∎ 11institutetext: S. Aoki22institutetext: Graduate School of Arts and
Sciences, University of Tokyo Komaba, Meguro-ku, Tokyo 153-8902, Japan
22email<EMAIL_ADDRESS>
33institutetext: H. Fukaya, T. Onogi, S. Yamaguchi44institutetext: Department
of Physics, Osaka University, Osaka, Japan
44email<EMAIL_ADDRESS>
http://www-het.phys.sci.osaka-u.ac.jp/~hfukaya/
44email<EMAIL_ADDRESS>
44email<EMAIL_ADDRESS>M. Furuta
66institutetext: Graduate School of Mathematical Sciences, The University of
Tokyo, Tokyo, Japan
66email<EMAIL_ADDRESS>77institutetext: S. Matsuo 88institutetext:
Graduate School of Mathematics, Nagoya University, Nagoya, Japan
88email<EMAIL_ADDRESS>
https://www.math.nagoya-u.ac.jp/~shinichiroh/

# The index of lattice Dirac operators and $K$-theory

Shoto Aoki  Hidenori Fukaya  Mikio Furuta  Shinichiroh Matsuo  Tetsuya Onogi
Satoshi Yamaguchi

(  Preprint numbers: OH-HET-1236 )

###### Abstract

We mathematically show an equality between the index of a Dirac operator on a
flat continuum torus and the $\eta$ invariant of the Wilson Dirac operator
with a negative mass when the lattice spacing is sufficiently small. Unlike
the standard approach, our formulation using $K$-theory does not require the
Ginsparg-Wilson relation or the modified chiral symmetry on the lattice. We
prove that a one-parameter family of continuum massive Dirac operators and the
corresponding Wilson Dirac operators belong to the same equivalence class of
the $K^{1}$ group at a finite lattice spacing. Their indices, which are
evaluated by the spectral flow or equivalently by the $\eta$ invariant at
finite masses, are proved to be equal.

##  1 Introduction

It is known in physics that the Atiyah-Singer (AS) index AS (68) can be
formulated in lattice gauge theory HLN (98); Neu (98). Employing a Dirac
operator which satisfies the Ginsparg-Wilson relation GW (82), and a modified
chirality (or $\mathbf{Z}_{2}$ grading) operator Lus (98), one can define the
index by the number of chiral zero modes, which is consistent with that in the
continuum theory, where the Dirac operator is Fredholm. Until recently,
mathematically rigorous formulation has, however, been limited to those
expressed by the heat-kernel regularization Ada (02), where the index is
evaluated by the geometrical index or winding number.

The goal of this work is to reinforce the mathematical relation between the
index of a continuum Dirac operator on a flat torus and that defined on a
discrete lattice, without passing through the geometric expression. We employ
a formulation of $K$-theory Kar (68); AS69a  in order to directly compare the
equivalence classes of the continuum and lattice Dirac operators. We will
prove that for a given continuum Dirac operator, there exists a finite lattice
spacing $a_{1}$ such that the lattice Dirac operator with any lattice spacing
$a<a_{1}$ belongs to the same equivalent class, or the same element of $K^{1}$
groups, which is a sufficient condition for having the same index.

The difficulty in lattice formulation is summarized by the Nielsen-Ninomiya
theorem NN81a ; NN81c ; NN81b . The theorem claims that if one requires
existence of the $\mathbf{Z}_{2}$ grading chirality operator which
anticommutes with a lattice Dirac operator, the Dirac operator must have
unphysical zero modes called doublers. Mathematically, these doubler modes
indicate that the lattice Dirac operator is no more elliptic. In fact, we will
see that the Wilson Dirac operator Wil (77) recovers the ellipticity, and an
associated a priori estimate is valid, at the cost of the $\mathbf{Z}_{2}$
grading structure. Once the chirality operator is lost, it is difficult to
identify the zero modes of the Dirac operator which are needed to define the
index. As mentioned above, a modified chirality operator was introduced in the
literature. But in our formulation, we pursue a different mathematical
formulation where the chirality operator is not essential from the beginning.

In this work, we first describe the $K$ groups in such a way that one can
combine the continuum Hilbert space and that on a lattice. We show that the
elements of $K^{0}$ group defined on a massless point, which is taken as a
base manifold of the Hilbert bundle, are characterized by the index defined in
the standard manner with the $\mathbf{Z}_{2}$ grading operator. We discuss
that the elements of $K^{1}$ defined on a finite interval of mass parameter,
are characterized by the spectral flow of the corresponding Dirac operator
with the mass term. By the suspension isomorphism, where the interval of the
mass is the suspended base space of the Hilbert bundle, it is guaranteed that
the index characterizing the $K^{0}$ group is equal to the spectral flow
related to the $K^{1}$ group.

Next we define a lattice approximation of the target Hilbert bundle in the
continuum space. We introduce the generalized link variables which
simultaneously determine a gauge connection field in the continuum theory, and
the lattice link variables at arbitrary lattice spacing. Note that the
chirality operator is not required at all in the definition of the $K^{1}$
group. Therefore the Wilson Dirac operator qualifies to be a mathematical
object as a natural representative group element of $K^{1}$.

Then the spectral flow of the continuum Dirac operator and its lattice version
are compared. We construct a piece-wise linear map from the Hilbert bundle on
a discretized lattice to the continuum bundle, as well as its conjugate. With
these maps, we can give a gap in the spectrum of the lattice-continuum
combined Dirac operator. Our main theorem shows by contradiction that there
exists a lattice spacing $a_{1}$ such that for any lattice spacing $a<a_{1}$
the $K^{1}$ element constructed is in the same equivalence class as the
original continuum element, and thus, shares the same spectral flow. It should
be noted in the proof that we do not need to evaluate the geometrical index.
We also remark here that the spectral flow can be evaluated essentially by the
finite rank subbundle even when the dimension of the original Hilbert bundle
is infinite.

The equivalence between the index of the overlap Dirac operator and the
spectral flow of the Wilson Dirac operator was also known in physics. It was
found in a rather empirical way at the early stage IIY (87) but later a
mathematically rigorous equivalence to the overlap Dirac operator was
established Ada (02) (see also the related works KY (99); Lus (99); Fuj (99);
Suz (99)). But as far as we know, the mathematical relevance of the Wilson
Dirac operator as the element of the $K$ group has not been discussed.

We should also mention two works on the lattice index theorem by
mathematicians. In Kub (22), the lattice approximation of analytic indices
through the higher index theory of almost flat vector bundles was established.
The Atiyah-Singer index theorem on lattices was directly formulated and proved
in Yam (21) using the algebraic index theorem of Nest and Tsygan.

One advantage of our method is that essentially exactly the same approach can
be applied to various systems. It is straightforward to generalize our
approach to the Hilbert space having real or quaternion structures, to define
the corresponding $KO^{i}$ or $KSp^{i}$ groups with general degree $i$. We can
also apply our method to the case with a group acting on the base manifold
like orbifolds ISTT (23), as well as the case with symmetries with anti-linear
operation. It is also interesting to consider the family version of this work.
See Appendix A.

Another interesting application is that to the systems with boundaries. In the
recent publications FOY (17); FFM+ (20, 22); Fuk (21) by some of the authors
and their collaborators it was shown in continuum theory that the spectral
flow of the massive domain-wall fermion Dirac operator is related to the index
defined on a manifold with boundary APS (75), with the Atiyah-Patodi-Singer
condition imposed. We do not expect difficulty, either, in this generalization
to the APS-type indices. In fact, a positive result was already obtained in a
perturbative analysis FKM+ (20). It is in contrast to the case of the index
through the Ginsparg-Wilson relation, since the relation is known to be
essentially broken by any kind of boundary conditions Lus (06).

The rest of the paper is organized as follows. In Sec. 2, we give a
formulation of the $K^{0}$ and $K^{1}$ groups such that the infinite rank
Hilbert bundle and the finite rank vector bundle are combined. These are
rather well known but we take detailed steps to explain the $K^{0}$ and
$K^{1}$ groups to make the paper self-contained and easy to understand for
non-mathematician readers. We still stress that our approach for showing the
equivalence between the two $K$ group elements is not the standard one. Then
we discuss possible generalization to the system with higher symmetries. In
Sec. 3, we explain how to construct the lattice Wilson Dirac operator from the
given continuum data. We introduce the generalized link variables to relate
the gauge field configurations in the two Dirac operators. Then we prove our
main theorem in Sec. 4. In Sec. 5, we discuss relation between our work and
the known results with the overlap Dirac operator. In Sec. 6, we give summary
and discussion on possible applications of this work.

##  2 Hilbert bundles and $K$ groups

In this work, we use a classification of Dirac operators acting on Hilbert
spaces in terms of $K$-theory. To compare the continuum Dirac operator and its
lattice approximation, we need a formulation in such a way that the Hilbert
spaces with infinite and finite dimensions can be combined Kar (68); AS69a .
In Sec. 2.1 and 2.2, we describe the $K^{0}$ and $K^{1}$ groups necessary for
our main theorem. We take rather detailed steps to define the $K^{0}$ and
$K^{1}$ groups in order to make the paper self-contained and easy to
understand for the readers who are not familiar with $K$-theory. Although the
expert mathematicians can skip these sections, we emphasize that our approach
to show the equivalence between the two $K$ group elements is not the standard
one (see Definition 2). We discuss possible generalizations to those with
higher degrees in Sec. 2.3.

###  2.1 $K^{0}(X,A)$

First, we give a definition of the $K^{0}$ group.

####  2.1.1 Representative elements of $K^{0}(X,A)$

Let $X$ be a compact Hausdorff space and $A$ be a closed subspace of $X$. We
denote the pair by $(X,A)$.

###### Definition 1

A triple $({\mathcal{H}},h,\gamma)$ of the pair $(X,A)$ is a collection of the
data ${\mathcal{H}},h$ and $\gamma$ which satisfies the following conditions.

  1. 1.

${\mathcal{H}}$ is a complex Hilbert bundle111The structure group is the group
of unitary operators endowed with the operator norm. over $X$. We denote the
fiber at $x\in X$ by ${\mathcal{H}}_{x}$.

  2. 2.

$h:{\mathcal{H}}\to{\mathcal{H}}$ is a family of bounded self-adjoint Fredholm
operators $\\{h_{x}:{\mathcal{H}}_{x}\to{\mathcal{H}}_{x}\\}_{x\in X}$, which
is continuous with respect to the operator norm. Namely, when it is expressed
by a local trivialization in the neighborhood of $x$, $h_{x}$ is continuous in
$x$ with respect to the operator norm.

  3. 3.

$\gamma:{\mathcal{H}}\to{\mathcal{H}}$ is a family of unitary and self-adjoint
operators $\\{\gamma_{x}:{\mathcal{H}}_{x}\to{\mathcal{H}}_{x}\\}_{x\in X}$
which is continuous with respect to the operator norm. In particular, each
$\gamma_{x}$ satisfies $\gamma_{x}^{2}={\rm id}$ and keeps the inner product
of the vectors in ${\mathcal{H}}_{x}$.

  4. 4.

$h$ and $\gamma$ anticommute: $\\{\gamma,h\\}=0$.

  5. 5.

For $x\in A$, $\operatorname{Ker}h_{x}=\\{0\\}$.

In our work, we need to consider unbounded operators, with which necessary
prescription is given later in Sec. 2.1.5.

In this paper, we introduce a relation $\sim$ as follows.

###### Definition 2

For the pair $(X,A)$, two triples $({\mathcal{H}},h,\gamma)$ and
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$ are in the relation

$({\mathcal{H}},h,\gamma)\sim({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime}),$

when there exist another triple $(\hat{{\mathcal{H}}},\hat{h},\hat{\gamma})$
for the same pair $(X,A)$ and a continuous one-parameter family of a bounded
self-adjoint Fredholm operators

$\tilde{h}_{t}:{\mathcal{H}}\oplus{\mathcal{H}}^{\prime}\oplus\hat{{\mathcal{H}}}\to{\mathcal{H}}\oplus{\mathcal{H}}^{\prime}\oplus\hat{{\mathcal{H}}}$

where the parameter space is labeled by $0\leq t\leq 1$, which satisfy the
following conditions.

  1. 1.

$(\hat{{\mathcal{H}}},\hat{h},\hat{\gamma})$ is a triple of $(X,X)$. Namely,
for arbitrary $x\in X$, $\operatorname{Ker}\hat{h}_{x}=\\{0\\}$.

  2. 2.

$\tilde{h}_{0}=-h\oplus h^{\prime}\oplus\hat{h}.$

  3. 3.

For any $(x,t)\in A\times[0,1]\cup X\times\\{1\\}$,
$\operatorname{Ker}(\tilde{h}_{t})_{x}=\\{0\\}$. This is equivalent to state
that for the projection $p:X\times[0,1]\to X$ and
$\tilde{h}=\\{\tilde{h}_{t}\\}$ in the range $t\in[0,1]$,
$(p^{*}({\mathcal{H}}\oplus{\mathcal{H}}^{\prime}\oplus\hat{{\mathcal{H}}}),\tilde{h},p^{*}(-\gamma\oplus\gamma^{\prime}\oplus\hat{\gamma}))$
is a triple of the pair $(X\times[0,1],A\times[0,1]\cup X\times\\{1\\})$
222The same (or essentially equivalent) condition can be stated as follows.
For the pair $(X\times[0,1],A\times[0,1]\cup X\times\\{1\\}),$ there exists a
triple whose restriction to $(X\times\\{0\\},A\times\\{0\\})$ is isomorphic to
$-h\oplus h^{\prime}\oplus\hat{h}$. Since the proof of the equivalence will be
rather technical, we do not employ this condition..

In the following, when ${\mathcal{H}}$ and $\gamma$ are obvious, we use a
simplified notation $h\sim h^{\prime}$ instead of
$({\mathcal{H}},h,\gamma)\sim({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$.

###### Proposition 1

For the pair $(X,A)$, the relation $\sim$ is an equivalence relation.

###### Proof

We show that the relation is symmetric, reflective and transitive in the next
subsections by Lemmas 1, 2, and 3.

With the above proposition, we give the definition

###### Definition 3

$K^{0}(X,A)$ is the whole set of the equivalence classes by $\sim$ for the
triples of the pair $(X,A)$333 Note that the triples themselves do not form a
set since the cardinality of a set consisting of the triples is not bounded.
But the equivalence classes form a set. One sees this by checking for each
equivalence class the existence of a representation of th form
$({\mathcal{H}},h,\gamma)$ such that ${\mathcal{H}}_{x}$ is a separable
Hilbert space, whose cardinality is bounded from above..

####  2.1.2 Equivalence relation $\sim$

In this subsection, we show that the relation $\sim$ between two triples is an
equivalence relation.

###### Lemma 1

(symmetry) For two triples $({\mathcal{H}},h,\gamma)$ and
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$ of the pair $(X,A)$, the
relation
$({\mathcal{H}},h,\gamma)\sim({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$
implies
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})\sim({\mathcal{H}},h,\gamma).$

###### Proof

Changing the sign of $\tilde{h}_{t}\to-\tilde{h}_{t}$ and $\hat{h}\to-\hat{h}$
gives the relation
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})\sim({\mathcal{H}},h,\gamma)$.

###### Lemma 2

(reflexivity) For a triple $({\mathcal{H}},h,\gamma)$ of the pair $(X,A)$, the
relation

$({\mathcal{H}},h,\gamma)\sim({\mathcal{H}},h,\gamma)$

holds.

###### Proof

Let us take $(\hat{{\mathcal{H}}},\hat{h},\hat{\gamma})=(0,0,\rm{id})$444Here
$\hat{{\mathcal{H}}}=0$ means the vector space consisting the zero vector
only. In this case, ${\rm Ker}\hat{h}=\\{0\\}$ is trivially true.. Then it is
enough to show that setting

$\tilde{h}_{t}:=(1-t)\left(\begin{array}[]{cc}-h&0\\\
0&h\end{array}\right)+t\left(\begin{array}[]{cc}0&{\rm id}\\\ {\rm
id}&0\end{array}\right),$

$(p^{*}({\mathcal{H}}\oplus{\mathcal{H}}),\tilde{h},p^{*}(-\gamma\oplus\gamma))$
is a triple with respect to $(X\times[0,1],A\times[0,1]\cup X\times\\{1\\})$.
Since the three operators on
${\mathcal{H}}\oplus{\mathcal{H}}={\mathcal{H}}\otimes{\mathbf{C}}^{2}$

$\left(\begin{array}[]{cc}-h&0\\\
0&h\end{array}\right),\quad\left(\begin{array}[]{cc}0&{\rm id}\\\ {\rm
id}&0\end{array}\right),\quad\left(\begin{array}[]{cc}-\gamma&0\\\
0&\gamma\end{array}\right)$

all anticommute, so do $\tilde{h}$ and $p^{*}(-\gamma\oplus\gamma)$.

When $t=0$ and $x\in A$ the kernel of $\tilde{h}_{t,x}=-h\oplus h$ is trivial.
It is sufficient to show that for arbitrary $t>0$ and $x\in X$ the kernel of
$\tilde{h}_{t,x}$ is trivial. From the anticommutation relation of the two
terms in $\tilde{h}_{t}$,

$\tilde{h}_{t}^{2}:=(1-t)^{2}\left(\begin{array}[]{cc}-h&0\\\
0&h\end{array}\right)^{2}+t^{2}\left(\begin{array}[]{cc}0&{\rm id}\\\ {\rm
id}&0\end{array}\right)^{2}=\\{(1-t)^{2}h^{2}+t^{2}\\}\otimes\left(\begin{array}[]{cc}{\rm
id}&0\\\ 0&{\rm id}\end{array}\right)$

holds. This is always a positive self-adjoint operator for $t>0$ and thus the
kernel is $\\{0\\}$.

###### Lemma 3

(transitivity) For three triples $({\mathcal{H}}_{1},h_{1},\gamma_{1})$,
$({\mathcal{H}}_{2},h_{2},\gamma_{2})$, $({\mathcal{H}}_{3},h_{3},\gamma_{3})$
of the pair $(X,A)$, when
$({\mathcal{H}}_{1},h_{1},\gamma_{1})\sim({\mathcal{H}}_{2},h_{2},\gamma_{2})$
and
$({\mathcal{H}}_{2},h_{2},\gamma_{2})\sim({\mathcal{H}}_{3},h_{3},\gamma_{3})$,
the relation
$({\mathcal{H}}_{1},h_{1},\gamma_{1})\sim({\mathcal{H}}_{3},h_{3},\gamma_{3})$
also holds.

###### Proof

In Definition 2 we have introduced
$(\hat{{\mathcal{H}}},\hat{h},\hat{\gamma})$ and $\tilde{h}$ to define the
relation
$({\mathcal{H}},h,\gamma)\sim({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$.
Let $(\hat{{\mathcal{H}}}_{ij},\hat{h}_{ij},\hat{\gamma}_{ij})$ and
$\tilde{h}_{ij}$ be the corresponding data for the relation
$({\mathcal{H}}_{i},h_{i},\gamma_{i})\sim({\mathcal{H}}_{j},h_{j},\gamma_{j})$
where $(i,j)=(1,2)$ or $(2,3)$. We write $\tilde{h}_{ij}(t)$ for
$\tilde{h}_{ij,t}(t\in[0,1])$.

Let us take another triple
$(\hat{{\mathcal{H}}}_{13},\hat{h}_{13},\hat{\gamma}_{13})$ of the pair
$(X,A)$ given by

$\displaystyle\hat{{\mathcal{H}}}_{13}$ $\displaystyle:=$
$\displaystyle\hat{{\mathcal{H}}}_{12}\oplus\hat{{\mathcal{H}}}_{23}\oplus({\mathcal{H}}_{2}\otimes{\mathbf{C}}^{2})$
$\displaystyle\hat{h}_{13}$ $\displaystyle:=$
$\displaystyle\hat{h}_{12}\oplus\hat{h}_{23}\oplus\left(\begin{array}[]{cc}0&{\rm
id}\\\ {\rm id}&0\end{array}\right)$ $\displaystyle\hat{\gamma}_{13}$
$\displaystyle:=$
$\displaystyle\hat{\gamma}_{12}\oplus\hat{\gamma}_{23}\oplus\left(\begin{array}[]{cc}-\gamma_{2}&0\\\
0&\gamma_{2}\end{array}\right).$

Then $(\hat{{\mathcal{H}}}_{13},\hat{h}_{13},\hat{\gamma}_{13})$ is actually a
triple with respect to $(X,X)$ since the kernel of $(\hat{h}_{13})_{x}$ is
trivial for any $x\in X$. As is in the proof of Lemma 2, let us introduce a
one-parameter family $\tilde{h}_{13}(t)$ with $t\in[0,1]$ of the operators on
${\mathcal{H}}_{1}\oplus{\mathcal{H}}_{3}\oplus\hat{{\mathcal{H}}}_{13}$ in
the following way.

  1. 1.

For $t\in[0,1/2]$ put

$\hat{h}_{13}(t):=2t\left[\hat{h}_{12}\oplus\hat{h}_{23}\oplus\left(\begin{array}[]{cc}-h_{2}&0\\\
0&h_{2}\end{array}\right)\right]+(1-2t)\hat{h}_{13}.$

Then the operator

$\tilde{h}_{13}(t):=-h_{1}\oplus h_{3}\oplus\hat{h}_{13}(t)$

on ${\mathcal{H}}_{1}\oplus{\mathcal{H}}_{3}\oplus\hat{{\mathcal{H}}}_{13}$ is
a triple for the pair $(X,A)$ at any value of $t\in[0,1/2]$.

  2. 2.

Note that $\hat{h}_{13}(1/2)$ can be interpreted as an operator
$\hat{h}_{12}\oplus\hat{h}_{23}\oplus(-h_{2}\oplus h_{2})$ on
$(\hat{{\mathcal{H}}}_{12}\oplus\hat{{\mathcal{H}}}_{23})\oplus({\mathcal{H}}_{2}\oplus{\mathcal{H}}_{2})$.
Therefore, we can rewrite

$\displaystyle\tilde{h}_{13}(1/2)=-h_{1}\oplus h_{3}\oplus\hat{h}_{13}(1/2)$
$\displaystyle=-h_{1}\oplus
h_{3}\oplus(\hat{h}_{12}\oplus\hat{h}_{23})\oplus(-h_{2}\oplus h_{2})$
$\displaystyle\cong(-h_{1}\oplus h_{2}\oplus\hat{h}_{12})\oplus(-h_{2}\oplus
h_{3}\oplus\hat{h}_{23}).$

The last form acts on
$({\mathcal{H}}_{1}\oplus{\mathcal{H}}_{2}\oplus\hat{{\mathcal{H}}}_{12})\oplus({\mathcal{H}}_{2}\oplus{\mathcal{H}}_{3}\oplus\hat{{\mathcal{H}}}_{23})$
and is equal to the operator $\tilde{h}_{12}(0)\oplus\tilde{h}_{23}(0)$.

  3. 3.

For $t\in[1/2,1]$ put

$\tilde{h}_{13}(t):=\tilde{h}_{12}(2t-1)\oplus\tilde{h}_{23}(2t-1),$

which acts on
$({\mathcal{H}}_{1}\oplus{\mathcal{H}}_{2}\oplus\hat{{\mathcal{H}}}_{12})\oplus({\mathcal{H}}_{2}\oplus{\mathcal{H}}_{3}\oplus\hat{{\mathcal{H}}}_{23})$.
Since $\tilde{h}_{12}(1)\oplus\tilde{h}_{23}(1)$ is a triple of the pair
$(X,X)$, Lemma 3 follows.

Generalizing Lemma 2, we would like to add another lemma below.

###### Lemma 4

Let $({\mathcal{H}},h,\gamma)$ and
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$ be two isomorphic
triples of $(X,A)$, in the sense that there exists an isomorphism
$S:{\mathcal{H}}\stackrel{{\scriptstyle\cong}}{{\to}}{\mathcal{H}}^{\prime}$
of the Hilbert bundles, satisfying $h^{\prime}=ShS^{-1}$ and
$\gamma^{\prime}=S\gamma S^{-1}$. Then we have the relation
$({\mathcal{H}},h,\gamma)\sim({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$.

###### Proof

The proof goes in parallel to that of Lemma 2 but with

$\tilde{h}_{t}:=(1-t)\left(\begin{array}[]{cc}-h&0\\\
0&h^{\prime}\end{array}\right)+t\left(\begin{array}[]{cc}0&S^{-1}\\\
S&0\end{array}\right).$

It is not difficult to show that $\tilde{h}_{t}^{2}$ is positive for $t>0$ and
its kernel is trivial.

####  2.1.3 Sum, product and functoriality of $K^{0}$

#### Pullback functor

Let $X$ and $Y$ be compact Hausdorff spaces and $A$ and $B$ be closed subsets
of $X$ and $Y$, respectively. For a continuous map $f:X\to Y$ satisfying
$f(A)\subset B$ we abbreviate it by $f:(X,A)\to(Y,B)$.

For a continuous map $f:(X,A)\to(Y,B)$, we have a pullback from a triple with
respect to $(Y,B)$ to the one for $(X,A)$. Let $f^{*}$ denote the pullback for
the Hilbert bundles and the maps on them. It is not difficult to show that
$f^{*}$ maintains the equivalence relation $\sim$ where each element of the
triple for $(Y,B)$ is pullbacked to that for $(X,A)$. From this we obtain the
map

$f^{*}:K^{0}(Y,B)\to
K^{0}(X,A),\qquad[({\mathcal{H}},h,\gamma)]\mapsto[(f^{*}{\mathcal{H}},f^{*}h,f^{*}\gamma)].$

It is also obvious that for two continuous maps $f:(X,A)\to(Y,B)$ and
$g:(Y,B)\to(Z,C)$ as well as their composite $gf:(X,A)\to(Z,C)$,
$(gf)^{*}=f^{*}g^{*}$ holds by construction.

Besides we can show that the sum and product structures defined below are also
kept by the pullbacks. The above observation is summarized as follows.

###### Proposition 2

$K^{0}$ is a contravariant functor from the category with pairs $(X,A)$ as the
objects and continuous maps between them as the morphisms to the category with
rings as the objects and homomorphisms as the morphisms.

#### Sum

For two triples $({\mathcal{H}},h,\gamma)$ and
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$ of the pair $(X,A)$, we
define their direct sum by

$({\mathcal{H}},h,\gamma)\oplus({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime}):=({\mathcal{H}}\oplus{\mathcal{H}}^{\prime},h\oplus
h^{\prime},\gamma\oplus\gamma^{\prime}).$

The equivalence class of this direct sum depends only on those of the original
two triples.

###### Definition 4

We define the sum on $K^{0}(X,A)$ by

$[({\mathcal{H}},h,\gamma)]+[({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})]:=[({\mathcal{H}},h,\gamma)\oplus({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})].$

###### Lemma 5

  1. 1.

$[({\mathcal{H}},h,\gamma)]+[(0,0,{\rm id})]=[({\mathcal{H}},h,\gamma)]$

  2. 2.

The sum $+$ satisfies the commutative and associative laws.

  3. 3.

$[({\mathcal{H}},h,\gamma)]+[({\mathcal{H}},-h,-\gamma)]=[(0,0,{\rm id})]$

###### Proof

The first two claims follow from the reflexivity of $\sim$. In the proof of
Lemma 2, a triple with respect to $(X,X)$ is constructed from
$({\mathcal{H}},h,\gamma)\oplus({\mathcal{H}},-h,-\gamma)$ for $(X,A)$ with a
continuous deformation of the operator $h\oplus(-h)$. The last claim follows
from this.

Now $K^{0}(X,A)$ forms an Abelian group with respect to the sum $+$. We denote
the identity element $[(0,0,{\rm id})]$ by $0$. The inverse element of
$[({\mathcal{H}},h,\gamma)]$ is $[({\mathcal{H}},-h,-\gamma)]$ From the
definition of $\sim$, the Lemma below follows.

###### Lemma 6

For a triple $({\mathcal{H}},h,\gamma)$ of the pair $(X,A)$, the necessary and
sufficient condition for $[({\mathcal{H}},h,\gamma)]=0$ is that there exists a
triple $(\hat{{\mathcal{H}}},\hat{h},\hat{\gamma})$ with respect to $(X,X)$
such that the triple

$({\mathcal{H}}\oplus\hat{{\mathcal{H}}},h\oplus\hat{h},\gamma\oplus\hat{\gamma})$

for $(X,A)$ can be continuously deformed to a triple with respect to $(X,X)$
555This statement indicates that $\alpha=0$ when $\alpha+0=0$..

There is another expression for the inverse.

###### Lemma 7

$[({\mathcal{H}},h,\gamma)]+[({\mathcal{H}},h,-\gamma)]=0$ holds. Namely,
$[({\mathcal{H}},h,-\gamma)]$ is the inverse element of
$[({\mathcal{H}},h,\gamma)]$.

###### Proof

We use Lemma 6. The operator in a triple

$({\mathcal{H}},h,\gamma)\oplus({\mathcal{H}},h,-\gamma)=\left({\mathcal{H}}\oplus{\mathcal{H}},\left(\begin{array}[]{cc}h&0\\\
0&h\end{array}\right),\left(\begin{array}[]{cc}\gamma&0\\\
0&-\gamma\end{array}\right),\right)$

with respect to $(X,A)$ has the following one-parameter continuous deformation
with $0\leq t\leq 1$,

$\tilde{h}_{t}:=(1-t)\left(\begin{array}[]{cc}h&0\\\
0&h\end{array}\right)+t\left(\begin{array}[]{cc}0&\gamma\\\
\gamma&0\end{array}\right).$

Noting

$\left(\begin{array}[]{cc}h&0\\\
0&h\end{array}\right),\quad\left(\begin{array}[]{cc}0&\gamma\\\
\gamma&0\end{array}\right),\quad\left(\begin{array}[]{cc}\gamma&0\\\
0&-\gamma\end{array}\right)$

all anticommute, we can continuously deform the triple
$({\mathcal{H}},h,\gamma)+({\mathcal{H}},h,-\gamma)$ to that of $(X,X)$, since
the kernel of $\tilde{h}_{1}$ is trivial.

###### Remark 1

From Lemma 5 and Lemma 7,

$[({\mathcal{H}},h,\gamma)]=[({\mathcal{H}},-h,\gamma)]\in K^{0}(X,A)$

follows.

#### Product

###### Lemma 8

Let $A$ and $B$ closed subsets of compact Hausdorff space $X$. For a triple
$({\mathcal{H}},h,\gamma)$ of the pair $(X,A)$ and
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$ with respect to $(X,B)$,
we define the product below as a triple with respect to $(X,A\cup B)$ 666The
tensor product ${\mathcal{H}}\otimes{\mathcal{H}}^{\prime}$ between the
Hilbert spaces ${\mathcal{H}}$ and ${\mathcal{H}}^{\prime}$ is defined by a
completion of their algebraically defined tensor product: the completion is
done with the inner product the algebraic tensor product naturally has. The
algebraic tensor product of the bounded operators on ${\mathcal{H}}$ and
${\mathcal{H}}^{\prime}$ is canonically extended to the completion. We simply
denote it by $\otimes$ between the operators..

$({\mathcal{H}}\otimes{\mathcal{H}}^{\prime},(h\otimes{\rm id})+(\gamma\otimes
h^{\prime}),\gamma\otimes\gamma^{\prime}).$

Moreover, its equivalence class in $K^{0}(X,A\cup B)$ depends only on the
equivalence classes in $K^{0}(X,A)$ and $K^{0}(X,B)$ before taking the
product.

###### Proof

The above product is a triple of the pair $(X,\emptyset)$, since the three
operators

$h\otimes{\rm id},\qquad\gamma\otimes
h^{\prime},\qquad\gamma\otimes\gamma^{\prime}$

all anticommute. We can also show that

$\\{(h\otimes{\rm id})+(\gamma\otimes h^{\prime})\\}^{2}=(h\otimes{\rm
id})^{2}+({\rm id}\otimes h^{\prime})^{2}$

and the right-hand side is a sum of semi-positive self-adjoint operators. The
operator has a kernel only when both $h\otimes{\rm id}$ and ${\rm id}\otimes
h^{\prime}$ have kernels. Therefore, the kernel is always trivial on $A\cup
B$. The uniqueness of the product is shown as follows. Fixing
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$ we replace
$({\mathcal{H}},h,\gamma)$ by another in the same equivalent class. For
$\hat{h}$’s and $\tilde{h}$’s in definition of the relation $\sim$, we can
directly confirm the equivalence of the two products. The same applies when we
replace $({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$ by another
element with $({\mathcal{H}},h,\gamma)$ fixed.

The above Lemma defines the product of the $K^{0}$ groups.

###### Definition 5

Let $A$ and $B$ be closed subsets of a compact Hausdorff space $X$. The
product

$K^{0}(X,A)\times K^{0}(X,B)\to K^{0}(X,A\cup B)$

is defined by

$[({\mathcal{H}},h,\gamma)]\cdot[({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})]=[({\mathcal{H}}\otimes{\mathcal{H}}^{\prime},(h\otimes{\rm
id})+(\gamma\otimes h^{\prime}),\gamma\otimes\gamma^{\prime})].$

###### Lemma 9

  1. 1.

The product is associative.

  2. 2.

The product and sum are left- and right-distributive.

  3. 3.

The element $[({\mathbf{C}},0,{\rm id})]\in K^{0}(X,\emptyset)$ is the
identity: $[({\mathcal{H}},h,\gamma)]\cdot[({\mathbf{C}},0,{\rm
id})]=[({\mathbf{C}},0,{\rm
id})]\cdot[({\mathcal{H}},h,\gamma)]=[({\mathcal{H}},h,\gamma)]$.

###### Proof

For $\alpha=[({\mathcal{H}},h,\gamma)]\in K^{0}(X,A)$,
$\alpha^{\prime}=[({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})]\in
K^{0}(X,A^{\prime})$ and
$\alpha^{\prime\prime}[({\mathcal{H}}^{\prime\prime},h^{\prime\prime},\gamma^{\prime\prime})]\in
K^{0}(X,A^{\prime\prime})$, both of
$(\alpha\cdot\alpha^{\prime})\cdot\alpha^{\prime\prime}$ and
$\alpha\cdot(\alpha^{\prime}\cdot\alpha^{\prime\prime})$ are given by the same
element

$[({\mathcal{H}}\otimes{\mathcal{H}}^{\prime}\otimes{\mathcal{H}}^{\prime},h\otimes{\rm
id}\otimes{\rm id}+\gamma\otimes h^{\prime}\otimes{\rm
id}+\gamma\otimes\gamma^{\prime}\otimes
h^{\prime\prime},\gamma\otimes\gamma^{\prime}\otimes\gamma^{\prime\prime})].$

The left- and right-distributivity can also be directly checked. It is also
obvious from definition that $[({\mathbf{C}},0,{\rm id})]$ is the identity.

###### Lemma 10

The product $K^{0}(X,A)\times K^{0}(X,B)\to K^{0}(X,A\cup B)$ is commutative.
Namely, the two maps $K^{0}(X,A)\times K^{0}(X,B)\to K^{0}(X,B)\times
K^{0}(X,A)$ and $K^{0}(X,B)\times K^{0}(X,A)\to K^{0}(X,A\cup B)$ are
identical777 In generalization in Sec. 2.3 to $K^{i}(X,A)$ and $K^{j}(X,B)$
with nonzero degrees, we need an additional sign depending on $(-1)^{ij}$.
This sign comes from the orientation flip of the finite-dimensional vector
space $V$ which determined the Clifford algebra $Cl(V)$. .

###### Proof

Take $\alpha=[({\cal H},h,\gamma)]\in K^{0}(X,A)$ and $\alpha^{\prime}=[({\cal
H}^{\prime},h^{\prime},\gamma^{\prime})]\in K^{0}(X,B)$. Let ${\cal H}={\cal
H}^{0}\oplus{\cal H}^{1}$ and ${\cal H}^{\prime}={{\cal
H}^{\prime}}^{0}\oplus{{\cal H}^{\prime}}^{1}$ be the ${\bf Z}_{2}$-gradings
associated with $\gamma$ and $\gamma^{\prime}$ respectively. We define an
isomorphism $S:{\cal H}\otimes{\cal H}^{\prime}\to{\cal
H}^{\prime}\otimes{\cal H}$ by switching the left and the right in the tensor
product together with the multiplication by $(-1)^{ij}$ on ${\cal
H}^{i}\otimes{{\cal H}^{\prime}}^{j}$. Then we have

$S(h^{\prime}\otimes{\rm id})S^{-1}=\gamma\otimes h^{\prime},\quad
S(\gamma^{\prime}\otimes h)S^{-1}=h\otimes{\rm 1},\quad
S(\gamma^{\prime}\otimes\gamma)S^{-1}=\gamma\otimes\gamma^{\prime}$

which and Lemma 4 imply $\alpha^{\prime}\alpha=\alpha\alpha^{\prime}$.

###### Definition 6

Consider two pairs of $(X,A)$ and $(Y,B)$ where $X$ and $Y$ are compact
Hausdorff spaces and $A$ and $B$ are their closed subsets $A\subset X$ and
$B\subset Y$, respectively. The projections

$p_{X}:X\times Y\to X,\qquad p_{Y}:X\times Y\to Y$

induce their pullbacks

$p_{X}^{*}:K^{0}(X,A)\to K^{0}((X,A)\times(Y,B)),\qquad
p_{Y}^{*}:K^{0}(Y,B)\to K^{0}((X,A)\times(Y,B)),$

by which we define the exterior product

$K^{0}(X,A)\times K^{0}(Y,B)\to K^{0}((X,A)\times(Y,B)).$

Specifically for $\omega_{X}\in K^{0}(X,A)$ and $\omega_{Y}\in K^{0}(Y,B)$,

$\omega_{X}\cdot\omega_{Y}=p_{X}^{*}\omega_{X}\cdot p_{Y}^{*}\omega_{Y}.$

Here $(X,A)\times(Y,B):=(X\times Y,A\times Y\cup X\times B)$.

####  2.1.4 $K^{0}(\\{{\rm pt}\\},\emptyset)$ group and the Fredholm index

Let us consider the case where $X$ is a point $\\{{\rm pt}\\}$ and $A$ is an
empty set $\emptyset$. When $({\mathcal{H}},h,\gamma)$ is a triple of the pair
$(\\{{\rm pt}\\},\emptyset)$, ${\mathcal{H}}$ is a single Hilbert space. As is
shown below, the elements of the $K^{0}(\\{{\rm pt}\\},\emptyset)$ are
classified by the Fredholm operator index, which is topological and stable
against perturbation.

The next statement is well known as a starting point of the notion of the
index.

###### Theorem 2.1

(Fredholm index) The following isomorphism holds.

$K^{0}(\\{{\rm
pt}\\},\emptyset)\cong{\mathbf{Z}},\qquad[({\mathcal{H}},h,\gamma)]\mapsto\operatorname{trace}\gamma|_{\operatorname{Ker}h}.$

The generator is the identity $[({\mathbf{C}},0,{\rm id})]$ of the product.

###### Proof

Here we give an argument according to our formulation of the $K^{0}$ group.
Since the constructions in the parts 3 and 4 are the prototype of our main
theorem (Theorem 3.1), we give a rather detailed argument.

Let $({\mathcal{H}},h,\gamma)$ be a triple of the pair $(\\{{\rm
pt}\\},\emptyset)$, where ${\mathcal{H}}$ is a single Hilbert space. We show
by the following (outlines of) four steps that for
$\nu=\operatorname{trace}\gamma|_{\operatorname{Ker}h}$ we can express
$[({\mathcal{H}},h,\gamma)]=\nu[({\mathbf{C}},0,{\rm id})]$.

  1. 1.

First we show that when
$({\mathcal{H}},h,\gamma)\sim({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$,
we have
$\operatorname{trace}\gamma|_{\operatorname{Ker}h}=\operatorname{trace}\gamma^{\prime}|_{\operatorname{Ker}h^{\prime}}$
by the following three steps.

    1. (a)

For a triple $(\hat{{\mathcal{H}}},\hat{h},\hat{\gamma})$ with respect to the
pair $(X,X)$, we can claim that
$\operatorname{trace}\hat{\gamma}|_{\operatorname{Ker}\hat{h}}=0$ since
$\operatorname{Ker}\hat{h}=\\{0\\}$888For any $\phi\in\operatorname{Ker}({\rm
id}\pm\hat{\gamma})$, we have $\hat{h}\phi\in\operatorname{Ker}({\rm
id}\mp\hat{\gamma})$ and contribution from $\phi$ and $\hat{h}\phi$ cancel in
the trace..

    2. (b)

We consider
$\check{{\mathcal{H}}}:={\mathcal{H}}\oplus{\mathcal{H}}^{\prime}\oplus\hat{{\mathcal{H}}}$
and operators on it. We will see below that for arbitrary bounded self-adjoint
Fredholm operator $\check{h}$ on $\check{{\mathcal{H}}}$ which anticommutes
with $\check{\gamma}:=-\gamma\oplus\gamma^{\prime}\oplus\hat{\gamma}$ and
arbitrary positive real number $\Lambda$ under which $\check{h}^{2}$ has no
spectrum other than finite eigenvalues with finite degeneracies,
$\operatorname{trace}\check{\gamma}|_{\operatorname{Ker}\check{h}}$ is
invariant against perturbation.

    3. (c)

From the above perturbative invariance, we can show that for the operator
$-h\oplus h^{\prime}\oplus\hat{h}$ on $\check{{\mathcal{H}}}$, the following
trace is zero:

$\operatorname{trace}\check{\gamma}|_{\operatorname{Ker}(-h\oplus
h^{\prime}\oplus\hat{h})}=-\operatorname{trace}\gamma|_{\operatorname{Ker}h}+\operatorname{trace}\gamma^{\prime}|_{\operatorname{Ker}h^{\prime}}+\operatorname{trace}\hat{\gamma}|_{\operatorname{Ker}\hat{h}}=0.$

  2. 2.

The perturbative invariance is shown as follows. When the domain of $h$ is
limited to $\operatorname{Ker}(\gamma-{\rm id})$, the range is in
$\operatorname{Ker}(\gamma+{\rm id})$. For this restriction:
$h_{-+}:\operatorname{Ker}(\gamma-{\rm id})\to\operatorname{Ker}(\gamma+{\rm
id})$, we have

$\operatorname{trace}\gamma|_{\operatorname{Ker}h}=\dim\operatorname{Ker}h_{-+}-\dim\operatorname{Coker}h_{-+}.$

One can show that this quantity is invariant under perturbation of the
Fredholm operator $h_{-+}$.

  3. 3.

Since $h$ is a Fredholm operator, $\operatorname{Ker}h$ is finite-dimensional.
Let us denote the dimension of the $\gamma=\pm 1$ subeigenspace in
$\operatorname{Ker}h$ by $\nu_{\pm}$. We consider another triple
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$ defined by

$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime}):=({\mathbf{C}}^{\nu_{+}}\oplus{\mathbf{C}}^{\nu_{-}},0,(+{\rm
id})\oplus(-{\rm id})).$

When $\nu=\operatorname{trace}\gamma|_{\operatorname{Ker}h}=\nu_{+}-\nu_{-}$,
we can show that
$[({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})]=\nu[({\mathbf{C}},0,{\rm
id})]$.

Then if we can show that
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})\sim({\mathcal{H}},h,\gamma)$,
we obtain $[({\mathcal{H}},h,\gamma)]=\nu[({\mathbf{C}},0,{\rm id})]$. For
this, it is sufficient to show that the bounded self-adjoint Fredholm operator
$-h\oplus 0$ on ${\mathcal{H}}\oplus{\mathcal{H}}^{\prime}$ can be
continuously deformed to another operator whose kernel is $\\{0\\}$999 There
is a more intuitive discussion to show
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})\sim({\mathcal{H}},h,\gamma)$.
Let us denote the restriction of domain of $\gamma$ onto $\operatorname{Ker}h$
and $(\operatorname{Ker}h)^{\perp}$ by $\gamma^{0}$ and $\gamma^{\perp}$,
respectively. Also let $h^{\perp}$ be the restriction of domain of $h$ onto
$(\operatorname{Ker}h)^{\perp}$. In this case,
$[({\mathcal{H}},h,\gamma)]\sim[(\operatorname{Ker}h,0,\gamma^{0})]+[((\operatorname{Ker}h)^{\perp},h^{\perp},\gamma^{\perp})]$
holds. As $\operatorname{Ker}(h)^{\perp}=\\{0\\}$, the second term is zero.
Since $(\operatorname{Ker}h,0,\gamma^{0})$ is isomorphic to
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})$, the equivalence
relation
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})\sim({\mathcal{H}},h,\gamma)$
holds. This argument is easier than what we mention in the main text but its
application to the general $K$ groups with nonzero degrees is limited. .

  4. 4.

Since the dimension of the eigen subspace of the restriction of
$\operatorname{Ker}h$ onto $\gamma=\pm 1$ is $\nu_{\pm}$, there exist
isomorphisms $f_{+}$ and $f_{-}$ from ${\mathbf{C}}^{\nu+}$ and
${\mathbf{C}}^{\nu_{-}}$ to the respective subspaces. Let us choose $f_{+}$
and $f_{-}$ which keep the metric. Their sum gives an injective map,

$f:=f_{+}+f_{-}:{\mathcal{H}}^{\prime}\to{\mathcal{H}},$

which preserves the metric and $f\gamma^{\prime}=\gamma f$ holds. For $0\leq
t\leq 1$, the following deformation is what we need to show
$({\mathcal{H}}^{\prime},h^{\prime},\gamma^{\prime})\sim({\mathcal{H}},h,\gamma)$:

$\left(\begin{array}[]{cc}-h^{\prime}&0\\\
0&h\end{array}\right)+t\left(\begin{array}[]{cc}0&f^{*}\\\
f&0\end{array}\right).$

The perturbative invariance is also explained as follows. Let us denote the
direct product of the eigenspaces of $\check{h}^{2}$ whose eigenvalue are
lower than $\Lambda$ by $E_{\check{h}^{2}<\Lambda}$. When $\check{h}^{2}$ has
no spectrum below $\Lambda$ other than finite eigenvalues with finite
degeneracies, the following holds101010This is a consequence of linear algebra
on a finite-dimensional vector space : for a finite vector space $E$ with an
inner product and an orthogonal transformation
$\gamma_{E}\in{\mathrm{E}nd}\,E$ which satisfies $\gamma_{E}^{2}={\rm id}$,
when a symmetric transformation $h_{E}\in{\mathrm{E}nd}\,E$ anticommutes with
$\gamma_{E}$,
$\operatorname{trace}\gamma_{E}=\operatorname{trace}\gamma_{E}|_{\operatorname{Ker}h_{E}}$.
Note that $h_{E}^{2}$ and $\gamma_{E}$ commute and the simultaneous
orthogonalization is possible. Since $h_{E}$ is symmetric, the zero mode space
of $h_{E}^{2}$ or $\operatorname{Ker}h_{E}^{2}$ is identical to
$\operatorname{Ker}h_{E}$. For nonzero eigenspace with $\Lambda>0$ of
$h_{E}^{2}$, the operation of $h_{E}$ gives an isomorphism between the $+1$
and $-1$ subeigenspaces of $\gamma_{E}$. Therefore, these nonzero eigenvalues
do not contribute to the trace of $\gamma_{E}$. .

$\operatorname{trace}\check{\gamma}|_{E_{\check{h}^{2}<\Lambda}}=\operatorname{trace}\check{\gamma}|_{\operatorname{Ker}\check{h}}.$

Suppose $\Lambda$ is not any eigenvalue of $\check{h}$. Then the dimension of
the subeigenspace of $\check{h}^{2}$ below $\Lambda$ is invariant against any
infinitesimal perturbation of $\check{h}$ with respect to the operator norm.
Moreover, the subeigenspace forms a finite rank vector bundle on the parameter
space of the perturbation. The eigen decomposition with $\gamma$ leads to a
direct-sum decomposition of the vector bundle. Therefore, the left-hand side
of the above equation is invariant against the perturbation, and so does the
right-hand side.

####  2.1.5 Some remarks on the use of Hilbert bundles

Here we give some remarks on our formulation with Hilbert bundles whose
dimension is infinite.

###### Remark 2

  1. 1.

In a standard formulation of the K theory in the literature Ati (89); Seg
(68), one takes a finite rank vector bundle to define $K^{0}(X,A)$ rather than
an infinite-dimensional Hilbert bundle. The discussion in this section goes in
parallel with such finite-dimensional expressions, replacing the bounded self-
adjoint Fredholm operator by a continuous self-adjoint operator on the finite
rank vector bundle. The propositions and discussions in this section are also
valid. We can prove that the standard finite-dimensional definition of
$K^{0}(X,A)$ is isomorphic to that in our infinite-dimensional definition.

  2. 2.

In particular, we can take a representative element using ${\mathcal{H}}$ with
a finite rank for any equivalence class. This indicates that $K^{0}(X,A)$ can
be “approximated” by finite-dimensional vector bundles.

  3. 3.

One advantage of introducing Hilbert bundles is that we can easily define the
index of differential operators which anti-commute with $\gamma$.

  4. 4.

Another advantage of introducing Hilbert bundles is found in the extension to
$K^{i}(X,A)$ with $i\neq 0$. In such a formulation, there exists a case that
the group element cannot be represented by data with finite rank
${\mathcal{H}}$111111 For example, the nontrivial elements of
$K^{1}(S^{1},\emptyset)$ cannot be represented by any finite rank vector
bundle. . Therefore, it is useful to introduce the infinite-dimensional
Hilbert space from the beginning. See Sec. 2.3 for the details.

So far, we have employed a bounded self-adjoint Fredholm operator family
denoted by $h$. Here we extend the definition of the $K^{0}$ group so that
unbounded operators like differential operators are allowed. Note that with
this extension, the $K^{0}(X,A)$ is unchanged and each equivalence class still
has a representative element with a finite rank ${\mathcal{H}}$.

We first explain which topology we use for the space of operators.

Fix a Hilbert space ${\cal H}_{0}$ and we write ${\cal B}_{F}^{sa}({\cal
H}_{0})$ be the set of bounded self-adjoint Fredholm operators on ${\cal
H}_{0}$ endowed with the norm topology. Let ${\cal C}_{F}^{sa}({\cal H}_{0})$
be the set of closed self-adjoint Fredholm operators with dense domain, which
are not necessary bounded. We make use of the injective map $\rho:{\cal
C}_{F}^{sa}({\cal H}_{0})\to{\cal B}_{F}^{sa}({\cal H}_{0})$ given by
$\rho(h):=h/(1+h^{2})^{1/2}$, which is bounded and its parity is odd such that
$\rho(-h)=-\rho(h)$. The Riesz topology of ${\cal C}_{F}^{sa}({\cal H}_{0})$
is the weakest topology for $\rho$ to be continuous121212The function $\rho$
could be any bounded continuous monotonically increasing function with odd
parity such that $\rho(-h)=-\rho(h)$. . We endow ${\cal C}_{F}^{sa}({\cal
H}_{0})$ with this Riesz topology to formulate the continuity of operators
131313 Let ${\cal U}_{F}({\cal H}_{0})$ be the set of unitary operators $u$
such that $u-1$ is Fredholm endowed with the norm topology. If we use
$\kappa:{{\cal C}_{F}^{sa}({\cal H}_{0})}\to{{\cal U}_{F}({\cal H}_{0})}$
defined by the Cayley transform $\kappa(h)=(1+ih)/(1-ih))$, instead of $\rho$,
we could endow ${{\cal C}_{F}^{sa}({\cal H}_{0})}$ with the weakest topology
for $\kappa$ to be continuous. This is the gap topology, which is strictly
weaker than the Riesz topology. As we explain in Appendix we could use ${\cal
U}_{F}({\cal H}_{0})$ to give a formulation of $K^{i}$ groups. The two
formulations, one using ${\cal B}_{F}^{sa}({\cal H}_{0})$ and the other using
${\cal U}_{F}({\cal H}_{0})$ give the canonically isomorphic $K$ groups (under
correct choice of Clifford symmetries). If we formulate the $K$ groups using
${\cal U}_{F}({\cal H}_{0})$, it would be natural to use the gap topology for
${\cal C}_{F}^{sa}({\cal H}_{0})$. We, however, chose a formulation using
${\cal B}_{F}^{sa}({\cal H}_{0})$ in this paper, and it seems natural to
choose the Riesz topology for ${\cal C}_{F}^{sa}({\cal H}_{0})$ here. .

###### Proposition 3

In the definition of $K^{0}(X,A)$ if we replace “bounded self-adjoint Fredholm
operators” with “closed self-adjoint Fredholm operators with dense domain”,
then we obtain the same $K^{0}(X,A)$ group. Here we use the Riesz topology to
formulate the continuity of families of closed self-adjoint Fredholm operators
with dense domains.

###### Proof

From a standard argument of functional analysis, the restriction map
$\rho|_{{\cal B}_{F}^{sa}({\cal H}_{0})}:{\cal B}_{F}^{sa}({\cal
H}_{0})\to{\cal B}_{F}^{sa}({\cal H}_{0})$ is continuous and homotopic to the
identity of ${\cal B}_{F}^{sa}({\cal H}_{0})$ linearly141414It implies that
the inclusion map ${\cal B}_{F}^{sa}({\cal H}_{0})\to{\cal C}_{F}^{sa}({\cal
H}_{0})$ is a homotopy equivalent map for the Riesz topology.. In particular
for any triple $({\cal H},h,\gamma)$ with $h$ bounded, we have another triple
$({\cal H},\rho(h),\gamma)$ with $\rho(h)$ bounded again. Note that $[({\cal
H},\rho(h),\gamma)]=[({\cal H},h,\gamma)]$ in the $K^{0}$ group. Moreover,
even when $h$ is not bounded, we always have a triple $({\cal
H},\rho(h),\gamma)$ with $\rho(h)$ bounded, and the class $[({\cal
H},h,\gamma)]$ is a well-defined element of the $K^{0}$ group. It implies that
using the class $[({\cal H},h,\gamma)]$ we can extend the definition to
unbounded operators $h$.

###  2.2 $K^{1}(X,A)$

In this section, we formulate the $K^{1}(X,A)$ group.

####  2.2.1 Definition of $K^{1}(X,A)$ and its properties

We can define $K^{1}(X,A)$ by the same steps in the definition of $K^{0}(X,A)$
shown in the previous section but forgetting about existence of $\gamma$.

###### Definition 7

$K^{1}(X,A)$ is defined by

$K^{1}(X,A):=\\{({\mathcal{H}},h)\\}/\sim$

  * •

$({\mathcal{H}},h)$ is a data which satisfies the following
conditions151515Note that $\\{({\mathcal{H}},h)\\}$ is not a set but its
equivalence class is a set..

    1. 1.

${\mathcal{H}}$ is a Hilbert bundle over $X$.

    2. 2.

$h:{\mathcal{H}}\to{\mathcal{H}}$ is a family of bounded self-adjoint Fredholm
operators $h_{x}:{\mathcal{H}}_{x}\to{\mathcal{H}}_{x}$ ($x\in X$), which is
continuous with respect to the operator norm

    3. 3.

$\operatorname{Ker}h_{x}=\\{0\\}$ for $x\in A$.

  * •

The equivalence relation $\sim$ is defined as follows. The relation
$({\mathcal{H}},h)\sim({\mathcal{H}}^{\prime},h^{\prime})$ holds when there
exist $(\hat{{\mathcal{H}}},\hat{h})$ and a continuous family of bounded self-
adjoint Fredholm operators with the parameter $0\leq t\leq 1$ given by

$\tilde{h}_{t}:{\mathcal{H}}\oplus{\mathcal{H}}^{\prime}\oplus\hat{{\mathcal{H}}}\to{\mathcal{H}}\oplus{\mathcal{H}}^{\prime}\oplus\hat{{\mathcal{H}}}$

and satisfy

    1. 1.

For arbitrary $x\in X$, $\operatorname{Ker}\hat{h}_{x}=\\{0\\}$.

    2. 2.

$\tilde{h}_{0}=-h\oplus h^{\prime}\oplus\hat{h}$

    3. 3.

For any $(x,t)\in A\times[0,1]\cup X\times\\{1\\}$,
$\operatorname{Ker}(\tilde{h}_{t})_{x}=\\{0\\}$.

We can prove that $\sim$ is an equivalence class in a similar manner to
$K^{0}(X,A)$.

For a continuous map $f:(X,A)\to(Y,B)$, the pullback is defined as

$f^{*}:K^{1}(Y,B)\to
K^{1}(X,A),\qquad[({\mathcal{H}},h)]\mapsto[(f^{*}{\mathcal{H}},f^{*}h)].$

This $f^{*}$ is compatible with the sum and product defined below.

The sum on $K^{1}(X,A)$ is defined in the same way as for $K^{0}(X,A)$.

######  Definition 8 (Sum)

The sum on $K^{1}(X,A)$ is defined by

$[({\mathcal{H}},h)]+[({\mathcal{H}}^{\prime},h^{\prime})]:=[({\mathcal{H}},h)\oplus({\mathcal{H}}^{\prime},h^{\prime})].$

The following product is also well defined.

######  Definition 9 (Product)

For closed subsets $A$ and $B$ in a compact Hausdorff space $X$, the product

$K^{0}(X,A)\times K^{1}(X,B)\to K^{1}(X,A\cup B)$

is defined by

$[({\mathcal{H}},h,\gamma)]\cdot[({\mathcal{H}}^{\prime},h^{\prime})]=[({\mathcal{H}}\otimes{\mathcal{H}}^{\prime},(h\otimes{\rm
id})+(\gamma\otimes h^{\prime}))].$

From this definition, the proposition below follows.

###### Proposition 4

  1. 1.

$(X,A)\mapsto K^{1}(X,A)$ is a contravariant functor with respect to the
pullback.

  2. 2.

When $B\supset A$, the above product

$K^{0}(X,A)\times K^{1}(X,B)\to K^{1}(X,B)$

indicates that $K^{1}(X,B)$ becomes a module with the action of $K^{0}(X,A)$,
which forms a commutative ring.

######  Definition 10 (Exterior product)

Consider two pairs of $(X,A)$ and $(Y,B)$ where $X$ and $Y$ are compact
Hausdorff spaces and $A$ and $B$ are their closed subsets $A\subset X$ and
$B\subset Y$, respectively. The projections

$p_{X}:X\times Y\to X,\qquad p_{Y}:X\times Y\to Y$

induce their pullbacks

$p_{X}^{*}:K^{0}(X,A)\to K^{0}((X,A)\times(Y,B)),\qquad
p_{Y}^{*}:K^{1}(Y,B)\to K^{1}((X,A)\times(Y,B))$

by which the exterior product

$K^{0}(X,A)\times K^{1}(Y,B)\to K^{1}((X,A)\times(Y,B))$

is defined. Specifically for $\omega_{X}\in K^{0}(X,A)$ and $\omega^{1}_{Y}\in
K^{1}(Y,B)$,

$\omega_{X}\cdot\omega^{1}_{Y}=p_{X}^{*}\omega_{X}\cdot
p_{Y}^{*}\omega^{1}_{Y}.$

Here $(X,A)\times(Y,B):=(X\times Y,A\times Y\cup X\times B)$.

####  2.2.2 The suspension isomorphism between $K^{0}$ and $K^{1}$

######  Definition 11 (Bott element)

We define the Bott element in $K^{1}(D^{1},S^{0})$ by

$\beta:=[(\underline{{\mathbf{C}}},x\cdot)]\in K^{1}(D^{1},S^{0}),$

where $\underline{{\mathbf{C}}}$ denotes a trivial line bundle over $D^{1}$
with the fiber ${\mathbf{C}}$, and the operator $x\cdot$ is multiplication of
$x$, which is the coordinate for $D^{1}=\\{x:-1\leq x\leq 1\\}$.

######  Definition 12 (Suspension)

For $\alpha\in K^{0}(X,A)$, the exterior product with the Bott element
$\alpha\cdot\beta\in K^{1}((X,A)\times(D^{1},S^{0}))$ is called the
suspension.

The explicit form of the exterior product is given by 161616Note that
$h+x\gamma$ in the right-hand side is an abbreviation. Precisely it is written
as $p_{X}^{*}h+({p_{D^{1}}}^{*}x)p_{X}^{*}\gamma$ making the pullback
explicit.,

$[({\mathcal{H}},h,\gamma)]\cdot\beta=[(p^{*}_{X}{\mathcal{H}},h+x\gamma)]\in
K^{1}((X,A)\times(D^{1},S^{0})).$

The next theorem is a part of the Bott periodicity theorem.

######  Theorem 2.2 (Suspension isomorphism/Bott periodicity)

The following suspension map is an isomorphism with respect to the
$K^{0}(X,A)$ module.

$K^{0}(X,A)\to
K^{1}((X,A)\times(D^{1},S^{0})),\qquad\alpha\to\alpha\cdot\beta$

Here $\beta$ is the Bott element.

We will give a detailed description for the case with $(X,A)=(\\{{\rm
pt}\\},\emptyset)$ later171717 The full statement of Theorem 2.2 for $(X,A)$
will be necessary when we extend the discussion on “comparison between the
Fredholm index of a Dirac operator and its discrete approximation” to a family
version. .

####  2.2.3 $K^{1}(D^{1},S^{0})$ and spectral flow

Consider $K^{1}(X,A)$ for the case where $X$ is $D^{1}=\\{x:-1\leq x\leq
1\\}$, and $A$ is $S^{0}=\\{-1,+1\\}$181818 Since $D^{1}$ is contractible, the
complex Hilbert bundle over $D^{1}$ is trivial. Note in our formulation that a
finite rank Hilbert bundle is allowed, to which the Kuiper theorem Kui (65)
guaranteeing that the space of unitary operators on any infinite-dimensional
Hilbert space is contractible, does not apply. We also note that it is not
difficult to construct a global trivialization of the Hilbert bundle over
$D^{1}$ from its local trivializations, without using the Kuiper theorem. When
one trivialization is fixed, $h:{\mathcal{H}}\to{\mathcal{H}}$ is nothing but
a one-parameter family of the bounded self-adjoint Fredholm operators on a
unique complex Hilbert space, which is continuous with respect to the operator
norm. .

From Theorem 2.2 and Theorem 2.1 we obtain

$K^{1}(D^{1},S^{0})\cong{\mathbf{Z}}$

with the Bott element $\beta$ as its generator.

Let us discuss its geometrical aspects. In our formulation of
$K^{1}(D^{1},S^{0})$, it is sufficient to characterize it by the spectral flow
of a one parameter family of the self-adjoint Fredholm operators. Recall that
the spectral flow $\operatorname{sf}({\mathcal{H}},h)\in{\mathbf{Z}}$ is
defined as follows.

  1. 1.

Since $h$ is a bounded self-adjoint Fredholm operator and $D^{1}$ is compact,
there exists $\Lambda_{0}$ such that for any $x\in D^{1}$, $h_{x}$ has no
spectrum in the range $\\{\lambda:-\Lambda_{0}\leq\lambda\leq\Lambda_{0}\\}$
except for finite eigenvalues with finite degeneracies.

  2. 2.

Note that $h_{x}$ has no zero mode at $x=\pm 1$. Let us introduce a finite
number of points $x_{0}=-1<x_{1}<\cdots x_{n}=+1$ in $D^{1}$ for which we
assign the values $\lambda_{1},\ldots,\lambda_{n}$ in the range
$\\{\lambda:-\Lambda_{0}<\lambda<\Lambda_{0}\\}$ such that

    1. (a)

$\lambda_{1}=\lambda_{n}=0$.

    2. (b)

For any $x\in D^{1}$ in the range $x_{k-1}\leq x\leq x_{k}$, $\lambda_{k}$ is
not an eigenvalue of $h_{x}$.

  3. 3.

For the $k$-th set $(x_{k},\lambda_{k})$ for $0<k<n$, we assign
$\operatorname{sgn}_{k}$ and $d_{k}$ as follows. For
$\lambda_{k}\neq\lambda_{k+1}$, we set

$\operatorname{sgn}_{k}=\frac{\lambda_{k}-\lambda_{k+1}}{|\lambda_{k}-\lambda_{k+1}|},$

and $d_{k}$ by sum of dimensions of the eigenspace with the eigenvalues in the
range between $\lambda_{k}$ and $\lambda_{k+1}$. When
$\lambda_{k}=\lambda_{k+1}$, we assign that $\operatorname{sgn}_{k}=0$ and
$d_{k}=0$.

######  Definition 13 (Spectral flow)

Let ${\mathcal{H}}$ be a complex Hilbert bundle over $D^{1}$ and $h$ be a
continuous family of bounded self-adjoint Fredholm operators on
${\mathcal{H}}$. The spectral flow of $h$ is defined by

$\operatorname{sf}({\mathcal{H}},h)=\sum_{0<k<n}\operatorname{sgn}_{k}d_{k}.$

we denote $\operatorname{sf}\alpha:=\operatorname{sf}({\mathcal{H}},h)$ for
$\alpha=[({\mathcal{H}},h)]\in K^{1}(D^{1},S^{0})$.

###### Lemma 11

The spectral flow is independent of the choice of the sets $\\{x_{k}\\}$ and
$\\{\lambda_{k}\\}$.

###### Remark 3

When $h$ is an unbounded self-adjoint Fredholm operator such that
$h/(h^{2}+1)^{1/2}$ is a continuous family with respect to the operator norm,
the spectral flow is defined in the same way as above.

For the Bott element $\beta$,

$\operatorname{sf}\beta=1.$

For the operator $x\cdot$ in $\beta$, we can set
$x_{0}=-1<x_{1}=-1/2<x_{2}=1/2<x_{3}=+1$ and assign
$\lambda_{1}=0,\lambda_{2}=2/3,\lambda_{3}=0$. Then we have
$\operatorname{sgn}_{1}=-1,N_{1}=0$ and $\operatorname{sgn}_{2}=+1,N_{2}=1$.

The next theorem is well known.

###### Theorem 2.3

The isomorphism below holds.

$K^{1}(D^{1},S^{0})\cong{\mathbf{Z}},\qquad[({\mathcal{H}},h)]\mapsto\operatorname{sf}({\mathcal{H}},h).$

###### Proof

We give an argument according to our formulation of the $K^{1}$ group. It is
sufficient to show that $[({\mathcal{H}},h)]=\nu\beta$ when
$\operatorname{sf}({\mathcal{H}},h)=\nu$. Here we summarize the outline of the
proof.

  1. 1.

First, we confirm that $\operatorname{sf}$ is well-defined as a map from
$K^{1}(D^{1},S^{0})$. This is shown by the invariance of $\operatorname{sf}h$
against a continuous deformation of $h$. It is not difficult to show that for
an infinitesimal variation of $h$, the same set $\\{x_{k}\\}$ and
$\\{\lambda_{k}\\}$ can be used to compute the spectral flow, which is
invariant.

  2. 2.

Next, we deform $h$ in $({\mathcal{H}},h)$ as follows.

The data for computing the spectral flow is taken as $\Lambda_{0}$,
$x_{0}=-1<x_{1}<\cdots x_{n}=+1$ and $\lambda_{1},\ldots,\lambda_{n}$. We
introduce a continuous function $\mu_{1}(x)$ which sufficiently approximates
the piece-wise constant function $\mu_{0}(x)=\lambda_{k}(x_{k-1}\leq
x<x_{k})$. We also introduce a monotonically increasing continuous function
which also continuously depends on $x$, $\rho_{x}:{\mathbf{R}}\to{\mathbf{R}}$
such that $\rho_{x}(\lambda)=\lambda$ for $|\lambda|\geq\Lambda$ and
$\rho_{x}(\mu_{1}(x))=0$.

We can continuously deform the family $\\{h_{x}\\}$ to
$\\{\rho_{x}(h_{x})\\}_{x}$. For a sufficiently precise approximation of
$\mu_{0}$ by $\mu_{1}$,

    1. (a)

Any $x$ such that $\operatorname{Ker}\rho_{x}(h_{x})\neq 0$ is located at an
sufficiently close point to one of $x_{k}$’s.

    2. (b)

In the vicinity of $x_{k}$, the eigenvalues of $\rho_{x}(h_{x})$ change as
follows. For $\operatorname{sgn}_{k}=\pm 1$, $d_{k}$ eigenvalues change their
sign from $\pm$ to $\mp$. When $\operatorname{sgn}_{k}=0$ no eigenvalue goes
to zero.

  3. 3.

There is a representative element $({\mathcal{H}}^{\prime},h^{\prime})$ which
has a similar $x$-dependence of the eigenspectrum to that of
$({\mathcal{H}},h)$:

$({\mathcal{H}}^{\prime},h^{\prime}):=\left(\bigoplus_{0<k<n}\underline{{\mathbf{C}}^{d_{k}}},\bigoplus_{0<k<n}\operatorname{sgn}_{k}(x-x_{k}){\rm
id}_{d_{k}\times d_{k}}\cdot\right).$

For $k$ with nonzero $\operatorname{sgn}_{k}=\pm 1$, the $d_{k}$ eigenvalues
of $h^{\prime}$ crosses zero at $x=x_{k}$ from positive/negative to
negative/positive. When $\operatorname{sf}({\mathcal{H}},h)=\nu$,
$[({\mathcal{H}}^{\prime},h^{\prime})]=\nu\beta$ holds.

  4. 4.

The last step is to show that
$({\mathcal{H}}^{\prime},h^{\prime})\sim({\mathcal{H}},\\{\rho_{x}(h_{x})\\})$
for the proof of $[({\mathcal{H}},h)]=\nu\beta$. For this, we consider a
deformation of the continuous family of the bounded self-adjoint Fredholm
operators $\\{-h^{\prime}_{x}\oplus\rho_{x}(h_{x})\\}_{x\in D^{1}}$ on
${\mathcal{H}}^{\prime}\oplus{\mathcal{H}}$ to another operator family whose
kernel is trivial everywhere in $D^{1}$.

Note that nontrivial kernel appears only in the neighborhood of the discrete
points $x_{k}\in D^{1}$. It is, thus, sufficient to deform the operators so
that its kernel becomes trivial in the neighborhood of $x_{k}$. The direct sum
of the near-zero eigenspaces of $\rho_{x}(h_{x})$ near $x_{k}$ which
potentially develop the kernel has $d_{k}$ dimensions. Let $f_{k}$ be an
isomorphism from ${\mathbf{C}}^{d_{k}}$ to the direct sum of the $d_{k}$
eigensections such that $f_{k}$ is continuous with respect to $x$ and keeps
the metric. Note that $f_{k}$ is defined in the neighborhood of $x_{k}$ only
but we can naturally extend it by zero with a continuous cut-off function to
the whole range of $x\in D^{1}$. There is no overlap of the support of
$f_{k}$’s at different $k$’s. We put the sum

$f:=\sum_{0<k<n}f_{k}:{\mathcal{H}}^{\prime}\to{\mathcal{H}}.$

For $0\leq t\leq 1$, we can define the operator family

$\left(\begin{array}[]{cc}-h^{\prime}_{x}&0\\\
0&\rho_{x}(h_{x})\end{array}\right)+t\left(\begin{array}[]{cc}0&f^{*}\\\
f&0\end{array}\right),$

which achieves the claim.

###  2.3 $K$ and $KO$ groups with general degrees

In the previous subsections, we have defined the $K^{0}$ and $K^{1}$ groups,
which are sufficient to give a proof of our main theorem. Here we discuss
possible generalizations to Hilbert bundles with higher symmetries.

Note that the operator $\gamma$ in the triple $({\mathcal{H}},h,\gamma)$ can
be regarded as a single Clifford generator which satisfies $\\{\gamma,h\\}=0$.
In order to define $K^{i}(X,A)$ with a general degree $i$, one can use more
general Clifford symmetry on ${\mathcal{H}}$ which satisfies appropriate anti-
commutativity with $h$, and their appropriate equivalence relation $\sim$.

Next we consider the case where the Hilbert bundle is real. In this case, the
$KO^{0}$ and $KO^{1}$ groups can be formulated exactly in the same way as in
the previous subsections with the simple replacement of ${\mathbf{C}}$ by
${\mathbf{R}}$ in the definitions of the identity element or the Bott element.
We also denote $KO^{0,0}(X,A)=KO^{0}(X,A)$ and $KO^{0,-1}(X,A)=KO^{1}(X,A)$.

As the Bott periodicity becomes more nontrivial than the complex $K$ groups,
it is worth giving the explicit forms of the $KO$ groups with higher degrees.
Here we employ a formulation in Kar (68).

Let us first formulate the $KO^{1,0}$ group, which turns out to be isomorphic
to $KO^{1}=KO^{0,-1}$.

###### Definition 14

$KO^{1,0}(X,A):=\\{({\mathcal{H}}^{{\mathbf{R}}},h,(\epsilon,e))\\}/\sim$

where

  1. 1.

${\mathcal{H}}^{{\mathbf{R}}}$ is a real Hilbert bundle over $X$.

  2. 2.

$h:{\mathcal{H}}^{{\mathbf{R}}}\to{\mathcal{H}}^{{\mathbf{R}}}$ is a family of
bounded self-adjoint Fredholm operators parametrized by $x\in X$ which is
continuous with respect to the operator norm.

  3. 3.

$\epsilon,e:{\mathcal{H}}^{{\mathbf{R}}}\to{\mathcal{H}}^{{\mathbf{R}}}$ are
continuous families of orthogonal operators which satisfy $\epsilon^{2}={\rm
id},e^{2}=-{\rm id}$. $h,\epsilon,e$ all anticommute among each other.

  4. 4.

For arbitrary $x\in A$, $\operatorname{Ker}h_{x}=\\{0\\}$.

The definition of the equivalence relation $\sim$ is given in the same way as
that of the $K$ groups. In particular, any deformation keeps the condition
that $h$ anticommutes with both of $\epsilon$ and $e$.

###### Lemma 12

The following natural isomorphism exists.

$KO^{0,-1}(X,A)\cong KO^{1,0}(X,A).$

###### Proof

The isomorphism is given by the map

$KO^{0,-1}(X,A)\longrightarrow
KO^{1,0}(X,A),\quad[({\mathcal{H}}^{{\mathbf{R}}}_{0},h^{\prime})]\mapsto[({\mathcal{H}}^{{\mathbf{R}}}_{1},h,(\epsilon,e))]$
${\mathcal{H}}^{{\mathbf{R}}}_{1}:={\mathcal{H}}^{{\mathbf{R}}}_{0}\otimes{\mathbf{R}}^{2},\quad
h:=h^{\prime}\otimes\left(\begin{array}[]{cc}{\rm id}&0\\\ 0&-{\rm
id}\end{array}\right),\quad\epsilon:=\left(\begin{array}[]{cc}0&{\rm id}\\\
{\rm id}&0\end{array}\right),\quad e:=\left(\begin{array}[]{cc}0&-{\rm id}\\\
{\rm id}&0\end{array}\right),$

and its inverse,

$KO^{0,-1}(X,A)\longleftarrow
KO^{1,0}(X,A),\quad[({\mathcal{H}}^{{\mathbf{R}}}_{0},h^{\prime})]\leftarrow[({\mathcal{H}}^{{\mathbf{R}}}_{1},h,(\epsilon,e))]$
${\mathcal{H}}^{{\mathbf{R}}}_{0}:=\operatorname{Ker}(\epsilon e-{\rm
id}:{\mathcal{H}}^{{\mathbf{R}}}_{1}\to{\mathcal{H}}^{{\mathbf{R}}}_{1}),\quad
h:=h^{\prime}|_{{\mathcal{H}}^{{\mathbf{R}}}_{0}}.\quad$

Now let us generalize the definition in the following way.

######  Definition 15 ($KO^{p,q}(X,A)$ for general $p$ and $q$)

For $p\geq 0,q\geq-1$, we can define191919 When $q=-1$, we do not have any
$\epsilon_{i}$, i.e.,
$KO^{p,-1}(X,A):=\\{({\mathcal{H}}^{{\mathbf{R}}},h,(e_{1},\ldots,e_{p}))\\}/\sim$.

$KO^{p,q}(X,A):=\\{({\mathcal{H}}^{{\mathbf{R}}},h,(\epsilon_{0},\epsilon_{1},\ldots,\epsilon_{q},e_{1},\ldots,e_{p}))\\}/\sim.$

###### Lemma 13

There is an isomorphism

$KO^{p,q}(X,A)\stackrel{{\scriptstyle\cong}}{{\to}}KO^{p+1,q+1}(X,A).$

With this isomorphism, we can define $KO^{i}$ group202020To be precise, the
definition is given as a direct limit, $p,q\to\infty$ with $p-q=i$ fixed. in a
similar way to Lemma 12.

###### Definition 16

$KO^{i}(X,Y):=KO^{p,q}(X,Y)\qquad(i=p-q).$

The product for $KO^{*}$ is constructed in a similar way as that for $K^{0}$
Kar (78). The well-definedness and the associativity of the product are shown
straightforwardly. The exterior product is also defined via pullbacks:

$KO^{i}(X,A)\times KO^{j}(Y,B)\to KO^{i+j}((X,A)\times(Y,B)).$

The commutativity of the product holds with sign $(-1)^{ij}$.

Note that the inverse element of
$[({\mathcal{H}}^{{\mathbf{R}}},h,(\epsilon_{0},\epsilon_{1},\ldots,\epsilon_{q},e_{1},\ldots,e_{p}))]$
is given by

$\displaystyle[({\mathcal{H}}^{{\mathbf{R}}},-h,(-\epsilon_{0},-\epsilon_{1},-\epsilon_{2},\ldots,-\epsilon_{q},-e_{1},\ldots,-e_{p}))].$

For $q\geq 0$, the inverse element is also described in a similar way to Lemma
7 by

$\displaystyle[({\mathcal{H}}^{{\mathbf{R}}},h,(-\epsilon_{0},\epsilon_{1},\epsilon_{2},\ldots,\epsilon_{q},e_{1},\ldots,e_{p}))].$

The suspension isomorphism for the $KO$ groups is given as follows Kar (78).

######  Definition 17 (Bott element)

We define the Bott element in $KO^{0,-1}(D^{1},S^{0})=KO^{1}(D^{1},S^{0})$ by

$\beta_{{\mathbf{R}}}:=[(\underline{{\mathbf{R}}},x\cdot)]\in
KO^{0,-1}(D^{1},S^{0})=KO^{1}(D^{1},S^{0}),$

where $\underline{{\mathbf{R}}}$ denotes a trivial line bundle over $D^{1}$
with the fiber ${\mathbf{R}}$, and the operator $x\cdot$ is multiplication of
$x$, which is the coordinate for $D^{1}=\\{x:-1\leq x\leq 1\\}$.

######  Definition 18 (Suspension)

For $\alpha\in KO^{i}(X,A)$, the exterior product with the Bott element
$\alpha\cdot\beta_{{\mathbf{R}}}\in KO^{i+1}((X,A)\times(D^{1},S^{0}))$ is
called the suspension.

The explicit form of the exterior product for $\alpha\in KO^{p,q}(X,A)$ with
$q\geq 0$ is given by

$\displaystyle[({\mathcal{H}}^{{\mathbf{R}}},h,\epsilon_{0},\cdots,\epsilon_{q},e_{1},\cdots,e_{p})]\cdot\beta_{{\mathbf{R}}}=$
$\displaystyle[(p^{*}_{X}{\mathcal{H}}^{{\mathbf{R}}},h+x\epsilon_{0},\epsilon_{1},\cdots,\epsilon_{q},e_{1},\cdots,e_{p})]$
$\displaystyle\in KO^{p,q-1}((X,A)\times(D^{1},S^{0})).$

The next theorem is a part of the Bott periodicity theorem.

######  Theorem 2.4 (Suspension isomorphism/Bott periodicity)

The following suspension map is an isomorphism with respect to the left
$KO^{*}(X,A)$ module.

$KO^{i}(X,A)\to
KO^{i+1}((X,A)\times(D^{1},S^{0})),\qquad\alpha\to\alpha\cdot\beta_{{\mathbf{R}}}.$

Here $\beta_{{\mathbf{R}}}$ is the Bott element.

The definitions of $K^{i}(X,A)$ or $KO^{i}(X,A)$ for $i\neq 0$ with Clifford
algebra symmetry also allow unbounded operators. Note that each equivalence
class does NOT always have representative elements with a finite rank
${\mathcal{H}}$ or ${\mathcal{H}}^{{\mathbf{R}}}$.

##  3 Lattice approximation of Dirac operator and the main theorem

In this section, we present our main theorem in this work where we compare two
Dirac operators, one is defined on a continuum torus, and the other is a
lattice approximation. We construct the two Dirac operators and treat them
together with the associated Hilbert bundles as the elements of the $K^{1}$
group. The main theorem states that they are in the same equivalence class to
represent the same $K^{1}$ group element when the lattice spacing is
sufficiently small.

###  3.1 Set up in continuum theory

We consider an $n$-dimensional torus $T^{n}={\mathbf{R}}^{n}/{\mathbf{Z}}^{n}$
and a complex vector bundle $E\to T^{n}$ equipped with a Hermitian fiber
metric. Therefore, the structure group $G$ is unitary. This bundle $E$ can be
twisted and the projection to the base space is denoted by $\pi:E\to T^{n}$.
Note that $E$ may contain not only the color degrees of freedom, which is
taken in some representation of the structure group $G$, but also those of
spinors and flavors in physics. Here we do not specify them unless they
potentially cause a confusion.

We summarize the details of the setup in the continuum theory below.

  * •

$T^{n}={\mathbf{R}}^{n}/{\mathbf{Z}}^{n}$. We set the size $L=1$ in every
direction and assign a flat metric.

  * •

$E\to T^{n}$: a complex rank $r$ vector bundle with a smooth Hermitian fiber
metric. We denote its structure group by $G$ which is unitary. For smooth
sections $u,v$ of $E$ and the inner product $(*,*)$ in the fiber space, we
denote their inner product by

$\langle u|v\rangle:=\int_{T^{n}}(u,v)d^{n}x,\quad||v||_{L^{2}}:={\langle
v|v\rangle}^{1/2},$

where $d^{n}x$ denotes the volume element of $T^{n}$ with the standard flat
metric.

  * •

$\gamma:E\to E$: ${\mathbf{Z}}/2$-grading self-adjoint orthogonal operator
satisfying $\gamma^{2}={\rm id}$.

  * •

$c_{i}:E\to E$ $(i=1,\ldots,n)$ : the self-adjoint orthogonal operators with
the condition $c_{i}^{2}={\rm id}$, which represent the Clifford algebra. They
satisfy $\\{\gamma,c_{i}\\}=0$ and $\\{c_{i},c_{j}\\}=0\,(i\neq j)$.

  * •

A fixed smooth connection on $E$ which preserves the above structure. The
connection determines the covariant derivative that we denote in the $i$-th
direction by $\nabla_{i}^{\rm cont.}$. Its explicit form will be given later.

  * •

A complex Hilbert space ${\mathcal{H}}$ defined by a completion of the smooth
sections of $E$ by the norm $||*||_{L^{2}}$,

${\mathcal{H}}:=L^{2}(E).$

This is an infinite-dimensional complex Hilbert space212121 The elements of
${\mathcal{H}}$ coincide with the equivalence classes of the $L^{2}$-finite
functions where the functions are equivalent when they match almost
everywhere. This coincidence can be understood as follows. Let us take an
element in our Hilbert space ${\mathcal{H}}$ defined by completion, which is
defined by a Cauchy series $u_{1},u_{2},u_{3},\ldots$ of smooth sections with
respect to the $L^{2}$ norm. When we take a subseries of it, they converge
almost everywhere. Namely, the set of non-convergent points has measure zero
and the subseries converges to a measurable function with a finite $L^{2}$
norm. When we take two such subseries, they match almost everywhere.
Conversely, for a $L^{2}$-finite measurable function, there exists a Cauchy
series $u_{1},u_{2},u_{3},\ldots$ as above. For another series
$v_{1},v_{2},v_{3},\ldots$, the two series represent the same element of the
completion. Namely, the alternate series
$u_{1},v_{1},u_{2},v_{2},u_{3},v_{3}\ldots$ is also a Cauchy series. .

  * •

The $L_{1}^{2}$ norm defined as follows.

$||v||_{L^{2}_{1}}:=\Big{(}||v||^{2}_{L^{2}}+\frac{1}{m_{0}^{2}}\sum_{i=1}^{n}\int_{T^{n}}|\nabla_{i}^{\rm
cont.}v|^{2}d^{n}x\Big{)}^{1/2},$

where $m_{0}$ is an arbitrary real number, which is often taken as a typical
scale of the physical system we focus on. Note that there are non-smooth
elements in ${\mathcal{H}}$ whose $L_{1}^{2}$ norm is not finite222222 To be
precise, the following statement is known to be not always true. For a Cauchy
series $u_{1},u_{2},u_{3},\ldots$ of smooth sections in ${\mathcal{H}}$, there
exists a Cauchy series $v_{1},v_{2},v_{3},\ldots$ with a finite $L^{2}_{1}$
norm such that the alternate series
$u_{1},v_{1},u_{2},v_{2},u_{3},v_{3}\ldots$ is a Cauchy series with respect to
the $L^{2}$ norm (When this statement is true, the series
$u_{1},u_{2},u_{3},\ldots$ converges to an element with a finite $L^{2}_{1}$
norm.). . Therefore, the $L_{1}^{2}$ norm is necessary in order to define the
domain of the Dirac operator. Here we have made a simplest choice from several
different definitions232323 The other definitions are given, for example, by
local parallel transports with respect to the fixed connection, or by
differentials of a flat trivial bundle into which $E$ is embedded. .

  * •

$L^{2}_{1}(E)$: the completion of smooth sections of $E$ with respect to the
$L_{1}^{2}$ norm $||*||_{L^{2}_{1}}$. There exists a natural injection map

$L^{2}_{1}(E)\to{\mathcal{H}}=L^{2}(E).$

###### Definition 19

The Dirac operator on $T^{n}$ is defined by

$\gamma D_{T^{n}}:L^{2}_{1}(E)\to{\mathcal{H}},\qquad\gamma
D_{T^{n}}v=\gamma\sum_{i}c_{i}\nabla_{i}^{\rm cont.}v,$

for $v\in L_{1}^{2}(E)$.

Since $D_{T^{n}}$ is an unbounded operator, Remark 3 applies in the following.
Note that $D_{T^{n}}$ anticommutes with a self-adjoint operator $\gamma:E\to
E$ which satisfies $\gamma^{2}={\rm id}$. Here we also introduce a self-
adjoint mass term $m{\rm id}:E\to E$ with $m\in{\mathbf{R}}$. In the
following, we denote the operator by $m$ for simplicity. To make a $K$-group
element, we consider $\gamma(D_{T^{n}}+m)$, which is self-adjoint242424 Note
that the $K$-groups can be defined with skew-adjoint operators AS69b , but we
employ a physicist-friendly notation where the Dirac operator is skew-adjoint
while the mass term is self-adjoint and only $\gamma(D_{T^{n}}+m)$ is self-
adjoint. . The standard Fredholm index of $D_{T^{n}}$ is given by

${\rm index}:=\operatorname{trace}\gamma|_{\operatorname{Ker}(\gamma
D_{T^{n}})}\in{\mathbf{Z}}.$

This index characterizes the element

$[(L^{2}(E),\gamma D_{T^{n}},\gamma)]\in K^{0}(\\{{\rm
pt}\\},\emptyset)\cong{\mathbf{Z}}.$

In the same way as in our discussion in the previous section, the index is
equal to the spectral flow, characterizing

$[(p^{*}(L^{2}(E)),\gamma(D_{T^{n}}+m))]]\in
K^{1}(D^{1},S^{0})\cong{\mathbf{Z}},$

where the parameter $m$ denotes the coordinate for $D^{1}=[-M,M]\;(M>0)$ and
$p^{*}$ is the pullback of the projection to the massless point,
$p:[-M,M]\to\\{m=0\\}$.

###  3.2 Wilson Dirac operator in lattice theory

Here we summarize the setup on a lattice. All the quantities below depend on
the lattice spacing $a$. But the $a$ dependence is suppressed for simplicity
except for those we need to manifestly distinguish from the corresponding
continuum quantities.

##### Data in lattice theory

We construct a lattice data for each $N\in{\mathbf{Z}}_{>0}$ from those in
continuum theory defined above.

  * •

A flat square lattice $T_{a}^{n}=a{\mathbf{Z}}^{n}/{\mathbf{Z}}^{n}\subset
T^{n}$ with the lattice spacing $a=\frac{1}{N}$. Note that the lattice size
$Na=1$ is fixed.

  * •

$E_{a}:=E|_{T^{n}_{a}}$.

  * •

$\gamma:E_{a}\to E_{a}$: the restriction of the continuum
${\mathbf{Z}}/2$-grading operator onto $E\to E_{a}$.

  * •

$c_{i}:E_{a}\to E_{a}$ $(i=1,\ldots n)$ : the restriction of the continuum
$c_{i}$’s onto $E\to E_{a}$.

  * •

$U_{i}$ $(i=1,\ldots,n)$ defined by the parallel transport on $E$ from each
lattice point to its nearest-neighbor site in the negative $i$-th direction.
The explicit form of the parallel transport is presented in Sec. 3.3. We
assume that $U_{i}$ is compatible with the Clifford operator $c_{j}$ (for any
$j=1,\ldots n$) and $\gamma$.

Differential operation is approximated by the forward difference operator

$\nabla_{i}:=\frac{U_{i}-{\rm id}}{a}:\Gamma(E_{a})\to\Gamma(E_{a}),$

or its conjugate,

$-\nabla^{*}_{i}=\frac{{\rm id}-U_{i}^{-1}}{a}:\Gamma(E_{a})\to\Gamma(E_{a}),$

which is the backward difference operator.

  * •

${\mathcal{H}}_{a}:=L^{2}(E_{a})$ where the $L^{2}$ norm is defined by

$||v_{a}||_{L^{2}}:=\Big{(}a^{n}\sum_{z\in
T^{n}_{a}}|v_{a}(z)|^{2}\Big{)}^{1/2},$

for $v_{a}\in{\mathcal{H}}_{a}$.

Using $\nabla_{i}$, we also define the $L_{1}^{2}$ norm on the lattice by

$||v_{a}||_{L^{2}_{1}}:=\left[||v_{a}||^{2}_{L^{2}}+\frac{a^{n}}{m_{0}^{2}}\sum_{z\in
T^{n}_{a}}\sum_{i=1}^{n}|(\nabla_{i}v_{a})(z)|^{2}\right]^{1/2}.$

With a fixed lattice spacing, there is no essential difference between the
$L^{2}$-finite space and the $L^{2}_{1}$-finite space. But in the continuum
limit $a\to 0$, the two converge to the different continuum counterparts.

Note that $\nabla_{i}$ and $-\nabla^{*}_{i}$ are not skew-adjoint, while their
continuum counterpart $\nabla^{\rm cont.}_{i}$ is. In order to make a skew-
adjoint discretization of $\nabla^{\rm cont.}_{i}$, we take their “average” to
define a naive skew-adjoint lattice Dirac operator,

$D_{a}:=\sum_{i}D_{i},\quad D_{i}:=c_{i}\frac{\nabla_{i}-\nabla^{*}_{i}}{2}.$

As is mentioned in the introduction, the operator $D_{a}$ has doubler modes.
To avoid their appearance, we introduce the Wilson term Wil (77) defined by

$W:=\sum_{i}W_{i},\quad W_{i}:=-\frac{\nabla_{i}+\nabla^{*}_{i}}{2}$

For $W_{i}$, the following lemma holds.

###### Lemma 14

$\nabla_{i}^{*}\nabla_{i}=\nabla_{i}\nabla_{i}^{*}=\frac{1}{a^{2}}({\rm
id}-U_{i}-U_{i}^{-1}+{\rm id})=-\frac{1}{a}(\nabla_{i}+\nabla_{i}^{*})$

or we can write

$W_{i}=\frac{a}{2}\nabla_{i}^{*}\nabla_{i},$

which is semi-positive and so does the Wilson term $W=\sum_{i}W_{i}$.

###### Definition 20

The Wilson-Dirac operator Wil (77) is defined by

$D_{W,a}:=D_{a}+W.$

Note that $\gamma D_{W,a}$ is Hermitian but the ${\mathbf{Z}}/2$ grading
structure is lost: $\\{D_{W,a},\gamma\\}\neq 0$.

For the modes with small momenta, the Wilson term is a good approximation of
the Laplacian operator with a coefficient proportional to the lattice spacing
$a$, whose effect vanishes in the $a\to 0$ limit. However, it is essential to
eliminate the doubler modes with wave numbers near the cutoff $\sim 1/a$.

Mathematically, the Wilson term guarantees the “ellipticity”. In fact, we will
prove a theorem, which corresponds to the Gårding’s inequality in the next
section.

Since it is difficult to maintain the ${\mathbf{Z}}/2$-grading operator
$\gamma$ on the lattice, the $K^{1}(D^{1},S^{0})$ group is more useful than
$K^{0}(\\{\rm pt\\},\emptyset)$ in the following analysis.

###  3.3 Generalized link variables

So far we have considered a general set of link variables given by
$\\{U_{i}\\}$. To identify $E_{a}$ as an approximation of the continuum vector
bundle $E$ with the rank $r$, we need to relate these link variables and
connection in $E$. In the literature, one often gives a connection vector
field, i.e., a distribution of the horizontal directions on $E$ first, and
then the link variables are given by the Wilson line (or parallel transport)
between the two lattice points. Here we take an opposite direction. We first
introduce the link variables between two points on the continuum torus, which
we call the generalized link variables, and then we determine the connection
from them. With this approach, we can avoid details of the local
trivializations of $E$ and ambiguity in the paths to define the Wilson lines.

Let us fix a small constant $a_{0}>0$ and consider a thin stripe $W\subset
T^{n}\times T^{n}$: $W=\\{(x,y)|\;x\in T^{n},y\in T^{n},|x-y|<a_{0}\\}$. Let
$\pi_{1}:W\to T^{n}$ be the projection map onto the first component:
$(x,y)\mapsto x$ and $\pi_{2}:W\to T^{n}$ be that onto the second component:
$(x,y)\mapsto y$.

We define the general link variable by a smooth section $U$ of ${\rm
Hom}(\pi^{*}_{2}E,\pi_{1}^{*}E)$, i.e.,

$U(x,y)\in{\rm Hom}(E_{y},E_{x}),$

which depends smoothly on $x$ and $y$. We assume that $U(x,y)$ satisfies the
conditions252525The standard parallel transport or Wilson line given by a
smooth connection on the shortest path between $x$ and $y$ satisfies the both
conditions. For our main theorem, the second condition $U(y,x)=U(x,y)^{-1}$ is
not essential but simplify the analysis. $U(x,x)={\rm id}$ and
$U(y,x)=U(x,y)^{-1}$. We also assume that the action of link variables are
compatible with the fiber metric.

The generalized link variables $U(x,y)$ must commute with all the symmetry
operations on $E$ such as the ${\mathbf{Z}}_{2}$ grading operator, the
Clifford generators, and so on. Let us consider a symmetry group $G_{\rm
sym.}$ and the operation of the element $\alpha\in G_{\rm sym.}$ The operation
is represented by a set of operators acting on $E$, for example,
$O_{\alpha}=\\{O_{\alpha}(x)\in End(E_{x})\\}$ which smoothly depends on $x$
and preserves the fiber metric (and thus the operation is unitary), to which
we require

$U(x,y)O_{\alpha}(y)=O_{\alpha}(x)U(x,y).$

We can fix the connection of $E$ by identifying $U(x,y)$ as the generator of
the parallel transport. The local connection field can be obtained from
$U(x,y)$ taking $y\to x$ limit. Choosing a local trivialization
$\varphi:\Omega_{\alpha}\times\mathbf{C}^{r}\to\pi^{-1}(U_{\alpha})$ on an
open patch $x,y\in\Omega_{\alpha}\subset T^{n}$, the logarithm of $U(x,y)$ can
be expanded as

$\varphi^{-1}(x)U(x,y)\varphi(y)=\exp\left[-A_{i}\left(\frac{x+y}{2}\right)(x-y)^{i}+R(x-y)\right],$

where we identify the skew-Hermitian matrix $A_{i}(x)=\lim_{y\to
x}A_{i}((x+y)/2):{\mathbf{R}}^{n}\to M_{r}(\bf{C})$ as the local connection
gauge field and the residual262626 The residual $R(x-y)$ is a series
$R(x-y)=\sum_{i,j}A^{(2)}_{ij}((x+y)/2)(x-y)^{i}(x-y)^{j}+\sum_{i,j,k}A_{ijk}^{(3)}((x+y)/2)(x-y)^{i}(x-y)^{j}(x-y)^{k}+\cdots$
where $A^{(l)}_{ij\cdots}:{\mathbf{R}}^{n}\to M_{r}(\bf{C})$ are skew-
Hermitian matrices because $U(x,y)$ is unitary. The condition
$U(y,x)=U(x,y)^{-1}$ implies that the sign flip
$A^{(l)}_{ij\cdots}\to-A^{(l)}_{ij\cdots}$ and the coordinate swap
$x\leftrightarrow y$ are equivalent. Therefore, $A^{(l)}_{ij\cdots}$ with even
$l$ must be zero.  is suppressed to $|R(x-y)|\leq const.|x-y|^{3}$. The
covariant derivative is defined as $\varphi^{-1}(x)\nabla^{\rm
cont.}_{i}\varphi(x)=\partial_{x_{i}}+A_{i}(x)$, with which the connection is
globally determined on $E$.

Changing the local trivialization from $\varphi(x)$ to $\varphi^{\prime}(x)$
corresponds to the gauge transformation. The gauge variable
$g(x)=\varphi^{\prime-1}(x)\varphi(x)$ is in some representation of the
structure group $G$. The link variables, local operators, and the connection
gauge field transform as

$\displaystyle\varphi^{-1}(x)U(x,y)\varphi(y)$ $\displaystyle\to
g(x)\varphi^{-1}(x)U(x,y)\varphi(y)g^{-1}(y),$
$\displaystyle\;\;\varphi^{-1}(x)O_{\alpha}(x)\varphi(x)$ $\displaystyle\to
g(x)\varphi^{-1}(x)O_{\alpha}(x)\varphi(x)g^{-1}(x),$ $\displaystyle A_{i}(x)$
$\displaystyle\to g(x)[A_{i}(x)+\partial_{x_{i}}]g(x)^{-1}.$

In the same way, the triangle Wilson loop $U_{W}(x,y,z)=U(x,y)U(y,z)U(z,x)$ is
evaluated as

$\displaystyle\varphi^{-1}(x)U_{W}(x,y,z)\varphi(x)$
$\displaystyle=e^{-A_{i}\left(\frac{x+y}{2}\right)(x-y)^{i}+\cdots}e^{-A_{i}\left(\frac{y+z}{2}\right)(y-z)^{i}+\cdots}e^{-A_{i}\left(\frac{z+x}{2}\right)(z-x)^{i}+\cdots}$
$\displaystyle=\exp\left[-\frac{1}{2}\sum_{i,j}\partial_{x_{j}}A_{i}(x)\\{(x^{i}-y^{i})(y^{j}-x^{j})\right.$
$\displaystyle+(y^{i}-z^{i})(y^{j}-x^{j})+(y^{i}-z^{i})(z^{j}-x^{j})+(z^{i}-x^{i})(z^{j}-x^{j})\\}$
$\displaystyle+\frac{1}{2}[A_{i}(x),A_{j}(x)]\\{(x^{i}-y^{i})(y^{j}-z^{j})+(x^{i}-y^{i})(z^{j}-x^{j})$
$\displaystyle\left.+(y^{i}-z^{i})(z^{j}-x^{j})\\}+\cdots\right]$
$\displaystyle=\exp\left[\frac{1}{2}F_{ij}(x)(x-y)^{i}(x-z)^{j}+R^{\prime}(x,y,z)\right]$

where $|R^{\prime}(x,y,z)|\leq const.(|x-y|^{3}+|y-z|^{3}+|z-x|^{3})$. Here
$F_{ij}(x)$ is the curvature tensor at $x$.

It is also important to note that the continuum covariant derivative of the
link variable is suppressed to $O(|x-y|)$:

$\displaystyle\varphi^{-1}(x)\nabla^{\rm cont.}_{i}U(x,y)\varphi(y)$
$\displaystyle=\lim_{\Delta x_{i}\to 0}\frac{U(x,x+\Delta
x_{i}e_{i})U(x+\Delta x_{i}e_{i},y)-U(x,y)}{\Delta x_{i}}$
$\displaystyle=\lim_{\Delta x_{i}\to 0}\frac{U_{W}(x,x+\Delta
x_{i}e_{i},y)-{\rm id}}{\Delta x_{i}}U(x,y)$
$\displaystyle=-\frac{1}{2}F_{ij}(x)(x-y)^{j}+R^{\prime\prime}(x,y),$

where $|R^{\prime\prime}(x,y)|\leq const.|x-y|^{3}$.

Therefore, it is useful to evaluate the following bound:

###### Lemma 15

There exists a positive constant $F$ such that the inequality

$|{\rm id}-U_{W}(x,y,z)|\leq F|x-y||x-z|,$

is satisfied for any $\\{(x,y,z)|(x,y),(y,z),(z,x)\in W\\}$ and so is

$|\nabla^{\rm cont.}_{i}U(x,y)|\leq F|x-y|,$

for any $(x,y)\in W$.

Any link variables on a finite lattice with the lattice spacing $a\leq a_{0}$
are simply determined by choosing $x$ to be one site $z$ and $y$ to be its
nearest neighbor $z+e_{i}a$ in the $i$-th direction:

$U(z,z+e_{i}a).$

We can identify the operator $U_{i}$ in the Wilson Dirac operator as the
parallel transport determined by $\\{U(z,z+e_{i}a)\\}$, and $U_{i}^{-1}$ as
that by $\\{U(z-e_{i}a,z)^{-1}\\}$. Specifically the forward and backward
covariant differences on $\phi\in L^{2}_{1}(E_{a})$ are given by

$\displaystyle\nabla_{i}\phi(z)$ $\displaystyle=\frac{U_{i}-{\rm
id}}{a}\phi(z)=\frac{1}{a}\left[U(z,z+e_{i}a)\phi(z+e_{i}a)-\phi(z)\right],$
$\displaystyle-\nabla_{i}^{*}\phi(z)$ $\displaystyle=\frac{{\rm
id}-U_{i}^{-1}}{a}\phi(z)=\frac{1}{a}\left[\phi(z)-U(z-e_{i}a,z)^{-1}\phi(z-e_{i}a)\right],$

respectively. It is then not difficult to confirm that $D_{W,a}$ transforms
covariantly:

$\varphi^{-1}(z)D_{W,a}(z,z^{\prime})\varphi(z^{\prime})\to
g(z)\varphi^{-1}(z)D_{W,a}(z,z^{\prime})\varphi(z^{\prime})g^{-1}(z^{\prime})\;\;\;z,z^{\prime}\in
T^{n}_{a},$

under the gauge transformation.

So far we have not looked into the finer structure of the vector bundle $E$
(and $E_{a}$). In physics on a flat torus, $E$ is given by a tensor product
bundle,

$E=E_{c}\otimes S\otimes\underline{F},$

where $E_{c}$, which is generally twisted, denotes the color degrees of
freedom in some representation of the structure group $G$. As a simple
example, let us take $G=SU(N_{c})$ and choose its fundamental representation
with $N_{c}$ dimensions. The other two bundles $S$ and $\underline{F}$ are
those of spinor (with rank $r_{S}=2^{[n/2]}$) and flavor ($r_{F}$) degrees of
freedom, respectively. On a flat torus, both bundles $S$ and $\underline{F}$
are trivial. The fermion field at $x\in T^{n}$ with the total
$r=N_{c}r_{S}r_{F}$ degrees of freedom is often expressed by
$\psi^{c}_{sf}(x)$ in a local trivialization of $E$ around $x$ where $c,s,f$
are color, spinor and flavor indices, respectively.

In this tensor product bundle case, the generalized link variables are given
by

$U(x,y)=U_{c}(x,y)\otimes{\rm id}_{S}\otimes{\rm id}_{\underline{F}},$

where

$U_{c}(x,y)\in{\rm Hom}((E_{c})_{y},(E_{c})_{x}).$

With the local trivialization
$\varphi_{c}:\Omega_{\alpha}\times\mathbf{C}^{r_{c}}\to\pi_{c}^{-1}(\Omega_{\alpha})$
on a patch $\Omega_{\alpha}\subset T^{n}$ with the projection
$\pi_{c}:E_{c}\to T^{n}$, $\varphi_{c}(x)^{-1}U_{c}(x,y)\varphi_{c}(y)\in
M_{N_{c}}(\mathbf{C})$ corresponds to the standard link variables in physics.
In the same way, the gauge field in the same local trivialization is

$A_{i}(x)=A^{c}_{i}(x)\otimes{\rm id}_{S}\otimes{\rm id}_{\underline{F}},$

where $A^{c}_{i}(x)$ is a anti-Hermitian $N_{c}\times N_{c}$ matrix, as well
as the gauge transformation function is given by

$g(x)=g_{c}(x)\otimes{\rm id}_{S}\otimes{\rm id}_{\underline{F}},$

with $g_{c}(x)\in M_{N_{c}}(\mathbf{C})$. Finally we express the Clifford
generator by

$c_{i}={\rm id}_{c}\otimes\gamma_{i}\otimes{\rm id}_{\underline{F}},$

where $\gamma_{i}$ is the standard $r_{S}\times r_{S}$ Dirac matrix,
satisfying $\gamma_{i}^{2}={\rm id}_{S}$.

###  3.4 Main theorem

We fix a constant $M>0$ and put

$X:=[-M,M],\qquad A:=\partial X=\\{-M,M\\}.$

Note that $X$ includes the massless point $m=0$ where the continuum Dirac
operator may develop zero modes.

Taking $m\in X$ as a parameter, we consider the following two one-parameter
families

$\gamma D^{m}:=\gamma(D_{T^{n}}+m),\qquad\gamma
D_{W,a}^{m}:=\gamma(D_{W,a}+m),$

whose domains are

$\gamma D^{m}:L^{2}_{1}(E)\to{\mathcal{H}},\qquad\gamma
D_{W,a}^{m}:L^{2}_{1}(E_{a})\to{\mathcal{H}}_{a}.$

###### Lemma 16

$\gamma D^{m}$ has the following properties.

  * •

$\gamma D^{m}$ is a Fredholm operator for any $m\in X=[-M,M]$.

  * •

$\operatorname{Ker}(\gamma D^{m})=\\{0\\}$ for $m\in A=\\{-M,M\\}$272727
$\operatorname{Ker}(\gamma D^{m})$ is defined as the set of all smooth
solutions to the massive Dirac equation. In general, the weak solutions of
$\gamma D^{m}$ are smooth. Here the weak solution means a measurable section
$\phi$ of the Hilbert bundle which satisfies $\langle(\gamma
D^{m})^{*}\beta|\phi\rangle=0$ for arbitrary smooth section $\beta$, where
$(\gamma D^{m})^{*}$ is a formal adjoint of $\gamma D^{m}$. Since $(\gamma
D^{m})^{*}=\gamma D^{m}$ in our set up, the set of all smooth solutions equals
to the set of all weak solutions..

###### Proof

For any $m\in{\mathbf{R}}$, $\gamma D^{m}$ is Fredholm from a general
argument. Since $\\{c_{i},\gamma\\}=0$, $\gamma D_{T^{n}}$ and $\gamma$
anticommute and $(\gamma D^{m})^{2}=(\gamma D_{T^{n}})^{2}+(\gamma
m)^{2}=(\gamma D_{T^{n}})^{2}+m^{2}$. For any
$\phi\in\operatorname{Ker}(\gamma D^{m})$,

$0=\langle\phi|(\gamma D^{m})^{2}|\phi\rangle=\langle\phi|(\gamma
D_{T^{n}})^{2}+m^{2}|\phi\rangle=||\gamma
D_{T^{n}}\phi||_{L^{2}}^{2}+m^{2}||\phi||_{L^{2}}^{2}.$

Noting that each term above is positive, we conclude that $\phi=0$ for $m\in
A$.

###### Theorem 3.1

Let us define a lattice-continuum combined Dirac operator $H_{\rm com}(m,t)$
which acts on ${\mathcal{H}}_{a}\oplus L^{2}_{1}(E)$ by

$H_{\rm com}(m,t):=\left(\begin{array}[]{cc}-\gamma D_{W,a}^{m}&tf_{a}^{*}\\\
tf_{a}&\gamma D^{m}\end{array}\right)=\left(\begin{array}[]{cc}-\gamma
D_{W,a}&0\\\ 0&\gamma
D_{T^{n}}\end{array}\right)+m\left(\begin{array}[]{cc}-\gamma&0\\\
0&\gamma\end{array}\right)+t\left(\begin{array}[]{cc}0&f_{a}^{*}\\\
f_{a}&0\end{array}\right).$

There exist $a_{1}>0$ and a linear map
$f_{a}:{\mathcal{H}}_{a}\to{\mathcal{H}}$ such that for arbitrary lattice
spacing $a=1/N$ satisfying $0<a\leq a_{1}$, the followings hold.

  * •

For any $m\in X=[-M,M]$ the kernel of $H_{\rm com}(m,t=1)$ is trivial.

  * •

For any $m\in A=\\{-M,M\\}$, the kernel of $H_{\rm com}(m,t)$ is trivial at
any $t\in[0,1]$.

We illustrate in Fig. 1 the staple-shaped parameter region in the $m$-$t$
plane where we will prove that $H_{\rm com}(m,t)$ defined above is invertible.

Figure 1:  The stape-shaped parameter region in the $m$-$t$ plane where we
prove that the lattice-continuum combined Dirac operator $H_{\rm com}(m,t)$ is
invertible.

Then the theorem below follows 282828Here, $\gamma D_{T^{n}}$ is an unbounded
operator. We can replace it by, for instance, $\gamma D_{T^{n}}/[(\gamma
D_{T^{n}})^{2}+M_{0}^{2}]^{1/2}$ with some mass parameter $M_{0}$ which is
independent of $a$ to make a bounded operator to which the theorems directly
apply..

###### Theorem 3.2

(Index of lattice Dirac operator) For a sufficiently small lattice spacing
$a=1/N$, the following equivalence holds.

$[(p^{*}{\mathcal{H}},\gamma(D_{T^{n}}+m)]=[(p^{*}{\mathcal{H}}_{a},\gamma(D_{W,a}+m)]\in
K^{1}(D^{1},S^{0}),$

where $p^{*}$ is the pullback of the projection map $p:[-M,M]\to\\{m=0\\}$.

Note here by the suspension isomorphism that

$K^{1}(D^{1},S^{0})\cong
K^{0}(\\{*\\},\emptyset)\quad[(p^{*}{\mathcal{H}},\gamma(D_{T^{n}}+m)]\leftrightarrow[({\mathcal{H}},\gamma
D_{T^{n}},\gamma)]$

holds. The original Fredholm index of the Dirac operator for the given
continuum data characterizes the right-hand side. We denote it by ${\rm
Ind}[\gamma D_{T^{n}}]$.

Note here that the spectral flow of the Hermitian massive Dirac operator can
be expressed by the variation of the $\eta$ invariant. Therefore, we have

${\rm sf}({\mathcal{H}}_{a},\gamma D^{m}_{W,a})=\frac{1}{2}\left[\eta(\gamma
D^{m}_{W,a}(m=+M))-\eta(\gamma D^{m}_{W,a}(m=-M))\right].$

Also, we can show that $\eta(\gamma D^{m}_{W,a}(m))=0$ for any $m>0$ (see
Appendix C ). Therefore we obtain a theorem below.

###### Theorem 3.3

(Index and eta invariant of Wilson Dirac operator) For a sufficiently small
lattice spacing $a=1/N$, the following equality holds.

${\rm Ind}[\gamma D_{T^{n}}]=-\frac{1}{2}\eta(\gamma D^{m}_{W,a}(m=-M)).$

Before going into the proof, let us discuss why $K^{1}(D^{1},S^{0})$ works
while $K^{0}(\\{{\rm pt}\\},\emptyset)$ does not. Suppose the original
continuum operator $\gamma D_{T^{n}}$ has $\nu_{+}$ zero modes with the
chirality $\gamma=1$ and $\nu_{-}$ zeros with $\gamma=-1$. These $\nu_{\pm}$
zero modes are also the eigenstates of $\gamma(D_{T^{n}}+m)$ with the
eigenvalue $\pm m$. Every other eigenvalue $\lambda_{m}$ whose absolute value
is greater than $|m|$ appears in a pair with $-\lambda_{m}$, since the
eigenmode $\psi_{\lambda_{m}}$ satisfies

$\displaystyle\gamma(D_{T^{n}}+m)\psi_{\lambda_{m}}$
$\displaystyle=\lambda_{m}\psi_{\lambda_{m}},$
$\displaystyle\gamma(D_{T^{n}}+m)(D_{T^{n}})\psi_{\lambda_{m}}$
$\displaystyle=-\lambda_{m}(D_{T^{n}})\psi_{\lambda_{m}},$

which follows from the anti-commutation relation
$\\{\gamma(D_{T^{n}}+m),D_{T^{n}}\\}=0$. Since
$[\gamma(D_{T^{n}}+m)]^{2}=(\gamma D_{T^{n}})^{2}+m^{2}$, The eigenvalues are
$\lambda_{m}=\pm\sqrt{\lambda_{1}^{2}+m^{2}}$.

The Dirac eigenvalue spectrum as a function of $m$ is illustrated in Fig. 2.
The upper panel shows the original Dirac eigenvalues in the continuum theory,
while the lower panel shows that of a lattice approximation of it, where the
chiral symmetry is broken and the Dirac operator does not anticommutes with
$\gamma$. While it is difficult to define the massless point and to identify
the number of zero modes, the spectral flow can still be identified by the
lines crossing zero between $m\in[-M,M]$. $K^{0}(\\{{\rm pt}\\},\emptyset)$
corresponds to counting the points at $m=0$, while $K^{1}(D^{1},S^{0})$ counts
the lines between $-M$ and $+M$. The latter is easier and stabler against
variation of the Dirac operators and violation of the chiral symmetry.

Figure 2:  Dirac eigenvalue spectrum as a function of $m$. The upper panel
illustrates the original one in the continuum theory, while the lower panel
shows that of a lattice approximation where the chiral symmetry is broken. The
spectral flow can still be identified while it is difficult to count the zero
eigenvalues at $m=0$.

##  4 Proof of the main theorem

###  4.1 Maps between continuum and lattice Hilbert spaces

Let us construct a linear map between the infinite-dimensional Hilbert bundle
${\mathcal{H}}$ in continuum theory and the finite-dimensional Hilbert bundle
${\mathcal{H}}_{a}$ on the lattice.

We denote the coordinate on $T^{n}$ by $x=(x_{1},\ldots x_{n})$ with the
standard flat metric where each component is in the range $0\leq x_{i}\leq 1$
and $x_{i}=0$ and $x_{i}=1$ are identified. On the lattice, we denote the
lattice coordinate by $z=(z_{1},\ldots z_{n})$ where each component $z_{i}$ is
an integer multiplied by $a$ less than $Na=1$. Let $E\to T^{n}$ be a complex
rank-$r$ vector bundle equipped with a Hermitian metric. We may set an open
covering of $T^{n}$ by $\\{U_{\alpha}\\}$ on which we fix a local
trivialization of the bundle but the properties below do not depend on details
of $\\{U_{\alpha}\\}$.

Let us define a map $\rho_{a}:[0,1]\to{\mathbf{R}}$ by
$\rho_{a}(t):=\frac{1}{a}\max\\{0,1-t/a,1-(1-t)/a\\}$ and consider its multi-
dimensional product as a function of $x\in T^{n}$,

$\rho_{a}(x):=\prod_{i=1}^{n}\rho_{a}(x_{i}).$

We will use this $\rho_{a}(x)$ as a cut-off function and the following
properties are essential for the proof of the main theorem.

Denoting the partial derivative with respect to $x_{i}$ by $\partial_{i}$, we
have for the lattice site $z\in T_{a}^{n}$

$\partial_{i}\rho_{a}(x-z)=\left\\{\begin{array}[]{ll}+\frac{1}{a^{2}}\prod_{j\neq
i}\rho_{a}(x_{j}-z_{j})&\qquad(-a<x_{i}-z_{i}<0)\\\
-\frac{1}{a^{2}}\prod_{j\neq
i}\rho_{a}(x_{j}-z_{j})&\qquad(0<x_{i}-z_{i}<a)\\\
0&\qquad(|x_{i}-z_{i}|>a)\end{array}.\right.$

###### Lemma 17

For arbitrary $x\in T^{n}$,

$a^{n}\sum_{z\in T^{n}_{a}}\rho_{a}(x-z)=1.$

and for arbitrary $z\in T^{n}_{a}$,

$\int_{x\in T^{n}}\rho_{a}(x-z)d^{n}x=1.$

###### Lemma 18

For $B=\\{0\\}\cup\\{\pm e_{k}\mid 1\leq k\leq n\\}$,

$a^{n}\sum_{e\in B}\int_{x\in T^{n}}\rho_{a}(x)\rho_{a}(x-ae)d^{n}x=1.$

Using the above cut-off function, we construct the lattice field from the
continuum counterpart in the neighborhood. For a given continuum field
$\psi_{1}(x)\in L^{2}(E)$, we define the lattice field $\phi_{1}$ by

$\phi_{1}(z):=\int_{x\in T^{n}}\rho_{a}(z-x)U(x,z)^{-1}\psi_{1}(x)d^{n}x.$

Note that $\rho_{a}(z-x)$ is nonzero only when $x$ is inside a unit hypercube
of the lattice where $z$ is one of the vertices. We denote this map by
$f_{a}^{*}:\psi_{1}\to\phi_{1}$, which is mathematically easier to evaluate
than its conjugate defined below.

Next we define the map from a lattice field $\phi\in L^{2}(E_{a})$ to the
continuum Hilbert space,

$f_{a}:\phi\to\psi$

where $\psi$ is given by

$\psi(x):=a^{n}\sum_{z\in T^{n}_{a}}\rho_{a}(x-z)U(x,z)\phi(z).$

Note again that $\rho_{a}(x-z)$ is nonzero only when $|x_{i}-z_{i}|<a$ for all
$i$.

Since the following two are equal

$\displaystyle a^{n}\sum_{z\in T^{n}_{a}}\phi(z)^{*}\phi_{1}(z)$
$\displaystyle=a^{n}\sum_{z\in T^{n}_{a}}\int_{x\in
T^{n}}\phi(z)^{*}\rho_{a}(z-x)U(x,z)^{-1}\psi_{1}(x)d^{n}x,\quad$
$\displaystyle\int_{x\in T^{n}}\psi(x)^{*}\psi_{1}(x)d^{n}x$
$\displaystyle=a^{n}\int_{x\in T^{n}}\sum_{z\in
T^{n}_{a}}\phi(z)^{*}U(x,z)^{-1}\rho_{a}(x-z)\psi_{1}(x)d^{n}x,$

we confirm that $f_{a}$ and $f_{a}^{*}$ are conjugates to each other. From the
above equality indicating $\langle\phi|f_{a}^{*}\psi_{1}\rangle=\langle
f_{a}\phi|\psi_{1}\rangle$, $f_{a}$ and $f_{a}^{*}$ have the same finite
operator norm $||f_{a}||=||f_{a}^{*}||$.

###  4.2 Key properties of $f_{a},f_{a}^{*}$ and $D_{W,a},D_{T^{n}}$

Here we summarize the key properties of $f_{a},f_{a}^{*}$ and
$D_{W,a},D_{T^{n}}$ used in the proof of our main theorem. As the proofs of
the key propositions are rather technical, we separate them in Appendix D.

######  Proposition 5 (Boundedness of $f_{a}$ and $f_{a}^{*}$)

For $k=0,1$, the operator norm of $f_{a}:L^{2}_{k}(E_{a})\to L^{2}_{k}(E)$ is
uniformly bounded with respect to $a$. Likewise, the operator norm of
$f_{a}^{*}:L^{2}_{k}(E)\to L^{2}_{k}(E_{a})$ is uniformly bounded with respect
to $a$.

######  Proposition 6 (Continuum limit of $f_{a}^{*}f_{a}$)

There exists $C>0$ which is independent of $a$ such that for any $a$ and
$v_{a}\in{\mathcal{H}}_{a}$

$||f_{a}^{*}f_{a}v_{a}-v_{a}||_{L^{2}}^{2}\leq Ca||v_{a}||_{L^{2}_{1}}^{2}.$

Intuitively, the continuum limit of $f_{a}^{*}f_{a}$ is the identity for
$v_{a}\in L^{2}_{1}(E_{a})$.

######  Proposition 7 (Convergence of $f_{a}f_{a}^{*}\to id$)

For arbitrary $\psi\in C^{\infty}(E)\subset{\mathcal{H}}$, in the limit $a\to
0$, $f_{a}f_{a}^{*}\psi$ strongly converges to $\psi$:

$f_{a}f_{a}^{*}\psi\longrightarrow\psi\quad(L^{2}).$

In particular, for arbitrary $\psi,\psi^{\prime}\in
C^{\infty}(E)\subset{\mathcal{H}}$,

$\langle
f_{a}^{*}\psi^{\prime}|f_{a}^{*}\psi\rangle\to\langle\psi^{\prime}|\psi\rangle$

in the $a\to 0$ limit.

We note that $f_{a}f_{a}^{*}$ is mathematically easier to handle than
$f_{a}^{*}f_{a}$. For the latter, a precise evaluation of the scaling with
respect to the lattice spacing $a$ is essential.

######  Proposition 8 (Convergence of $D_{W,a}\to D_{T^{n}}$)

For arbitrary $\psi\in C^{\infty}(E)\subset{\mathcal{H}}$, in the limit $a\to
0$,

$f_{a}(D_{W,a})^{*}f_{a}^{*}\psi\longrightarrow(D_{T^{n}})^{*}\psi\quad(L^{2}).$

Namely, for arbitrary $\psi,\psi^{\prime}\in C^{\infty}(E)$,

$\langle f_{a}^{*}\psi^{\prime}|(\gamma
D_{W,a})^{*}|f_{a}^{*}\psi\rangle\to\langle\psi^{\prime}|(\gamma
D_{T^{n}})^{*}|\psi\rangle$

in the $a\to 0$ limit.

###  4.3 A priori estimate for lattice Dirac operator

The following theorem corresponds to the elliptic estimate or Gårding’s
inequality in the continuum theory.

######  Theorem 4.1 (A priori estimate for the Wilson Dirac operator)

There exist two positive $a$-independent constants $a_{2}>0$ and $C>0$ such
that the following inequality uniformly holds for any finite lattice spacing
satisfying $0<a\leq a_{2}$ and arbitrary $\phi\in\Gamma(E_{a})$.

$\sum_{i}||\nabla_{i}\phi||^{2}\leq 2||\gamma
D_{W,a}\phi||^{2}+C||\phi||^{2}.$

###### Proof

Let us decompose $(\gamma D_{W,a})^{2}$ as follows.

$(\gamma D_{W,a})^{2}=\sum_{i}[(\gamma
D_{i})^{2}+W_{i}^{2}]+R_{DD}+R_{WD}+R_{WW},$

where

$R_{DD}=\sum_{i<j}\\{\gamma D_{i},\gamma D_{j}\\},\quad
R_{WD}=\gamma\sum_{i,j}[W_{i},\gamma D_{j}],\quad R_{WW}=\sum_{i\neq
j}W_{i}W_{j}.$

In the simplest case where all $U_{i}$ are the identity, $R_{DD}=R_{WD}=0$ and
$W_{i}$ and $W_{j}$ commute. Moreover, $R_{WW}$ becomes a semi-positive
operator, which leads to an inequality

$(\gamma
D_{W,a})^{2}\geq\sum_{i}W_{i}^{2}=\sum_{i}[\nabla_{i}-(-\nabla^{*}_{i})]^{2}.$

From this we have

$||(\gamma
D_{W,a})\phi||^{2}\geq\sum_{i}||\nabla_{i}\phi-(-\nabla^{*}_{i})\phi||^{2}.$

This indicates that when the left-hand side for $\phi$ is small, the forward
difference operator $\nabla_{i}$ and the backward one $(-\nabla^{*}_{i})$ are
close to each other, with respect to the $L^{2}$ norm.

Our goal is to achieve a similar inequality for a general case. The following
qualities which hold for any non-flat $U_{i}$’s are useful.

$\displaystyle(\gamma D_{i})^{2}+W_{i}^{2}$
$\displaystyle=-\Big{\\{}\frac{\nabla_{i}-\nabla_{i}^{*}}{2}\Big{\\}}^{2}+\Big{\\{}\frac{\nabla_{i}+\nabla_{i}^{*}}{2}\Big{\\}}^{2}=\frac{1}{2}(\nabla_{i}\nabla_{i}^{*}+\nabla_{i}^{*}\nabla_{i}),$
(3) $\displaystyle\\{\gamma D_{i},\gamma D_{j}\\}$
$\displaystyle=-\frac{c_{i}c_{j}}{4}[\nabla_{i}-\nabla_{i}^{*},\nabla_{j}-\nabla_{j}^{*}]\;\;\;\mbox{for
any $i,j$,}$ (4) $\displaystyle[W_{i},\gamma D_{j}]$
$\displaystyle=-\frac{\gamma
c_{j}}{4}[\nabla_{i}+\nabla_{i}^{*},\nabla_{j}-\nabla_{j}^{*}]\;\;\;\mbox{for
any $i,j$,}$ (5) $\displaystyle W_{i}W_{j}$
$\displaystyle=\frac{a}{2}\nabla_{i}^{*}\nabla_{i}\frac{1}{2}(-\nabla_{j}-\nabla_{j}^{*})$
$\displaystyle=\frac{a}{2}\nabla_{i}^{*}[\nabla_{i},\frac{1}{2}(-\nabla_{j}-\nabla_{j}^{*})]+\frac{a}{2}\nabla_{i}^{*}\frac{1}{2}(-\nabla_{j}-\nabla_{j}^{*})\nabla_{i}$
$\displaystyle=-\frac{a}{4}R_{ij}+\frac{a^{2}}{4}\nabla_{i}^{*}\nabla_{j}^{*}\nabla_{j}\nabla_{i}.$
(6)

Here we have used a definition

$R_{ij}:=\nabla_{i}^{*}([\nabla_{i},\nabla_{j}]+[\nabla_{i},\nabla_{j}^{*}])$

and a formula $\nabla_{j}+\nabla_{j}^{*}=-(a/2)\nabla_{j}^{*}\nabla_{j}$.

Note in the above equations that (3) and the second term of (4.3) are semi-
positive and other parts are all expressed by the commutators among the
difference operators which vanishes in the flat case.

The following lemma holds.

###### Lemma 19

There exists an $a$-independent constant $C_{0}>0$ such that the operator
norms are uniformly bounded by $C_{0}$:

$||[\nabla_{i},\nabla_{j}]||,||[\nabla_{i},\nabla_{j}^{*}]||,||[\nabla_{i}^{*},\nabla_{j}^{*}]||\leq
C_{0},\;\;\;i,j=1,2,\cdots n.$

From (4) (5) and (4.3), the following corollary holds.

###### Corollary 1

For arbitrary $\phi\in\Gamma(E_{a})$,

$\displaystyle|\langle\phi|\\{\gamma D_{i},\gamma D_{j}\\}|\phi\rangle|$
$\displaystyle\leq$ $\displaystyle C_{0}||\phi||^{2}\quad(i<j),$
$\displaystyle|\langle\phi|[W_{i},\gamma D_{j}]|\phi\rangle|$
$\displaystyle\leq$ $\displaystyle C_{0}||\phi||^{2},$
$\displaystyle|\langle\phi|R_{ij}|\phi\rangle|$ $\displaystyle\leq$
$\displaystyle 2C_{0}||\nabla_{i}\phi||||\phi||\quad(i\neq j)$

hold.

The first two immediately follow from (4) and (5). For the last inequality, we
use a partial integration together with (4.3).

Let us summarize the computation so far done.

###### Lemma 20

Let us decompose $(\gamma D_{W,a})^{2}$ as

$(\gamma
D_{W,a})^{2}=\frac{1}{2}\sum_{i}(\nabla_{i}^{*}\nabla_{i}+\nabla_{i}\nabla_{i}^{*})+\frac{a^{2}}{4}(\nabla_{i}\nabla_{j})^{*}(\nabla_{i}\nabla_{j})+\tilde{R}$

where

$\tilde{R}=R_{DD}+R_{WD}-(a/4)\sum_{i\neq j}R_{ij},$

which satisfies the following inequalities. There exist two positive constants
$a_{1}>0$ and $C>0$ which were determined by $n$ and $C_{0}$ only such that
for any $0<a\leq a_{1}$

$\langle\phi|\tilde{R}|\phi\rangle\leq\frac{1}{2}\Big{(}\sum_{i}||\nabla_{i}\phi||^{2}+C||\phi||^{2}\Big{)}.$

###### Proof

From Collorary 1 we have

$\langle\phi|\tilde{R}|\phi\rangle\leq\frac{n(n-1)}{2}C_{0}||\phi||^{2}+n^{2}C_{0}||\phi||^{2}+\frac{(n-1)a}{2}C_{0}\sum_{i}||\nabla_{i}\phi||||\phi||.$

For the last term, we can use $2a||\nabla_{i}\phi||||\phi||\leq
a^{2}||\nabla_{i}\phi||^{2}+||\phi||^{2}$. Setting $a_{1}$ such that
$nC_{0}a_{1}^{2}\leq 2$ and taking $C>(7n^{2}/2-3n/2)C_{0}$, the lemma holds.

###### Proof

From Lemma 20,

$||\gamma
D_{W,a}\phi||^{2}=\frac{1}{2}\sum_{i}(||\nabla_{i}\phi||^{2}+||\nabla_{i}^{*}\phi||^{2})+\frac{a^{2}}{4}||\nabla_{i}\nabla_{j}\phi||^{2}+\langle\phi|\tilde{R}|\phi\rangle.$

Noting that $\nabla_{i}^{*}=-U_{i}^{*}\nabla_{i}$,
$||\nabla_{i}^{*}\phi||=||\nabla_{i}\phi||$ and the second term is non-
negative, we have

$||\gamma
D_{W,a}\phi||^{2}\geq\sum_{i}||\nabla_{i}\phi||^{2}-|\langle\phi|\tilde{R}|\phi\rangle|$

Using the estimate given in Lemma 20, we obtain

$||\gamma
D_{W,a}\phi||^{2}\geq\sum_{i}||\nabla_{i}\phi||^{2}-\frac{1}{2}\big{(}\sum_{i}||\nabla_{i}\phi||^{2}+C||\phi||^{2}\big{)}.$

###  4.4 Assumptions for the proof by contradiction

Suppose that Theorem 3.1 does not hold. Then there should be a series labeled
by $i=1,2,\ldots$ composed by

  * •

$a_{i}=1/N_{i}\to 0$,

  * •

$(m_{i},t_{i})\in\\{(m,t):m\in X,t=1\\}\cup\\{(m,t):m\in A,0\leq t\leq 1\\}$,
which is the staple-shaped region shown in Fig. 1,

  * •

$(v_{a_{i}},w_{a_{i}})\in{\mathcal{H}}_{a_{i}}\oplus L^{2}_{1}(E)$292929From a
general argument of regularity, $w_{a_{i}}$ is an element of $L^{2}_{1}(E)$.,

which satisfy

  1. 1.

$||v_{a_{i}}||_{L^{2}}^{2}+||w_{a_{i}}||_{L^{2}}^{2}=1$

and

  2. 2.

$\left(\begin{array}[]{cc}-\gamma D_{W,{a_{i}}}^{m_{i}}&t_{i}f_{a_{i}}^{*}\\\
t_{i}f_{a_{i}}&\gamma
D_{T^{n}}\end{array}\right)\left(\begin{array}[]{c}v_{a_{i}}\\\
w_{a_{i}}\end{array}\right)=0.$

Taking subsequences, we can assume without loss of generality that
$(m_{i},t_{i})$ converges to a point $(m_{\infty},t_{\infty})\neq(0,0)$:

$m_{i}\to m_{\infty},\qquad t_{i}\to t_{\infty}.$

From Proposition 5 and Theorem 4.1,
$f_{a_{i}}v_{a_{i}},w_{a_{i}}\in{\mathcal{H}}$ are uniformly $L^{2}_{1}$
finite. Therefore taking subsequences, we can assume without loss of
generality that $f_{a_{i}}v_{a_{i}}$ weakly converges in $L^{2}_{1}(E)$ to a
vector $v_{\infty}\in{\mathcal{H}}$ and $w_{a_{i}}$ weakly converges in
$L^{2}_{1}(E)$ to another vector $w_{\infty}\in{\mathcal{H}}$303030 Note in
any finite sequence in a Hilbert space that there exists a weakly converging
subsequence. . From the Rellich theorem, the weakly converging series in the
$L_{1}^{2}$ space is a strongly converging series in $L^{2}$.

We also note that $t_{i}f_{a_{i}}^{*}w_{a_{i}}$ has a finite norm and so does
$\gamma D_{W,{a_{i}}}^{m_{i}}v_{a_{i}}$. This indicates that $v_{a_{i}}$ has a
finite $L^{2}_{1}$ norm.

###  4.5 Weak limit of the equation

###### Lemma 21

With the assumptions made in the previous subsection for the proof by
contradiction, the following equation must hold in the continuum limit.

$\left(\begin{array}[]{cc}-\gamma D^{m_{\infty}}&t_{\infty}\\\
t_{\infty}&\gamma
D^{m_{\infty}}\end{array}\right)\left(\begin{array}[]{c}v_{\infty}\\\
w_{\infty}\end{array}\right)=0.$

###### Proof

Our goal is to show the two equations,

$\displaystyle\left\\{\begin{array}[]{rlcc}-\gamma
D^{m_{\infty}}v_{\infty}&+t_{\infty}w_{\infty}&=&0\\\
t_{\infty}v_{\infty}&+\gamma D^{m_{\infty}}w_{\infty}&=&0\end{array}\right..$
(9)

Note that

$\left\\{\begin{array}[]{rlcc}-\gamma
D^{m_{i}}_{W,a_{i}}v_{a_{i}}&+t_{i}f_{a_{i}}^{*}w_{a_{i}}&=&0\\\
t_{i}f_{a_{i}}v_{a_{i}}&+\gamma D^{m_{i}}w_{a_{i}}&=&0\end{array}\right.$

holds as shown in the assumptions in Sec.4.4.

The weak limit in $L^{2}$ of the second equation is that of Eq. (9).

Multiplying $f_{a_{i}}$ to the first equation, we obtain an equation in
${\mathcal{H}}$,

$-f_{a_{i}}\gamma
D_{W,a_{i}}^{m_{i}}v_{a_{i}}+t_{i}f_{a_{i}}f_{a_{i}}^{*}w_{a_{i}}=0.$ (10)

For arbitrary $\beta\in C^{\infty}(E)$, we have

$\langle\beta|t_{i}f_{a_{i}}f_{a_{i}}^{*}w_{a_{i}}\rangle=t_{i}\langle
f_{a_{i}}f_{a_{i}}^{*}\beta|w_{a_{i}}\rangle\to
t_{\infty}\langle\beta|w_{\infty}\rangle,$

where the last arrow indicates the limit $t_{i}\to t_{\infty}$, strong
convergence of $f_{a_{i}}f_{a_{i}}^{*}\beta\to\beta$ shown in the Proposition
7, and strong convergence of $w_{a_{i}}\to w_{\infty}$ in $L^{2}$. Thus, the
second term of Eq. (10), $t_{i}f_{a_{i}}f_{a_{i}}^{*}w_{a_{i}}$ weakly
converges to $t_{\infty}w_{\infty}$.

Note for any $\beta_{i}\in L^{2}(E_{a_{i}})$ that

$|\langle\beta_{i}|\left(f_{a_{i}}^{*}f_{a_{i}}|v_{a_{i}}\rangle-|v_{a_{i}}\rangle\right)|\leq\sqrt{||\beta_{i}||_{L^{2}}^{2}||f_{a_{i}}^{*}f_{a_{i}}v_{a_{i}}-v_{a_{i}}||_{L^{2}}^{2}}\leq
C\sqrt{a_{i}||\beta_{i}||_{L^{2}}^{2}||v_{a_{i}}||_{L_{1}^{2}}^{2}}$

from the Proposition 6 with a positive constant $C$. Choosing
$\beta_{i}=(\gamma D_{W,a_{i}}^{m_{i}})^{*}f_{a_{i}}^{*}\beta\in
L^{2}(E_{a_{i}})$ for a $\beta\in C^{\infty}(E)$, we have

$\displaystyle\langle(\gamma
D_{W,a_{i}}^{m_{i}})^{*}f_{a_{i}}^{*}\beta|f_{a_{i}}^{*}f_{a_{i}}v_{a_{i}}\rangle-\langle(\gamma
D_{W,a_{i}}^{m_{i}})^{*}f_{a_{i}}^{*}\beta|v_{a_{i}}\rangle\to 0$

in the $a_{i}\to 0$ limit. Since

$\displaystyle\langle(\gamma
D_{W,a_{i}}^{m_{i}})^{*}f_{a_{i}}^{*}\beta|f_{a_{i}}^{*}f_{a_{i}}v_{a_{i}}\rangle$
$\displaystyle=\langle f_{a_{i}}(\gamma
D_{W,a_{i}}^{m_{i}})^{*}f_{a_{i}}^{*}\beta|f_{a_{i}}v_{a_{i}}\rangle$
$\displaystyle\to\langle(\gamma
D^{m_{i}})^{*}\beta|v_{\infty}\rangle=\langle\beta|\gamma
D^{m_{i}}v_{\infty}\rangle,$

from the Proposition 8 and strong convergence of $f_{a_{i}}v_{a_{i}}\to
v_{\infty}$, and

$\langle(\gamma
D_{W,a_{i}}^{m_{i}})^{*}f_{a_{i}}^{*}\beta|v_{a_{i}}\rangle=\langle\beta|f_{a_{i}}\gamma
D_{W,a_{i}}^{m_{i}}v_{a_{i}}\rangle,$

the first term of Eq. (10) weakly converges to $-\gamma D^{m_{i}}v_{\infty}$.
Thus, Eq. (9) holds.

###### Lemma 22

Under the assumptions in Sec. 4.4,

$||v_{\infty}||_{L^{2}}^{2}+||w_{\infty}||_{L^{2}}^{2}=1$

holds.

###### Proof

Under the assumptions in Sec. 4.4,

$||v_{a_{i}}||_{L^{2}}^{2}+||w_{a_{i}}||_{L^{2}}^{2}=1.$

From the strong convergence of $w_{a_{i}}\to w_{\infty}$ in $L^{2}$, we
have$||w_{a_{i}}||^{2}_{L^{2}}\to||w_{\infty}||_{L^{2}}^{2}$. Since
$f_{a_{i}}v_{a_{i}}$ is uniformly finite in $L^{2}_{1}$ and from the
Proposition 6 we observe

$|||f_{a_{i}}v_{a_{i}}||_{L^{2}}^{2}-||v_{a_{i}}||_{L^{2}}^{2}\to 0$

and from the strong convergence of $f_{a_{i}}v_{a_{i}}\to v_{\infty}$ in
$L^{2}$,

$||f_{a_{i}}v_{a_{i}}||_{L^{2}}^{2}\to||v_{\infty}||^{2}$

holds.

The proof is achieved by contradiction.

From Lemma 21 the vector $(v_{\infty},w_{\infty})^{T}$ in
$L^{2}_{1}(E)\otimes{\mathbf{C}}^{2}$ is an element of the kernel of the
operator

$\left(\begin{array}[]{cc}-\gamma D^{m_{\infty}}&-t_{\infty}\\\
t_{\infty}&\gamma
D^{m_{\infty}}\end{array}\right)=\left(\begin{array}[]{cc}-\gamma
D_{T^{n}}&0\\\ 0&\gamma
D_{T^{n}}\end{array}\right)+\left(\begin{array}[]{cc}-m_{\infty}\gamma&0\\\
0&m_{\infty}\gamma\end{array}\right)+\left(\begin{array}[]{cc}0&-t_{\infty}\\\
t_{\infty}&0\end{array}\right),$

whose three terms all anticommute among one another.

From a general argument that solutions of a smooth elliptic operator are
smooth, $(v_{\infty},w_{\infty})^{T}\in C^{\infty}(E)\otimes{\mathbf{C}}^{2}$.
On the solution, the above operator squared gives zero, too. Since

$\left(\begin{array}[]{cc}-\gamma D^{m_{\infty}}&-t_{\infty}\\\
t_{\infty}&\gamma D^{m_{\infty}}\end{array}\right)^{2}=\gamma
D_{T^{n}}^{2}\otimes\left(\begin{array}[]{cc}{\rm id}&0\\\ 0&{\rm
id}\end{array}\right)+m_{\infty}^{2}\otimes\left(\begin{array}[]{cc}{\rm
id}&0\\\ 0&{\rm
id}\end{array}\right)+t_{\infty}^{2}\otimes\left(\begin{array}[]{cc}{\rm
id}&0\\\ 0&{\rm id}\end{array}\right)$

we have

$\displaystyle 0$ $\displaystyle=$
$\displaystyle(v_{\infty},w_{\infty})\left(\begin{array}[]{cc}-\gamma
D^{m_{\infty}}&-t_{\infty}\\\ t_{\infty}&\gamma
D^{m_{\infty}}\end{array}\right)^{2}\left(\begin{array}[]{c}v_{\infty}\\\
w_{\infty}\end{array}\right)$ $\displaystyle=$ $\displaystyle||\gamma
D_{T^{n}}v_{\infty}||_{L^{2}}^{2}+||\gamma
D_{T^{n}}w_{\infty}||_{L^{2}}^{2}+(m_{\infty}^{2}+t_{\infty}^{2})(||v_{\infty}||_{L^{2}}^{2}+||w_{\infty}||_{L^{2}}^{2})$
$\displaystyle=$ $\displaystyle||\gamma
D_{T^{n}}v_{\infty}||_{L^{2}}^{2}+||\gamma
D_{T^{n}}w_{\infty}||_{L^{2}}^{2}+(t_{\infty}^{2}+m_{\infty}^{2}),$

where we have used Lemma 22. This requires that $t_{\infty}=0$ and
$m_{\infty}=0$, which contradicts the assumption
$(m_{\infty},t_{\infty})\neq(0,0)$, and proves Theorem 3.1, from which
Theorems 3.2 and 3.3 immediately follow.

##  5 Relation to the index of the overlap Dirac operator

In lattice gauge theory, it has been known in even dimensions that the index
can be formulated by the overlap Dirac operator Neu (98) in the same manner as
the Fredholm index. In this section, we compare our new formulation with the
overlap Dirac index, and discuss the consistency.

The overlap Dirac operator is defined by

$D_{\rm ov}:=\frac{1}{a}\left[{\rm id}+\gamma{\rm sgn}(H_{W}(-M))\right],$

where $H_{W}(m)$ is the Hermitian massive Wilson Dirac operator
$H_{W}(m)=\gamma(D_{W}+m)$ and we have set its mass to a negatively large
value $-M=-1/a$.

From the definition, it is easy to see that $D_{\rm ov}$ satisfies the
Ginsparg-Wilson relation GW (82), and a “modified” chiral symmetry Lus (98)
with the relation

$\gamma D_{\rm ov}+D_{\rm ov}\gamma({\rm id}-aD_{\rm ov})=0,$

which corresponds to the anticommutation relation of the continuum Dirac
operator. See Appendix B.3 for a relation to Karoubi’s formulation of the $K$
groups.

The index is defined by a modified ${\mathbf{Z}}_{2}$ grading operator

$\Gamma:=\gamma({\rm id}-aD_{\rm ov}/2)=\frac{\gamma}{2}-\frac{1}{2}{\rm
sgn}(H_{W}(-M)).$

Since $\Gamma$ anticommutes with $H_{\rm ov}=\gamma D_{\rm ov}$, every non-
zero eigenmode $\phi_{i}$ of $H_{\rm ov}$ satisfies
$\langle\phi_{i}|\Gamma\phi_{i}\rangle=0$, including the case
$\Gamma\phi_{i}=0$ (when $D_{\rm ov}\phi_{i}=(2/a)\phi_{i}$), which eliminates
the doubler contributions. Then the index

${\rm Tr}\Gamma={\rm Tr}\Gamma|_{{\rm Ker}D_{\rm ov}},$

converges to the standard Fredholm index in the continuum theoryHLN (98); Neu
(98), where the factor $({\rm id}-aD_{\rm ov}/2)$ plays a similar role to the
heat-kernel regulator of the trace in the conventional analysis. It is
interesting to note that $D_{\rm ov}$ is defined by a unitary operator
$u=\gamma{\rm sgn}(H_{W}(-M))$, which may be useful to consider this index as
an analog of Karoubi’s formulation of the $K$ group Kar (78). See Appendix B.

Noting that ${\rm Tr}\gamma=0$ holds in the finite lattice Hilbert space, the
index can be expressed by

${\rm Tr}\Gamma=-\frac{1}{2}{\rm Tr}\;{\rm
sgn}[H_{W}(-M)]=-\frac{1}{2}\eta(H_{W}(-M)),$

where the right-hand side is nothing but that of Theorem 3.3, which equals to
the spectral flow of the Hermitian Wilson Dirac operator $H_{W}(m)$ from
$m=-M$ to $m=M$. Namely, our formulation of the lattice Dirac operator index
in this work is consistent with the overlap Dirac index.

This equivalence between the overlap Dirac index and the spectral flow of the
massive Wilson Dirac operator has been known in physics Ada (02); KY (99); Lus
(99); Fuj (99); Suz (99) but its mathematical relevance was not rigorously
discussed. In this work, we have given a mathematical role to the massive
Wilson Dirac operator and its Hilbert space on the lattice as an element of
$K^{1}$ group. In the $K^{1}$ group, unlike the standard definition in the
$K^{0}$ group, the ${\mathbf{Z}}_{2}$ grading operator or chiral symmetry play
no role but the Dirac operator spectrum is still able to describe the index
through the suspension isomorphism to the $K^{0}$ group.

##  6 Summary and discussion

In this work, we have shown that the massive Wilson Dirac operator can be
treated as a mathematically well-defined object. Taking its mass term as the
base space of the Hilbert bundle, we have proved that the one parameter family
of the continuum Dirac operator on a flat torus $T^{n}$ and the lattice
approximation belong to the same equivalence class as an element of the
$K^{1}$ group, when the lattice spacing is sufficiently small. This has lead
to the equality between the index of the original continuum massless Dirac
operator and the $\eta$ invariant of the Wilson Dirac operator with a negative
mass.

Our formulation does not require the Ginsparg-Wilson relation or the chirality
operator to evaluate the index, since the $K^{1}$ group does not require the
chiral symmetry from the beginning. For the proof, we have directly compared
the continuum and lattice Dirac operator without passing through the geometric
index. We expect that this physicist-friendly expression using the $K^{1}$
group has wider applications than the standard Fredholm-like formulation of
the index. Here we summarize our outlook in the two possible directions.

The first possible applications is the case with higher symmetries. It will be
straightforward to apply our formulation to the Hilbert space having real or
quaternion structures. The corresponding $KO$ or $KSp$ groups will be defined
in a similar way as discussed in Sec. 2.3. It will also be interesting to
consider the case with a group acting on the base manifold, the case with
symmetries with anti-linear operation, and the family version of the index.

The second application is the case with boundary. In continuum theory some of
the authors and their collaborators have shown in the previous works FOY (17);
FFM+ (20, 22); Fuk (21) that the spectral flow of a Dirac operator with a
position-dependent mass term can describe the index of the massless Dirac
operator on a manifold with boundary. The generalization is nontrivial both
physically and mathematically in that the boundary of the target manifold is
capped to form a closed manifold, and the domain-wall of the mass term is put
instead of the original boundary. The domain-wall is given by the sign
function $\epsilon$ so that the extended part has a positive mass, while the
negative mass is assigned for the original manifold region. The spectral flow
from a trivially massive Dirac operator (with a positive mass $+M$) to that
for the domain-wall fermion was proven to be equal to the original index with
the APS boundary condition. In particular, in even dimensions, the spectral
flow can be given by a difference of the $\eta$ invariant of the massive Dirac
operator at $m=\pm M$:

${\rm Ind}(\gamma D)|_{\rm
APS}=-\left[\frac{1}{2}\eta(\gamma(D+M\epsilon))-\frac{1}{2}\eta(\gamma(D+M))\right],$

where $D$ is the original massless Dirac operator.

We call it a “physicist-friendly” formulation of the index since it does not
require any unphysical boundary condition nor exact chiral symmetry, in
contrast to the original Fredholm index defined with a unphysical boundary
condition by Atiyah-Patodi-Singer to maintain the chiral symmetry. It will be
interesting to study the lattice version of the extension to the case with
boundary. We refer the readers to FKM+ (20) in which a promising result was
already obtained in a perturbative analysis.

###### Acknowledgements.

We thank Mayuko Yamashita for her significant contribution at the early stage
of this work. We thank Sinya Aoki, Yoshio Kikukawa, and Yosuke Kubota for
useful discussions. This work was partly supported by JSPS KAKENHI Grant
Numbers JP21K03222, JP21K03574, JP22H01219, JP23K03387, JP23K22490,
JP23KJ1459.

##  Appendix A Generalizations: $KO$, equivariant, and family versions

We explain how to generalize our argument to various settings.

To formulate our main theorem Theorem 3.7 we fixed our setup in Sections 3.1,
3.2 and 3.3. The setup amounts to the following data
$(E,\gamma,\\{c_{i}\\}_{1\leq i\leq n},U)$, which we will call “$K^{0}({\rm
pt})$ version” because Theorem 3.7 is an equality in $K^{1}(D^{1},S^{0})\cong
K^{0}({\rm pt})$.

##### Data for $K^{0}({\rm pt})$ version

  * 1

$E$ is a smooth complex vector bundle over $T^{n}={\bf R}^{n}/{\bf Z}^{n}$
equipped with inner product.

  * 2

$\gamma$ is a self-adjoint operator on $E$ satisfying $\gamma^{2}=1$.

  * 3

$\\{c_{k}\\}_{1\leq k\leq n}$ are self-adjoint operators on $E$ satisfying
$\\{\gamma,c_{k}\\}=0$, $c_{k}^{2}=1$ and $\\{c_{k},c_{l}\\}=0(k\neq l)$.

  * 4

$U$ is a smooth section of ${\rm Hom}(\pi_{2}^{*}E,\pi_{1}^{*}E)$ on an open
neighborhood $W$ of the diagonal subset of $T^{n}\times T^{n}$ satisfying
$U(x,x)={\rm id}$ and $U(y,x)=U(x,y)^{-1}$ if $(x,y),(y,x)\in W$. We assume
that $U(x,y):E_{y}\to E_{x}$ preserves the inner products.

From this data, for sufficiently small $a=1/N$ $(N\in{\bf N}$), we have the
associated link variables $\\{U_{i}\\}_{i=1,\ldots,n}$ on the lattice
$T^{n}_{a}=a{\bf Z}^{n}/{\bf Z}^{n}\subset T^{n}$, and the associated
continuum covariant derivatives $\\{\nabla_{i}^{\rm cont}\\}_{1\leq i\leq n}$
at the same time.

We can generalize or modify the setting to various directions. For instance,
we have:

##### Data for $KO^{p,q}({\rm pt})$ version

($p,q\geq 0$)

  * 1

$E$ is a smooth real vector bundle over $T^{n}={\bf R}^{n}/{\bf Z}^{n}$
equipped with inner product.

  * 2

$\\{\epsilon^{\prime}_{j}\\}_{0\leq j\leq p}$ are self-adjoint operators on
$E$ and $\\{e^{\prime}_{i}\\}_{1\leq i\leq q}$ are skew-adjoint operators on
$E$ satisfying ${\epsilon^{\prime}_{j}}^{2}=1$ and ${e^{\prime}_{j}}^{2}=-1$.
We assume that they all anticommute each other.

  * 3

$\\{c_{k}\\}_{1\leq k\leq n}$ are self-adjoint operators on $E$ satisfying
$\\{c,c_{k}\\}=0$ for $c\in\\{\epsilon^{\prime}_{j}\\}_{0\leq j\leq
p}\cup\\{e^{\prime}_{i}\\}_{1\leq i\leq q}$, $c_{k}^{2}=1$ and
$\\{c_{k},c_{l}\\}=0\;(k\neq l)$.

  * 4

$U$ is a smooth section of ${\rm Hom}(\pi_{2}^{*}E,\pi_{1}^{*}E)$ on an open
neighborhood $W$ of the diagonal subset of $T^{n}\times T^{n}$ satisfying
$U(x,x)={\rm id}$ and $U(y,x)=U(x,y)^{-1}$ if $(x,y),(y,x)\in W$. We assume
that $U(x,y):E_{y}\to E_{x}$ preserves the inner products.

We put $\gamma:=\epsilon^{\prime}_{0}$ and

$e_{j}:=\gamma\epsilon^{\prime}_{j}\,(1\leq j\leq p),\quad\epsilon_{i}:=\gamma
e^{\prime}_{i}\,(1\leq i\leq q).$

Note that $\gamma^{2}=1$, $e_{j}^{2}=-1$, $\epsilon_{i}^{2}=1$ and they all
anticommute each other 313131 The conversion from
$\epsilon^{\prime}_{j},e^{\prime}_{i}$ to $e_{j},\epsilon_{i}$ looks
artificial and technical, which reflects a notational difference between
mathematics and physics. In this paper we use skew-adjoint Dirac operators
following the standard convention of lattice gauge theory while we use self-
adjoint Fredholm operators to formulate the $K$ groups following the standard
convention in mathematics. If we used self-adjoint Dirac operator $\sum
c^{\prime}_{k}\nabla_{k}$ constructed from mutually anticommuting skew-adjoint
operators $\\{c^{\prime}_{k}\\}$ satisfying ${c^{\prime}_{k}}^{2}=-1$, then we
could directly use our formulation of the $KO$ groups in terms of the data
$({\cal H}^{\bf R},h,\epsilon_{i},e_{j})$. Or if we used the formulation of
the $KO$ group by the data $({\cal H}^{\bf
R},s,\epsilon^{\prime}_{j},e^{\prime}_{i})$ with a skew adjoint Fredholm
operator $s$, then we could directly use our skew-adjoint Dirac operator
$D_{T^{n}}$ and the Wilson Dirac operator $D_{W,q}$ without $\gamma$ factor. .
We write ${\cal H}^{\rm R}$ for the $L^{2}$-sections of $E$, and ${\cal
H}_{a}^{\rm R}$ for the sections of $E_{a}:=E|_{T_{a}^{n}}$. We define
$D_{T^{n}}$ and $D_{W,a}$ by the same formula in the $K^{0}({\rm pt})$
version. Then both $\gamma D_{T^{n}}$ and $\gamma D_{W,a}$ anticommute with
each of $\\{\epsilon_{i}\\}_{1\leq i\leq q}$ and $\\{e_{j}\\}_{1\leq j\leq
p}$. Then our whole arguments go through for this real case with the extra
Clifford action of $\\{\epsilon_{i}\\}_{1\leq i\leq q}$ and
$\\{e_{j}\\}_{1\leq j\leq p}$, and we obtain:

###### Theorem A.1

($KO^{p,q}({\rm pt})$ version for $p,q\geq 0$)

For a sufficiently small lattice spacing $a=1/N$, we have the equality

$[(p^{*}{\cal H}^{\rm
R}),\gamma(D_{T^{n}}+m),\epsilon_{1},\ldots,\epsilon_{q},e_{1},\ldots,e_{p}]=[(p^{*}{\cal
H}_{a}^{\rm
R}),\gamma(D_{W,a}+m),\epsilon_{1},\ldots,\epsilon_{q},e_{1},\ldots,e_{p}]$

in the group $KO^{p,q-1}(D^{1},S^{0})\cong KO^{p,q}({\rm pt})$.

In the above statement we used the suspension isomorphism.

Using the construction of Lemma 2.36, we have a simple bijective
correspondence between the data for the $KO^{p,q}({\rm pt})$ version and that
for the $KO^{p+1,q+1}({\rm pt})$ version in the following way. If
$(E,\\{\epsilon^{\prime}_{j}\\}_{0\leq j\leq p},\\{e^{\prime}_{i}\\}_{1\leq
i\leq q},\\{c_{k}\\}_{1\leq k\leq n},U)$ is a data for $KO^{p,q}({\rm pt})$
version, then the next $(\hat{E},\\{{\hat{\epsilon}}^{\prime}_{j}\\}_{0\leq
j\leq p+1},\\{{\hat{e}}^{\prime}_{i}\\}_{1\leq i\leq
q+1},\\{\hat{c}_{k}\\}_{1\leq k\leq n},U)$ is a data for $KO^{p+1,q+1}({\rm
pt})$ version.

$\hat{E}:=E\otimes{\bf R}^{2},\,{\hat{\epsilon}}^{\prime}_{p+1}:={\rm
id}_{E}\otimes\begin{pmatrix}0&1\\\ 1&0\\\
\end{pmatrix},\,{\hat{e}}^{\prime}_{q+1}:=:{\rm
id}_{E}\otimes\begin{pmatrix}0&-1\\\ 1&0\\\ \end{pmatrix},\,$
${\hat{\epsilon}}^{\prime}_{j}:=\epsilon^{\prime}_{j}\otimes\begin{pmatrix}1&0\\\
0&-1\\\
\end{pmatrix},\,{\hat{e}}^{\prime}_{i}:=e^{\prime}_{i}\otimes\begin{pmatrix}1&0\\\
0&-1\\\ \end{pmatrix},\,\hat{c}_{k}:=c_{k}\otimes\begin{pmatrix}1&0\\\ 0&-1\\\
\end{pmatrix}.$

The converse correspondence is given by taking the invariant part of the
action of $\epsilon^{\prime}_{p+1}e^{\prime}_{q+1}$. It is straightforward to
see that the two mutually corresponding data give two mutually corresponding
equalities in Theorem A.1: one in $KO^{p,q}({\rm pt})$ and the other in
$KO^{p+1,q+1}({\rm id})$.

We remark three consequences of this simple algebraic correspondences.

###### Remark 4

  1. 1.

For any integer $n$ we can regard the “$KO^{p,q}({\rm pt})$ versions” for
$p,q\geq 0$ satisfying $p-q=n$ as a single “$KO^{n}(\rm pt)$ version”.

  2. 2.

The $KO^{p,-1}({\rm pt})$ version, which is without any
$\epsilon^{\prime}_{j}$, can be treated via the $KO^{p+1,0}({\rm pt})$
version.

  3. 3.

The $K^{1}({\rm pt})$ version, whose data is $(E,\\{c_{k}\\}_{1\leq k\leq n})$
without any symmetries $\epsilon^{\prime}_{j},e^{\prime}_{i}$, is treated as
the complex version of the $KO^{0,-1}({\rm pt})$ version.

Using quaternionic vector bundle and quaternionic Hilbert bundle, we have
$KSp^{p,q}$ groups or $KSp^{n}$ groups for $n=p-q$. “The $KSp$ version” and
the equality of indices are formulated and given just by replacing real vector
bundle with quaternionic vector bundle, and real Hilbert bundle with
quaternionic Hilbert bundle. Since there exists a simple algebraic isomorphism
$KSp^{n}\cong KO^{n+4}$, the formulation and statement could be reduced to the
$KO$ versions.

As generalizations of these, when an isometric involution on $T^{n}$ is given,
we could also use two variants : Atiyah’s $KR$ theory and Dupont’s $KQ$ theory
Ati (66); Dup (69).

Similarly we have an equivariant version and family version for the data and
the equality of indices. We can combine various versions to produce more
variants. Since the statements of the theorem about the equality are quite
parallel to the other versions, we just give what kind of data are necessary
to formulate the equivariant version and the family version.

Let $G$ be a finite group acting on the flat torus $T^{n}$. Suppose $G$
preserves the lattices $T^{n}_{a}$, as sets, for infinitely many $a=1/N$. For
example, the group generated by the rotation around a coordinate axis by angle
$\pi/2$, or some nonsymmorphic space group actions satisfy this condition.

##### Data for $K_{G}^{0}({\rm pt})$ version

($G$ preserves infinitely many $T^{n}_{a}$’s)

  * 1

$E$ is a $G$-equivariant smooth complex vector bundle over $T^{n}={\bf
R}^{n}/{\bf Z}^{n}$ equipped with inner product.

  * 2

$\gamma$ is a self-adjoint $G$-equivariant operator on $E$ satisfying
$\gamma^{2}=1$.

  * 3

$\\{c_{k}\\}_{1\leq k\leq n}$ are self-adjoint $G$-equivariant operators on
$E$ satisfying $\\{\gamma,c_{k}\\}=0$, $c_{k}^{2}=1$ and
$\\{c_{k},c_{l}\\}=0(k\neq l)$

  * 4

$U$ is a $G$-invariant smooth section of ${\rm
Hom}(\pi_{2}^{*}E,\pi_{1}^{*}E)$ on an open neighborhood $W$ of the diagonal
subset of $T^{n}\times T^{n}$ satisfying $U(x,x)={\rm id}$ and
$U(y,x)=U(x,y)^{-1}$ if $(x,y),(y,x)\in W$. We assume that $U(x,y):E_{y}\to
E_{x}$ preserves the inner products.

Let $\Omega$ be a compact Hausdorff space. Suppose we have a $T^{n}$-bundle
${\cal T}$ over $\Omega$ whose structure group is $G$ satisfying the condition
for $K_{G}^{0}({\rm pt})$ version. We write $T^{n}_{\omega}$ for the fiber at
$\omega\in\Omega$.

If we have a family of Data for $K^{0}({\rm pt})$ on the each fiber
$T^{n}_{\omega}$ which vary continuously with respect to $\Omega$, then we
have the $K^{0}(\Omega)$ version.

##### Data for $K^{0}(\Omega)$ version

($\Omega$ is compact Hausdorff)

  * 1

$E$ is a continuous complex vector bundle over the total space ${\cal T}$
equipped with inner product.

  * 2

$\gamma$ is a self-adjoint operator on $E$ satisfying $\gamma^{2}=1$.

  * 3

$\\{c_{k}\\}_{1\leq k\leq n}$ are self-adjoint operators on $E$ satisfying
$\\{\gamma,c_{k}\\}=0$, $c_{k}^{2}=1$ and $\\{c_{k},c_{l}\\}=0(k\neq l)$

  * 4

$U$ is a section of ${\rm Hom}(\pi_{2}^{*}E,\pi_{1}^{*}E)$ on an open
neighborhood $W$ of the diagonal subset of ${\cal T}\times_{\Omega}{\cal T}$
satisfying $U(x,x)={\rm id}$ and $U(y,x)=U(x,y)^{-1}$ if $(x,y),(y,x)\in