclusions (70) using the norm topology on bounded operators.

######  Remark 4.17.

Since a tensor product of ${\rm Cl}_{n}$ and ${\rm Cl}_{m}$-modules produces a
${\rm Cl}_{n+m}$-module and any pair of stable ${\rm Cl}_{n}$-modules are
isomorphic, there are continuous functors

(71)
$\displaystyle{\mathcal{M}}_{n}\times{\mathcal{M}}_{m}\stackrel{{\scriptstyle\otimes}}{{\to}}{\mathcal{M}}_{n+m},$

unique up to a contractible space of choices. The topological category
${\mathcal{M}}_{n}$ also has a partially-defined sum operation, defined for
orthogonal subspaces $W,W^{\prime}\subset\mathcal{H}_{n}$.

###  4.5. The families index and the Atiyah–Bott–Shapiro category

We will compare the classifying space of the $n$th Atiyah–Bott–Shapiro
category with the space ${\rm Inf}_{-n}$ through an index map, similar to the
families index described in §2. Define an open cover with components indexed
by $\lambda\in{\mathbb{R}}_{>0}$

(72) $\displaystyle{\rm Inf}_{n,\lambda}:=\\{D\in{\rm
Inf}_{-n}\mid\lambda\notin{\rm Spec}(D^{2})\\}\subset{\rm Inf}_{-n}.$

Let $\check{{\rm C}}({\rm Inf}_{n,\lambda})$ denote the Čech category of this
cover of ${\rm Inf}_{-n}$. Explicitly, the objects of this category are given
by an unbounded operator $D$ and a $(k+1)$-tuple of positive real numbers,

$\check{{\rm C}}({\rm
Inf}_{n,\lambda})_{0}=\\{D,\lambda_{0}\leq\lambda_{1}\leq\cdots\leq\lambda_{k}\mid
D\in{\rm Inf}_{-n},\ \lambda_{i}\in{\mathbb{R}}_{>0},\ \lambda_{i}\notin{\rm
Spec}(D^{2})\\},$

satisfying the indicated property. Morphisms are given by the same data
together with a specified subset
$\\{\lambda_{i_{0}},\dots,\lambda_{i_{l}}\\}\subseteq\\{\lambda_{0},\dots,\lambda_{k}\\}$
with $\lambda_{i_{0}}\leq\lambda_{i_{1}}\leq\cdots\leq\lambda_{i_{l}}$.

Define a continuous functor

(73) $\displaystyle{\rm Ind}\colon\check{{\rm C}}({\rm
Inf}_{n,\lambda})^{\sf{op}}\to{\mathcal{M}}_{n}$

as follows. On objects, ${\rm Ind}$ assigns to
$(D,\lambda_{0}\leq\lambda_{1}\leq\cdots\leq\lambda_{k})$ the subspace
$\mathcal{H}^{<\lambda_{k}}\subset\mathcal{H}_{n}$ given by the direct sum of
eigenspaces of $D^{2}$ of eigenvalue less than $\lambda_{k}$,

$\mathcal{H}^{<\lambda_{k}}:=\bigoplus_{\mu<\lambda}\mathcal{H}_{\mu}.$

The subspace $\mathcal{H}^{<\lambda_{k}}\subset\mathcal{H}_{n}$ is finite-
dimensional because $D$ has compact resolvent. Furthermore,
$\mathcal{H}^{<\lambda_{k}}$ is a graded ${\rm Cl}_{n}$-submodule because $D$
is a Clifford linear operator. On morphisms, the functor (73) assigns the
inclusion of Clifford submodules
$\mathcal{H}^{<\lambda_{i_{l}}}\subset\mathcal{H}^{<\lambda_{k}}\subset\mathcal{H}_{n}$,
where on the orthogonal complement to the inclusion the operator $D$ is used
to construct a ${\rm Cl}_{-1}$-action as follows. First decompose the
orthogonal complement to the inclusion into $\mu$-eigenspaces for the operator
$D^{2}$,

$(\mathcal{H}^{<\lambda_{i_{l}}})^{\perp}\simeq\bigoplus_{\lambda_{i_{l}}<\mu<\lambda_{k}}\mathcal{H}_{\mu}.$

The sum is finite because $D^{2}$ has compact resolvent and is nonnegative,
and the eigenspaces are orthogonal because $D^{2}$ is self-adjoint. Now define
an odd operator

$e\colon(\mathcal{H}^{<\lambda_{i_{l}}})^{\perp}\to(\mathcal{H}^{<\lambda_{i_{l}}})^{\perp},\qquad
e:=\bigoplus_{\mu}(\frac{1}{\sqrt{\mu}}D|_{\mathcal{H}_{\mu}}).$

By construction, $e^{2}=+1$, and hence determines a ${\rm Cl}_{-1}$-action
(see Remark 4.14). This action is self-adjoint and graded commutes with the
${\rm Cl}_{n}$-action because $D$ is self-adjoint and ${\rm Cl}_{n}$-linear.

######  Lemma 4.18.

The continuous functor (73) induces a weak equivalence

(74) $\displaystyle{\rm{B}}{\rm Ind}\colon{\rm{B}}\check{{\rm C}}({\rm
Inf}_{n,\lambda})\stackrel{{\scriptstyle\sim}}{{\to}}{\rm{B}}{\mathcal{M}}_{n},$

and hence ${\rm{B}}{\mathcal{M}}_{n}$ represents the functor ${\rm
KO}^{n}(-)$.

###### Proof.

In [HST10, page 53], there is a topological category denoted ${\sf D}_{n}$
with the same objects as ${\mathcal{M}}_{n}$ and whose morphisms are denoted

${\rm Mor}_{{\sf D}_{n}}(W_{0},W_{1})=\left\\{\begin{array}[]{cl}\\{R\in
O_{C_{n}}(W_{2}^{\perp})\mid R\ {\rm odd}\ R^{2}=1\\}&{\rm if}\ W_{0}\subset
W_{1}\\\ \emptyset&{\rm else.}\end{array}\right.$

Since $R$ is orthogonal and $R^{2}=1$, it generates a self-adjoint action of
${\rm Cl}_{-1}$. From this we observe that ${\sf D}_{n}$ and
${\mathcal{M}}_{n}$ are isomorphic as topological categories: their spaces of
objects and morphisms are homeomorphic. Then [HST10, Proposition 3.11,
Proposition 4.6, and Theorem 7.1] construct a homotopy equivalence
${\rm{B}}{\mathcal{M}}_{n}\stackrel{{\scriptstyle\sim}}{{\to}}{\rm Inf}_{-n}$
that sits in the homotopy commutative triangle,

where the weak equivalence ${\rm{B}}\check{C}({\rm Inf}_{n,\lambda})\to{\rm
Inf}_{-n}$ follows from (63), and homotopy commutativity follows from the
argument in [HST10, Proposition 4.6]. By the 2-out-of-3 property, this shows
that the map (74) is a weak equivalence. The second statement in the lemma
then follows from Theorem 4.7. ∎

######  Remark 4.19.

The fact that ${\rm{B}}{\mathcal{M}}_{n}$ represents ${\rm KO}^{n}(-)$ follows
directly from [HST10, Theorem 7.1]. An independent proof is given in [Che07,
§1.3-1.4], using a comparison between ${\rm{B}}{\mathcal{M}}_{n}$ and Clifford
linear Fredholm operators. The only new content in Lemma 4.18 is the
observation that the index map implements this weak equivalence.

######  Remark 4.20.

We revisit the Atiyah–Bott–Shapiro description of compactly supported ${\rm
KO}$-class $[V,Q]\in{\rm KO}_{c}^{0}(M)$ from (54). Choose an open cover
$U_{0},U_{1}$ of $M$ with the property that the closure of $U_{0}$ is a
compact subset of $M$ and $Q|_{U_{1}}$ is invertible. Next choose an embedding
$V|_{U_{0}}\subset U_{0}\times\mathcal{H}_{0}$ as a summand of the trivial
Hilbert bundle; such an embedding is unique up to a contractible space of
choices by [Kui65]. Next consider the constant map $U_{1}\to{\mathcal{M}}_{0}$
to the basepoint of ${\mathcal{M}}_{0}$, i.e., the zero module
$\\{0\\}\subset\mathcal{H}_{n}$. On the overlap $U_{0}\bigcap U_{1}$, the odd
operator $Q$ determines a continuous family of morphisms in
${\mathcal{M}}_{0}$ between $V|_{U_{0}}\subset U_{0}\times\mathcal{H}_{0}$ and
the zero object. All together, we obtain a continuous functor
$(V,Q)\colon{\rm{B}}\check{C}(U_{i})\to{\mathcal{M}}_{0}$ of topological
categories. After realization, this determines a map

$M\simeq{\rm{B}}\check{C}(U_{i})\to{\rm{B}}{\mathcal{M}}_{0}$

representing the compactly supported class $[V,Q]\in{\rm KO}_{c}^{0}(M)$; see
Proposition 5.12 and Corollary 5.15.

##  5\. ${\rm KO}$-classes from ${\rm Cl}_{n}$-bundles

In this section we define categories of ${\rm Cl}_{n}$-bundles over a smooth
manifold $M$ that provides a cocycle model for ${\rm KO}$-theory and recovers
the Atiyah–Bott–Shapiro category when $M={\rm pt}$. All super vector bundles
are assumed to be finite-rank. A _metric_ on a super vector bundle is a metric
on its underlying (ungraded) vector bundle such that the even and odd
subspaces in each fiber are orthogonal.

###  5.1. ${\rm Cl}_{n}$-bundles

We begin with a naive candidate for the category of ${\rm Cl}_{n}$-bundles.

######  Definition 5.1.

A _basic ${\rm Cl}_{n}$-bundle_ over $M$ is a metrized super vector bundle
$V\to M$ with ${\rm Cl}_{n}$-action for which the fibers are self-adjoint left
${\rm Cl}_{n}$-modules. A _stable isomorphism_ between basic ${\rm
Cl}_{n}$-bundles is a ${\rm Cl}_{n}$-equivariant inclusion $V\hookrightarrow
V^{\prime}$ over $M$ together with the data of a self-adjoint ${\rm
Cl}_{n}\otimes{\rm Cl}_{-1}$-action on $V^{\perp}\subset V^{\prime}$ extending
the given ${\rm Cl}_{n}$-action.

Definition 5.1 recovers a category equivalent to the (discrete)
Atiyah–Bott–Shapiro category when $M={\rm pt}$. Furthermore, a ${\rm
Cl}_{n}$-bundle in the above sense determines a map
$M\to{\rm{B}}{\mathcal{M}}_{n}$; see the proof of Proposition 5.12. However,
there are continuous maps $M\to{\rm{B}}{\mathcal{M}}_{n}$ that are not
realized by a basic ${\rm Cl}_{n}$-bundle, and hence ${\rm KO}$-classes that
do not admit a cocycle description using Definition 5.1. The easiest such
example is a map $S^{1}\to{\rm{B}}{\mathcal{M}}_{1}$ whose class in
$\widetilde{{\rm KO}}{}^{1}(S^{1})\simeq{\mathbb{Z}}$ is nontrivial. The
following definition repairs this defect. We use the notation from Example 4.8
for open subsets of a manifold $M$ and their intersections.

######  Definition 5.2.

A _${\rm Cl}_{n}$ -bundle_ over a manifold $M$ is data:

  1. (1)

an open cover $\\{U_{i}\\}_{i\in I}$ of $M$;

  2. (2)

for each nonempty finite subset $\sigma\subset I$, a metrized super vector
bundle $V_{\sigma}\to U_{\sigma}$ with a ${\rm Cl}_{n}$-action on $V_{\sigma}$
for which the fibers are self-adjoint left ${\rm Cl}_{n}$-modules;

  3. (3)

for each inclusion of nonempty finite subsets $\sigma\subset\tau\subset I$, a
${\rm Cl}_{n}$-equivariant, injective map of metrized super vector bundles
$g_{\sigma\tau}\colon V_{\sigma}|_{U_{\tau}}\hookrightarrow V_{\tau}$ over
$U_{\tau}$, together with a ${\rm Cl}_{n}\otimes{\rm Cl}_{-1}$-action on the
orthogonal complement of the image $V_{\sigma}^{\perp}\subset V_{\tau}$ that
extends the ${\rm Cl}_{n}$-action.

For $\sigma\subset\tau\subset\rho$, we require an equality of maps
$g_{\sigma\tau}\circ g_{\tau\rho}=g_{\sigma\rho}\colon
V_{\sigma}|_{U_{\rho}}\hookrightarrow V_{\rho}$, and we require the ${\rm
Cl}_{-1}$-action on $V_{\sigma}^{\perp}\subset V_{\tau}\hookrightarrow
V_{\rho}$ restricts from the ${\rm Cl}_{-1}$-action on
$V_{\tau}^{\perp}\subset V_{\rho}$. We use the notation
$e_{\sigma\tau}\in\Gamma(U_{\tau},{\rm End}(V_{\tau}^{\perp})_{\rm odd})$ for
the generator of the ${\rm Cl}_{-1}$-action on $V_{\sigma}^{\perp}\subset
V_{\tau}$ and $V=\\{V_{\sigma},g_{\sigma\tau},e_{\sigma\tau}\\}$ when
referring to a ${\rm Cl}_{n}$-bundle over $M$.

######  Example 5.3.

Real vector bundles are examples of ${\rm Cl}_{0}$-modules. However, a ${\rm
Cl}_{0}$-bundle need not be a vector bundle. For example, for a ${\rm
Cl}_{0}$-bundle defined relative to an open cover $U_{0},U_{1}\subset M$,
Definition 5.2 requires data

(75) $\displaystyle V_{0}\to U_{0},\ \ V_{1}\to U_{1},\qquad
V_{0}|_{U_{01}}\hookrightarrow V_{01}\hookleftarrow V_{1}|_{U_{01}}$

for real vector bundles $V_{0}$ and $V_{1}$, and a stable equivalences over
$U_{01}=U_{0}\bigcap U_{1}$. Vector bundles $V_{0}$ and $V_{1}$ that are
stably isomorphic over $U_{01}$ but have different ranks therefore determine a
${\rm Cl}_{0}$-bundle that is not a vector bundle.

######  Definition 5.4.

For a smooth map $\phi\colon N\to M$ and a ${\rm Cl}_{n}$-bundle
$V=\\{V_{\sigma},g_{\sigma\tau},e_{\sigma\tau}\\}$, the _pullback_
$\phi^{*}V:=\\{\phi^{*}V_{\sigma},\phi^{*}g_{\sigma\tau},\phi^{*}e_{\sigma\tau}\\}$
is a ${\rm Cl}_{n}$-bundle on $N$ defined relative to the open cover
$\\{\phi^{-1}(U_{i})\\}$. In particular, ${\rm Cl}_{n}$-bundles can be
restricted to submanifolds $N\subset M$.

We require the following versions of equivalence between ${\rm
Cl}_{n}$-modules. Each of these generalizes a standard notion of equivalence
between vector bundles; see Example 5.8.

######  Definition 5.5.

An _isomorphism_ between ${\rm Cl}_{n}$-bundles
$V=\\{V_{\sigma},g_{\sigma\tau},e_{\sigma\tau}\\}$ and
$V^{\prime}=\\{V^{\prime}_{\alpha},g_{\alpha\beta}^{\prime},e_{\alpha\beta}^{\prime}\\}$
defined relative to open covers $\\{U_{i}\\}_{i\in I}$ and
$\\{U_{i}^{\prime}\\}_{i\in I^{\prime}}$, respectively, consists of ${\rm
Cl}_{n}$-equivariant metrized vector bundles isomorphisms
$\varphi_{\alpha\sigma}\colon
V_{\sigma}|_{U_{\alpha}^{\prime}}\stackrel{{\scriptstyle\sim}}{{\to}}V_{\alpha}|_{U_{\sigma}}$
for $\sigma\subset I$ and $\alpha\subset I^{\prime}$ with
$U_{\alpha}^{\prime}\bigcap U_{\sigma}$ nonempty. The isomorphisms
$\varphi_{\alpha\sigma}$ are required to be compatible with the restrictions
of
$g_{\sigma\tau},g^{\prime}_{\alpha\beta},e_{\sigma\tau},e_{\alpha\beta}^{\prime}$:
for $\sigma\subset\tau$ and $\alpha\subset\beta$, the above data are required
to fit into a commuting diagram

(76)

and the ${\rm Cl}_{-1}$-actions on orthogonal complements of the images are
required to be compatible. We write $\varphi\colon
V\stackrel{{\scriptstyle\sim}}{{\to}}V^{\prime}$ for an isomorphism of ${\rm
Cl}_{n}$-bundles.

######  Definition 5.6.

Using the notation from Definition 5.5, a _stable equivalence_ between ${\rm
Cl}_{n}$-bundles $V$ and $V^{\prime}$ is the data of ${\rm
Cl}_{n}$-equivariant isometric inclusion $\varphi_{\alpha\sigma}\colon
V_{\sigma}|_{U_{\alpha}^{\prime}}\hookrightarrow
V^{\prime}_{\alpha}|_{U_{\sigma}}$ with a ${\rm Cl}_{n}\otimes{\rm
Cl}_{-1}$-action on $V_{\sigma}^{\perp}\subset
V^{\prime}_{\alpha}|_{U_{\sigma}}$ for each $\sigma\subset I$ and
$\alpha\subset I^{\prime}$ with $U_{\sigma}\bigcap
U^{\prime}_{\alpha}\neq\emptyset$. These data are required to satisfy
compatibility properties analogous to (76).

######  Definition 5.7.

A _concordance_ between ${\rm Cl}_{n}$-bundles $V$ and $V^{\prime}$ on $M$ is
a ${\rm Cl}_{n}$-bundle $\widetilde{V}$ on $M\times{\mathbb{R}}$ together with
isomorphisms $i_{0}^{*}\widetilde{V}\cong V$ and $i_{1}^{*}\widetilde{V}\cong
V^{\prime}$, where $i_{0},i_{1}\colon M\hookrightarrow M\times{\mathbb{R}}$
are the inclusions at $0$ and $1$ respectively. When such a concordance
exists, we say that $V$ and $V^{\prime}$ are _concordant._ Concordance defines
an equivalence relation on isomorphism classes of ${\rm Cl}_{n}$-bundles over
$M$ whose equivalence classes are called _concordance classes_.

######  Example 5.8.

For vector bundles $V,V^{\prime}\to M$ regarded as ${\rm Cl}_{0}$-modules, an
isomorphism in the sense of Definition 5.5 is the same data as an isomorphism
of vector bundles. If vector bundles $V,V^{\prime}\to M$ are stably equivalent
${\rm Cl}_{0}$-bundles in the sense of Definition 5.6, then
$[V]=[V^{\prime}]\in{\rm KO}^{0}(M)$; indeed,
$[V]-[V^{\prime}]=[V^{\perp}]\in{\rm KO}^{0}(M)$ and the existence of an odd
isomorphism $e\colon V^{\perp}_{\pm}\to V_{\mp}^{\perp}$ implies
$[V^{\perp}]=[V^{\perp}_{\rm ev}]-[V^{\perp}_{\rm odd}]=0$. In fact, a stable
equivalence determines a concordance between $V$ and $V^{\prime}$ as ${\rm
Cl}_{0}$-modules, as constructed in the proof of the following.

######  Lemma 5.9.

If ${\rm Cl}_{n}$-bundles over $M$ are stable equivalent, then they are
concordant.

###### Proof.

Let $V$ and $V^{\prime}$ be stable equivalent ${\rm Cl}_{n}$-bundles over $M$.
By refining the cover, we can assume that ${\rm Cl}_{n}$-bundles $V$,
$V^{\prime}$, and the stable equivalence between them are defined subordinate
to a fixed cover $\\{U_{i}\\}$ of $M$. So let
$V=\\{V_{\sigma},g_{\sigma\tau},e_{\sigma\tau}\\}$ and
$V^{\prime}=\\{V^{\prime}_{\sigma},g_{\sigma\tau}^{\prime},e_{\sigma\tau}^{\prime}\\}$.
Consider the open cover of ${\mathbb{R}}$ given by
$\\{(-\infty,1),(0,\infty)\\}$. The concordance $\tilde{V}$ is defined via the
basic ${\rm Cl}_{n}$-modules,

$V_{\sigma}\times(-\infty,1)\to U_{\sigma}\times(-\infty,1),\quad
V_{\sigma}^{\prime}\times(0,1)\to U_{\sigma}\times(0,1),\quad
V^{\prime}_{\sigma}\times(0,\infty)\to U_{\sigma}\times(0,\infty).$

For inclusions $U_{\sigma}\times J\hookrightarrow U_{\tau}\times J$ with fixed
$J=(-\infty,1)$, $(0,\infty)$ or $(0,1)$, we have the evident data
$(g_{\sigma\tau},e_{\sigma\tau})$ or
$(g_{\sigma\tau}^{\prime},e_{\sigma\tau}^{\prime})$ from $V$ or $V^{\prime}$.
It remains to specify the compatibility data (3) in Definition 5.2 for the
inclusions

$U_{\sigma}\times(-\infty,1)\hookleftarrow
U_{\sigma}\times(0,1)\hookrightarrow U_{\sigma}\times(0,\infty).$

For the right inclusion, the bundles are related by restriction, so the
orthogonal complements are the zero bundle and there is no additional data to
specify. For the left inclusion, we use the stable equivalence datum for
$\varphi_{\sigma}\colon V_{\sigma}\times(0,1)\hookrightarrow
V_{\sigma}^{\prime}\times(0,1)$ to specify the compatibility data. This
completes the construction of the concordance. ∎

######  Remark 5.10.

By composing concordances, a zig-zag of stable equivalences also determines a
concordance between ${\rm Cl}_{n}$-bundles.

######  Definition 5.11.

Let $V=\\{V_{\sigma},g_{\sigma\tau},e_{\sigma\tau}\\}$ be a ${\rm
Cl}_{n}$-bundle on $M$ defined relative to an open cover $\\{U_{i}\\}$, and
let
$V^{\prime}=\\{V^{\prime}_{\alpha},g_{\alpha\beta}^{\prime},e_{\alpha\beta}^{\prime}\\}$
be a ${\rm Cl}_{m}$-bundle on $M^{\prime}$ defined relative to an open cover
$\\{U_{k}^{\prime}\\}$. Define the following addition and multiplication
operations on ${\rm Cl}_{n}$-bundles.

  1. (1)

If $n=m$, the _external direct sum_ is a ${\rm Cl}_{n}$-bundle $V\boxplus
V^{\prime}:=\\{V_{\sigma}\boxplus V_{\alpha}^{\prime},g_{\sigma\tau}\boxplus
g_{\alpha\beta}^{\prime},e_{\sigma\tau}\boxplus e_{\alpha\beta}^{\prime}\\}$
on $M\times M^{\prime}$ defined relative to the product open cover
$\\{U_{i}\times U_{k}^{\prime}\\}$, where $V_{\sigma}\boxplus
V_{\alpha}^{\prime}$ denotes the ${\mathbb{Z}}/2$-graded external direct sum
of super vector bundles over $U_{\sigma}\times U^{\prime}_{\alpha}$.

  2. (2)

The _external tensor product_ is a ${\rm Cl}_{n}\otimes{\rm Cl}_{m}$-bundle
$V\boxtimes V^{\prime}=\\{V_{\sigma}\boxtimes
V_{\alpha}^{\prime},g_{\sigma\tau}\boxtimes
g_{\alpha\beta}^{\prime},e_{\sigma\tau}\boxtimes e_{\alpha\beta}^{\prime}\\}$
on $M\times M^{\prime}$ defined relative to the open cover $\\{U_{i}\times
U_{k}^{\prime}\\}$, where $V_{\sigma}\boxtimes V_{\alpha}^{\prime}$ denotes
the ${\mathbb{Z}}/2$-graded external tensor product of super vector bundles
over $U_{\sigma}\times U^{\prime}_{\alpha}$.

Direct sum and tensor product operations for ${\rm Cl}_{n}$-bundles on $M$ are
defined by pulling back the external direct sum and external tensor product
along the diagonal $M\hookrightarrow M\times M$.

###  5.2. From ${\rm Cl}_{n}$-bundles to ${\rm KO}^{n}$-classes

######  Proposition 5.12.

There is a surjective map from ${\rm Cl}_{n}$-bundles over $M$ to ${\rm
KO}^{n}(M)$ that reduces to the Atiyah–Bott–Shapiro map (57) when $M={\rm
pt}$. A pair of ${\rm Cl}_{n}$-bundles determine the same class in ${\rm
KO}^{n}(M)$ if and only if they are concordant. The external product and
external sum of ${\rm Cl}_{n}$-bundles is compatible with the external product
and external sum in ${\rm KO}$.

###### Proof.

Given a ${\rm Cl}_{n}$-bundle $V$ over $M$ defined relative to an open cover
$\\{U_{i}\\}_{i\in I}$, consider the topological category $\check{C}(U_{i})$
from Example 4.8. The data in Definition 5.2 determines a map
$\check{C}(U_{i})_{0}\to({\mathcal{M}}_{n})_{0}$ to the space of objects in
the $n$th Atiyah–Bott–Shapiro category that is unique up to contractible
choice, defined as follows. Choose ${\rm Cl}_{n}$-equivariant embeddings
$\epsilon_{\sigma}\colon V_{\sigma}\subset\mathcal{H}_{n}\times U_{\sigma}$
over $U_{\sigma}$ for all $\sigma$. By [Kui65], the space of embeddings is
contractible, so these choices are indeed unique up to contractible choice.
Further specify homotopies between the embeddings

$\displaystyle\mathcal{H}_{n}\times
U_{\tau}\stackrel{{\scriptstyle\epsilon_{\sigma}|_{U_{\tau}}}}{{\hookleftarrow}}V_{\sigma}|_{U_{\tau}}\stackrel{{\scriptstyle
g_{\sigma\tau}}}{{\hookrightarrow}}V_{\tau}\stackrel{{\scriptstyle\epsilon_{\tau}}}{{\hookrightarrow}}\mathcal{H}_{n}\times
U_{\tau}$

and higher homotopies between nested inclusions (again by [Kui65]). This
results in a map between realizations that is unique up to contractible choice

(77) $\displaystyle
M\simeq|N_{\bullet}\check{C}(U_{i})^{\sf{op}}|\to|N_{\bullet}{\mathcal{M}}_{n}|={\rm{B}}{\mathcal{M}}_{n}.$

By Lemma 4.18, this determines a class $[V]\in{\rm KO}^{n}(M)$. By the nerve
theorem [Bor48], any continuous map $M\to{\rm{B}}{\mathcal{M}}_{n}$ is
homotopic to the realization of a map
$N_{\bullet}\check{C}(U_{i})^{\sf{op}}\to N_{\bullet}{\mathcal{M}}_{n}$ where
$\\{U_{i}\\}$ is taken to be a good open cover; hence by Theorem 4.18 any
class in ${\rm KO}^{n}(M)$ can be represented by a ${\rm Cl}_{n}$-bundle.
Concordant ${\rm Cl}_{n}$-bundles determine homotopic maps to
${\rm{B}}{\mathcal{M}}_{n}$ by construction, and therefore represent the same
${\rm KO}$-class. Choosing a good open cover of $M\times{\mathbb{R}}$,
homotopic maps can be lifted to concordant ${\rm Cl}_{n}$-bundles.

The compatibility with external tensor product follows from Theorem 4.18
together with the fact that realization commutes with finite products;
explicitly, the map (77) associated with an external tensor product of ${\rm
Cl}_{n}$\- and ${\rm Cl}_{m}$-modules is homotopic to the composition

$\displaystyle M\times M^{\prime}$ $\displaystyle\simeq$
$\displaystyle|N_{\bullet}\check{C}(U_{i}\times
U_{j}^{\prime})^{\sf{op}}|\simeq|N_{\bullet}\check{C}(U_{i})^{\sf{op}}\times
N_{\bullet}\check{C}(U_{i}^{\prime})^{\sf{op}}|\simeq|N_{\bullet}\check{C}(U_{i})^{\sf{op}}|\times|N_{\bullet}\check{C}(U_{i}^{\prime})^{\sf{op}}|$
$\displaystyle\to$
$\displaystyle{\rm{B}}{\mathcal{M}}_{n}\times{\rm{B}}{\mathcal{M}}_{m}\to{\rm{B}}{\mathcal{M}}_{n+m},$

where the first arrow is determined by the maps (77) associated with the
constituent ${\rm Cl}_{n}$\- and ${\rm Cl}_{m}$-bundles, and the second arrow
is the realization of (71). Hence, this agrees with the product map ${\rm
KO}^{n}(M)\times{\rm KO}^{m}(M^{\prime})\to{\rm KO}^{n+m}(M\times
M^{\prime})$.

The external sum is handled similarly, up to one subtlety coming from the fact
that the sum operation on ${\mathcal{M}}_{n}$ is only partially defined (see
Remark 4.17). To ensure that the desired sums are defined, when choosing the
maps to ${\rm{B}}{\mathcal{M}}_{n}$ associated with ${\rm Cl}_{n}$-bundles $V$
and $V^{\prime}$, one chooses subspaces
$V_{\sigma}\subset\mathcal{H}_{n}\times U_{\sigma}$ and
$V_{\alpha}^{\prime}\subset\mathcal{H}_{n}\times U_{\alpha}$ that are
orthogonal to one another. Then the collection of subspaces
$V_{\sigma}\boxplus V_{\alpha}^{\prime}\subset{\rm H}_{n}\times
U_{\sigma}\times U_{\alpha}^{\prime}$ with the obvious compatibility data
determines a map $M\times M^{\prime}\to{\rm{B}}{\mathcal{M}}_{n}$ associated
with the ${\rm Cl}_{n}$-bundle $V\boxplus V^{\prime}$. ∎

For a ${\rm Cl}_{n}$-bundle $V$ over $M$, we use the notation $[V]\in{\rm
KO}^{n}(M)$ to denote the associated ${\rm KO}$-class from Proposition 5.12.

######  Corollary 5.13.

If ${\rm Cl}_{n}$-bundles $V$ and $V^{\prime}$ over $M$ are stable equivalent,
then $[V]=[V^{\prime}]\in{\rm KO}^{n}(M)$.

###### Proof.

This follows immediately from Lemma 5.9. ∎

###  5.3. A representative of the suspension class

######  Definition 5.14.

A ${\rm Cl}_{n}$-bundle $V$ on $M$ has _compact support_ if on the complement
of a compact subset $K\subset M$, $V$ has the zero ${\rm Cl}_{n}$-bundle as
its underlying super vector bundle. Similarly, for a fiber bundle $\pi\colon
M\to N$, a ${\rm Cl}_{n}$-bundle $V$ on $M$ has _compact vertical support_ if
for each $y\in N$, there exists a neighborhood $U\subset N$ such that $V$ has
compact support on $\pi^{-1}(U)$.

######  Corollary 5.15.

There is a surjective map from compactly supported ${\rm Cl}_{n}$-bundles over
$M$ to ${\rm KO}^{n}_{c}(M)$. A pair of ${\rm Cl}_{n}$-bundles determine the
same class in ${\rm KO}^{n}_{c}(M)$ if and only if they are concordant. The
external product of ${\rm Cl}_{n}$-bundles is compatible with the external
product in compactly supported ${\rm KO}$.

###### Proof.

This follows from the fact that the zero ${\rm Cl}_{n}$-bundle uniquely
determines the constant map to basepoint of ${\rm{B}}{\mathcal{M}}_{n}$ under
(77). ∎

######  Definition 5.16.

The _suspension class_ is a choice of generator ${\rm
KO}^{1}_{c}({\mathbb{R}})\simeq\widetilde{{\rm
KO}}{}^{1}(S^{1})\simeq{\mathbb{Z}}$.

######  Lemma 5.17.

The suspension class is represented by the following ${\rm Cl}_{1}$-bundle.
Consider

(80) $\displaystyle\begin{array}[]{c}\\{U_{i}\\}=\\{(-1,1),\
{\mathbb{R}}\setminus\\{0\\}\\}\\\ V_{{\mathbb{R}}\setminus\\{0\\}}={\bf
0},\quad V_{(-1,1)}=\underline{{\rm Cl}}_{1}\end{array}$

where ${\rm Cl}_{1}\simeq{\mathbb{R}}^{1|1}$ is a left module over itself,
together with the stable isomorphism ${\bf 0}\hookrightarrow\underline{{\rm
Cl}}_{1}$ over $(-1,0)\coprod(0,1)$ from the ${\rm Cl}_{1}\otimes{\rm
Cl}_{-1}$-action on ${\mathbb{R}}^{1|1}$ from Example 3.9.

###### Proof.

Let $\sigma_{\mathbb{R}}$ denote the ${\rm Cl}_{1}$-bundle (80), and
$[\sigma_{\mathbb{R}}]\in{\rm KO}^{1}_{c}({\mathbb{R}})$ the corresponding
${\rm KO}$-class using Corollary 5.15. The open cover
$\\{(-\infty,0),(-1,1),(0,\infty)\\}$ is good, so (again by Corollary 5.15)
the suspension class can be expressed in terms of basic ${\rm Cl}_{n}$-bundles
on this cover together with stable isomorphisms on overlaps. The compact
support condition requires that we take the zero module on $(-\infty,0)$ and
$(0,\infty)$. On $(-1,1)$, a ${\rm Cl}_{1}$-module must be the trivial bundle
with fiber the super vector space
${\mathbb{R}}^{k|k}\simeq({\mathbb{R}}^{1|1})^{\oplus k}$ for some
$k\in{\mathbb{N}}$; this is because $f\in{\rm Cl}_{1}$ gives an isomorphism
between the even and odd subspaces of the bundle of ${\rm Cl}_{1}$-modules. On
$(-1,0)$ and $(0,1)$, we require a ${\rm Cl}_{1}\otimes{\rm Cl}_{-1}\simeq{\rm
Cl}_{1,1}$-module structure on ${\mathbb{R}}^{k|k}$. Up to parity reversal,
${\mathbb{R}}^{1|1}$ with ${\rm Cl}_{1,1}$-module structure from Example 3.9
is the unique irreducible ${\rm Cl}_{1,1}$-module. We conclude that any class
in ${\rm KO}^{1}_{c}({\mathbb{R}})$ is of the form
$[(\sigma_{\mathbb{R}})^{\oplus k}]$ for some $k\in{\mathbb{Z}}$, where
negative $k$ takes the parity reversed module. Hence, $[\sigma_{\mathbb{R}}]$
is a generator of ${\rm KO}^{1}_{c}({\mathbb{R}})\simeq{\mathbb{Z}}$, and
therefore a representative of the suspension class. ∎

##  6\. The Pontryagin class of a ${\rm Cl}_{n}$-bundle as a Čech–de Rham
cocycle

For an ordinary vector bundle $V\to M$, a choice of connection $\nabla$ on $V$
determines a closed differential form that refines the Pontryagin character
${\rm Ch}(V)\in{\rm H}(M;{\mathbb{R}}[u^{\pm 1}])$ of $[V]\in{\rm KO}^{0}(M)$,

${\sf sTr}(e^{-u\nabla^{2}})\in\Omega^{\bullet}(M;{\mathbb{R}}[u^{\pm
1}]),\qquad[{\sf sTr}(e^{-u\nabla^{2}})]={\rm Ch}(V)\in{\rm
H}(M;{\mathbb{R}}[u^{\pm 1}]).$

In this section we construct an analogous refinement for ${\rm
Cl}_{n}$-bundles: after choosing connection data, a ${\rm Cl}_{n}$-bundle
$V\to M$ determines a cocycle in the Čech–de Rham complex of $M$ that refines
the Pontryagin character of $[V]$; see Propositions 6.15 and 6.16.

###  6.1. Review of Chern forms of superconnections

For a real super vector bundle $V\to M$, let $V_{\rm ev}\subset V$ and $V_{\rm
odd}\subset V$ denote its even and odd subbundles, respectively. Consider the
space of $V$-valued differential forms
$\Omega^{\bullet}(M;V)=\Gamma(M;\Lambda^{\bullet}(T^{*}M)\otimes V)$ as a
super vector space.

######  Definition 6.1 ([Qui85]).

A _superconnection_ ${\mathbb{A}}$ on $V$ is an odd ${\mathbb{R}}$-linear map

(81)
$\displaystyle{\mathbb{A}}\colon\Omega^{\bullet}(M;V)\to\Omega^{\bullet}(M;V),\quad{\mathbb{A}}(\alpha\cdot
v)=d\alpha\cdot v+(-1)^{|\alpha|}\alpha\cdot{\mathbb{A}}v$

satisfying the indicated Leibniz rule, where $v\in\Omega^{\bullet}(M;V)$ and
$\alpha\in\Omega^{\bullet}(M)$.

On an $n$-manifold, one can express a superconnection as sum

(82) $\displaystyle{\mathbb{A}}=\sum_{j=0}^{n}{\mathbb{A}}_{(j)},$

where ${\mathbb{A}}_{(1)}=\nabla$ is an ordinary connection on $V$,
${\mathbb{A}}_{(2j)}\in\Omega^{2j}(M;{\rm End}(V)_{\rm odd})$ are valued in
odd endomorphisms, and ${\mathbb{A}}_{(2j+1)}\in\Omega^{2j+1}(M;{\rm
End}(V)_{\rm ev})$ are valued in even endomorphisms.

######  Definition 6.2 ([BGV92], page 117).

Let $V\to M$ be a metrized real super vector bundle equipped with a
superconnection ${\mathbb{A}}$. The _adjoint_ superconnection is

(83)
$\displaystyle{\mathbb{A}}^{*}=\sum_{j}(-1)^{j(j+1)/2}{\mathbb{A}}_{(j)}^{\dagger}$

where for $j\neq 1,$ ${\mathbb{A}}_{(j)}^{\dagger}$ is the adjoint of an
endomorphism-valued form using the metric on $V$, and
${\mathbb{A}}_{(1)}^{\dagger}=\nabla^{\dagger}$ is the adjoint of the ordinary
connection characterized by

$d\langle v,w\rangle=\langle\nabla v,w\rangle+\langle
v,\nabla^{\dagger}w\rangle.$

A superconnection is _self-adjoint_ if ${\mathbb{A}}^{*}={\mathbb{A}}$.

######  Remark 6.3.

Self-adjointness is equivalent to ${\mathbb{A}}_{(j)}$ being self-adjoint in
degrees $j=4k,4k+3$, and skew-adjoint in degrees $j=4k+1,4k+2$, $j\neq 1$.

For $u^{1/2}$ a formal variable of degree $-1$, define

(84)
$\displaystyle{\mathbb{A}}(u):=u^{-1/2}{\mathbb{A}}_{(0)}+{\mathbb{A}}_{(1)}+u^{1/2}{\mathbb{A}}_{(2)}+\dots+u^{j/2}{\mathbb{A}}_{(j)}+\dots\quad$

Then the _Pontryagin form of ${\mathbb{A}}$_ is the differential form

(85) $\displaystyle{\rm Ch}({\mathbb{A}}):={\sf
sTr}(e^{-u{\mathbb{A}}(u)^{2}})\in\Omega^{\bullet}(M;{\mathbb{R}}[u,u^{-1}])$

of total degree 0, representing the Pontryagin character of $V$. The absence
of fractional powers of $u$ in ${\rm Ch}({\mathbb{A}})$ follows from the fact
that the super trace of an odd endomorphism is zero. Using Remark 6.3, self-
adjointness further implies ${\rm Ch}({\mathbb{A}})$ takes values in
differential forms in degrees 0 mod 4; see Lemma 6.10 below. The equalities of
differential forms

(86) $\displaystyle{\rm Ch}({\mathbb{A}}\oplus{\mathbb{A}}^{\prime})={\rm
Ch}({\mathbb{A}})+{\rm Ch}({\mathbb{A}}^{\prime}),\quad{\rm
Ch}({\mathbb{A}}\otimes{\mathbb{A}}^{\prime})={\rm Ch}({\mathbb{A}})\cdot{\rm
Ch}({\mathbb{A}}^{\prime})\in\Omega^{\bullet}(M;{\mathbb{R}}[u,u^{-1}])$

refine the direct sum and tensor product formulas for the Pontryagin character
in cohomology.

######  Definition 6.4.

Suppose we are given super vector bundles and superconnections
$(V_{0},{\mathbb{A}}_{0})$ and $(V_{1},{\mathbb{A}}_{1})$ over $M$. A
_concordance_ between the superconnections ${\mathbb{A}}_{0}$ and
${\mathbb{A}}_{1}$ is a superconnection $\tilde{{\mathbb{A}}}$ on a super
vector bundle $\tilde{V}\to M\times{\mathbb{R}}$ with the data of isomorphisms
$i_{0}^{*}\tilde{V}\simeq V_{0}$ and $i_{1}^{*}\tilde{V}\simeq V_{1}$ and the
property that

$i_{0}^{*}\tilde{{\mathbb{A}}}={\mathbb{A}}_{0},\qquad
i_{1}^{*}\tilde{{\mathbb{A}}}={\mathbb{A}}_{1}$

for inclusions $i_{j}\colon M\times\\{j\\}\hookrightarrow
M\times{\mathbb{R}}$, $j=0,1$.

######  Definition 6.5.

The _Chern–Simons form_ of a concordance $\tilde{{\mathbb{A}}}$ of
superconnections is

(87) $\displaystyle{\rm CS}(\tilde{{\mathbb{A}}}):=\int_{[0,1]}{\rm
Ch}(\tilde{{\mathbb{A}}})\in\Omega^{\bullet}(M;{\mathbb{R}}[u,u^{-1}]).$

By Stokes’ theorem, the Chern–Simons form satisfies

(88) $\displaystyle d{\rm CS}(\tilde{{\mathbb{A}}})={\rm
Ch}({\mathbb{A}}_{1})-{\rm Ch}({\mathbb{A}}_{0}).$

######  Remark 6.6.

Modulo exact forms ${\rm CS}(\tilde{{\mathbb{A}}})$ only depends on
${\mathbb{A}}_{1}$ and ${\mathbb{A}}_{0}$ and not on the choice of concordance
$\tilde{{\mathbb{A}}}$.

######  Example 6.7.

For superconnections ${\mathbb{A}}_{0},{\mathbb{A}}_{1}$ on the same super
vector bundle $V\to M$, affine interpolation

(89)
$\displaystyle\tilde{{\mathbb{A}}}=dt\partial_{t}+(1-t){\mathbb{A}}_{0}+t{\mathbb{A}}_{1}\qquad
i_{0}^{*}\tilde{{\mathbb{A}}}={\mathbb{A}}_{0},\ \
i_{1}^{*}\tilde{{\mathbb{A}}}={\mathbb{A}}_{1}$

gives a concordance on $p^{*}V\to M\times{\mathbb{R}}$ for $p\colon
M\times{\mathbb{R}}\to M$ the projection. We use the notation

(90) $\displaystyle{\rm CS}({\mathbb{A}}_{0},{\mathbb{A}}_{1}):={\rm
CS}(\tilde{{\mathbb{A}}})$

for concordances of the form (89).

The existence of a concordance implies that $V_{0}\simeq V_{1}$, so often we
shall implicitly assume that concordant superconnections ${\mathbb{A}}_{0}$
and ${\mathbb{A}}_{1}$ are defined on the same bundle. However, the
concordance need not be of the form (89).

Using (86), Chern–Simons forms have the following behavior under direct sums
and tensor products

(94) $\displaystyle\begin{array}[]{ccl}{\rm
CS}(\tilde{{\mathbb{A}}}\oplus\tilde{{\mathbb{B}}})&=&{\rm
CS}(\tilde{{\mathbb{A}}})+{\rm CS}(\tilde{{\mathbb{B}}})\\\ {\rm
CS}(\tilde{{\mathbb{A}}}\otimes\tilde{{\mathbb{B}}})&=&{\rm
CS}(\tilde{{\mathbb{A}}}){\rm Ch}({\mathbb{B}})+{\rm
Ch}({\mathbb{A}}^{\prime}){\rm CS}(\tilde{{\mathbb{B}}})\mod{\rm image}(d)\\\
&=&{\rm CS}(\tilde{{\mathbb{A}}}){\rm Ch}({\mathbb{B}})+{\rm
Ch}({\mathbb{A}}){\rm CS}(\tilde{{\mathbb{B}}})+d{\rm
CS}(\tilde{{\mathbb{A}}}){\rm CS}(\tilde{{\mathbb{B}}}).\end{array}$

We require a generalization of (87) to $(k+1)$-tuples of concordant
superconnections. First define the affine smooth simplex ${\mathbb{A}{\rm
ff}}^{k}$ and the closed $k$-simplex $\Delta^{k}\subset{\mathbb{A}{\rm
ff}}^{k}$ as

${\mathbb{A}{\rm
ff}}^{k}=\\{(t_{0},t_{1},\dots,t_{k})\in{\mathbb{R}}^{k+1}\mid\sum
t_{i}=1\\},\qquad\Delta^{k}=\\{(t_{0},t_{1},\dots,t_{k})\in{\mathbb{A}{\rm
ff}}^{k}\mid t_{i}\geq 0\\}.$

Then given a super vector bundle $V\to M$ with $k+1$ superconnections
${\mathbb{A}}_{0},{\mathbb{A}}_{1},{\mathbb{A}}_{2},\dots,{\mathbb{A}}_{k}$,
define a superconnection $\tilde{{\mathbb{A}}}$ on $p^{*}V$ by

$\tilde{{\mathbb{A}}}:=\sum_{j=0}^{k}dt_{j}\partial_{t_{j}}+\sum_{j=0}^{k}t_{j}p^{*}{\mathbb{A}}_{j}$

where $p\colon M\times{\mathbb{A}{\rm ff}}^{k}\to M$ is the projection. Then
the differential forms

(95) $\displaystyle{\rm CS}_{k}(\tilde{{\mathbb{A}}})=\int_{\Delta^{k}}{\rm
Ch}(\tilde{{\mathbb{A}}})\in\Omega^{\bullet}(M;{\mathbb{R}}[u^{\pm 1}])$

have total degree $-k$ and (by Stokes Theorem) satisfy

(96) $\displaystyle d{\rm
CS}_{k}(\tilde{{\mathbb{A}}})=\int_{\partial\Delta^{k}}{\rm
Ch}(\tilde{{\mathbb{A}}})=\sum_{j=0}^{k}(-1)^{j}\int_{\Delta^{k-1}}{\rm
Ch}(\tilde{{\mathbb{A}}}_{j})$

where $\tilde{{\mathbb{A}}}_{j}$ is the restriction of $\tilde{{\mathbb{A}}}$
to the $j$th face of ${\mathbb{A}{\rm ff}}^{k}$. When $k=1$, (95) reproduces
the Chern–Simons form (87), where the property (96) reduces to (88).

###  6.2. Clifford linear superconnections

In [Qui85, §5], Quillen uses superconnections to construct a differential form
refinement of the Chern character for complex super vector bundles with
$\mathbb{C}{\rm l}_{1}$-actions as an odd degree counterpart to (85). The
following is an elaboration on Quillen’s ideas that treats real bundles with
${\rm Cl}_{n}$-actions.

######  Definition 6.8.

Let $V\to M$ be a real super vector bundle with fiberwise ${\rm
Cl}_{n}$-action. A superconnection ${\mathbb{A}}$ on $V$ is _${\rm Cl}_{n}$
-linear_ if ${\mathbb{A}}$ graded commutes with the ${\rm Cl}_{n}$-action.

######  Definition 6.9.

Let $V\to M$ be a real, metrized super vector bundle with fiberwise self-
adjoint ${\rm Cl}_{n}$-action. The _Chern form_ of a ${\rm Cl}_{n}$-linear
self-adjoint superconnection ${\mathbb{A}}$ on $V$ is

(97) $\displaystyle{\rm Ch}_{n}({\mathbb{A}}):={\sf
sTr}(u^{-n/2}\Gamma_{n}\circ\exp(-u{\mathbb{A}}(u)^{2}))\in\Omega^{\bullet}(M;{\mathbb{R}}[u^{1/2},u^{-1/2}]),$

using the super trace on ${\rm Cl}_{n}$-modules (48) extended
$\Omega^{\bullet}(M)$-linearly.

######  Lemma 6.10.

For a self-adjoint ${\rm Cl}_{n}$-linear superconnection ${\mathbb{A}}$, the
differential form ${\rm Ch}_{n}({\mathbb{A}})$ lies in the subalgebra
$\Omega^{\bullet}(M;{\mathbb{R}}[\alpha,\alpha])\subset\Omega^{\bullet}(M;{\mathbb{R}}[u^{1/2},u^{-1/2}])$
where $\alpha=2u^{2}$. Furthermore, ${\rm Ch}_{n}({\mathbb{A}})$ is closed and
of total degree $n$.

###### Proof.

The fact that ${\rm
Ch}_{n}({\mathbb{A}})\in\Omega^{\bullet}(M;{\mathbb{R}}[u,u])\subset\Omega^{\bullet}(M;{\mathbb{R}}[u^{1/2},u^{-1/2}])$
follows from the super trace of an odd endomorphism being zero. In more
detail, (82) specifies whether a component of a superconnection takes values
in even or odd endomorphisms; the odd endomorphisms correspond to the half-
integer coefficients in (84). The endomorphism $\Gamma_{n}$ is even or odd
depending on the parity of $n$, and carries a factor of $u^{-n/2}$ from the
definition of the Clifford super trace. Hence, the half-integer powers of $u$
in the expansion of ${\rm Ch}_{n}({\mathbb{A}})$ are coefficients of odd
endomorphisms and therefore sent to zero under the super trace. Self-adjoint
superconnections give a further refinement to even powers of $u$ using that
the super trace of a skew-adjoint operator is zero, see Remark 6.3. This
verifies that ${\rm
Ch}_{n}({\mathbb{A}})\in\Omega^{\bullet}(M;{\mathbb{R}}[\alpha,\alpha])$.

Since $u{\mathbb{A}}(u)^{2}$ has total degree zero and $u^{-n/2}\Gamma_{n}$
has total degree $n$, ${\rm Ch}_{n}({\mathbb{A}})$ has total degree $n$.
Finally, to see that ${\rm Ch}_{n}({\mathbb{A}})$ is closed, one applies the
same argument as in the case ordinary superconnections, e.g., [BGV92,
Proposition 1.41]. The key fact is

(98) $\displaystyle{\sf sTr}_{{\rm Cl}_{n}}([{\mathbb{A}},\omega])=d{\sf
sTr}_{{\rm Cl}_{n}}(\omega).$

This follows from a local computation, writing ${\mathbb{A}}=d+\gamma$ and
using that the Clifford super trace vanishes on super commutators [Qui85,
Proposition 2]. Then the fact that ${\rm Ch}_{n}({\mathbb{A}})$ is closed
follows from (98) and the identity $[{\mathbb{A}},\exp(-{\mathbb{A}}^{2})]=0$.
∎

######  Definition 6.11.

A _concordance_ between self-adjoint ${\rm Cl}_{n}$-linear superconnections
${\mathbb{A}}_{0},{\mathbb{A}}_{1}$ on $V_{0},V_{1}\to M$ is a self-adjoint
${\rm Cl}_{n}$-linear superconnection $\tilde{{\mathbb{A}}}$ on $\tilde{V}\to
M\times{\mathbb{R}}$ together with the data of isomorphisms

$i_{0}^{*}\widetilde{V}\simeq V_{0},\qquad i_{1}^{*}\widetilde{V}\simeq V_{1}$

with the property that $i_{0}^{*}\tilde{{\mathbb{A}}}={\mathbb{A}}_{0}$ and
$i_{1}^{*}\tilde{{\mathbb{A}}}={\mathbb{A}}_{1}$ for inclusions $i_{j}\colon
M\times\\{j\\}\hookrightarrow M\times{\mathbb{R}}$, $j=0,1$.

######  Definition 6.12.

The _Chern–Simons form_ of a concordance between ${\rm Cl}_{n}$-linear self-
adjoint superconnections is a differential form with total degree $n-1$,

(99) $\displaystyle{\rm CS}(\tilde{{\mathbb{A}}}):=\int_{[0,1]}{\rm
Ch}_{n}(\tilde{{\mathbb{A}}})\in\Omega^{\bullet}(M;{\mathbb{R}}[\alpha,\alpha^{-1}]).$

The standard arguments show that the Chern–Simons form satisfies (88), and
only depends on ${\mathbb{A}}_{0}$ and ${\mathbb{A}}_{1}$ up to exact forms.
We have the same formulas (94) for Chern–Simons forms involving direct sums
and tensor products.

######  Lemma 6.13.

Suppose we are given the data of a stable equivalence of ${\rm
Cl}_{n}$-bundles

(100) $\displaystyle V_{0}\hookrightarrow V_{1},\qquad e\in\Gamma(M;{\rm
End}(V^{\perp})_{\rm odd}),\ e^{2}={{\rm id}}$

together with ${\rm Cl}_{n}$-linear self-adjoint superconnections
${\mathbb{A}}_{0}$ on $V_{0}$ and ${\mathbb{A}}_{1}$ on $V_{1}$ (that need not
be compatible with the inclusion $V_{0}\hookrightarrow V_{1}$). Then there is
a uniquely determined self-adjoint ${\rm Cl}_{n}$-linear superconnection
${\mathbb{A}}_{0}^{\prime}$ on $V_{1}$ with the equalities of differential
forms of total degree $n$

(101) $\displaystyle{\rm Ch}_{n}({\mathbb{A}}_{1})-{\rm
Ch}_{n}({\mathbb{A}}_{0})=d{\rm
CS}({\mathbb{A}}_{0}^{\prime},{\mathbb{A}}_{1}),\qquad{\rm
Ch}_{n}({\mathbb{A}}_{0})={\rm
Ch}_{n}({\mathbb{A}}_{0}^{\prime})\in\Omega^{\bullet}(M;{\mathbb{R}}[u^{\pm
1}])$

refining the equality in cohomology ${\rm Ch}(V_{0})={\rm Ch}(V_{1})$ of
Pontryagin characters. In particular, a stable equivalence $V_{0}={\bf
0}\hookrightarrow V_{1}$ with a choice of super connection on $V_{1}$
determines a differential form $\zeta$ satisfying ${\rm
Ch}({\mathbb{A}}_{1})=d\zeta$.

###### Proof.

Let $V_{0}^{\perp}\subset V_{1}$ denote the orthogonal complement to the
inclusion $V_{0}\hookrightarrow V_{1}$. The existence of the ${\rm
Cl}_{n}$-linear endomorphism $e\in\Gamma({\rm End}(V_{0}^{\perp})_{\rm odd})$
with $e^{2}={{\rm id}}_{V_{0}^{\perp}}$ implies that there is a direct sum
decomposition of basic ${\rm Cl}_{n}$-bundles
$V_{0}^{\perp}\simeq(V_{0}^{\perp})_{+}\oplus(V_{0}^{\perp})_{-}$ where
$e\colon(V_{0}^{\perp})_{\pm}\to(V_{0}^{\perp})_{\mp}$ is a grading-reversing
isomorphism that exchanges the summands. Let

$p^{+}\colon V_{1}\to(V_{0}^{\perp})_{+}$

denote the orthogonal projection. Using the isomorphism of basic ${\rm
Cl}_{n}$-bundles,

$V_{1}\simeq V_{0}\oplus V_{0}^{\perp}\simeq
V_{0}\oplus(V_{0}^{\perp})_{+}\oplus(V_{0}^{\perp})_{-}$

define the self-adjoint ${\rm Cl}_{n}$-linear superconnection on $V_{1}$

(102)
$\displaystyle{\mathbb{A}}_{0}^{\prime}={\mathbb{A}}_{0}\oplus(p^{+}\circ\nabla_{1}\circ
p^{+})\oplus e^{*}(p^{+}\circ\nabla_{1}\circ p^{+})$

where $\nabla_{1}$ is the ordinary connection part of ${\mathbb{A}}_{1}$, see
(82). Using (86) and properties of the Clifford supertrace, we compute

$\displaystyle{\rm Ch}_{n}({\mathbb{A}}_{0}^{\prime})$ $\displaystyle=$
$\displaystyle{\rm Ch}_{n}({\mathbb{A}}_{0})+{\rm
Ch}_{n}((p^{+}\circ\nabla_{1}\circ p^{+}))+{\rm
Ch}_{n}(e^{*}(p^{+}\circ\nabla_{1}\circ p^{+}))$ $\displaystyle=$
$\displaystyle{\rm Ch}_{n}({\mathbb{A}}_{0})+{\rm
Ch}_{n}((p^{+}\circ\nabla_{1}\circ p^{+}))-{\rm
Ch}_{n}((p^{+}\circ\nabla_{1}\circ p^{+}))$ $\displaystyle=$
$\displaystyle{\rm Ch}_{n}({\mathbb{A}}_{0})$

where the second line uses that $e$ is an odd automorphism. The result
follows. ∎

######  Example 6.14.

The modules in Example 3.15 can be regarded as basic ${\rm Cl}_{n}$-bundles
over $M={\rm pt}$. The Clifford super traces then compute the well-known
Pontryagin characters of the associated classes in ${\rm KO}^{\bullet}({\rm
pt})$. Specifically,

(103) $\displaystyle{\rm Ch}_{8}({}_{{\rm
Cl}_{8}}{\mathbb{R}}^{8|8})=u^{-4}=\beta^{-1}=\frac{1}{4}\alpha^{-2}\qquad{\rm
Ch}_{4}({}_{{\rm
Cl}_{4}}{\mathbb{R}}^{4|4})=2u^{-2}=\beta^{-1}\alpha=\frac{1}{4}\alpha^{-1},$

using the notation (58) for elements of ${\rm KO}^{\bullet}({\rm pt})$ and the
identifications

$\Omega^{\bullet}({\rm
pt};{\mathbb{R}}[\alpha,\alpha^{-1}])\simeq{\mathbb{R}}[\alpha,\alpha^{-1}]\subset{\mathbb{R}}[u,u^{-1}],\qquad
2u^{2}=\alpha.$

###  6.3. The Pontryagin form of a ${\rm Cl}_{n}$-bundle

The Pontryagin form of a ${\rm Cl}_{n}$-bundle is locally determined by (97),
where the dependence of the differential form representative on the choice of
superconnection is measured by (99). Our present goal is to show that patching
these local values together results in a cocycle in the Čech–de Rham complex,
$(\Omega^{\bullet}(U_{*}),d+\delta)$. Our conventions for the Čech–de Rham
double complex follow Bott and Tu [BT82, §8], and (in particular) the Čech
differential $\delta$ requires an ordering on the indexing set $I$ of the
cover.

######  Proposition 6.15.

Given a ${\rm Cl}_{n}$ bundle
$V=\\{V_{\sigma},g_{\sigma\tau},e_{\sigma\tau}\\}$ and an ordering on $I$,
choices of self-adjoint superconnections ${\mathbb{A}}_{\sigma}$ on
$V_{\sigma}\to U_{\sigma}$ for each $\sigma\subset I$ determine a Čech–de Rham
cocycle denoted

(104) $\displaystyle{\rm
Ch}_{n}(\\{U_{i}\\},{\mathbb{A}}_{i})\in\Omega^{\bullet}(U_{*};{\mathbb{R}}[u^{\pm
1}]).$

###### Proof.

Starting with the degree $n$ Chern forms for each $i\in I$,

(105) $\displaystyle{\rm
Ch}_{n}({\mathbb{A}}_{i})\in\Omega^{\bullet}(U_{i};{\mathbb{R}}[u^{\pm 1}])$

we construct an extension to a cocycle in the Čech–de Rham complex. For
$\sigma=\\{i_{0}<i_{1}<\cdots<i_{k}\\}$, the ${\rm Cl}_{n}$-bundle gives the
data of stable isomorphisms $g_{i_{j}\sigma}\colon
V_{i_{j}}|_{U_{\sigma}}\hookrightarrow V_{\sigma}$ over $U_{\sigma}$. Using
the chosen superconnections ${\mathbb{A}}_{i_{j}}$ on $V_{i_{j}}$ and
${\mathbb{A}}_{\sigma}$ on $V_{\sigma}\to U_{\sigma}$, Lemma 6.13 produces
superconnections ${\mathbb{A}}_{i_{j}}^{\prime}$ on $V_{\sigma}$ for each
$i_{j}\in\sigma\subset I$,

${\mathbb{A}}_{i_{0}}^{\prime},{\mathbb{A}}_{i_{1}}^{\prime},\dots,{\mathbb{A}}_{i_{k}}^{\prime},\qquad\tilde{{\mathbb{A}}}_{\sigma}=\sum_{j=0}^{k}dt_{j}\partial_{t_{j}}+\sum_{j=0}^{k}t_{j}{\mathbb{A}}_{j}^{\prime},\qquad
i_{0}\leq i_{1}\leq\cdots\leq i_{k}$

which determine the indicated superconnection $\tilde{{\mathbb{A}}}_{\sigma}$
on $p^{*}V_{\sigma}$ where $k+1=|\sigma|$ and $p\colon
U_{\sigma}\times{\mathbb{A}{\rm ff}}^{k}\to U_{\sigma}$ is the projection.
Above, we order the superconnections
${\mathbb{A}}_{i_{0}}^{\prime},{\mathbb{A}}_{i_{1}}^{\prime},\dots,{\mathbb{A}}_{i_{k}}^{\prime}$
using the ordering on $I$. Following (95), define the higher Chern–Simons
forms

(106) $\displaystyle{\rm
CS}_{k}(\tilde{{\mathbb{A}}}_{\sigma})=\int_{\Delta^{k}}{\rm
Ch}_{n}(\tilde{{\mathbb{A}}}_{\sigma})\in\Omega^{\bullet-k}(U_{\sigma};{\mathbb{R}}[u^{\pm
1}])$

of total degree $n-k$. We note that changing the ordering of $I$ can change
the sign of the form (106). The Chern forms (105) and higher Chern–Simons
forms (106) provide the necessary data for a degree $n$ cocycle in the Čech–de
Rham complex; it remains to show that they satisfy the required properties.
For $\delta$ the Čech differential and $d$ the de Rham differential in the
Čech–de Rham double complex, the first condition to check is

$\delta\\{{\rm Ch}_{n}({\mathbb{A}}_{i})\\}=\delta\\{{\rm
Ch}_{n}(\widetilde{{\mathbb{A}}}_{i})\\}=\\{d{\rm
CS}({\mathbb{A}}_{i},{\mathbb{A}}_{j})\\}=\\{d{\rm
CS}(\tilde{{\mathbb{A}}}_{ij})\\}\in\prod_{i<j}\Omega^{n}(U_{i}\bigcap U_{j})$

where the first and second equalities use (101), and the third is the
definition of (106). The conditions on higher intersections follow similarly
from

$\displaystyle d\int_{\Delta^{k}}{\rm
Ch}_{n}({\mathbb{A}}_{i_{0},\dots,i_{k}})$ $\displaystyle=$
$\displaystyle\int_{\partial\Delta^{k}}{\rm
Ch}_{n}({\mathbb{A}}_{i_{0},\dots,i_{k}})$ $\displaystyle=$
$\displaystyle\sum(-1)^{j}\int_{\Delta^{k-1}}{\rm
Ch}_{n}({\mathbb{A}}_{i_{0},\dots,\widehat{i_{j}},\dots i_{k}})$
$\displaystyle=$ $\displaystyle\delta\int_{\Delta^{k-1}}{\rm
Ch}_{n}({\mathbb{A}}_{i_{0},\dots,i_{k}}).$

This shows we have produced a cocycle in the double complex, completing the
proof. ∎

######  Proposition 6.16.

The Čech–de Rham cocycle (104) represents the Pontryagin character of the
${\rm Cl}_{n}$-bundle $V=\\{V_{\sigma},g_{\sigma\tau},e_{\sigma\tau}\\}$.

###### Proof.

Since the space of superconnections is affine, $[{\rm
Ch}_{n}(\\{U_{i}\\},{\mathbb{A}}_{i})]$ only depends on $[V]\in{\rm
KO}^{n}(M)$. So by Proposition 5.12, ${\rm Ch}_{n}$ determines a map

(107) $\displaystyle[{\rm Ch}_{n}]\colon{\rm KO}^{n}(M)\to{\rm
H}^{n}(M;{\mathbb{R}}[\alpha,\alpha^{-1}]).$

The local description (97) along with the properties (49) and (50) of the
Clifford supertrace imply that the map (107) is additive and multiplicative.

When $n=0$ (where ${\rm Cl}_{0}={\mathbb{R}}$) the formula (97) coincides with
the standard differential form refinement of the Pontryagin character for real
vector bundles, verifying the lemma in this case. The statement for general
$n$ follows from the fact that ${\rm Ch}_{n}$ is compatible with the
suspension class and Bott element, as we now explain. Let
$V=\\{V_{\sigma},g_{\sigma\tau},e_{\sigma\tau}\\}$ be a ${\rm Cl}_{j}$-module,
and choose $k\in{\mathbb{Z}}$ such that $8k-j\geq 0$. Then consider the
diagram,

The upper left horizontal arrow comes from the external product with the ${\rm
Cl}_{n}$-bundle representing the suspension class
$\sigma_{{\mathbb{R}}^{8k-j}}=(\sigma_{\mathbb{R}})^{\boxtimes(8k-j)}$, where
$\sigma_{\mathbb{R}}$ is defined in (80). The image of this class under ${\rm
Ch}_{n}$ is a Čech–de Rham representative of the suspension class in ordinary
cohomology, being compactly supported and nonzero in cohomology; compare Lemma
7.16 below. This implies that the horizontal arrows on the left are both
(suspension) isomorphisms. The middle horizontal arrows are the Bott
isomorphisms, where Example 6.14 computes the image of the Bott class under
${\rm Ch}_{n}$. The vertical arrow on the far right is the usual Pontryagin
character in degree 0. Since all the horizontal arrows are isomorphisms,
compatibility of the Pontryagin character in cohomology with suspension
implies that the other vertical arrows are also given by the Pontryagin
character. ∎

##  7\. Differential cocycles

Our next goal is to refine the ${\rm KO}$-cocycles from Proposition 5.12 to
differential cocycles (see Definition 7.3). The specifics are motivated by a
generators and relations description of differential complex K-theory reviewed
in §7.1 below. We then prove Theorem 1.2, showing that $\widehat{\rm
KO}{}^{\bullet}(M)$ from Definition 7.6 is a model for differential ${\rm
KO}$-theory.

###  7.1. Review of differential complex K-theory

We review the geometric construction of differential complex K-theory from
[Klo08, Chapter 2]. This is an extension of [FL10, Definition 2.14] to super
vector bundles with superconnection; see Remark 7.2. For a smooth manifold $M$
consider triples $(V,{\mathbb{A}},\phi)$ where $V\to M$ is a hermitian vector
bundle, ${\mathbb{A}}$ is a self-adjoint superconnection on $V$, and
$\phi\in\Omega^{\rm odd}(M)$. Define an equivalence relation

(110)
$\displaystyle(V,{\mathbb{A}},\phi)\sim(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})\iff\begin{array}[]{l}{\rm
there\ exists\ an\ isomorphism\ }f\colon V\xrightarrow{\sim}V^{\prime}\\\ {\rm
with}\ \ \phi^{\prime}={\rm
CS}({\mathbb{A}},f^{*}{\mathbb{A}}^{\prime})+\phi\in\Omega^{\rm odd}(M)/{\rm
image}(d).\end{array}$

i.e., an isomorphism of hermitian vector bundles with a condition on the
resulting Chern–Simons form.

######  Definition 7.1.

Define $\widehat{\rm K}{}^{0}(M)$ as the quotient of the free abelian group
generated by equivalence classes (110), modulo the subgroup generated by

(111)
$\displaystyle(V,{\mathbb{A}},\phi)+(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})-(V\oplus
V^{\prime},{\mathbb{A}}\oplus{\mathbb{A}}^{\prime},\phi+\phi^{\prime})\quad{\rm
and}\quad(W\oplus\Pi W,\nabla\oplus\Pi\nabla,0)$

where $\Pi W$ is the parity reversal of a purely even vector bundle $W$,
$\nabla$ is an ordinary connection on $W$, and $\Pi\nabla$ is the
corresponding connection on $\Pi W$. A triple $(V,\nabla,\eta)$ is a
_differential cocycle_ and $[V,{\mathbb{A}},\phi]\in\widehat{\rm K}{}^{0}(M)$
denotes its class in differential K-theory.

######  Remark 7.2.

In any given equivalence class (110) there is a representative involving an
ordinary connection: every superconnection is concordant to an ordinary
connection.

We note that (110) implies there is a well-defined map

$\widehat{\rm K}{}^{0}(M)\to\Omega^{\rm
even}(M),\qquad[V,\nabla,\phi]\mapsto{\rm Ch}(V,\nabla,\phi):={\sf
sTr}(e^{-\nabla^{2}})+d\phi,$

valued in closed, even differential forms. For the generalization below, we
emphasize that $\phi$ is a coboundary mediating between the differential forms
${\sf sTr}(e^{-\nabla^{2}})$ and ${\rm Ch}(V,\nabla,\phi)$.

###  7.2. Differential refinements of ${\rm Cl}_{n}$-bundles

######  Definition 7.3.

A _differential refinement_ of a ${\rm Cl}_{n}$-bundle
$V=\\{V_{\sigma},g_{\sigma\tau},e_{\sigma\tau}\\}$ defined relative to an open
cover $\\{U_{i}\\}_{i\in I}$ is data

  1. (i)

for each $\sigma\subset I$, a self-adjoint ${\rm Cl}_{n}$-linear
superconnection ${\mathbb{A}}_{\sigma}$ on $V_{\sigma}\to U_{\sigma}$, and

  2. (ii)

a closed global form ${\rm
Ch}(V,{\mathbb{A}},\phi)\in\Omega^{\bullet}(M;{\mathbb{R}}[\alpha^{\pm 1}])$
of total degree $n$;

  3. (iii)

a coboundary
$\\{\phi_{\sigma}\in\Omega^{\bullet}(U_{\sigma};{\mathbb{R}}[\alpha^{\pm
1}])\\}$ in the Čech–de Rham complex mediating between the global form ${\rm
Ch}(V,{\mathbb{A}},\phi)$ and the cocycle ${\rm
Ch}_{n}(\\{U_{i}\\},{\mathbb{A}}_{i})\in\Omega^{\bullet}(U_{*};{\mathbb{R}}[u^{\pm
1}])$ constructed in Proposition 6.15 (which depends on a choice of ordering
on $I$).

A differential refinement of a ${\rm Cl}_{n}$-bundle is a _differential
cocycle_ denoted by
$(V,{\mathbb{A}},\phi)=\\{(V_{\sigma},{\mathbb{A}}_{\sigma},\phi_{\sigma}),(g_{\sigma\tau},e_{\sigma\tau})\\}$.

We unpack the data and property of the Čech–de Rham coboundary in (iii). The
data of this coboundary are differential forms
$\phi_{i}\in\Omega^{\bullet}(U_{i};{\mathbb{R}}[\alpha^{\pm 1}])$ of total
degree $n-1$ for each $i\in I$. These forms satisfy two properties. First, the
locally defined quantities

(112) $\displaystyle{\rm
Ch}({\mathbb{A}}_{i})+d\phi_{i}\in\Omega^{\bullet}(U_{i};{\mathbb{R}}[\alpha^{\pm
1}]),\qquad\delta({\rm Ch}({\mathbb{A}}_{i})+d\phi_{i})=0$

determine the global form in (ii), i.e.,

${\rm Ch}(V,{\mathbb{A}},\phi)|_{U_{i}}-{\rm
Ch}({\mathbb{A}}_{i})=d\phi_{i}\in\Omega^{\bullet}(U_{i};{\mathbb{R}}[\alpha^{\pm
1}]),\qquad{\rm
Ch}(V,{\mathbb{A}},\phi)\in\Omega^{\bullet}(M;{\mathbb{R}}[\alpha^{\pm 1}]).$

Second, for all $i<j$,

(113) $\displaystyle{\rm
CS}(\tilde{{\mathbb{A}}}_{ij})=\phi_{j}-\phi_{i}\in\Omega^{\bullet}(U_{ji};{\mathbb{R}}[\alpha,\alpha^{-1}]),$

for the Chern–Simons form defined in (106).

######  Definition 7.4.

A _concordance_ between differential cocycles
$(V_{0},{\mathbb{A}}_{0},\phi_{0}),(V_{1},{\mathbb{A}}_{1},\phi_{1})$ is a
differential cocycle $(\tilde{V},\tilde{{\mathbb{A}}},\widetilde{\phi})$ over
$M\times{\mathbb{R}}$ together with isomorphisms

$i_{0}^{*}\widetilde{V}\cong V_{0},\
i_{0}^{*}(\tilde{{\mathbb{A}}},\widetilde{\phi})=({\mathbb{A}}_{0},\phi_{0}),\quad
i_{1}^{*}\widetilde{V}\cong V_{1},\
i_{1}^{*}(\tilde{{\mathbb{A}}},\widetilde{\phi})=({\mathbb{A}}_{1},\phi_{1}).$

satisfying the indicated compatibility with the superconnections and
differential form data.

######  Definition 7.5.

Given a concordance between differential cocycles, define the _total
Chern–Simons form_

(114) $\displaystyle{\rm
CS}((V_{0},{\mathbb{A}}_{0},\phi_{0}),(V_{1},{\mathbb{A}}_{1},\phi_{1}))=\int_{[0,1]}{\rm
Ch}_{n}(\widetilde{V},\tilde{{\mathbb{A}}},\widetilde{\phi})\in\Omega^{\bullet}(M;{\mathbb{R}}[\alpha,\alpha^{-1}])$

of total degree $n-1$. Again by Stokes’ theorem, the total Chern–Simons
measures the difference between the Chern characters of concordant
differential cocycles as in (88).

For a smooth manifold $M$, consider the set of degree $n$ differential
cocycles modulo the following equivalence relation

(117)
$\displaystyle(V,{\mathbb{A}},\phi)\sim(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})\iff\begin{array}[]{l}{\rm
there\ exists\ a\ concordance\
(\tilde{V},\tilde{{\mathbb{A}}},\tilde{\phi})}\\\ {\rm with}\ \ {\rm
CS}((V,{\mathbb{A}},\phi),(V,{\mathbb{A}},\phi))\in{\rm image}(d).\end{array}$

######  Definition 7.6.

For a smooth manifold $M$, define $\widehat{\rm KO}{}^{n}(M)$ as the quotient
of the free abelian group generated by equivalence classes (117) modulo the
subgroup generated by

(118)
$\displaystyle(V,{\mathbb{A}},\phi)+(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})-(V\oplus
V^{\prime},{\mathbb{A}}\oplus{\mathbb{A}}^{\prime},\phi+\phi^{\prime}).$

Let $[V,{\mathbb{A}},\phi]\in\widehat{\rm KO}{}^{n}(M)$ denote the equivalence
class of a differential cocycle in differential ${\rm KO}$-theory.

######  Remark 7.7.

Suppose $(V,{\mathbb{A}},\phi)$ is a differential cocycle for which $V$ is a
basic ${\rm Cl}_{n}$-bundle. Then by Stokes’ theorem

$\displaystyle{\rm
CS}((V,{\mathbb{A}},\phi),(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime}))$
$\displaystyle=$ $\displaystyle\int_{[0,1]}({\rm
Ch}_{n}(\widetilde{{\mathbb{A}}})+d\widetilde{\phi})={\rm
CS}({\mathbb{A}},{\mathbb{A}}^{\prime})+\phi^{\prime}-\phi.$

The condition (117) asks that the right hand side be an exact form, which
directly generalizes the equivalence relation (110) in differential complex
K-theory.

###  7.3. A ring structure on $\widehat{\rm KO}{}^{\bullet}(M)$

We define the following product structure on differential cocycles; compare
[FL10, Equation 2.20] or [Klo08, §4.2].

######  Definition 7.8.

Given a differential cocycle $(V,{\mathbb{A}},\phi)$ on $M$ and a differential
cocycle $(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})$ on $M^{\prime}$
define a differential cocycle on $V\boxtimes V^{\prime}$ (see Definition 5.11)
as

(119) $\displaystyle\\{V_{\sigma}\boxtimes
V_{\alpha}^{\prime},{\mathbb{A}}_{\sigma}\boxtimes
1+1\boxtimes{\mathbb{A}}_{\alpha}^{\prime},\phi_{\sigma}{\rm
Ch}_{n}({\mathbb{A}}_{\alpha}^{\prime})+(-1)^{n}{\rm
Ch}_{n}({\mathbb{A}}_{\sigma})\phi_{\alpha}^{\prime}+\phi_{\sigma}d\phi_{\alpha}^{\prime}\\}$

where ${\mathbb{A}}_{\sigma}\boxtimes
1+1\boxtimes{\mathbb{A}}_{\alpha}^{\prime}$ is a superconnection on
$V_{\sigma}\boxtimes V_{\alpha}^{\prime}$ and $\phi_{\sigma}{\rm
Ch}_{n}({\mathbb{A}}_{\alpha}^{\prime})+\phi_{\alpha}^{\prime}{\rm
Ch}_{m}({\mathbb{A}}_{\sigma})+\phi_{\sigma}d\phi_{\alpha}^{\prime}\in\Omega^{\bullet}(U_{\sigma}\times
U_{\alpha}^{\prime};{\mathbb{R}}[\alpha,\alpha^{-1}])$. We use the shorthand
$(V\boxtimes
V^{\prime},{\mathbb{A}}\boxtimes{\mathbb{A}}^{\prime},\phi\boxtimes\phi^{\prime})$
for the resulting differential cocycle.

######  Lemma 7.9.

The assignment $(V,{\mathbb{A}},\phi)\mapsto{\rm Ch}_{n}(V,{\mathbb{A}},\phi)$
is compatible with sums and products:

  1. (1)

For differential cocycles $(V,{\mathbb{A}},\phi)$ and
$(V,{\mathbb{A}}^{\prime},\phi^{\prime})$ of degree $n$ we have ${\rm
Ch}_{n}(V\oplus
V^{\prime},{\mathbb{A}}\oplus{\mathbb{A}}^{\prime},\phi+\phi^{\prime})={\rm
Ch}_{n}(V,{\mathbb{A}},\phi)+{\rm
Ch}_{n}(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})$.

  2. (2)

For differential cocycles $(V,{\mathbb{A}},\phi)$ and
$(V,{\mathbb{A}}^{\prime},\phi^{\prime})$ of degree $n$ and $m$, we have ${\rm
Ch}_{n+m}(V\boxtimes
V^{\prime},{\mathbb{A}}\boxtimes{\mathbb{A}}^{\prime},\phi\boxtimes\phi^{\prime})={\rm
Ch}_{n}(V,{\mathbb{A}},\phi)\cdot{\rm
Ch}_{m}(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})$.

###### Proof.

The statement can be checked locally, where it follows from the definition and
properties (49) and (50) of the Clifford super trace. ∎

The following verifies that tensor products of differential cocycles endows
$\widehat{\rm KO}{}^{\bullet}(M)$ with a graded ring structure.

######  Lemma 7.10.

If degree $n$ differential cocycles $(V,{\mathbb{A}},\phi)$ and
$(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})$ over $M$ are concordant,
and degree $m$ differential cocycles $(W,{\mathbb{B}},\psi)$ and
$(W^{\prime},{\mathbb{B}}^{\prime},\psi^{\prime})$ over $N$ are concordant,
then $(V\boxtimes W,{\mathbb{A}}\boxtimes{\mathbb{B}},\phi\boxtimes\psi)$ is
concordant to $(V^{\prime}\boxtimes
W^{\prime},{\mathbb{A}}^{\prime}\boxtimes{\mathbb{B}}^{\prime},\phi^{\prime}\boxtimes\psi^{\prime})$
over $M\times N$. Furthermore, up to exact forms the total Chern–Simons form
of the concordance is

$\displaystyle{\rm CS}((V\boxtimes
W,{\mathbb{A}}\boxtimes{\mathbb{B}},\phi\boxtimes\psi),$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!(V^{\prime}\boxtimes
W^{\prime},{\mathbb{A}}^{\prime}\boxtimes{\mathbb{B}}^{\prime},\phi^{\prime}\boxtimes\psi^{\prime}))$
$\displaystyle=$ $\displaystyle{\rm
CS}((V,{\mathbb{A}},\phi),(V^{\prime},{\mathbb{A}}^{\prime},\phi^{\prime})){\rm
Ch}_{m}(W,{\mathbb{B}},\psi)$ $\displaystyle+(-1)^{n}{\rm
Ch}_{n}(V,{\mathbb{A}},\phi){\rm
CS}((W,{\mathbb{B}},\psi),(W^{\prime},{\mathbb{B}}^{\prime},\psi^{\prime}))$
$\displaystyle+{\rm
CS}((V,{\mathbb{A}},\phi),(V^{\prime},{\mathbb{A}}^{\prime},\phi))d{\rm
CS}((W,{\mathbb{B}},\psi),(W^{\prime},{\mathbb{B}}^{\prime},\psi^{\prime})).$

###### Proof.

Let $\tilde{W}$ be the concordance over $M\times{\mathbb{R}}$ and $\tilde{V}$
be the concordance over $N\times{\mathbb{R}}$. Then we pullback
$\tilde{W}\boxtimes\tilde{V}$ along the map

$M\times N\times{\mathbb{R}}\stackrel{{\scriptstyle{{\rm
id}}\times\Delta}}{{\longrightarrow}}M\times
N\times{\mathbb{R}}\times{\mathbb{R}}\simeq(M\times{\mathbb{R}})\times(N\times{\mathbb{R}})$

where $\Delta\colon{\mathbb{R}}\to{\mathbb{R}}\times{\mathbb{R}}$ is the
diagonal map. This pullback gives the desired concordance between $(W\boxtimes
V,{\mathbb{A}}\boxtimes{\mathbb{B}},\phi\boxtimes\psi)$ and
$(V^{\prime}\boxtimes
W^{\prime},{\mathbb{A}}^{\prime}\boxtimes{\mathbb{B}}^{\prime},\phi^{\prime}\boxtimes\psi^{\prime})$.
By Lemma 7.9 part (2) and a short computation using (94), the Chern–Simons
form is as claimed. ∎

###  7.4. Differential refinements of stable equivalences

######  Definition 7.11.

Let $V_{0},V_{1}\to M$ be basic ${\rm Cl}_{n}$-modules. A _differential stable
equivalence_ between differential cocycles $(V_{0},{\mathbb{A}}_{0},\phi_{0})$
and $(V_{1},{\mathbb{A}}_{1},\phi_{1})$ is an inclusion $g\colon
V_{0}\hookrightarrow V_{1}$ with the data of a ${\rm Cl}_{n}\otimes{\rm
Cl}_{-1}$-action on $V_{0}^{\perp}\subset V_{1}$ extending the given ${\rm
Cl}_{n}$-action such that

(120) $\displaystyle{\rm
CS}({\mathbb{A}}_{0}^{\prime},{\mathbb{A}}_{1})=\phi_{0}-\phi_{1}\in\Omega^{\bullet}(M;{\mathbb{R}}[u,u])^{n-1}/{\rm
image}(d)$

for ${\rm CS}({\mathbb{A}}_{0}^{\prime},{\mathbb{A}}_{1})$ the differential
form constructed in Lemma 6.13.

######  Lemma 7.12.

Let $V_{0},V_{1}\to M$ be ${\rm Cl}_{n}$-modules, and suppose that
differential cocycles $(V_{0},{\mathbb{A}}_{0},\phi_{0})$ and
$(V_{1},{\mathbb{A}}_{1},\phi_{1})$ are differential stable equivalent in the
sense of Definition 7.11. Then $(V_{0},{\mathbb{A}}_{0},\phi_{0})$ and
$(V_{1},{\mathbb{A}}_{1},\phi_{1})$ are concordant with vanishing total
Chern–Simons form.

###### Proof.

Lemma 5.9 constructs a concordance between the ${\rm Cl}_{n}$ bundles $V$ and
$V^{\prime}$, so it remains to construct a differential refinement of this
concordance with the claimed properties. The superconnection data is specified
by

$(p_{-}^{*}V_{0},p_{-}^{*}{\mathbb{A}}_{0},p_{-}^{*}\phi_{0})\to
M\times(-\infty,1),\quad(p^{*}V_{1},p^{*}{\mathbb{A}}_{1},p^{*}\phi_{1})\to
M\times(0,1),$
$(p_{+}^{*}V_{1},p_{+}^{*}{\mathbb{A}}_{1},p_{+}^{*}\phi_{1})\to
M\times(0,\infty)$

where $p\colon M\times(0,1)\to M$, $p_{-}\colon M\times(-\infty,1)\to M$ and
$p_{+}\colon M\times(0,\infty)\to M$. The Čech–de Rham coboundary is given by
$\phi_{0}$ and $\phi_{1}$: the condition (113) follows from the property (120)