#  New conjecture on species doubling of lattice fermions

Jun Yumoto  d8521007(at)s.akita-u.ac.jp Department of Mathematical Science,
Akita University, 1-1 Tegata-Gakuen-machi, Akita 010-8502, Japan  Tatsuhiro
Misumi  misumi(at)phys.kindai.ac.jp Department of Physics, Kindai University,
3-4-1 Kowakae, Higashi-osaka, Osaka 577-8502, Japan  Research and Education
Center for Natural Sciences, Keio University, 4-1-1 Hitoshi, Yokohama,
Kanagawa 223-8521, Japan

###### Abstract

We propose a new conjecture on the relation between the species doubling of
lattice fermions and the topology of manifold on which the fermion action is
defined. Our conjecture claims that the maximal number of fermion species on a
finite-volume and finite-spacing lattice defined by discretizing a
$D$-dimensional manifold is equal to the summation of the Betti numbers of the
manifold. We start with reconsidering species doubling of naive fermions on
the lattices whose topologies are torus ($T^{D}$), hyperball ($B^{D}$) and
their direct-product space ($T^{D}\times B^{d}$). We find that the maximal
number of species is in exact agreement with the sum of Betti numbers
$\displaystyle\sum^{D}_{r=0}\beta_{r}$ for these manifolds. Indeed, the $4D$
lattice fermion on torus has up to $16$ species while the sum of Betti numbers
of $T^{4}$ is $16$. This coincidence holds also for the $D$-dimensional
hyperball and their direct-product space $T^{D}\times B^{d}$. We study several
examples of lattice fermions defined on discretized hypersphere ($S^{D}$), and
find that it has up to $2$ species, which is the same number as the sum of
Betti numbers of $S^{D}$. From these facts, we conjecture the equivalence of
the maximal number of fermion species and the summation of Betti numbers. We
discuss a program for proof of the conjecture in terms of Hodge theory and
spectral graph theory.

###### Contents

  1. I Introduction
  2. II Maximal number of species on discretized manifolds
    1. II.1 $D$-dimensional torus $T^{D}$
    2. II.2 $D$-dimensional hyperball $B^{D}$
    3. II.3 ($D+d$)-dimensional product manifold $T^{D}\times B^{d}$
    4. II.4 $D$-dimensional sphere $S^{D}$
  3. III New conjecture
  4. IV Program for proof
    1. IV.1 One-dimensional lattices
    2. IV.2 Higher-dimensional lattices
  5. V Summary and Discussion
  6. A Numerical analysis for $D$-dimensional spheres
    1. A.1 Two-dimensional sphere
    2. A.2 Four-dimensional sphere

##  I Introduction

Quantum field theories are non-perturbatively investigated by use of the
numerical Monte-Carlo simulation based on lattice field theory Wilson:1974sk ;
Creutz:1980zw . One of the most delicate problems in the application is how to
introduce fermionic degrees of freedom Karsten:1980wd ; Nielsen:1980rz ;
Nielsen:1981xu ; Nielsen:1981hk , where we encounter serious problems such as
the reconcilement of a desirable number of flavors and chiral symmetry, the
realization of a single Weyl fermion, and the sign problem of the quark
determinant. The lattice fermion formulations including Wilson fermions
Wilson:1975id , Domain-wall or overlap fermions Kaplan:1992bt ; Shamir:1993zy
; Furman:1994ky ; Neuberger:1998wv ; Ginsparg:1981bj , and staggered fermions
Kogut:1974ag ; Susskind:1976jm ; Kawamoto:1981hw ; Sharatchandra:1981si ;
Golterman:1984cy ; Golterman:1985dz ; Kilcup:1986dg  have been proposed to
bypass the problems and have been broadly used in the lattice simulation.
Apart from them, relatively new approaches have been proposed, including the
generalized Wilson fermions Bietenholz:1999km ; Creutz:2010bm ; Durr:2010ch ;
Durr:2012dw ; Misumi:2012eh ; Cho:2013yha ; Cho:2015ffa ; Durr:2017wfi , the
staggered-Wilson fermions Golterman:1984cy ; Adams:2009eb ; Adams:2010gx ;
Hoelbling:2010jw ; deForcrand:2011ak ; Creutz:2011cd ; Misumi:2011su ;
Follana:2011kh ; deForcrand:2012bm ; Misumi:2012sp ; Misumi:2012eh ;
Durr:2013gp ; Hoelbling:2016qfv ; Zielinski:2017pko , the minimally doubled
fermion Karsten:1981gd ; Wilczek:1987kw ; Creutz:2007af ; Borici:2007kz ;
Bedaque:2008xs ; Bedaque:2008jm ; Capitani:2009yn ; Kimura:2009qe ;
Kimura:2009di ; Creutz:2010cz ; Capitani:2010nn ; Tiburzi:2010bm ;
Kamata:2011jn ; Misumi:2012uu ; Misumi:2012ky ; Capitani:2013zta ;
Capitani:2013iha ; Misumi:2013maa ; Weber:2013tfa ; Weber:2017eds ;
Durr:2020yqa  and the central-branch Wilson fermion Kimura:2011ik ;
Creutz:2011cd ; Misumi:2012eh ; Chowdhury:2013ux .

The non-torus lattice fermion formulation is also an interesting avenue, where
we consider lattices with nonzero-genus topologies. In the previous work of
ours, we introduced spectral graph theory to clarify the number of zero-
eigenvalues of lattice Dirac operators YM2022 . By use of this tool, we made
progress in understanding the number of fermion species on lattices with
arbitrary topologies and no discrete translational symmetry111In the other
context, the Dirac-Kahler fermions on generic manifolds have been
investigated, where the relation between the fermion zero-modes and the Euler
characteristics of the manifold are discussed Catterall:2018lkj ; Butt:2021brl
..

In this work we propose a new conjecture on the relation between the topology
of a manifold and the number of species (doublers) of lattice fermions defined
on a lattice-discretized version of the manifold. All through this work, we
identify the number of "species" as the number of Dirac zero-modes of free
lattice fermions. The claim of our conjecture is that the maximal number of
fermion species on a finite-volume and finite-spacing lattice defined by
discretizing a $D$-dimensional manifold ${\mathcal{M}}$ is identical to the
summation of the Betti numbers of the manifold
$\displaystyle\sum_{r=0}^{D}\beta_{r}({\mathcal{M}})$ as long as the
formulation has several basic properties, including locality,
$\gamma_{5}$-hermiticity and central difference. The $r$-th Betti number is
defined as the rank of the $r$-th homology group $H_{r}({\mathcal{M}})$. We
show that our conjecture holds for lattice fermions on torus ($T^{D}$),
hyperball ($B^{D}$) and their product space $T^{D}\times B^{d}$. We also
investigate lattice fermions defined on discretized hypersphere ($S^{2}$) and
find a empirical fact that they have up to $2$ species, while the the sum of
Betti numbers for $S^{D}$ is $2$ irrespective of dimensions. This explicit
investigation indicates that the conjecture holds even for spherical and non-
hypercubic lattices. In the end of this paper, we discuss the way how to prove
this coincidence in terms of Hodge theory Rabin:1981qj ; Hod41 ; Eck45 ; Dod76
; DP76  and spectral graph theory west2001introduction ; bondy1976graph ;
mieghem_2010 ; Watts1998 ; YM2022 . We there introduce a new viewpoint ithat
the lattice Wilson term corresponds to a sort of Laplacian operator giving
Betti numbers.

The reason we use the terminology, “the maximal number of fermion species", is
two-folded: The number of species depends on the number of lattice sites when
we consider a finite-volume and finite-spacing lattice. Thus, we focus just on
the case that the number of species are “maximal" in this sense given a
lattice-discretized manifold. It also depends on lattice fermion formulations.
Thus, we focus on the case of the “maximal" number of species given a certain
lattice, which is usually realized by adopting the naive fermion formulation.

This paper is constructed as follows: In Sec. II, we present empirical
evidences for our conjecture. In Sec. III, we propose the main conjecture. In
Sec. IV, we discuss a program for the proof of the conjecture. Sec. V is
devoted to the summary and discussion.

##  II Maximal number of species on discretized manifolds

In this section, we present empirical evidences that the maximal number of
species on the discretized manifold is equal to the summation of the Betti
numbers of the continuum manifold. We firstly focus on three types of
manifolds, $D$-dimensional torus $T^{D}$, $D$-dimensional hyperball $B^{D}$,
and ($D+d$)-dimensional cylinder $T^{D}\times B^{d}$. We also present explicit
calculations of fermion species on the discretized sphere $S^{D}$. As
discussed in the introduction, the definition of “the number of fermion
species" is the number of Dirac zero-modes of the lattice Dirac operator
(Dirac matrix) in a free theory, where we have to pay attention to the
difference of sizes of $\gamma$-matrices in different dimensions.

###  II.1 $D$-dimensional torus $T^{D}$

The discretized $D$-dimensional torus is realized as the $D$-dimensional
square (hypercubic) lattice with the periodic boundary condition (PBC) imposed
in each dimension. Hereafter, we denote the discretized $D$-dimensional torus
as ${}^{\ast}T^{D}$ in order to distinguish it from the continuum manifold
$T^{D}$. The discretized torus in two dimensions is schematically depicted in
Fig. 1.

Figure 1: Two-dimensional square lattice with opposite boundaries identified
corresponds to the discretized two-dimensional torus.

The lattice naive-fermion action on ${}^{\ast}T^{D}$, or on $D$-dimensional
square lattice with PBC, is

$S_{T^{D}}=\sum_{n}\sum_{\mu=1}^{D}\bar{\psi}_{n}\gamma_{\mu}D_{\mu}^{({\rm
PBC})}\psi_{n}\equiv\bar{\bm{\psi}}\mathcal{D}^{({\rm PBC})}\bm{\psi}\,,$ (1)

where $D_{\mu}\equiv(T_{+\mu}-T_{-\mu})/2$ with
$T_{\pm\mu}\psi_{n}=U_{n,\pm\hat{\mu}}\psi_{n\pm\hat{\mu}}$ and $\hat{\mu}$ is
a unit vector. In a free theory, we just set $U_{n,\pm\hat{\mu}}=\bm{1}$.
Unless otherwise specified, we consider a free theory hereafter. The sum
$\sum_{n}$ stands for the summation over lattice sites
$n=(n_{1},n_{2},\cdots,n_{D})\in\mathbb{N}^{D}$, whose intervals are $1\leq
n_{\mu}\leq N$. The vector $\bm{\psi}$ is defined as
$\bm{\psi}\equiv\sum_{n}\psi_{n}\ket{n}$ with
$\ket{n}\equiv{\displaystyle\bigotimes_{\mu=1}^{D}}\ket{n_{\mu}}=\ket{n_{1}}\otimes\ket{n_{2}}\otimes\cdots\otimes\ket{n_{D}}$.
Then, the diagonalized Dirac matrix $\mathcal{D}$ is obtained as

$\mathcal{U}^{\dagger}\mathcal{D}^{({\rm
PBC})}\mathcal{U}=\sum_{k}\sum_{\mu}^{D}i\sin\left[\frac{2\pi}{N}\left(k_{\mu}-1\right)\right]\gamma_{\mu}\ket{k}\bra{k}\,,$
(2)

where $k=\left(k_{1},k_{2},\cdots,k_{D}\right)\in[1,N]^{D}$ with
$k_{\mu}\in\mathbb{Z}$. The symbol $\mathcal{U}$ is the unitary matrix defined
as
$\mathcal{U}\equiv{\displaystyle\sum_{n,k}}\left\\{\prod_{\mu}\exp\left[\frac{2\pi
i}{N}\left(n_{\mu}-1\right)\left(k_{\mu}-1\right)\right]\right\\}\ket{n}\bra{k}$
with $\ket{k}\equiv{\displaystyle\bigotimes_{\mu=1}^{D}}\ket{k_{\mu}}$. The
condition that the Dirac matrix has zero-eigenvalues (zero-modes) is simply
given by

$\sum_{\mu}^{D}i\gamma_{\mu}\sin\left[\frac{2\pi}{N}\left(k_{\mu}-1\right)\right]=0\quad\Longrightarrow\quad\sin\left[\frac{2\pi}{N}\left(k_{\mu}-1\right)\right]=0\,,$
(3)

as a consequence of the linear independence of $\gamma$-matrices. It shows
that we have up to $2^{D}$ fermion species on ${}^{\ast}T^{D}$ as long as we
take an even $N$. For an odd $N$, the number of species can decrease, but
never increases. Even if we adopt another fermion formulation such as Wilson
and staggered fermions on the discretized torus, but the number of species for
these cases are certainly smaller than $2^{D}$. Thus, we argue that the
maximal number of species is $2^{D}$ on the discretized torus. Note that we
only consider a finite-volume and finite-spacing lattice here. We never take a
continuum or a thermodynamic limit in this paper.

Meanwhile, the $r$-th Betti number $\beta_{r}(T^{D})$ of the $D$-dimensional
torus $T^{D}$ is obtained as

$\beta_{r}(T^{D})={}_{D}C_{r}\,,$ (4)

where $r$ is an integer from 1 to $D$. For instance, Betti numbers of four-
dimensional torus $T^{4}$ are $\beta_{0}(T^{4})=1,\ \beta_{1}(T^{4})=4,\
\beta_{2}(T^{4})=6,\ \beta_{3}(T^{4})=4,\ \beta_{0}(T^{4})=1$. Then, the
summation of the Betti number in Eq. (4) over $r$ is

$\sum_{r=0}^{D}\beta_{r}(T^{D})=\sum_{r=0}^{D}{}_{D}C_{r}=2^{D}.$ (5)

From these results, we argue that the summation of the Betti numbers for the
$D$-dimensional torus $T^{D}$ is equal to the maximal number of lattice
fermion species on the discretized $D$-dimensional torus ${}^{\ast}T^{D}$.

###  II.2 $D$-dimensional hyperball $B^{D}$

The discreized $D$-dimensional hyperball ${}^{\ast}B^{D}$ is realized as the
$D$-dimensional square lattice with Dirichlet boundary condition (DBC) imposed
in each dimension. The difference from ${}^{\ast}T^{D}$ is that there is a
pair of unconnected boundaries in each direction. The lattice naive-fermion
action on ${}^{\ast}B^{D}$ is

$S_{B^{D}}=\sum_{n}\sum_{\mu=1}^{D}\bar{\psi}_{n}\gamma_{\mu}D_{\mu}^{({\rm
DBC})}\psi_{n}\equiv\bar{\bm{\psi}}\mathcal{D}^{({\rm DBC})}\bm{\psi}.$ (6)

Note that this action is different from Eq. (1) in that there are no hoppings
between the boundaries. The diagonalized Dirac matrix by the unitary matrix
$\mathcal{V}\equiv\sum_{n,k}\left(\prod_{\mu}i^{k_{\mu}}\sin\left[\frac{n_{\mu}k_{\mu}\pi}{N+1}\right]\right)\ket{n}\bra{k}$
is

$\mathcal{V}^{\dagger}\mathcal{D}^{({\rm
DBC})}\mathcal{V}=\sum_{k}\sum_{\mu}^{D}i\cos\left[\frac{k_{\mu}\pi}{N+1}\right]\gamma_{\mu}\ket{k}\bra{k}\,,$
(7)

where $k=\left(k_{1},k_{2},\cdots,k_{D}\right)\in[1,N]^{D}$ with
$k_{\mu}\in\mathbb{Z}$. The Dirac matrix has zero-eigenvalues (zero-modes)
when Eq. (2) satisfies

$\sum_{\mu}^{D}i\gamma_{\mu}\cos\left[\frac{k_{\mu}\pi}{N+1}\right]=0\quad\Longrightarrow\quad\cos\left[\frac{k_{\mu}\pi}{N+1}\right]=0.$
(8)

It means that there is one Dirac zero-mode, or equivalently equivalently a
single species, if we take an odd number as $N$. Eq. (8) has no solution since
$k_{\mu}=\frac{N+1}{2}$ is not an integer if $N$ is even. As with the case of
torus, other fermion formulations never increase the number species. Hence, we
argue that the maximal number of fermion species on ${}^{\ast}B^{D}$ is one.
We also note that we just consider the bulk fermionic degrees of freedom, not
an edge mode.

On the other hand, the $r$-th Betti numbers for the hyperball are given as

$\beta_{r}(B^{D})=\delta_{r0}\,,$ (9)

where $\delta$ stands for the Kronecker delta. The summation of the Betti
numbers in Eq. (9) is

$\sum_{r=0}^{D}\beta_{r}(B^{D})=\sum_{r=0}^{D}\delta_{r0}=1.$ (10)

From these results, we claim that the maximal number of fermion species on
$D$-dimensional lattice-discretized hyperball ${}^{\ast}B^{D}$ is equal to the
summation of the Betti numbers over $0\leq r\leq D$ for the $D$-dimensional
hyperball $B^{D}$.

If we take an infinite-volume limit, the number of naive-fermion species on
$D$-dimensional discretized hyperball ${}^{\ast}B^{D}$ approaches $2^{D}$,
which is the same as that on $D$-dimensional discretized torus. It is notable
that our conjecture is applicable only to the finite-volume lattice.

###  II.3 ($D+d$)-dimensional product manifold $T^{D}\times B^{d}$

The ($D+d$)-dimensional product manifold $T^{D}\times B^{d}$ consists of two
manifolds, the $D$-dimensional torus $T^{D}$ and the $d$-dimensional hyperball
$B^{d}$. It means that the discretized ($D+d$)-dimensional product manifold
${}^{\ast}T^{D}\times{}^{\ast}B^{d}$ consists of the discretized
$D$-dimensional torus ${}^{\ast}T^{D}$ and the discretized $d$-dimensional
hyperball ${}^{\ast}B^{D}$. The ($D+d$)-dimensional square lattice is
identified as ${}^{\ast}T^{D}\times{}^{\ast}B^{d}$ by imposing PBC on the
$D$-dimensions and imposing DBC on the remaining $d$-dimensions. The manifold
in ($1+1$)-dimensions is depicted in Fig. 2.

Figure 2: ($1+1$)-dimensional square lattice with the left and right
boundaries connected is identified as the discretized hollow cylinder.

The lattice naive-fermion action on this lattice is

$S_{T^{D}\times
B^{d}}=\sum_{n}\bar{\psi}_{n}\left[\sum_{\mu=1}^{D}\gamma_{\mu}D_{\mu}^{({\rm
PBC})}+\sum_{\nu=D+1}^{d}\gamma_{\nu}D_{\nu}^{({\rm
DBC})}\right]\psi_{n}\equiv\bar{\bm{\psi}}\mathcal{D}^{(\mathrm{P}+\mathrm{D})}\bm{\psi}\,.$
(11)

We note that, if we add the Wilson term to this action in ($4+1$) dimensions,
it results in the action of Domain-wall fermion. For this case, the zero-modes
of the Dirac matrix emerge as edge modes. The Dirac matrix
$\mathcal{D}^{(\mathrm{P}+\mathrm{D})}$ in Eq. (11) can be diagonalized by the
unitary matrix
$\mathcal{W}\equiv\sum_{k}\left\\{\prod_{\mu}\exp\left[\frac{2\pi
i}{N}\left(n_{\mu}-1\right)\left(k_{\mu}-1\right)\right]\right\\}\left\\{\prod_{\nu}i^{k_{\nu}}\sin\left[\frac{n_{\nu}k_{\nu}\pi}{N+1}\right]\right\\}\ket{n}\bra{k}$
with $\ket{n}\equiv\bigotimes_{\lambda=1}^{D+d}\ket{n_{\lambda}}$ and
$\ket{k}\equiv\bigotimes_{\lambda=1}^{D+d}\ket{k_{\lambda}}$. The diagonalized
Dirac matrix is

$\mathcal{W}^{\dagger}\mathcal{D}^{(\mathrm{P}+\mathrm{D})}\mathcal{W}=i\left\\{\sum_{\mu=1}^{D}\gamma_{\mu}\sin\left[\frac{2\pi}{N}\left(k_{\mu}-1\right)\right]+\sum_{\nu=D+1}^{D+d}\gamma_{\nu}\cos\left[\frac{k_{\nu}\pi}{N+1}\right]\right\\}\ket{k}\bra{k}\,.$
(12)

The condition for Eq. (12) to have zero eigenvalues (zero-modes) is

$\begin{gathered}\sum_{\mu=1}^{D}\gamma_{\mu}\sin\left[\frac{2\pi}{N}\left(k_{\mu}-1\right)\right]+\sum_{\nu=D+1}^{D+d}\gamma_{\nu}\cos\left[\frac{k_{\nu}\pi}{N+1}\right]=0\\\
\quad\Longrightarrow\quad\sin\left[\frac{2\pi}{N}\left(k_{\mu}-1\right)\right]=0,\quad\cos\left[\frac{k_{\nu}\pi}{N+1}\right]=0\,.\end{gathered}$
(13)

It shows that the maximal number of fermion species on
${}^{\ast}T^{D}\times{}^{\ast}B^{d}$ is $2^{D}=2^{D}\times 1$, which is the
product of the maximum numbers of fermion species on ${}^{\ast}T^{D}$ and
${}^{\ast}B^{D}$. The other fermion formulations, including the Domain-wall
setup, never increase the number of species, but can decrease it.

The Betti numbers of the product manifold are

$\beta_{r}(T^{D}\times B^{d})=\begin{cases}{}_{D}C_{r}&(1\leq r\leq D)\\\
0&(r>D)\end{cases}\,.$ (14)

The summation of the $r$-th Betti numbers ($0\leq r\leq D+d$) for the
($D+d$)-dimensional product manifold $T^{D}\times B^{d}$ is

$\sum_{r=0}^{D+d}\beta_{r}(T^{D}\times
B^{d})=\sum_{r=0}^{D}{}_{D}C_{r}=2^{D}.$ (15)

These results indicate that the maximum number of species on
${}^{\ast}T^{D}\times{}^{\ast}B^{d}$ is equal to the summation of the Betti
numbers of $T^{D}\times B^{d}$.

We also note that the maximal number of naive-fermion species approaches
$2^{D+d}$ in an infinite-volume limit. Our statement of the conjecture holds
only for finite-volume cases.

###  II.4 $D$-dimensional sphere $S^{D}$

In the continuum field theory, the fermion action on spheres gives massive
fermionic degrees of freedom since the curvature works as effective mass. It
is, however, not the case on the discretized sphere.

In this subsection, we study the number of fermion species on the discretized
sphere. We empirically show that the maximal number of species on the
discretized sphere is equal to the sum of the Betti number of $D$-dimensional
sphere $S^{D}$.

We begin with the two-dimensional cases. We firstly consider the following
discretized spherical coordinate system for $2$-sphere, labeled by two
integers $(M,N)$:

$x_{3}=r\cos\theta_{2},\qquad x_{2}=r\sin\theta_{2}\cos\theta_{1},\qquad
x_{1}=r\sin\theta_{2}\sin\theta_{1},$ (16)

where $r$ is a radial distance and two angles are discretized as
$\theta_{2}\equiv\left(N-n_{2}\right)\pi/\left(N-1\right),\ \theta_{1}\equiv
2n_{1}\pi/M$ with $n_{1},n_{2}\in\mathbb{N}$ and $n_{1}\in[1,N],\
n_{2}\in[1,M]$. For simplicity, we fix a radial distance as $r=1$. We label
lattice sites as $(n_{1},n_{2})$. Note that there are two special points
$(n_{1},1)$, $(n_{1},N)$ who ignore the hopping in $n_{1}$-direction. We call
the two points the south pole, relabeled as $(0,1)$, and the north pole,
relabeled as $(0,N)$, respectively. By this definition, we consider the naive-
fermion action on the discretized 2-sphere labeled by $(M,N)$ as

$S_{(M,N)}=\sum_{n}\sum_{\mu=1}^{2}\bar{\psi}_{n}\sigma_{\mu}D_{\mu}\psi_{n}\equiv\bm{\bar{\psi}}\mathcal{D}^{(M,N)}\bm{\psi}\,,$
(17)

where $\bm{\psi}$ is a vector defined as
$\bm{\psi}=\sum_{n}\psi_{n}\bm{e}_{n}$ with the standard basis $\bm{e}_{n}$
and $\mathcal{D}^{(M,N)}$ is the Dirac matrix. The sum $\sum_{n}$ stands for
the summation over lattice site $n=(n_{1},n_{2})$. Here, we specify the order
of components $\psi_{n}$ in the vector as
$(0,1)\to(1,2)\to(2,2)\to\cdots\to(M,2)\to(1,3)\to\cdots\to(M,N-1)\to(0,N)$.
Namely, it is expressed as

$\bm{\psi}=\begin{pmatrix}\psi_{(0,1)}\\\ \psi_{(1,2)}\\\ \psi_{(2,2)}\\\
\vdots\\\ \psi_{(M,2)}\\\ \psi_{(1,3)}\\\ \vdots\\\ \psi_{(M,N-1)}\\\
\psi_{(0,N)}\end{pmatrix}\,.$ (18)

The Dirac matrix $\mathcal{D}^{(M,N)}$ is expressed as

$\mathcal{D}^{(M,N)}=\frac{1}{2}\begin{pmatrix}I_{N-2}\otimes P_{M}\\\
&O_{2}\end{pmatrix}\otimes\sigma_{1}+\frac{1}{2}\begin{pmatrix}D_{N-2}\otimes
I_{M}&V_{M(N-2),2}\\\
-V_{M(N-2),2}^{\dagger}&O_{2}\end{pmatrix}\otimes\sigma_{2}\,,$ (19)

where $\sigma_{i}$ is the Pauli matrix and $I_{k}$ and $O_{k}$ are the $k$
dimensional identity matrix and null matrix, respectively. $P_{M}$ is the
$M\times M$ difference matrix with the periodic boundary condition and
$D_{N-2}$ is the $(N-2)\times(N-2)$ difference matrix with the Dirichlet
boundary condition as

$P_{M}=\begin{pmatrix}0&1&0&&0&0&-1\\\ -1&0&1&\cdots&0&0&0\\\ 0&-1&0&&0&0&0\\\
&\vdots&&\ddots&&\vdots\\\ 0&0&0&&0&1&0\\\ 0&0&0&\cdots&-1&0&1\\\
1&0&0&&0&-1&0\end{pmatrix},\quad D_{N-2}=\begin{pmatrix}0&1&0\\\ -1&0&1\\\
0&-1&0\\\ &&&\ddots\\\ &&&&0&1&0\\\ &&&&-1&0&1\\\ &&&&0&-1&0\end{pmatrix}.$
(20)

$V_{M(N-2),2}$ is given by

$V_{M(N-2),2}\equiv\begin{pmatrix}1&0\\\ 0&0\\\
\lx@intercol\hfil\vdots\hfil\lx@intercol\\\ 0&0\\\
0&-1\end{pmatrix}_{N-2,2}\otimes\begin{pmatrix}1\\\ 1\\\ \vdots\\\ 1\\\
1\end{pmatrix}_{M,1}\,,$ (21)

where the subscripts $j,k$ of the matrix $(\ )_{j,k}$ stand for the row and
column sizes.

In the example of the discretized 2-sphere $(M,3)$, the action is

$S_{(M,3)}=\bm{\bar{\psi}}\mathcal{D}^{(M,3)}\bm{\psi}\,.$ (22)

The discretized 2-spheres $(4,3)$ and $(6,3)$ are depicted in Fig. 3. As seen
from the figures, these cases correspond to a sort of the triangulation of
2-sphere.

(a) Discretized 2-sphere $(4,3)$.

(b) Discretized 2-sphere $(6,3)$.

Figure 3:  The discretized 2-spheres $(4,3)$ and $(6,3)$ are depicted. Red
points stand for lattice sites and lines between the red points stand for
links.

The Dirac matirx $\mathcal{D}^{(M,3)}$ is

$\mathcal{D}^{(M,3)}=\frac{1}{2}\begin{pmatrix}P_{M}\\\
&O_{2}\end{pmatrix}\otimes\sigma_{1}+\frac{1}{2}\begin{pmatrix}I_{M}&V_{M,2}\\\
-V_{M,2}^{\dagger}&O_{2}\end{pmatrix}\otimes\sigma_{2}\,,$ (23)

where $V_{M,2}$ is

$V_{M,2}=\begin{pmatrix}1&-1\end{pmatrix}_{1,2}\otimes\begin{pmatrix}1\\\ 1\\\
\vdots\\\ 1\\\ 1\end{pmatrix}_{M,1}\,.$ (24)

This matrix is diagonalized by the unitary matrix

$\mathcal{U}^{(M,3)}\equiv U^{(M,3)}\otimes I_{2}\,,$ (25)

where $I_{2}$ is a $2\times 2$ identity matrix and $U^{(M,3)}$ is

$\begin{split}&U^{(M,3)}=\frac{1}{\sqrt{M}}\\\
&\times\begin{pmatrix}\sqrt{\frac{M}{2}}&0&0&0&\cdots&0&0&-\sqrt{M}|\chi|^{2}&-\sqrt{M}|\chi|^{2}\\\
0&1&1&1&\cdots&1&1&\chi&\bar{\chi}\\\
0&\xi&\xi^{2}&\xi^{3}&\cdots&\xi^{M-2}&\xi^{M-1}&\chi&\bar{\chi}\\\
0&\xi^{2}&\xi^{4}&\xi^{6}&\cdots&\xi^{2(M-2)}&\xi^{2(M-1)}&\chi&\bar{\chi}\\\
\vdots&\vdots&\vdots&\vdots&\ddots&\vdots&\vdots&\vdots&\vdots\\\
0&\xi^{M-3}&\xi^{2(M-3)}&\xi^{3(M-3)}&\cdots&\xi^{(M-3)(M-2)}&\xi^{(M-3)(M-1)}&\chi&\bar{\chi}\\\
0&\xi^{M-2}&\xi^{2(M-2)}&\xi^{3(M-2)}&\cdots&\xi^{(M-2)^{2}}&\xi^{(M-2)(M-1)}&\chi&\bar{\chi}\\\
0&\xi^{M-1}&\xi^{2(M-1)}&\xi^{3(M-1)}&\cdots&\xi^{(M-1)(M-2)}&\xi^{(M-1)^{2}}&\chi&\bar{\chi}\\\
\sqrt{\frac{M}{2}}&0&0&0&\cdots&0&0&\sqrt{M}|\chi|^{2}&\sqrt{M}|\chi|^{2}\end{pmatrix}\end{split}\,,$
(26)

with $\xi\equiv e^{\frac{2\pi}{M}\left(n_{1}-1\right)}$ and $\chi\equiv
i/\sqrt{2}$. Then, the spectra of the Dirac matrix $\mathcal{D}^{(M,3)}$ is
obtained as

$\begin{split}&\left(U^{(M,3)}\right)^{\dagger}\mathcal{D}^{(M,3)}U^{(M,3)}\\\
&=\mathrm{Diag}\begin{pmatrix}0,&\sin\left[\frac{2\pi}{M}\right]\sigma_{1},&\sin\left[\frac{4\pi}{M}\right]\sigma_{1},&\cdots,&\sin\left[\frac{2\pi\left(M-1\right)}{M}\right]\sigma_{1},&-i\sqrt{\frac{M}{2}}\sigma_{2},&i\sqrt{\frac{M}{2}}\sigma_{2}\end{pmatrix}\end{split}\,.$
(27)

From Eq. (27), one finds that the number of Dirac zero-modes of the Dirac
matrix is algebraically two for even $M$ since there is a certain $j$
satisfying $j=\frac{M}{2}+1\in\mathbb{N}$ and
$\sin\left[\frac{2\pi\left(j-1\right)}{M}\right]=0$ other than $j=1$. On the
other hand, the number of Dirac zero-modes is one for odd $M$ since only $j=1$
satisfies $\sin\left[\frac{2\pi\left(j-1\right)}{M}\right]=0$ for this case.

These results indicate that there are up to two fermion species on the
discretized 2-sphere $(M,3)$. We show that the maximal number of fermion
species are two also in other cases including
$(M,N)=(4,4),(5,4),(5,5),(4,5),(6,9)$ by numerical calculations as shown in
Appendix. A.1. The results are summarized in Table. 2.

The maximal number of fermion species, “two", is equal to the sum of the Betti
numbers for the 2-sphere

$\sum_{r=0}^{2}\beta_{r}(S^{2})=\sum_{r=0}^{2}\left(\delta_{r0}+\delta_{r2}\right)=2\,,$
(28)

where the $r$-th Betti number for the 2-sphere is
$\beta_{r}(S^{2})=\delta_{r0}+\delta_{r2}$.

In higher dimensions, we discuss the fermion action on the cellular-decomposed
sphere in a parallel manner. We show that there are up to two species on the
discretized 4-sphere labeled by four integers $(N_{1},N_{2},N_{3},N_{4})$ by
numerical calculation in Appendix. A.2. On the other hand, the $r$-th Betti
numbers ($0\leq r\leq D$) for $D$-dimensional sphere $S^{D}$ is given as

$\beta_{r}(S^{D})=\delta_{r0}+\delta_{rD}\,.$ (29)

Thus, the sum of the Betti numbers for the $D$-dimensional sphere is given as

$\sum_{r=0}^{D}\beta_{r}(S^{D})=\sum_{r=0}^{D}\left(\delta_{r0}+\delta_{rD}\right)=2\,.$
(30)

From these results, we argue that the maximal number of fermion species on the
$D$-dimensional discretized sphere ${}^{\ast}S^{D}$ is equal to the sum of the
Betti numbers for the $D$-dimensional sphere $S^{D}$.

It is notable that the lattice fermion action on the spherical lattice has
been studied in the literature in the different context Kamata:2016xmu ;
Kamata:2016rqi ; Brower2017 . Our result is consistent with the observations
obtained in the literature.

##  III New conjecture

In the previous section, we have discussed that the maximal number of lattice
fermion in terms of the Betti numbers of the continuum manifold. In Table. 1
we summarize the relation of the sum of the Betti numbers and the maximal
number of fermion species.

Table 1: Betti numbers and Maximal numbers of species manifold $M$ |  sum of $\beta_{r}(M)$ |  maximal $\\#$ of species
---|---|---
$1$-d torus |  $1+1$ |  $2$
$2$-d torus |  $1+2+1$ |  $4$
$3$-d torus |  $1+3+3+1$ |  $8$
$4$-d torus |  $1+4+6+4+1$ |  $16$
Torus $T^{D}$ |  $\left(1+1\right)^{D}$ |  $2^{D}$
Hyperball $B^{D}$ |  $1+0+0+\cdots$ |  $1$
Sphere $S^{D}$ |  $1+0+0+\cdots+1$ |  $2$
$T^{D}\times B^{d}$ |  $2^{D}\times 1$ |  $2^{D}$

From these facts, we propose a new conjecture on the number of fermion species
on the discretized manifold. The conjecture is as follows:

######  Conjecture. 1.

We firstly impose the following five conditions on the fermion action of the
lattice-discretized $D$-dimensional manifold $\mathcal{M}$:

  1. i.

Central difference; anti-hermiticity of the Dirac matrix in the action holds
due to this condition.

  2. ii.

$\gamma_{5}$ hermiticity; even the action with the mass term or the Wilson
term satisfies this condition.

  3. iii.

Four spinors; this condition assures the linear independence of the lattice
action for each direction.

  4. iv.

Locality; this condition leads to finite-hopping actions although it may be
unnecessary for our conjecture because non-locality usually decreases the
number of species.

  5. v.

Finite volume lattice; our conjecture claims that the fermion action on the
finite-volume lattice picks up the topology of the continuum manifold.

Our conjecture claims that, as long as these conditions hold, the maximal
number of fermion species on the lattice-discretized $D$-dimensional manifold
is equal to the summation of Betti numbers $\beta_{r}({\mathcal{M}})$ over
$0\leq r\leq D$ for the continuum manifold ${\mathcal{M}}$. It is expressed as

${\rm
max}[{\mathcal{N}}({}^{\ast}{\mathcal{M}})]\,=\,\sum_{r=0}^{D}\beta_{r}({\mathcal{M}})\,,$
(31)

where ${\mathcal{N}}({}^{\ast}{\mathcal{M}})$ is the number of fermion species
on the lattice-discretized manifold ${}^{\ast}{\mathcal{M}}$.

We consider that this conjecture holds for any orientable manifolds in any
dimensions, including $N$-holed tori.

##  IV Program for proof

In the previous section, we propose the conjecture on the equivalence between
the maximal number of fermion species on the discretized manifold and the
summation of Betti numbers of the continuum manifold, by studying the
$D$-dimensional torus-lattice ${}^{\ast}T^{D}$, the hyperball-lattice
${}^{\ast}B^{D}$, the $D\times d$ dimensional cylinder-lattice
${}^{\ast}T^{D}\times{}^{\ast}B^{d}$ and the $D$-dimensional sphere lattice
${}^{\ast}S^{D}$. We now propose a program for proof of the conjecture in
terms of spectral graph theory and Hodge theory. In Hodge theory Hod41 ; Eck45
; Dod76 ; DP76 , the number of zero-eigenvalues of a $r$-th Laplacian
($\Delta_{r}=\partial_{r+1}\partial_{r+1}^{*}+\partial_{r}^{*}\partial_{r}$)
defined on a complex chain coincides with the $r$-th Betti number, where
$\partial_{r}$ is a $r$-th boundary operator. Our program for the proof is
inspired by this fact.

In our program, we re-interpret lattice fermions as spectral graphs YM2022 .
We define a Laplacian operator $\mathcal{L}$ of a graph $G(V,E)$ having the
set of vertices $V$ and the set of edge $E$ as

$\mathcal{L}_{ij}\equiv\begin{cases}d_{i}&\mbox{if $i=j$}\\\ -1&\mbox{if
$i\neq j$ and $(i,j)$ are linked}\\\ 0&\mbox{if $i\neq j$ and $(i,j)$ are
linked}\end{cases}\,,$ (32)

where $d_{i}$ is the number of edges sharing the site $i$. Vertices in graph
theory correspond to lattice sites in lattice field theory while edges
correspond to links. The detailed correspondence between lattice field theory
and spectral graph theory is summarized in our previous work YM2022 .

###  IV.1 One-dimensional lattices

We begin with one-dimensional lattices. We consider two types of one-
dimensional lattices: the one with PBC ($1$-dim discretized torus
${}^{\ast}T^{1}$) and the other with DBC ($1$-dim discretized hyperball
${}^{\ast}B^{1}$).

For the one-dimensional lattice with $N$ vertices with PBC, the diagonal
elements of Laplacian matrix $d_{i}$ are $d_{i}=2$ for $i=1,2,...,N$, thus the
Laplacian in Eq. (32) is

$\mathcal{L}_{ij}=2\sum_{k=1}^{N}\delta_{ik}\delta{kj}-\left(\delta_{iN}\delta_{1j}+\delta_{i1}\delta_{Nj}+\sum_{k=1}^{N-1}\delta_{ik}\delta_{k+1\,j}+\sum_{k=2}^{N}\delta_{ik}\delta_{k-1\,j}\right)\,.$
(33)

For simplicity, we take an even $N$ for this case. It is notable that the
number of zero-eigenvalues of this Laplacian is one, which can be checked by
direct calculation. From the generic argument of Laplacian matrices in
homology theory, it is known that the number of zero-eigenvalues of
$\mathcal{L}$ of the graph $G(V,E)$ represents the number of connected
components, or equivalently the $0$-th Betti number $\beta_{0}(G)$ of
$G(V,E)$. Since an certain manifold in one dimension and its lattice-
discretized graph share the same Betti numbers ($\beta_{0},\beta_{1}$), the
number of zero-eigenvalues of $\mathcal{L}$ in Eq. (33) gives the $0$-th Betti
number of one-dimensional torus. Indeed, the $0$-th Betti number of one-
dimensional torus is one.

The Lagrangian constructed from the Laplacian matrix and the field vector
$\bm{\psi}=\sum_{i=1}^{N}\psi_{i}\bm{e}_{i}$ results in the Wilson term
$S_{\mathrm{Wt}}$ with the mass parameter $m$ set to zero and the Wilson-
fermion parameter $r$ set to one in a free theory, where it is expressed as

$\begin{split}\bm{\bar{\psi}}\mathcal{L}\bm{\psi}&=\sum_{n=1}^{N}\bar{\psi}_{n}\left[\psi_{n+1}+\psi_{n-1}-2\psi_{n}\right]\\\
&=2\sum_{n=1}^{N}\bar{\psi}_{n}\left.\left[m\psi_{n}-\frac{r}{2}\left(\psi_{n+1}+\psi_{n-1}-2\psi_{n}\right)\right]\right|_{m=0,\,r=1}\equiv\left.S_{\mathrm{Wt}}\right|_{m=0,\,r=1}\end{split}\,.$
(34)

This fact means that the number of zero-eigenvalues of the matrix
corresponding to the Wilson term on ${}^{\ast}T^{1}$ is equivalent to the
$0$-th Betti number $\beta_{0}=1$ for the continuum torus $T^{1}$. We will
later discuss that it also holds for ${}^{\ast}B^{1}$ and $B^{1}$.

We next consider another Laplacian operator defined as

$\mathcal{L}^{\prime}_{ij}\equiv\begin{cases}-d_{i}&\mbox{if $i=j$}\\\
-1&\mbox{if $i\neq j$ and $(i,j)$ are linked}\\\ 0&\mbox{if $i\neq j$ and
$(i,j)$ are linked}\end{cases}\,,$ (35)

with $d_{i}=2$ on ${}^{\ast}T^{1}$. This Laplacian corresponds to the case
with the mass parameter $m=-2$ in Eq. (34). The number of zero-eigenvalues of
this Laplacian $\mathcal{L}^{\prime}$ on ${}^{\ast}T^{1}$ is one, which is
equivalent to the first Betti number of the continuum torus $T^{1}$. From this
fact, we obtain one observation that the number of zero-eigenvalues of
$\mathcal{L}^{\prime}$ gives the first Betti number in one dimension, which we
have to prove in the future.

To convince readers, we study the one-dimensional lattice with $N$ vertices
with DBC (${}^{\ast}B^{1}$), where the diagonal elements of the Laplacian
matrices $d_{i}$ are $d_{1}=1$, $d_{N}=1$, and $d_{i}=2$ for $i=2,...,N-1$,
where we take an odd $N$ for this case. For the Laplacian $\mathcal{L}$ in Eq.
(33), the number of zero-eigenvalues of $\mathcal{L}$, which again corresponds
to the Wilson term on ${}^{\ast}B^{1}$, gives the number of connected
components, thus it is gives the $0$-th Betti number $\beta_{0}=1$ of $B^{1}$.
For the Laplacian $\mathcal{L}^{\prime}$ in Eq. (35), the number of its zero-
eigenvalues on ${}^{\ast}B^{1}$ is zero, which is equivalent to the first
Betti number of $B^{1}$. It is notable that our observation on the equivalence
of the number of zero eigenvalues of $\mathcal{L}^{\prime}$ and the first
Betti number also holds for ${}^{\ast}B^{1}$ and $B^{1}$.

So far, we study the Laplacian corresponding to the Wilson term on the one-
dimensional lattice (graph) and find the following facts:

  * •

The number of zero-eigenvalues of $\mathcal{L}$ corresponding to the Wilson
term with $m=0,r=1$ is equivalent to $\beta_{0}$ of the corresponding
manifold. It is a mathematically rigorous fact.

  * •

The number of zero-eigenvalues of $\mathcal{L}^{\prime}$ corresponding to the
Wilson term with $m=-2,r=1$ seems to be equivalent to $\beta_{1}$ of the
corresponding manifold.

Two zero-eigenvectors of the free naive Dirac matrix (Dirac operator) on
${}^{\ast}T^{1}$, which are found from the condition in Eq. (3), are identical
to the one of $\mathcal{L}$ and the one of $\mathcal{L}^{\prime}$,
respectively, while one zero-eigenvector of the naive Dirac matrix on
${}^{\ast}B^{1}$, which is found from the condition in Eq. (8), is identified
with that of $\mathcal{L}^{\prime}$. These results originate in the fact that
the naive Dirac matrix and the Laplacian operators ($\mathcal{L}$,
$\mathcal{L}^{\prime}$) are simultaneously diagonalizable.

It strongly indicates that the summation of Betti numbers on an one-
dimensional manifold ($T^{1}$ or $B^{1}$) is in exact agreement with the
number of zero-eigenvalues of the naive Dirac matrix on the lattice-
discretized manifold (${}^{\ast}T^{1}$ or ${}^{\ast}B^{1}$), which gives the
maximal number of species. To complete our proof for one dimension, however,
we have to prove that the number of zero-eigenvalues of $\mathcal{L}^{\prime}$
is $\beta_{1}$ of the corresponding manifold.

###  IV.2 Higher-dimensional lattices

For the higher-dimensional torus-lattice ${}^{\ast}T^{D}$, the hyperball-
lattice ${}^{\ast}B^{D}$ and the cylinder-lattice
${}^{\ast}T^{D}\times{}^{\ast}B^{d}$, we can utilize Kunneth theorem claiming
that the homology groups of two cellular chain complexes $C,C^{\prime}$ and
$C\times C^{\prime}$ have the following relation:

$\displaystyle H_{r}(C\times
C^{\prime})\,\cong\,\bigoplus_{p+q=r}H_{p}(C)\otimes H_{q}(C^{\prime})\,.$
(36)

This theorem means the homology group and its rank (Betti number) of a certain
two-dimensional product manifold is obtained from those of the one-dimensional
manifolds. By repeating this, we can obtain the homology groups and Betti
numbers for any higher-dimensional manifolds.

At least for ${}^{\ast}T^{D}$, ${}^{\ast}B^{D}$ and
${}^{\ast}T^{D}\times{}^{\ast}B^{d}$, we consider that we can extend the
argument in the previous subsection to higher-dimensional cases. In this
extension, the Laplacian operators giving Betti numbers $\beta_{r}$ are
expressed as the sum of tensor products of the Laplacians $\mathcal{L}$ in Eq.
(33) and $\mathcal{L}^{\prime}$ in Eq. (35).

We now give an example: the Laplacian giving $0$-th Betti number for four-
dimensional lattices is expressed as

$\begin{split}\mathcal{L}^{(D=4)}_{r=0}&=\Bigl{(}\mathcal{L}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}+\bm{1}_{N}\otimes\mathcal{L}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\\\
&\qquad+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\mathcal{L}\otimes\bm{1}_{N}+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\mathcal{L}\Bigr{)}\otimes\bm{1}_{4}\end{split}\,,$
(37)

where the lattice has $N^{4}$ vertices (lattice sites). The symbol
$\bm{1}_{N}$ is a $N\times N$ identity matrix. The Lagrangian constructed from
this Laplacian is nothing but the four-dimensional Wilson term with $m=0,r=1$.
The Laplacian giving $4$-th Betti number for four-dimensional lattices is
expressed as

$\begin{split}\mathcal{L}^{(D=4)}_{r=4}&=\Bigl{(}\mathcal{L}^{\prime}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}+\bm{1}_{N}\otimes\mathcal{L}^{\prime}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\\\
&\qquad+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\mathcal{L}^{\prime}\otimes\bm{1}_{N}+\bm{1}_{N}\otimes\bm{1}_{N}\otimes\bm{1}_{N}\otimes\mathcal{L}^{\prime}\Bigr{)}\otimes\bm{1}_{4}\end{split}\,,$
(38)

which corresponds to the four-dimensional Wilson term with $m=-8,r=1$. The
Laplacian giving other Betti numbers are also expressed as some tensor product
of $\mathcal{L}$ and $\mathcal{L}^{\prime}$, resulting in a certain Wilson
term for which one of the branches of Wilson Dirac spectrum is set to the
origin. As a by-product, this procedure of proof shows that the number of
fermion species on the well-known five branches of Wilson Dirac spectrum
$1,4,6,4,1$ stand for the Betti numbers $\beta_{r}(T^{4})$ for $r=0,1,2,3,4$.

This generalization to higher-dimensions is directly applicable to the
manifold obtained as a product space of $T^{D}$ and $B^{D}$. For the manifold
such as $S^{D}$, we have to develop a more generic way of generalization to
higher-dimensions.

In the end of this section, we comment on another avenue toward proof of the
conjecture: Squaring the free naive Dirac matrix leads to another Laplacian
operator. If we can prove that the number of zero-modes of this Laplacian is
the sum of the Betti numbers of the continuum manifold, we can easily give a
generic proof for any kind of manifolds including $S^{D}$.

##  V Summary and Discussion

Our conjecture claims that the maximal number of fermion species on a finite
lattice is equivalent to the summation of the Betti numbers of the
$D$-dimensional manifold from which the lattice is constructed. It is
summarized as

${\rm
max}[{\mathcal{N}}({}^{\ast}{\mathcal{M}})]\,=\,\sum_{r=0}^{D}\beta_{r}({\mathcal{M}})\,,$
(39)

where ${\mathcal{N}}({}^{\ast}{\mathcal{M}})$ is the number of fermion species
on the lattice, which is defined as a lattice-discretized version
${}^{\ast}{\mathcal{M}}$ of the manifold $\mathcal{M}$. In a sense that this
conjecture relates the number of lattice fermion species (doublers) to the
topology of spacetime manifold, it is complementary to the Nielsen-Ninomiya’s
no-go theorem which claims the emergence of pairs of fermion species as a
result of the cancellation of chiral charges on torus.

We note that the term, “the number of fermion species", we used in the paper
is defined as the number of Dirac zero-modes of fermions on finite lattices.
For infinite-volume lattices things drastically change: For example, the
number of species of naive fermion on the one-dimensional lattice hyperball
${}^{\ast}B^{1}$ approaches two in an infinite-volume limit, which is the same
number as that on ${}^{\ast}T^{1}$. It is of great importance that our
conjecture relates the topology of a continuum manifold and the fermion
species on a finite lattice defined by discretizing the manifold.

We believe that the program for the proof we have discussed in this paper is
applicable to lattice fermions defined on generic manifolds. Future works will
be devoted to the complete proof of the theorem.

###### Acknowledgements.

T. M. and J. Y. are grateful to S. Aoki, S. Matsuura and K. Ohta for the
fruitful discussion on the topics of the paper. The completion of this work is
owed to the discussion in the workshop "Lattice field theory and continuum
field theory" at Yukawa Institute for Theoretical Physics, Kyoto University
(YITP-W-22-02). This work of T. M. is supported by the Japan Society for the
Promotion of Science (JSPS) Grant-in-Aid for Scientific Research (KAKENHI)
Grant Numbers 19K03817.

##  Appendix A Numerical analysis for $D$-dimensional spheres

###  A.1 Two-dimensional sphere

In this appendix, we show that there are up to two species on the discretized
2-sphere labeled by $(M,N)$222Hereafter, we consider the number of sites on
the latitude direction $N$ is $N>3$. by numerical calculations:

  * •

If both the number of sites on the longitude direction $M$ and the number of
sites on the latitude direction $N$ are even, there is no fermion species, or
no Dirac zero-modes of the Dirac matrix. For instance, we consider the
discretized 2-sphere labeled by $(4,4)$, which is depicted in Fig. 4.

Figure 4: Discretized 2-sphere labeled by $(4,4)$.

The Dirac matrix on the $(4,4)$ sphere is

$\mathcal{D}^{(4,4)}=\frac{1}{2}\begin{pmatrix}I_{2}\otimes P_{4}\\\
&O_{2}\end{pmatrix}\otimes\sigma_{1}+\frac{1}{2}\begin{pmatrix}D_{2}\otimes
I_{4}&V_{8,2}\\\ -V_{8,2}^{\dagger}&O_{2}\end{pmatrix}\otimes\sigma_{2}\,,$
(40)

with

$V_{8,2}=\begin{pmatrix}1&0\\\
0&-1\end{pmatrix}_{2,2}\otimes\begin{pmatrix}1\\\ 1\\\ 1\\\ 1\end{pmatrix}\,.$
(41)

The eigenvalues of this Dirac matrix are depicted in Fig. 5. There is no zero-
eigenvalue in $\mathcal{D}^{(4,4)}$, which means that there is no fermion
species.

Figure 5: Eigenvalue distribution of the Dirac matrix $\mathcal{D}^{(4,4)}$.
There is no species, or equivalently no zero-eigenvalue.

  * •

If $M$ is odd and $N$ is even, there is again no species. For instance, we
take the discretized 2-sphere labeled by $(5,4)$. We numerically find that the
number of zero-modes or the number of species is zero as shown in Fig. 6.

Figure 6:  Eigenvalue distribution of the Dirac matrix $\mathcal{D}^{(5,4)}$.
There is no species (no zero-eigenvalue) in $\mathcal{D}^{(5,4)}$.

  * •

If both of $M$ and $N$ are odd, there is one species. For instance, we
consider the discretized 2-sphere labeled by $(5,5)$. Then, we numerically
find that the number of species is one as shown in Fig. 7 and Fig. 8. We note
that a pair of eigenvalues corresponds to a single Dirac mode since the two-
dimensional $\gamma$ matrices are $2\times 2$ matrices. Thus, the existence of
the pair of zero-eigenvalues shown in Fig. 7 and Fig. 8 means that there is a
single Dirac zero-mode.

Figure 7:  Eigenvalue distribution of the Dirac matrix $\mathcal{D}^{(5,5)}$.
The pair of zero eigenvalues corresponds to a single Dirac zero-mode.  Figure
8:  Eigenvalue distribution is depicted, where the vertical axis represents
the imaginary part of the Dirac matrix $\mathcal{D}^{(5,5)}$ and the
horizontal axis represents the serial number of eigenvalues. The pair of zero
eigenvalues corresponds to a single Dirac zero-mode.

  * •

If $M$ is even and $N$ is odd, there are two species on the discretized
2-sphere. For instance, we take two cases, $(4,5)$ and $(6,9)$. The
discretized 2-sphere $(4,5)$ is depicted in Fig. 9.

Figure 9: Discretized 2-sphere labeled by $(4,5)$.

The Dirac matrix on the $(4,5)$ sphere is

$\mathcal{D}^{(4,5)}=\frac{1}{2}\begin{pmatrix}I_{3}\otimes P_{4}\\\
&O_{2}\end{pmatrix}\otimes\sigma_{1}+\frac{1}{2}\begin{pmatrix}D_{3}\otimes
I_{4}&V_{12,2}\\\ -V_{12,2}^{\dagger}&O_{2}\end{pmatrix}\otimes\sigma_{2}\,,$
(42)

with

$V_{12,2}=\begin{pmatrix}1&0\\\ 0&0\\\
0&-1\end{pmatrix}_{3,2}\otimes\begin{pmatrix}1\\\ 1\\\ 1\\\ 1\end{pmatrix}\,.$
(43)

The eigenvalues of this Dirac matrix are depicted in Fig. 10. There are two
species on 2-sphere $(4,5)$ since there are four zero-eigenvalues as seen from
Fig. 11.

Figure 10: Eigenvalue distribution of the Dirac matrix $\mathcal{D}^{(4,5)}$.
There are four zero-eigenvalues corresponding to two species in two
dimensions.  Figure 11: Eigenvalue distribution is depicted, where the
vertical axis represents the imaginary part of the Dirac matrix
$\mathcal{D}^{(4,5)}$ and the horizontal axis represents the serial number of
eigenvalues. There are four zero-eigenvalues corresponding to two species in
two dimensions.

In the case of the $(6,9)$ 2-sphere, the Dirac matrix is

$\mathcal{D}^{(6,9)}=\frac{1}{2}\begin{pmatrix}I_{7}\otimes P_{6}\\\
&O_{2}\end{pmatrix}\otimes\sigma_{1}+\frac{1}{2}\begin{pmatrix}D_{7}\otimes
I_{6}&V_{42,2}\\\ -V_{42,2}^{\dagger}&O_{2}\end{pmatrix}\otimes\sigma_{2}\,,$
(44)

with

$V_{42,2}=\begin{pmatrix}1&0\\\ 0&0\\\ 0&0\\\ 0&0\\\ 0&0\\\ 0&0\\\
0&-1\end{pmatrix}_{7,2}\otimes\begin{pmatrix}1\\\ 1\\\ 1\\\ 1\\\ 1\\\
1\end{pmatrix}\,.$ (45)

The eigenvalues of this Dirac matrix are depicted in Fig. 12. There are again
two species as there are four zero-eigenvalues in $\mathcal{D}^{(6,9)}$ as
seen from from Fig. 13.

Figure 12: Eigenvalue distribution of the Dirac matrix $\mathcal{D}^{(6,9)}$.
There are two species (four zero-eigenvalues).  Figure 13: Eigenvalue
distribution is depicted, where the vertical axis represents the imaginary
part of the Dirac matrix $\mathcal{D}^{(6,9)}$ and horizontal axis represents
the serial number of eigenvalues. There are four zero-eigenvalues
corresponding to two species in two dimensions.

We summarize our results in Table. 2.

Table 2: Maximal number of species on 2-sphere $(M,N)$ the number of $M$ |  the number of $N$ |  maximal $\\#$ of species
---|---|---
even | even | 0
odd | even | 0
odd | odd | 1
even | odd | 2

###  A.2 Four-dimensional sphere

We discuss the discretized four-dimensional sphere. We first consider the
discretized four-dimensional spherical-coordinate system for $4$-sphere
labeled by four integers $(N_{1},N_{2},N_{3},N_{4})$ as

$\displaystyle x_{5}=r\cos\theta_{4},\qquad
x_{4}=r\sin\theta_{4}\cos\theta_{3},\qquad
x_{3}=r\sin\theta_{4}\sin\theta_{3}\cos\theta_{2},$ (46) $\displaystyle
x_{2}=r\sin\theta_{4}\sin\theta_{3}\sin\theta_{2}\cos\theta_{1},\qquad
x_{1}=r\sin\theta_{4}\sin\theta_{3}\sin\theta_{2}\sin\theta_{1}$ (47)

where $r$ is a radial distance and the four angles are discretized as
$\theta_{1}\equiv\frac{2\pi}{N_{1}}\left(n_{1}-1\right),\
\theta_{i}\equiv\frac{\pi}{N_{i}-1}\left(n_{i}-1\right)$ for $i=2,3,4$.
$n_{1},n_{i}\in\mathbb{N}$ run as $n_{1}\in[1,N_{1}],\ n_{i}\in[1,N_{i}]$. For
simplicity, we fix a radial distance as $r=1$. In a parallel manner to the
discussion for 2-sphere, we label the lattice sites as
$(n_{1},n_{2},n_{3},n_{4})$.

For instance, we take $N_{1}=4$ and $N_{i}=3$ for $i=2,3,4$. A graph
corresponding to the $(4,3,3,3)$ 4-sphere is depicted in Fig. 14.

Figure 14:  Graph corresponding to the $(4,3,3,3)$ 4-sphere is the 5-orthoplex
inside Petrie polygon.

The Dirac matrix is given as

$\begin{split}\mathcal{D}^{(4,3,3,3)}&=\begin{pmatrix}D_{4}\\\
&O_{6}\end{pmatrix}\otimes\gamma_{1}+\begin{pmatrix}O_{4}&V_{4,2}\\\
-V_{4,2}^{\dagger}&O_{2}\\\ &&O_{4}\end{pmatrix}\otimes\gamma_{2}\\\
&\quad+\begin{pmatrix}O_{6}&V_{6,2}\\\ -V_{6,2}^{\dagger}&O_{2}\\\
&&O_{2}\end{pmatrix}\otimes\gamma_{3}+\begin{pmatrix}O_{8}&V_{8,2}\\\
-V_{8,2}^{\dagger}&O_{2}\end{pmatrix}\otimes\gamma_{4}\end{split}\,.$ (48)

This Dirac matrix $\mathcal{D}^{(4,3,3,3)}$ is $10\times 10$ square matrix,
apart from the $\gamma$ matrix structure.

Figure 15: Eigenvalue distribution of the Dirac matrix
$\mathcal{D}^{(4,3,3,3)}$. There are two species in four dimensions, which
emerge as eight zero-eigenvalues in the figure.  Figure 16: Eigenvalues of the
Dirac matrix $\mathcal{D}^{(4,3,3,3)}$. There are eight zero-eigenvalues
corresponding to two species in four dimensions.

Fig. 15 and Fig. 16 shows the eigenvalue distributions of the Dirac matrix
$\mathcal{D}^{(4,3,3,3)}$. For the dicretized 4-sphere $(4,3,3,3)$, we find
that there are two species as there are eight zero-eigenvalues of the Dirac
matrix $\mathcal{D}^{(4,3,3,3)}$.

By studying other cases, we find that the number of species on the discretized
4-sphere $(N_{1},N_{2},N_{3},N_{4})$ is two when the number of sites $N_{1}$
is even and the number of sites $N_{i}$ for $i$ from $2$ to $4$ are odd, as
with the case on the discretized 2-sphere $(M,N)$. We could not find any
example where the number of species goes beyond two in four dimensions too.

## References

  * (1) K. G. Wilson, “Confinement of Quarks,” Phys. Rev. D10 (1974) 2445–2459.
  * (2) M. Creutz, “Monte Carlo Study of Quantized SU(2) Gauge Theory,” Phys. Rev. D21 (1980) 2308–2315.
  * (3) L. H. Karsten and J. Smit, “Lattice Fermions: Species Doubling, Chiral Invariance, and the Triangle Anomaly,” Nucl. Phys. B183 (1981) 103. [,495(1980)].
  * (4) H. B. Nielsen and M. Ninomiya, “Absence of Neutrinos on a Lattice. 1. Proof by Homotopy Theory,” Nucl. Phys. B185 (1981) 20.
  * (5) H. B. Nielsen and M. Ninomiya, “Absence of Neutrinos on a Lattice. 2. Intuitive Topological Proof,” Nucl. Phys. B193 (1981) 173–194.
  * (6) H. B. Nielsen and M. Ninomiya, “No Go Theorem for Regularizing Chiral Fermions,” Phys. Lett. 105B (1981) 219–223.
  * (7) K. G. Wilson, “Quarks and Strings on a Lattice,” in New Phenomena in Subnuclear Physics: Proceedings, International School of Subnuclear Physics, Erice, Sicily, Jul 11-Aug 1 1975. Part A.  1975\.
  * (8) D. B. Kaplan, “A Method for simulating chiral fermions on the lattice,” Phys. Lett. B288 (1992) 342–347, arXiv:hep-lat/9206013 [hep-lat].
  * (9) Y. Shamir, “Chiral fermions from lattice boundaries,” Nucl. Phys. B406 (1993) 90–106, arXiv:hep-lat/9303005 [hep-lat].
  * (10) V. Furman and Y. Shamir, “Axial symmetries in lattice QCD with Kaplan fermions,” Nucl. Phys. B 439 (1995) 54–78, arXiv:hep-lat/9405004.
  * (11) H. Neuberger, “More about exactly massless quarks on the lattice,” Phys. Lett. B427 (1998) 353–355, arXiv:hep-lat/9801031 [hep-lat].
  * (12) P. H. Ginsparg and K. G. Wilson, “A Remnant of Chiral Symmetry on the Lattice,” Phys. Rev. D25 (1982) 2649.
  * (13) J. B. Kogut and L. Susskind, “Hamiltonian Formulation of Wilson’s Lattice Gauge Theories,” Phys. Rev. D11 (1975) 395–408.
  * (14) L. Susskind, “Lattice Fermions,” Phys. Rev. D16 (1977) 3031–3039.
  * (15) N. Kawamoto and J. Smit, “Effective Lagrangian and Dynamical Symmetry Breaking in Strongly Coupled Lattice QCD,”.
  * (16) H. S. Sharatchandra, H. J. Thun, and P. Weisz, “Susskind Fermions on a Euclidean Lattice,” Nucl. Phys. B192 (1981) 205–236.
  * (17) M. F. Golterman and J. Smit, “Selfenergy and Flavor Interpretation of Staggered Fermions,” Nucl. Phys. B 245 (1984) 61–88.
  * (18) M. F. Golterman, “STAGGERED MESONS,” Nucl. Phys. B 273 (1986) 663–676.
  * (19) G. Kilcup and S. R. Sharpe, “A Tool Kit for Staggered Fermions,” Nucl. Phys. B 283 (1987) 493–550.
  * (20) W. Bietenholz and I. Hip, “The Scaling of exact and approximate Ginsparg-Wilson fermions,” Nucl. Phys. B 570 (2000) 423–451, arXiv:hep-lat/9902019.
  * (21) M. Creutz, T. Kimura, and T. Misumi, “Index Theorem and Overlap Formalism with Naive and Minimally Doubled Fermions,” JHEP 12 (2010) 041, arXiv:1011.0761 [hep-lat].
  * (22) S. Durr and G. Koutsou, “Brillouin improvement for Wilson fermions,” Phys. Rev. D 83 (2011) 114512, arXiv:1012.3615 [hep-lat].
  * (23) S. Durr, G. Koutsou, and T. Lippert, “Meson and Baryon dispersion relations with Brillouin fermions,” Phys. Rev. D 86 (2012) 114514, arXiv:1208.6270 [hep-lat].
  * (24) T. Misumi, “New fermion discretizations and their applications,” PoS LATTICE2012 (2012) 005, arXiv:1211.6999 [hep-lat].
  * (25) Y.-G. Cho, S. Hashimoto, J.-I. Noaki, A. Juttner, and M. Marinkovic, “$O(a^{2})$-improved actions for heavy quarks and scaling studies on quenched lattices,” PoS LATTICE2013 (2014) 255, arXiv:1312.4630 [hep-lat].
  * (26) Y.-G. Cho, S. Hashimoto, A. Juttner, T. Kaneko, M. Marinkovic, J.-I. Noaki, and J. T. Tsang, “Improved lattice fermion action for heavy quarks,” JHEP 05 (2015) 072, arXiv:1504.01630 [hep-lat].
  * (27) S. Durr and G. Koutsou, “On the suitability of the Brillouin action as a kernel to the overlap procedure,” arXiv:1701.00726 [hep-lat].
  * (28) D. H. Adams, “Theoretical foundation for the index theorem on the lattice with staggered fermions,” Phys.Rev.Lett. 104 (2010) 141602, arXiv:0912.2850 [hep-lat].
  * (29) D. H. Adams, “Pairs of chiral quarks on the lattice from staggered fermions,” Phys.Lett.B 699 (2011) 394–397, arXiv:1008.2833 [hep-lat].
  * (30) C. Hoelbling, “Single flavor staggered fermions,” Phys.Lett.B 696 (2011) 422–425, arXiv:1009.5362 [hep-lat].
  * (31) P. de Forcrand, A. Kurkela, and M. Panero, “Numerical properties of staggered overlap fermions,” PoS LATTICE2010 (2010) 080, arXiv:1102.1000 [hep-lat].
  * (32) M. Creutz, T. Kimura, and T. Misumi, “Aoki Phases in the Lattice Gross-Neveu Model with Flavored Mass terms,” Phys. Rev. D83 (2011) 094506, arXiv:1101.4239 [hep-lat].
  * (33) T. Misumi, M. Creutz, T. Kimura, T. Z. Nakano, and A. Ohnishi, “Aoki Phases in Staggered-Wilson Fermions,” PoS LATTICE2011 (2011) 108, arXiv:1110.1231 [hep-lat].
  * (34) E. Follana, V. Azcoiti, G. Di Carlo, and A. Vaquero, “Spectral Flow and Index Theorem for Staggered Fermions,” PoS LATTICE2011 (2011) 100, arXiv:1111.3502 [hep-lat].
  * (35) P. de Forcrand, A. Kurkela, and M. Panero, “Numerical properties of staggered quarks with a taste-dependent mass term,” JHEP 04 (2012) 142, arXiv:1202.1867 [hep-lat].
  * (36) T. Misumi, T. Z. Nakano, T. Kimura, and A. Ohnishi, “Strong-coupling analysis of parity phase structure in staggered-wilson fermions,” Phys.Rev.D 86 (2012) 034501, arXiv:1205.6545 [hep-lat].
  * (37) S. Durr, “Taste-split staggered actions: eigenvalues, chiralities and Symanzik improvement,” Phys. Rev. D 87 no. 11, (2013) 114501, arXiv:1302.0773 [hep-lat].
  * (38) C. Hoelbling and C. Zielinski, “Spectral properties and chiral symmetry violations of (staggered) domain wall fermions in the Schwinger model,” Phys. Rev. D 94 no. 1, (2016) 014501, arXiv:1602.08432 [hep-lat].
  * (39) C. Zielinski, Theoretical and Computational Aspects of New Lattice Fermion Formulations.  PhD thesis, Nanyang Technol. U., 2016.  arXiv:1703.06364 [hep-lat].
  * (40) L. H. Karsten, “Lattice Fermions in Euclidean Space-time,” Phys. Lett. B 104 (1981) 315–319.
  * (41) F. Wilczek, “ON LATTICE FERMIONS,” Phys. Rev. Lett. 59 (1987) 2397.
  * (42) M. Creutz, “Four-dimensional graphene and chiral fermions,” JHEP 04 (2008) 017, arXiv:0712.1201 [hep-lat].
  * (43) A. Borici, “Creutz fermions on an orthogonal lattice,” Phys. Rev. D 78 (2008) 074504, arXiv:0712.4401 [hep-lat].
  * (44) P. F. Bedaque, M. I. Buchoff, B. C. Tiburzi, and A. Walker-Loud, “Broken Symmetries from Minimally Doubled Fermions,” Phys. Lett. B 662 (2008) 449–455, arXiv:0801.3361 [hep-lat].
  * (45) P. F. Bedaque, M. I. Buchoff, B. C. Tiburzi, and A. Walker-Loud, “Search for Fermion Actions on Hyperdiamond Lattices,” Phys. Rev. D 78 (2008) 017502, arXiv:0804.1145 [hep-lat].
  * (46) S. Capitani, J. Weber, and H. Wittig, “Minimally doubled fermions at one loop,” Phys. Lett. B 681 (2009) 105–112, arXiv:0907.2825 [hep-lat].
  * (47) T. Kimura and T. Misumi, “Characters of Lattice Fermions Based on the Hyperdiamond Lattice,” Prog. Theor. Phys. 124 (2010) 415–432, arXiv:0907.1371 [hep-lat].
  * (48) T. Kimura and T. Misumi, “Lattice Fermions Based on Higher-Dimensional Hyperdiamond Lattices,” Prog. Theor. Phys. 123 (2010) 63–78, arXiv:0907.3774 [hep-lat].
  * (49) M. Creutz and T. Misumi, “Classification of Minimally Doubled Fermions,” Phys. Rev. D 82 (2010) 074502, arXiv:1007.3328 [hep-lat].
  * (50) S. Capitani, M. Creutz, J. Weber, and H. Wittig, “Renormalization of minimally doubled fermions,” JHEP 09 (2010) 027, arXiv:1006.2009 [hep-lat].
  * (51) B. C. Tiburzi, “Chiral Lattice Fermions, Minimal Doubling, and the Axial Anomaly,” Phys. Rev. D 82 (2010) 034511, arXiv:1006.0172 [hep-lat].
  * (52) S. Kamata and H. Tanaka, “Minimal Doubling Fermion and Hermiticity,” PTEP 2013 (2013) 023B05, arXiv:1111.4536 [hep-lat].
  * (53) T. Misumi, “Phase structure for lattice fermions with flavored chemical potential terms,” JHEP 08 (2012) 068, arXiv:1206.0969 [hep-lat].
  * (54) T. Misumi, T. Kimura, and A. Ohnishi, “QCD phase diagram with 2-flavor lattice fermion formulations,” Phys. Rev. D 86 (2012) 094505, arXiv:1206.1977 [hep-lat].
  * (55) S. Capitani, “Reducing the number of counterterms with new minimally doubled actions,” Phys. Rev. D 89 no. 1, (2014) 014501, arXiv:1307.7497 [hep-lat].
  * (56) S. Capitani, “New chiral lattice actions of the Borici-Creutz type,” Phys. Rev. D 89 no. 7, (2014) 074508, arXiv:1311.5664 [hep-lat].
  * (57) T. Misumi, “Fermion Actions extracted from Lattice Super Yang-Mills Theories,” JHEP 12 (2013) 063, arXiv:1311.4365 [hep-lat].
  * (58) J. H. Weber, S. Capitani, and H. Wittig, “Numerical studies of Minimally Doubled Fermions,” PoS LATTICE2013 (2014) 122, arXiv:1312.0488 [hep-lat].
  * (59) J. H. Weber, Properties of minimally doubled fermions.  PhD thesis, Mainz U., 2015.  arXiv:1706.07104 [hep-lat].
  * (60) S. Durr and J. H. Weber, “Dispersion relation and spectral range of Karsten-Wilczek and Borici-Creutz fermions,” arXiv:2003.10803 [hep-lat].
  * (61) T. Kimura, S. Komatsu, T. Misumi, T. Noumi, S. Torii, and S. Aoki, “Revisiting symmetries of lattice fermions via spin-flavor representation,” JHEP 01 (2012) 048, arXiv:1111.0402 [hep-lat].
  * (62) A. Chowdhury, A. Harindranath, J. Maiti, and S. Mondal, “Many avatars of the wilson fermion: A perturbative analysis,” JHEP 02 (2013) 037, arXiv:1301.0675 [hep-lat].
  * (63) J. Yumoto and T. Misumi, “Lattice fermions as spectral graphs,” Journal of High Energy Physics 2022 no. 2, (2022) 104, arXiv:2112.13501 [hep-lat].
  * (64) S. Catterall, J. Laiho, and J. Unmuth-Yockey, “Topological fermion condensates from anomalies,” JHEP 10 (2018) 013, arXiv:1806.07845 [hep-lat].
  * (65) N. Butt, S. Catterall, A. Pradhan, and G. C. Toga, “Anomalies and symmetric mass generation for Kähler-Dirac fermions,” Phys. Rev. D 104 no. 9, (2021) 094504, arXiv:2101.01026 [hep-th].
  * (66) J. M. Rabin, “Homology Theory of Lattice Fermion Doubling,” Nucl. Phys. B 201 (1982) 315–332.
  * (67) W. Hodge, “The Theory and Applications of Harmonic Integrals,”.
  * (68) B. Eckmann, “Harmonische Funktionen und Randvertanfgaben in einem Komplex,” Commentarii Math. Helvetici 17 (1945) 240–245.
  * (69) J. Dodziuk, “Finite-difference approach to the Hodge theory of harmonic forms,” American Journal of Mathematics 98 (1976) 79–104.
  * (70) J. Dodziuk and V. Patodi, “Riemannian structures and triangulations of manifolds,” J. of the Indian Math. Soc. 40 (1976) 1–52.
  * (71) D. West, Introduction to Graph Theory.  Featured Titles for Graph Theory. Prentice Hall, 2001.  https://books.google.co.jp/books?id=TuvuAAAAMAAJ.
  * (72) J. Bondy and U. Murty, Graph Theory with Applications.  American Elsevier Publishing Company, 1976.  https://books.google.co.jp/books?id=4bwrAAAAYAAJ.
  * (73) P. v. Mieghem, Graph Spectra for Complex Networks.  Cambridge University Press, 2010.
  * (74) D. J. Watts and S. H. Strogatz, “Collective dynamics of ‘small-world’ networks,” Nature 393 no. 6684, (1998) 440–442. https://doi.org/10.1038/30918.
  * (75) S. Kamata, S. Matsuura, T. Misumi, and K. Ohta, “Anomaly and sign problem in $\mathcal{N}=(2,2)$ SYM on polyhedra: Numerical analysis,” PTEP 2016 no. 12, (2016) 123B01, arXiv:1607.01260 [hep-th].
  * (76) S. Kamata, S. Matsuura, T. Misumi, and K. Ohta, “Numerical Analysis of Discretized ${\cal N}=(2,2)$ SYM on Polyhedra,” PoS LATTICE2016 (2016) 210, arXiv:1612.01968 [hep-lat].
  * (77) R. C. Brower, E. S. Weinberg, G. T. Fleming, A. D. Gasbarro, T. G. Raben, and C.-I. Tan, “Lattice dirac fermions on a simplicial riemannian manifold,” Phys. Rev. D 95 (Jun, 2017) 114510. https://link.aps.org/doi/10.1103/PhysRevD.95.114510.