# Light fermion masses in partially deconstructed models

Nahomi Kan<EMAIL_ADDRESS>National Institute of Technology, Gifu College,
Motosu-shi, Gifu 501-0495, Japan  Kiyoshi Shiraishi111Author to whom any
correspondence should be addressed<EMAIL_ADDRESS>Graduate School
of Sciences and Technology for Innovation, Yamaguchi University, Yamaguchi-
shi, Yamaguchi 753–8512, Japan  Maki Takeuchi<EMAIL_ADDRESS>Graduate School of Sciences and Technology for Innovation, Yamaguchi
University, Yamaguchi-shi, Yamaguchi 753–8512, Japan

###### Abstract

Considering a theory space consisting of a large number of five-dimensional
Dirac fermion field theories including background abelian gauge fields, we can
construct a theory similar to a continuous six-dimensional theory compactified
with two-dimensional manifolds with and without magnetic flux or orbifolds as
extra dimensions. This method, called dimensional deconstruction, can be used
to construct a model with one-dimensional discrete space, which represents
general graph structures. In this paper, we propose the models with two extra
dimensions, which resemble two-dimensional tori, cylinders, and rectangular
regions, as continuum limits. We also try to build a model that mimics one
with the two-dimensional orbifold compactification.
Keywords: deconstruction, orbifolds, compactification

##  I Introduction

Higher-dimensional models are promising candidates for theories beyond the
standard model of elementary particles, and it is known that the geometry of
the extra dimensions is significantly related to the characteristics of the
models KK . Numerous studies have been conducted on compactification of extra
dimensions, while early work on compactification into manifolds discussed the
problem of chiral fermions. The problem was that four-dimensional chiral
fermions do not appear after compactification to a simple manifold. In recent
years, however, it has been accepted that the chiral fermion problem can be
solved simply by considering a brane motivated by string theory and
considering fermions that exist only on it ADD .

However, since the number of massless fermions is related to the number of
generations in the standard model, the emergence of four-dimensional chiral
fermions due to compactification of extra dimensions (without/with magnetic
flux) remains of interest. There are many models with extra dimensions, but
compactification onto orbifolds has been actively examined because it is
widely used in string theory, and it is also known that compactification onto
orbifolds in field theory brings about the appearance mechanism of four-
dimensional chiral fermions.

On the other hand, a scheme of dimensional deconstruction was proposed about
twenty years ago as an idea for construction of analogs of compactified
theories ACG1 ; HPW ; Lane ; HL . Its original motivation was to construct
Higgsless models ACG2 ; Schmaltz  as alternatives to the higher-dimensional
gauge-Higgs unified theory Hosotani1 ; Hosotani2 ; HIL . The idea of
dimensional deconstruction is that by means of connecting multiple copies of a
four-dimensional field theory with a link field similar to a gauge field, one
can present a theory similar to a higher-dimensional model. In models based on
this idea, the degrees of freedom of the link fields play the role of the
Higgs fields. In addition, in the case of the special type of scalar theory
considered in the paper CKS , it was shown that a non-trivial link field
produce the same effect as a magnetic flux in extra dimensions.

In the present paper, we propose a new model that can be interpreted as a
hybrid of compactification and deconstruction of extra dimensions. Although
only simple examples will be given here, future study may address the
following prospects: One is to see, if there are small differences between the
“manifold/orbifold mimicker” which will be described in the following sections
and the corresponding continuous model, it will serve as an alternative to
confirm the degree of certainty of the model with minimum requirements from
observable physical quantities. Another is that, since the simple structure of
the deconstruction theory can be replaced with the structure of a general
graph, so further extensions of the model can be discussed in the future.
Other additional issues will be discussed in the final section. Although the
model treated in this paper is a toy model as a simple example, it is possible
to discuss models with which we can perform one-loop quantum calculations
including vacuum energy, because the masses of all excited states can be
obtained, in principle, in our model. For this reason, further analysis on
quantum effects can also be expected in further extended models.

The organization of this paper is as follows. In the next Section 2, we
propose the general form of the toy model with graph structure considered
below. In Section 3, we review the compactification of the five-dimensional
Dirac field theory, which is a component of the model, and prepare for the
construction of the entire model. The fermion mass matrix and its spectrum,
when a cycle graph is adopted as the graph, will be discussed in Section 4.
This is a model in which the extra dimensions are a two-dimensional torus or a
cylinder in the continuum limit. On the other hand, in Section 5, we consider
the fermion mass matrix and its spectrum, when a path graph is used. In this
case, it is a model that gives extra dimensions of a cylinder or a rectangle
in the continuum limit. We try to construct a model in which the continuum
limit looks like an orbifold $T^{2}/Z_{2}$ in the last part of the section.
The final section provides a summary and future prospects. We use the metric
convention $(-+\cdots+)$.

##  II The model

Let $G(\mathcal{V},\mathcal{E})$ be a graph with vertex set $\mathcal{V}$ and
edge set $\mathcal{E}$.222For the (spectral) graph theory, see Mohar1 ; Mohar2
; Mohar3 ; Merris . We here consider a directed graph. An oriented edge is
represented by $e=[u,v]$ ($u,v\in\mathcal{V}(G)$). One of these vertices, say
$u$, is the open of the edge, denoted by $o(e)$, while the other vertex is the
terminus of the edge, denoted by $t(e)$.

We consider two sets of five-dimensional Dirac fields $\Psi_{v}$ and
$\tilde{\Psi}_{e}$, which are assigned to vertices (sites) and edges (links),
respectively. The model that imitates the six-dimensional theory considered
here is described by the Lagrangian density

$\mathcal{L}=\sum_{v\in\mathcal{V}(G)}\mathcal{L}_{K}[\Psi_{v};A_{M,v}]+\sum_{e\in\mathcal{E}(G)}\mathcal{L}_{K}[\tilde{\Psi}_{e};A_{M,e}]+\mathcal{L}_{M}[\Psi_{v};\tilde{\Psi}_{e}]\,,$
(1)

where

$\mathcal{L}_{K}[\Psi;A_{M}]=i\overline{\Psi}\Gamma^{M}D_{M}\Psi\,,$ (2)

with

$D_{M}=\partial_{M}+iA_{M}\,,$ (3)

and

$\mathcal{L}_{M}[\Psi_{v},\tilde{\Psi}_{e}]=-m\sum_{e\in\mathcal{E}(G)}\left[\overline{\tilde{\Psi}}_{e}(\Psi_{t(e)}-\Psi_{o(e)})+h.c.\right]\,.$
(4)

Here, $A_{M}$ $(M=0,1,2,3,5)$ denotes a $U(1)$ gauge field and $m$ is a
constant with dimension of mass. The reason why we consider two types of Dirac
fields is that, in six dimensions, fermions have twice as many degrees of
freedom as in five dimensions. Under the $U(1)$ gauge transformation, it
follows that all fields undergo a common local phase transformation. This is
due to the term $\mathcal{L}_{M}$.

The gamma matrices for five dimensions satisfies
$\\{\Gamma^{M},\Gamma^{N}\\}=-2\eta^{MN}$ and
$\overline{\Psi}\equiv\Psi^{\dagger}\Gamma^{0}$. The five dimensional gamma
matrices can be expressed by the usual four dimensional gamma matrices
$\gamma^{\mu}$, as $\Gamma^{\mu}=\gamma^{\mu}$ $(\mu=0,1,2,3)$ and
$\Gamma^{5}=i\gamma^{5}$, where $(\gamma^{5})^{2}=1$ is satisfied.

The term $\mathcal{L}_{M}$ can be rewritten by

$\mathcal{L}_{M}[\Psi_{v},\tilde{\Psi}_{e}]=m\sum_{v\in\mathcal{V}(G)}\sum_{e\in\mathcal{E}(G)}\left(\overline{\tilde{\Psi}}_{e}E^{T}_{ev}(G)\Psi_{v}+h.c.\right)\,,$
(5)

where $E(G)$ is the incidence matrix for a directed graph $G$ defined by

$E_{ve}(G)=\left\\{\begin{array}[]{cc}1&\mbox{if~{}}v=o(e)\\\
-1&\mbox{if~{}}v=t(e)\\\ 0&\mbox{otherwise}\end{array}\right.\,.$ (6)

Figure 1: Left: the cycle graph $C_{4}$. Right: the path graph $P_{4}$. Each
suffix indicates the number of vertices in the graph.

In this paper, only $G=C_{N}$ and $G=P_{N+1}$ are used as graphs. The cycle
graph $C_{N}$ has a structure in which $N$ vertices are connected by $N$ edges
like a ring (left of Fig. 1). For example, the incidence matrix of $C_{4}$ is
expressed as

$E(C_{4})=\left[\begin{array}[]{rrrr}1&0&0&-1\\\ -1&1&0&0\\\ 0&-1&1&0\\\
0&0&-1&1\end{array}\right]\,,$ (7)

where sequence numbers of vertices and edges are supposed to be appropriately
assigned.

A path graph $P_{N}$ is a graph in which $N$ vertices are connected by $N-1$
edges like a single path (right of Fig. 1). For example, the incidence matrix
of $P_{4}$ is

$E(P_{4})=\left[\begin{array}[]{rrrr}1&0&0\\\ -1&1&0\\\ 0&-1&1\\\
0&0&-1\end{array}\right]\,,$ (8)

where sequence numbers of vertices and edges are supposed to be appropriately
assigned.

Incidentally, the graph Laplacian Mohar1 ; Mohar2 ; Mohar3 ; Merris  can be
defined as

$\Delta(G)=E(G)E^{T}(G)\,.$ (9)

For example, we find333Note that
$\Delta(C_{N})=E(C_{N})E^{T}(C_{N})=E^{T}(C_{N})E(C_{N})$ holds true only for
cycle graphs.

$\Delta(C_{4})=\left[\begin{array}[]{rrrr}2&-1&0&-1\\\ -1&2&-1&0\\\
0&-1&2&-1\\\ -1&0&-1&2\end{array}\right]\,,$ (10)

and

$\Delta(P_{4})=\left[\begin{array}[]{rrrrr}1&-1&0&0\\\ -1&2&-1&0\\\
0&-1&2&-1\\\ 0&0&-1&1\end{array}\right]\,.$ (11)

Although it can be intuitively anticipated that $C_{N}$ is a discretization of
$S^{1}$ (a circle) and $P_{N}$ is a discretization of $S^{1}/Z_{2}$ (a line
segment), the continuum limit of the current model will actually be clarified
in the following sections.

To obtain a four-dimensional theory, it is necessary to compactify one
dimension of each five-dimensional theory. For the fifth dimension, we
consider the manifold $S^{1}$ or the orbifold $S^{1}/Z_{2}$. The
compactification is reviewed in the next section. In the following sections,
the four-dimensional components of the gauge field are assumed to be zero, as
a trivial background field.

##  III compactification of the five-dimensional theory

In this section, we consider compactification of the five-dimensional fermion
theory as a component in “theory space”. The theme of this paper is to
construct a higher-dimensional model mimicker (imitator) with two extra
dimensions by stitching these $N$ copies together using a graph structure.

###  III.1 compactification on $S^{1}$

Let $y$ be the coordinate of the fifth dimension ($y=x^{5}$). Since we set a
periodic boundary condition for $y$,

$\Psi(x,y+2\pi R)=\Psi(x,y)\,,$ (12)

where $x$ denotes the four-dimensional coordinates, $x^{\mu}$ $(\mu=0,1,2,3)$,
and $R$ is a constant with length dimension. Then, the Dirac field can be
expanded in a Fourier series with respect to $y$ as

$\Psi(x,y)=\frac{1}{\sqrt{2\pi
R}}\sum_{n=-\infty}^{\infty}\psi_{n}(x)e^{i\frac{n}{R}y}\,,$ (13)

where $\psi_{n}(x)$ are regarded as four-dimensional Dirac fields.

We here assume that there is a zero mode of the gauge field $A_{y}$ in the
fifth direction, which can have a constant vacuum expectation value, $\langle
A_{y}\rangle$. Thus the kinetic part of the Lagrangian reads

$\mathcal{L}_{K}[\Psi;A_{y}]=i\overline{\Psi}\Gamma^{M}D_{M}\Psi=i\overline{\Psi}\left[\Gamma^{\mu}\partial_{\mu}+\Gamma^{5}(\partial_{y}+i\langle
A_{y}\rangle)\right]\Psi\,.$ (14)

Integrating this kinetic part of the Lagrangian over the extra coordinate $y$,
we obtain the Lagrangian for four-dimensional fields,

$\displaystyle\int_{0}^{2\pi
R}\mathcal{L}_{K}[\Psi;A_{y}]dy=\sum_{n=-\infty}^{\infty}\left[i\overline{\psi}_{n}\gamma^{\mu}\partial_{\mu}\psi_{n}-i\frac{n+\delta}{R}\overline{\psi}_{n}\gamma^{5}\psi_{n}\right]$
(15) $\displaystyle=$
$\displaystyle\sum_{n=-\infty}^{\infty}\left[i\overline{\psi}_{Ln}\gamma^{\mu}\partial_{\mu}\psi_{Ln}+i\overline{\psi}_{Rn}\gamma^{\mu}\partial_{\mu}\psi_{Rn}+i\frac{n+\delta}{R}\left(\overline{\psi}_{Rn}\psi_{Ln}-\overline{\psi}_{Ln}\psi_{Rn}\right)\right]\,,$

where $\delta=R\langle A_{y}\rangle$. In this expression, the decomposition
$\psi=\psi_{L}+\psi_{R}$ is done according to
$\psi_{L}=\frac{1-\gamma^{5}}{2}\psi$ and
$\psi_{R}=\frac{1+\gamma^{5}}{2}\psi$.

###  III.2 compactification on $S^{1}/Z_{2}$

Next, we consider compactification on $S^{1}/Z_{2}$. It is the simplest scheme
producing a four-dimensional chiral fermion from a five-dimensional Dirac
field PQ . On $S^{1}/Z_{2}$, in addition to the periodic condition
$\Psi(x,y+2\pi R)=\Psi(x,y)$, we require the conditions

$\Psi(x,-y)=P_{0}\gamma^{5}\Psi(x,y)\,,\quad\Psi(x,\pi
R-y)=P_{1}\gamma^{5}\Psi(x,\pi R+y)\quad(P_{0,1}=\pm 1)\,,$ (16)

which realize two fixed points of the orbifold $S^{1}/Z_{2}$, $y_{0}=0$ and
$y_{1}=\pi R$. If we choose $P_{0}=P_{1}=-1$, we find

$\displaystyle\Psi_{L}(x,-y)=+\Psi_{L}(x,y)\,,\quad\Psi_{L}(x,\pi
R-y)=+\Psi_{L}(x,\pi R+y)\,,$ (17)
$\displaystyle\Psi_{R}(x,-y)=-\Psi_{R}(x,y)\,,\quad\Psi_{R}(x,\pi
R-y)=-\Psi_{R}(x,\pi R+y)\,,$ (18)

and thus, the fermion field can be expanded as

$\displaystyle\Psi_{L}(x,y)$ $\displaystyle=$
$\displaystyle\frac{1}{\sqrt{2\pi R}}\psi_{L0}(x)+\sqrt{\frac{1}{\pi
R}}\sum_{n=1}^{\infty}\psi_{Ln}(x)\cos\frac{ny}{R}\,,$ (19)
$\displaystyle\Psi_{R}(x,y)$ $\displaystyle=$ $\displaystyle
i\sqrt{\frac{1}{\pi R}}\sum_{n=1}^{\infty}\psi_{Rn}(x)\sin\frac{ny}{R}\,.$
(20)

If we choose $P_{0}=P_{1}=+1$, the left and right handed fermions are swapped
from the above expressions. Note that a zero mode of the gauge field may exist
when an unusual, exotic condition in the fifth direction is assumed, but it is
not considered here, and so we suppose $\langle A_{y}\rangle=0$ for the
$S^{1}/Z_{2}$ compactification throughout this paper.

The kinetic part of the Lagrangian can be expressed in terms of the four-
dimensional fields, after integration over $y$, as

$\displaystyle\int_{0}^{2\pi R}\mathcal{L}_{K}[\Psi;0]dy$ (21)
$\displaystyle=$ $\displaystyle
i\overline{\psi}_{L0}\gamma^{\mu}\partial_{\mu}\psi_{L0}+\sum_{n=1}^{\infty}\left[i\overline{\psi}_{Ln}\gamma^{\mu}\partial_{\mu}\psi_{Ln}+i\overline{\psi}_{Rn}\gamma^{\mu}\partial_{\mu}\psi_{Rn}+i\frac{n}{R}\left(\overline{\psi}_{Rn}\psi_{Ln}-\overline{\psi}_{Ln}\psi_{Rn}\right)\right]\,.$

In the following sections, the model will be described when a graph and
compactification are specified. In the following sections, for instance, a
model that employs $S^{1}$ compactification and a $C_{N}$ graph will be
abbreviated as the $S^{1}\otimes C_{N}$ model.

##  IV deconstruction with $C_{N}$

In this section we adopt a cycle graph $C_{N}$ (Fig. 2) in our model. We label
the fields as $\Psi_{p}(x,y)$ $(p=0,1,\dots,N-1)$, and $\tilde{\Psi}_{p}(x,y)$
$(p=1,2,\dots,N)$. Then, we find

$-m\sum_{e\in\mathcal{E}(C_{N})}\left[\overline{\tilde{\Psi}}_{e}(\Psi_{t(e)}-\Psi_{o(e)})+h.c.\right]=-m\sum_{p=1}^{N}\left[\overline{\tilde{\Psi}}_{p}(\Psi_{p}-\Psi_{p-1})+h.c.\right]\,,$
(22)

where, in the sum, the identification $\Psi_{N}=\Psi_{0}$ is required.

Figure 2: A cycle graph $C_{8}$ as a Moose diagram of the model.

###  IV.1 $S^{1}\otimes C_{N}$

Figure 3: Graphical representation of the $S^{1}\otimes C_{N}$ model ($N=8$).

We assume that $\tilde{\Psi}_{p}$ are compactified on a common radius $R$ of
$S^{1}$, as well as ${\Psi}_{p}$ (Fig. 3). Then, we simply obtain

$\displaystyle\int_{0}^{2\pi
R}\mathcal{L}_{M}[\Psi_{p},\tilde{\Psi}_{p}]dy=-m\sum_{n=-\infty}^{\infty}\sum_{p=1}^{N}\left(\overline{\tilde{\psi}}_{p,n}(\psi_{p,n}-\psi_{p-1,n})+h.c.\right)$
$\displaystyle=-m\sum_{n=-\infty}^{\infty}\sum_{p=1}^{N}\left(\overline{\tilde{\psi}}_{R,p,n}(\psi_{L,p,n}-\psi_{L,p-1,n})+\overline{\tilde{\psi}}_{L,p,n}(\psi_{R,p,n}-\psi_{R,p-1,n})+h.c.\right)\,.$
(23)

Considering the mass terms that come out of compactification together, the
total mass term of component fields can be written as follows:

$-\sum_{n=-\infty}^{\infty}\begin{array}[]{rr}\Bigl{[}\overline{\psi}_{n}&\overline{\tilde{\psi}}_{n}\Bigr{]}_{R}\\\
&\end{array}\left[\begin{array}[]{rr}M_{n}&{M^{\prime}}^{T}_{n}\\\
{M^{\prime}_{n}}&\tilde{M}_{n}\end{array}\right]\left[\begin{array}[]{r}\psi_{n}\\\
\tilde{\psi}_{n}\end{array}\right]_{L}+h.c.\,,$ (24)

where $\psi_{n}=(\psi_{0,n},\dots\psi_{N-1,n})^{T}$ and
$\tilde{\psi}_{n}=(\tilde{\psi}_{1,n},\dots\tilde{\psi}_{N,n})^{T}$ with

$M_{n}=\left[\begin{array}[]{rrrrr}-i\frac{n+\delta_{0}}{R}&&&&\\\
&-i\frac{n+\delta_{1}}{R}&&&\\\ &&&\ddots&\\\
&&&&-i\frac{n+\delta_{N-1}}{R}\end{array}\right]\,,\quad\tilde{M}_{n}=\left[\begin{array}[]{rrrrr}-i\frac{n+\tilde{\delta}_{1}}{R}&&&&\\\
&-i\frac{n+\tilde{\delta}_{2}}{R}&&&\\\ &&&\ddots&\\\
&&&&-i\frac{n+\tilde{\delta}_{N}}{R}\end{array}\right]\,,$ (25)

and

${M^{\prime}}_{n}=m\left[\begin{array}[]{rrrrr}-1&1&0&\cdots&0\\\
0&-1&~{}1&\cdots&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots\\\
1&0&0&\cdots&-1\end{array}\right]=-mE^{T}(C_{N})\,,$ (26)

and the contributions of the background gauge fields come into play as
follows: $\delta_{p}\equiv R\langle(A_{y})_{v=p}\rangle$ and
$\tilde{\delta}_{p}\equiv R\langle(\tilde{A}_{y})_{e=p}\rangle$.

From the $n$th mass matrix

$\mathcal{M}_{n}=\left[\begin{array}[]{rr}M_{n}&{M^{\prime}}^{T}_{n}\\\
{M^{\prime}_{n}}&\tilde{M}_{n}\end{array}\right]\,,$ (27)

we obtain the $n$th mass-squared matrix

$\mathcal{M}^{2}_{n}\equiv\mathcal{M}^{\dagger}_{n}\mathcal{M}_{n}=\left[\begin{array}[]{rr}M_{n}^{\dagger}M_{n}+{M^{\prime}}^{T}_{n}M^{\prime}_{n}&M_{n}^{\dagger}{M^{\prime}}^{T}_{n}+{M^{\prime}}^{T}_{n}\tilde{M}_{n}\\\
{M^{\prime}}_{n}M_{n}+\tilde{M}^{\dagger}_{n}{M^{\prime}}_{n}&\tilde{M}_{n}^{\dagger}\tilde{M}_{n}+{M^{\prime}}_{n}{M^{\prime}}^{T}_{n}\end{array}\right]\,.$
(28)

If
$\delta_{0}=\delta_{1}=\cdots=\delta_{N-1}=\tilde{\delta}_{1}=\tilde{\delta}_{2}=\cdots=\tilde{\delta}_{N}\equiv\delta$
are assumed, we find

$\mathcal{M}^{2}_{n}=\left[\begin{array}[]{cc}\mathbf{M}^{2}_{n}&0\\\
0&\mathbf{M}^{2}_{n}\end{array}\right]\,,$ (29)

where

$\mathbf{M}^{2}_{n}=\frac{(n+\delta)^{2}}{R^{2}}+m^{2}\Delta(C_{N})\,.$ (30)

Therefore, the eigenvalues of the mass-squared of fermions are Mohar3

$\frac{(n+\delta)^{2}}{R^{2}}+4m^{2}\sin^{2}\frac{\pi
k}{N}\quad(n=\cdots,-1,0,1,\cdots,k=0,1,\cdots,N-1)\,.$ (31)

Two (for $\psi$ and $\tilde{\psi}$) Dirac zero modes are found for $n=0$ and
$k=0$, when $\delta=0$.

If we further replace $m\rightarrow\frac{N}{2\pi\tilde{R}}$ and taking the
limit of $N\rightarrow\infty$, the eigenvalues become

$\frac{(n+\delta)^{2}}{R^{2}}+\frac{k^{2}}{\tilde{R}^{2}}\,.$ (32)

We thus recognized that the continuum limit of the model in this section is a
theory compactified on a two-torus $T^{2}$ (topologically equivalent to a
surface of a doughnut). Fig. 4 shows the small eigenvalues of the mass-squared
matrix for $\delta=0$, $m=\frac{N}{2\pi\tilde{R}}$, $R=\tilde{R}=1$, and
$N=8$.

Figure 4: The first several eigenvalues of the mass-squared matrix of the
$S^{1}\otimes C_{N}$ model with $m=\frac{N}{2\pi\tilde{R}}$, $R=\tilde{R}=1$,
$\delta=0$, and $N=8$.

###  IV.2 $S^{1}\otimes C_{N}$ with flux

In a continuous theory with torus compactification, a homogeneous flux on the
two-torus, whose coordinates $(x,y)$ satisfy $x\sim x+2\pi\tilde{R}$ and
$y\sim y+2\pi R$, is described by the gauge field

$A_{y}=fx\quad(f\mbox{~{}is a constant})\,,$ (33)

and then, the constant magnetic flux density is given by

$F_{xy}=\partial_{x}A_{y}-\partial_{y}A_{x}=f\,.$ (34)

If the torus is expressed by $x\sim x+2\pi\tilde{R}$, $A_{y}$ and
$A_{y}+2\pi\tilde{R}f$ must be equivalent. On the other hand, the degree of
freedom of the gauge transformation

$\Psi\rightarrow e^{i\xi}\Psi\,,\quad A_{y}\rightarrow A_{y}+\partial_{y}\xi$
(35)

implies the identification of the gauge field on the torus described by $y\sim
y+2\pi R$, that is,

$A_{y}\sim A_{y}+\frac{q}{R}\quad(q\mbox{: integer}).$ (36)

Therefore, consistency on the torus requires $f=\frac{q}{2\pi R\tilde{R}}$ and
thus

$A_{y}=\frac{q}{2\pi R\tilde{R}}x\quad(q\mbox{~{}is a constant})\,.$ (37)

Now, in our $S^{1}\otimes C_{N}$ model, we think of $C_{N}$ as a discretized
circle with radius $\tilde{R}$, it can be interpreted as equating
$x/(2\pi\tilde{R})$ in the continuous theory, with an integer multiple of
$1/N$, so the background gauge fields on vertices $\delta_{p}=R\langle
A_{y}\rangle$ can be read as

$\delta_{p}=\frac{qp}{N}\quad(p=0,1,\dots,N-1)\,,$ (38)

for an integer $q$. Furthermore, if we set simply

$\tilde{\delta}_{p}=\frac{qp}{N}\quad(p=1,\dots,N)\,,$ (39)

for the background gauge fields on edges. These sets of configurations yield a
model whose continuum limit gives the torus compactification with constant
flux.

Figure 5: The first several eigenvalues of the mass-squared matrix in the case
that $\delta_{p}$ and $\tilde{\delta}_{p}$ are given by (38) and (39) with
$q=1$, respectively. The other settings are, $m=\frac{N}{2\pi\tilde{R}}$,
$R=\tilde{R}=1$, and $N=8$. The horizontal lines represent integer multiples
of $\frac{1}{\pi}$. Figure 6: The first several eigenvalues of the mass-
squared matrix in the case that $\delta_{p}$ and $\tilde{\delta}_{p}$ are
given by (38) and (39) with $q=2$, respectively. The other settings are,
$m=\frac{N}{2\pi\tilde{R}}$, $R=\tilde{R}=1$, and $N=8$. The horizontal lines
represent integer multiples of $\frac{2}{\pi}$. Figure 7: The first several
eigenvalues of the mass-squared matrix in the case that $\delta_{p}$ and
$\tilde{\delta}_{p}$ are given by (38) and (39) with $q=3$, respectively. The
other settings are, $m=\frac{N}{2\pi\tilde{R}}$, $R=\tilde{R}=1$, and $N=8$.
The horizontal lines represent integer multiples of $\frac{3}{\pi}$.

The first several eigenvalues of the mass-squared matrix for
$m=\frac{N}{2\pi\tilde{R}}$, $R=\tilde{R}=1$, and $N=8$ are shown in Fig. 5–7.
Fig. 5 shows the case with $q=1$, Fig. 6 shows the case with $q=2$, and Fig. 7
shows the case with $q=3$.

In the continuous theory with the background gauge field (37), the eigenvalues
are expressed by DS ; DSW

$\frac{q}{\pi R\tilde{R}}k\quad(k=0,1,2,\dots)\,,$ (40)

and the degeneracy of the zero modes is $2q$, because two species $\psi$ and
$\tilde{\psi}$ exist in our model.

In our result shown in Fig. 5–7, the first relatively large mass gap appears
approximately at the expected location in the continuum limit (however, in the
case of $q=1$, it is not so clear after that). The reason that each eigenvalue
that should be degenerate in the continuum limit is not so degenerate for
finite $N$ is due to the large anisotropy in the extra two (continuous and
discrete) directions, which is unavoidable. For example, it is known from
previous study that degeneracy is guaranteed to some extent in a lattice model
that is discrete in both directions TT .

###  IV.3 $S^{1}/Z_{2}\otimes C_{N}$

Figure 8: Graphical representation of the $S^{1}/Z_{2}\otimes C_{N}$ model
($N=8$).

In this case, we suppose that the fermions on the edges satisfy the reverse
condition $P_{0}=P_{1}=+1$, while the fermions on the vertices satisfy the
condition $P_{0}=P_{1}=-1$. Otherwise, we find that $\mathcal{L}_{M}\equiv 0$
if all fermions are governed by the same condition,and it results in a trivial
set of $N$ individual five-dimensional theories.

Now, we obtain, for the model consisting of $S^{1}/Z_{2}$ compactification
with $C_{N}$,

$\displaystyle\int_{0}^{2\pi
R}\mathcal{L}_{M}[\Psi_{p},\tilde{\Psi}_{p}]dy=-m\sum_{p=1}^{N}\left[\overline{\tilde{\psi}}_{R,p,0}(\psi_{L,p,0}-\psi_{L,p-1,0})+h.c.\right]$
$\displaystyle-m\sum_{n=1}^{\infty}\sum_{p=1}^{N}\left[\overline{\tilde{\psi}}_{R,p,n}(\psi_{L,p,n}-\psi_{L,p-1,n})+\overline{\tilde{\psi}}_{L,p,n}(\psi_{R,p,n}-\psi_{R,p-1,n})+h.c.\right]\,.$
(41)

The eigenvalues of the mass squared of fermions in this case are

$\frac{n^{2}}{R^{2}}+4m^{2}\sin^{2}\frac{\pi
k}{N}\quad(n=0,1,\cdots,k=0,1,\cdots,N-1)\,.$ (42)

One zero mode is expressed by
$\psi_{L,0,0}=\psi_{L,1,0}=\cdots=\psi_{L,N-1,0}$, and one mode is expressed
by $\tilde{\psi}_{R,1,0}=\tilde{\psi}_{R,2,0}=\cdots=\tilde{\psi}_{R,N,0}$.
These two modes are equivalent to one Dirac zero mode. If we replace
$m\rightarrow\frac{N}{2\pi\tilde{R}}$ and taking the limit of
$N\rightarrow\infty$, the eigenvalues become

$\frac{n^{2}}{R^{2}}+\frac{k^{2}}{\tilde{R}^{2}}\,.$ (43)

Further, if $\tilde{R}=R$, six (Dirac) modes belong to the second-smallest
eigenvalue. We thus recognized that the continuum limit of the model in this
section is a theory compactified on a cylinder.

##  V deconstruction with $P_{N+1}$

Figure 9: A path graph $P_{9}$ as a Moose diagram of the model.

In this section we adopt a path graph $P_{N+1}$ in the primary model. We label
the fields as $\Psi_{p}(x,y)$ $(p=0,1,\dots,N)$, and $\tilde{\Psi}_{p}(x,y)$
$(p=1,2,\dots,N)$ (Fig. 9). Then, we find

$-m\sum_{e\in\mathcal{E}(P_{N+1})}\left[\overline{\tilde{\Psi}}_{e}(\Psi_{t(e)}-\Psi_{o(e)})+h.c.\right]=-m\sum_{p=1}^{N}\left[\overline{\tilde{\Psi}}_{p}(\Psi_{p}-\Psi_{p-1})+h.c.\right]\,,$
(44)

###  V.1 $S^{1}\otimes P_{N+1}$

Figure 10: Graphical representation of the $S^{1}\otimes P_{N+1}$ model
($N=8$).

We assume that $\tilde{\Psi}_{p}$ are compactified on a common radius $R$ of
$S^{1}$, as well as ${\Psi}_{p}$, as before (Fig. 10). Then, we simply obtain

$\displaystyle\int_{0}^{2\pi
R}\mathcal{L}_{M}[\Psi_{p},\tilde{\Psi}_{p}]dy=-m\sum_{n=-\infty}^{\infty}\sum_{p=1}^{N}\left[\overline{\tilde{\psi}}_{p,n}(\psi_{p,n}-\psi_{p-1,n})+h.c.\right]$
$\displaystyle=-m\sum_{n=-\infty}^{\infty}\sum_{p=1}^{N}\left[\overline{\tilde{\psi}}_{R,p,n}(\psi_{L,p,n}-\psi_{L,p-1,n})+\overline{\tilde{\psi}}_{L,p,n}(\psi_{R,p,n}-\psi_{R,p-1,n})+h.c.\right]\,.$
(45)

This apparently describes non-chiral theory. The fermion mass term can be
written similarly to (24) in the $S^{1}\otimes C_{N}$ model. In the present
case, the $n$th mass matrix is given by

$\mathcal{M}_{n}=\left[\begin{array}[]{rr}M_{n}&{M^{\prime}}^{T}_{n}\\\
{M^{\prime}_{n}}&\tilde{M}_{n}\end{array}\right]\,,$ (46)

where

$M_{n}=\left[\begin{array}[]{rrrrr}-i\frac{n+\delta_{0}}{R}&&&&\\\
&-i\frac{n+\delta_{1}}{R}&&&\\\ &&&\ddots&\\\
&&&&-i\frac{n+\delta_{N}}{R}\end{array}\right]\,,\quad\tilde{M}_{n}=\left[\begin{array}[]{rrrrr}-i\frac{n+\tilde{\delta}_{1}}{R}&&&&\\\
&-i\frac{n+\tilde{\delta}_{2}}{R}&&&\\\ &&&\ddots&\\\
&&&&-i\frac{n+\tilde{\delta}_{N}}{R}\end{array}\right]\,,$ (47)

and

${M^{\prime}}_{n}=m\left[\begin{array}[]{rrrrrr}-1&1&0&\cdots&0&0\\\
0&-1&~{}1&\cdots&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\cdots&-1&~{}1\end{array}\right]=-mE^{T}(P_{N+1})\,.$ (48)

Further, if
$\delta_{0}=\delta_{1}=\cdots=\delta_{N-1}=\tilde{\delta}_{1}=\tilde{\delta}_{2}=\cdots=\tilde{\delta}_{N}\equiv\delta$
is assumed, we find

$\mathcal{M}^{2}_{n}=\left[\begin{array}[]{cc}\mathbf{M}^{2}_{n}&0\\\
0&\tilde{\mathbf{M}}^{2}_{n}\end{array}\right]\,,$ (49)

where

$\mathbf{M}^{2}_{n}=\frac{(n+\delta)^{2}}{R^{2}}+m^{2}\Delta(P_{N+1})\,,\quad\tilde{\mathbf{M}}^{2}_{n}=\frac{(n+\delta)^{2}}{R^{2}}+m^{2}L(P_{N+1})\,.$
(50)

Here, $L(G)\equiv E^{T}(G)E(G)$ is the Laplacian of the line graph of $G$
Mohar1 ; Merris .

The eigenvalues of $\mathbf{M}^{2}_{n}$ are Mohar1

$\frac{(n+\delta)^{2}}{R^{2}}+4m^{2}\sin^{2}\frac{\pi
k}{2(N+1)}\quad(n=\cdots,-1,0,1,\cdots,k=0,1,\cdots,N)\,,$ (51)

while the eigenvalues of $\tilde{\mathbf{M}}^{2}_{n}$ are444Note the fact that
the eigenvalues of matrices $AB$ and $BA$ are identical except for zero
eigenvalues.

$\frac{(n+\delta)^{2}}{R^{2}}+4m^{2}\sin^{2}\frac{\pi
k}{2(N+1)}\quad(n=\cdots,-1,0,1,\cdots,k=1,\cdots,N)\,.$ (52)

Therefore, one Dirac zero mode is found when $\delta=0$, since the matrix
$\tilde{\mathbf{M}}^{2}_{n}$ (for $\tilde{\psi}$) has no zero mode. If we
replace $m\rightarrow\frac{N+1}{\pi\tilde{R}}$ and taking the limit of
$N\rightarrow\infty$, the eigenvalues become

$\frac{(n+\delta)^{2}}{R^{2}}+\frac{k^{2}}{\tilde{R}^{2}}\,.$ (53)

Six Dirac modes belong to the second-smallest eigenvalue in this limit, when
$\delta=0$ and $R=\tilde{R}$. We thus recognized that the continuum limit of
the model in this section is a theory compactified on a cylinder.555In a crude
sense, it can be said that the $S^{1}/Z_{2}\otimes C_{N}$ model and the
$S^{1}\otimes P_{N+1}$ model are “dual” to each other.

###  V.2 $S^{1}/Z_{2}\otimes P_{N+1}$

Figure 11: Graphical representation of the $S^{1}/Z_{2}\otimes P_{N+1}$ model
($N=8$).

Next in this section, we consider the $S^{1}/Z_{2}\otimes P_{N+1}$ model (Fig.
11). As in the $S^{1}/Z_{2}\otimes C_{N}$ model, we assume that $\Psi$ and
$\tilde{\Psi}$ have opposite parity conditions. In this case, we find

$\displaystyle\int_{0}^{2\pi
R}\mathcal{L}_{M}[\Psi_{p},\tilde{\Psi}_{p}]dy=-m\sum_{p=1}^{N}\left[\overline{\tilde{\psi}}_{R,p,0}(\psi_{L,p,0}-\psi_{L,p-1,0})+h.c.\right]$
$\displaystyle\qquad-m\sum_{n=1}^{\infty}\sum_{p=1}^{N}\left[\overline{\tilde{\psi}}_{R,p,n}(\psi_{L,p,n}-\psi_{L,p-1,n})+\overline{\tilde{\psi}}_{L,p,n}(\psi_{R,p,n}-\psi_{R,p-1,n})+h.c.\right]\,.$
(54)

The mass-squared matrix $\mathcal{M}^{2}_{n}$ has the same structure as in the
$S^{1}\otimes P_{N+1}$ model in the previous subsection. The eigenvalues of
$\mathbf{M}^{2}_{n}$ are

$\frac{n^{2}}{R^{2}}+4m^{2}\sin^{2}\frac{\pi
k}{2(N+1)}\quad(n=0,1,\cdots,k=0,1,\cdots,N)\,,$ (55)

while the eigenvalues of $\tilde{\mathbf{M}}^{2}_{n}$ are

$\frac{n^{2}}{R^{2}}+4m^{2}\sin^{2}\frac{\pi
k}{2(N+1)}\quad(n=0,1,\cdots,k=1,\cdots,N)\,.$ (56)

Thus, one chiral zero mode is found. If we replace
$m\rightarrow\frac{N+1}{\pi\tilde{R}}$ and taking the limit of
$N\rightarrow\infty$, the eigenvalues become

$\frac{n^{2}}{R^{2}}+\frac{k^{2}}{\tilde{R}^{2}}\,.$ (57)

If further $R=\tilde{R}$, the number of the second-smallest Dirac eigenmodes
is three.

Judging from the geometric shape (an interval times an interval), the
continuum limit of the model in this subsection may be a theory compactified
on a rectangle, but the spectrum differs from the one for six-dimensional
Dirac fermions on a rectangle FHNST . This result may seem strange, but in the
current model, the numbers of $\Psi_{v}$ and $\tilde{\Psi}_{e}$, that is, the
numbers of vertices and edges, do not match, so a discrepancy occurs in the
number of eigenmodes.

###  V.3 mimicking $T^{2}/Z_{2}$

Figure 12: Graphical representation of the $T^{2}/Z_{2}$-mimicker model.

We replace zeroth and $N$th $S^{1}$-compactified theories in the $S^{1}\otimes
P_{N+1}$ model with the $S^{1}/Z_{2}$-compactified theories (Fig. 12). We
anticipate that a “mock” orbifold with four fixed points as $T^{2}/Z_{2}$ can
be obtained by geometrically blocking a cylinder with two line segments
$S^{1}/Z_{2}$.

The mass term in this case becomes

$-\overline{\psi}_{R,0}\mathcal{M}_{0}\psi_{L,0}-\sum_{n=1}^{\infty}\overline{\psi}_{R,n}\mathcal{M}_{n}\psi_{L,n}+h.c.\,,$
(58)

where
$\psi_{L0}=(\psi_{0,0},\dots,\psi_{N,0},\tilde{\psi}_{1,0},\dots,\tilde{\psi}_{N,0})^{T}_{L}$,
$\psi_{R0}=(\psi_{1,0},\dots,\psi_{N-1,0},\tilde{\psi}_{1,0},\dots,\tilde{\psi}_{N,0})^{T}_{R}$,
and
$\psi_{n}=(\psi_{0,n},\dots,\psi_{N,n},\psi_{1,-n},\dots,\psi_{N-1,-n},\tilde{\psi}_{1,n},\dots,\tilde{\psi}_{N,n},\tilde{\psi}_{1,-n},\dots,\tilde{\psi}_{N,-n})^{T}$,
for $n\geq 1$. The $(2N-1)\times(2N+1)$ matrix $\mathcal{M}_{0}$ is expressed
by

$\mathcal{M}_{0}=\left[\begin{array}[]{rr}M_{0}&{M^{\prime\prime}}_{0}\\\
{M^{\prime}_{0}}&\tilde{M}_{0}\end{array}\right]\,,$ (59)

and

${M}_{0}=\left[\begin{array}[]{rrrrr}0&-i\delta_{1}/R&&&0\\\
\vdots&&\ddots&&\vdots\\\
0&&&-i\delta_{N-1}/R&0\end{array}\right]\,,\quad\tilde{M}_{0}=\left[\begin{array}[]{rrr}-i\tilde{\delta}_{1}/R&&\\\
&\ddots&\\\ &&-i\tilde{\delta}_{N}/R\end{array}\right]\,,$ (60)
${M^{\prime}}_{0}=m\left[\begin{array}[]{rrrrrr}-1&1&0&\cdots&0&0\\\
0&-1&~{}1&\cdots&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\cdots&-1&~{}1\end{array}\right]=-mE^{T}(P_{N+1})\,,$ (61)
${M^{\prime\prime}}_{0}=m\left[\begin{array}[]{rrrrrr}1&-1&0&\cdots&0&0\\\
0&1&-1&\cdots&0&0\\\ \vdots&\vdots&\vdots&\ddots&\vdots&\vdots\\\
0&0&0&\cdots&1&-1\end{array}\right]=mE^{T}(P_{N})\,.$ (62)

The $4N\times 4N$ matrix $\mathcal{M}_{n}$ is expressed by

$\mathcal{M}_{n}=\left[\begin{array}[]{rr}M_{n}&{{M}^{\prime\prime}}_{n}\\\
{M^{\prime}_{n}}&\tilde{M}_{n}\end{array}\right]\,,$ (63)

and

${M}_{n}=\left[\begin{array}[]{rrrrrrrr}-i\frac{n}{R}&&&&&&&\\\
&-i\frac{(n+\delta_{1})}{R}&&&&&&\\\ &&\ddots&&&&&\\\
&&&-i\frac{(n+\delta_{N-1})}{R}&&&&\\\ &&&&-i\frac{n}{R}&&&\\\
&&&&&-i\frac{(-n+\delta_{1})}{R}&&\\\ &&&&&&\ddots&\\\
&&&&&&&-i\frac{(-n+\delta_{N-1})}{R}\\\ \end{array}\right]\,,$ (64)
$\tilde{M}_{n}=\left[\begin{array}[]{rrrrrrr}-i\frac{(n+\tilde{\delta}_{1})}{R}&&&&&&\\\
&\ddots&&&&&\\\ &&-i\frac{(n+\tilde{\delta}_{N})}{R}&&&&\\\
&&&-i\frac{(-n+\tilde{\delta}_{1})}{R}&&&\\\ &&&&&\ddots&\\\
&&&&&&-i\frac{(-n+\tilde{\delta}_{N})}{R}\\\ \end{array}\right]\,,$ (65)
${M^{\prime}}_{n}=m\left[\begin{array}[]{rrrrrrrrrr}-\frac{1}{\sqrt{2}}&1&&&&&&&&\\\
&-1&~{}1&&&&&&&\\\ &&\ddots&\ddots&&&&&&\\\ &&&-1&1&&&&&\\\
&&&&-1&\frac{1}{\sqrt{2}}&&&&\\\ -\frac{1}{\sqrt{2}}&&&&&&1&&&\\\
&0&&&&&-1&1&&\\\ &&&\ddots&&&&\ddots&\ddots&\\\ &&&&0&&&&-1&1\\\
&&&&&\frac{1}{\sqrt{2}}&&&&-1\end{array}\right]\,,$ (66)
${{M}^{\prime\prime}}_{n}=m\left[\begin{array}[]{rrrrrrrrrr}-\frac{1}{\sqrt{2}}&&&&\frac{1}{\sqrt{2}}&&&&&\\\
1&-1&&&&0&&&&\\\ &\ddots&\ddots&&&&&\ddots&&\\\ &&1&-1&&&&&0&\\\
&&&\frac{1}{\sqrt{2}}&&&&&&-\frac{1}{\sqrt{2}}\\\ &&&&1&-1&&&&\\\
&&&&&1&-1&&&\\\ &&&&&&\ddots&\ddots&&\\\ &&&&&&&1&-1&\\\
&&&&&&&&1&-1\end{array}\right]\,.$ (67)

Figure 13: The first several eigenvalues of the mass-squared matrix of the
“mock” orbifold model with vanishing gauge fields. The parameters are chosen
as: $m=\frac{N+1}{2\pi\tilde{R}}$, $R=\tilde{R}=1$, and $N=8$.

In Fig. 13, the small eigenvalues are shown in the case without background
gauge field. The spectrum resembles the spectrum of the torus compactification
plus two massless chiral modes.

Next, we consider, as in the $S^{1}\otimes C_{N}$ model, the flux-like
configuration of the gauge fields:

$\delta_{p}=\frac{qp}{N}\quad(p=1,\dots,N-1)\,,\quad\tilde{\delta}_{p}=\frac{qp}{N}\quad(p=1,\dots,N)\,.$
(68)

Figure 14: The first several eigenvalues of the mass-squared matrix in the
case that $\delta_{p}$ and $\tilde{\delta}_{p}$ are given by (68) with $q=1$,
respectively. The other settings are, $m=\frac{N+1}{2\pi\tilde{R}}$,
$R=\tilde{R}=1$, and $N=8$. The horizontal lines represent integer multiples
of $\frac{2}{\pi}$. Figure 15: The first several eigenvalues of the mass-
squared matrix in the case that $\delta_{p}$ and $\tilde{\delta}_{p}$ are
given by (68) with $q=2$, respectively. The other settings are,
$m=\frac{N+1}{2\pi\tilde{R}}$, $R=\tilde{R}=1$, and $N=8$. The horizontal
lines represent integer multiples of $\frac{4}{\pi}$. Figure 16: The first
several eigenvalues of the mass-squared matrix in the case that $\delta_{p}$
and $\tilde{\delta}_{p}$ are given by (68) with $q=3$, respectively. The other
settings are, $m=\frac{N+1}{2\pi\tilde{R}}$, $R=\tilde{R}=1$, and $N=8$. The
horizontal lines represent integer multiples of $\frac{6}{\pi}$.

For $N=8$, the eigenvalues are shown in Figs. 14, 15, and 16, when $q=1,2,3$,
respectively. It is known that the corresponding compactification yields the
eigenvalues located at integer multiples of $\frac{2q}{\pi R\tilde{R}}$.666The
explanation for the factor two is illustrated in FIG. 17. In our model, the
relatively large gap appears at the same location, though the degeneracies do
not really match them. Thus, unfortunately, the continuum limit of this model
may not be a “true ” $T^{2}/Z_{2}$. There are, in any case in the model, two
chiral zero modes which belong to the zero eigenvalues of $\mathcal{M}_{0}$.

Figure 17: We explain the factor two in the spectral gap when $q=1$, for
example. The $S^{1}$ structure is obtained by identifying the top and bottom
sides of the rectangle. The top and bottom halves of the left and right sides
are identified respectively, since we take the quotient with $Z_{2}$.
Meanwhile, the diagonal line in the figure schematically represents
$\delta=R\langle A_{y}\rangle$. $\delta$ changes by unity between the left and
right ends, and then, the gauge fields on both ends are gauge equivalent.
Since we take the quatient with $Z_{2}$, the flux is the twice the value
expected on a simple torus.

##  VI Summary and discussion

In this paper, the mass spectra of light fermions are obtained in our “half-
deconstructed” models. When compared with the spectrum for manifolds,
qualitative agreement was found when supposing the continuum limit. Rather,
from a particle physics perspective, it is an interesting toy model that
fermions with non-zero tiny mass appear in the models with graphs of finite
size. If we take the results into account, it can be realized that a very
small mass scale is generated in the partially deconstructed model, which is a
feature of this scheme. In that case, it is considered that the fermions in
this model are not genuine quarks or leptons. The scale which appears in the
model may have an indirect influence on the weak interaction scale or provides
a new scale of a model beyond the standard model. For example, the scale may
indicate that of the dark matter candidates, or may have the relevance to
additional intermediate sectors which interact only the fields at specific
sites. A clockwork mechanism CI ; KR ; GM ; FPRT 777See also Refs. CS ;
Sutherland  has already been proposed as a mechanism for generating such a
small hierarchy,888Note that in clockwork mechanisms, clear and relatively
large mass gap (as in our model) generally do not appear. so examining and
comparing it with our model, or constructing a fused model, will be an
important future challenge.

Since strict topological restrictions cannot be placed on the configuration of
our discrete fields, it will be necessary to study not only the gauge field as
a background field but also their dynamics. In other words, the additional
bilinear term corresponding to the kinetic term of the gauge field may be
important in the whole Lagrangian of the model.

A natural extension of the model is one with varying the compactification
scale $R$ of each five-dimensional theory depending on the site. It
corresponds to warped manifolds such as the Randall–Sundrum theory RS . In
that case, it will be necessary to consider mechanisms that give the dynamics
of such scales. It is possible to randomly connect five-dimensional theories
with different compactifications with random configurations of gauge fields or
Wilson loops, but some kind of symmetry or guiding principle would be required
in such cases.

The model discussed in this paper is characterized by not requiring the
introduction of new link fields corresponding to a component of a gauge field
on a discrete dimension. Although this is preferable from the standpoint of
simplicity, the feasibility of introducing such a link field in a modified
version of the model should be considered in the future, taking into account
of phenomenological models of particle physics.

In this paper, we focused on fermions and their modes of light mass
eigenvalues, but in future we should of course also discuss important research
topics such as theories involving degrees of freedom of gauge fields and
scalar fields, and models related to dark sectors. A model with supersymmetry
can be constructed relatively easily by replacing fields in the model here
with superfields KKS . Therefore, we will be able to explore various
possibilities for supersymmetric models and their spectra, including semi-
deconstructed models such as those introduced here.

In this paper, we mainly introduced toy models with reference to the continuum
limits, but when a general graph is used, there is no counterpart of
continuous theory, and novel pseudo-extra-dimensional models become possible.
The deconstruction of higher dimensional manifolds (or orbifolds) is an
interesting extension. In the future, we would also like to consider full
deconstruction, that is, a four-dimensional theory that represents the extra
dimensions by graphs. In addition, it may be feasible that some other
orbifold-like compactification can be constructed based on the deconstruction
(for example, SSK ) of a tilted torus (for a pedagogical introduction Dienes
), so we wish to pursue its mathematical structure and applications.

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# Differential Characterization of Quasi-Concave Functions without Twice
Differentiability

Yuhki Hosoya
Faculty of Economics, Chuo University TEL: +81-90-5525-5142, E-mail:
ukki(at)gs.econ.keio.ac.jp742-1, Higashinakano, Hachioji-shi, Tokyo, 192-0393,
Japan.

###### Abstract

This paper presents a necessary and sufficient condition for a real-valued
function defined on an open and convex subset of a Banach space to be quasi-
concave, and a sufficient condition for such a function to be strictly quasi-
concave. These conditions are applicable to continuously differentiable
functions that satisfy a mild additional assumption, and do not require the
functions to be twice differentiable. Because this additional assumption is
trivially satisfied for twice continuously differentiable functions, our
results are pure extensions to classical results.

Keywords: quasi-concavity, Banach space, continuously differentiable function.

JEL codes: C61, C65, D11.

MSC2020 codes: 91B08, 91B16

##  1 Introduction

The notion of quasi-concave functions is frequently used for optimization
problems in economics. However, it is difficult to characterize the quasi-
concavity of a function in terms of conditions for its derivatives. For
concavity, in contrast, there is a clear characterization: for example, a
continuously differentiable real-valued function defined on an open and convex
subset of the real line is concave if and only if its derivative is
nonincreasing. However, for quasi-concavity, such a clear characterization is
absent.

Otani (1983) treated this problem and provided a necessary and sufficient
condition for twice continuously differentiable non-degenerate functions
defined on an open and convex subset in $\mathbb{R}^{n}$ to be quasi-concave.
His condition is that the Hessian matrix of this function is negative semi-
definite on the kernel of the gradient vector. However, there are several
optimization problems in which twice continuous differentiability of the
objective function cannot be assumed. The main purpose of this paper is to
extend Otani’s result to functions that are not necessarily twice continuously
differentiable but are once continuously differentiable and satisfy an
additional property. Note that, any twice continuously differentiable function
satisfies this additional property, and so our result is a pure extension to a
known result.

Moreover, we extend some classical results for quasi-concave functions on
$\mathbb{R}^{n}$ to any Banach space. Specifically, we treat a continuously
differentiable real-valued function defined on an open and convex subset of a
Banach space, and present three results. First, we provide a necessary and
sufficient condition for a non-degenerate function to be quasi-concave
(Theorem 1). Second, we provide a sufficient condition for a function to be
strictly quasi-concave (Theorem 2). As we noted above, both results are pure
extensions to classical known results.

Note that our “additional property” introduced in this paper is deeply related
to Frobenius’ theorem. Debreu (1972) considered the following total
differential equation:

$\nabla f(x)=\lambda(x)g(x),$

where $g(x)$ is the given function and the pair of $f(x)$ and $\lambda(x)$ is
a solution. In Debreu’s theory, the vector field $g(x)$ exhibits the price
vector under which consumption plan $x$ is chosen. Thus, the above equation
concerns Lagrange’s multiplier rule, and $f(x)$ can be treated as the
objective function for the consumer. Debreu expected that if $g(x)$ is
continuously differentiable, then the solution $f(x)$ must be twice
continuously differentiable. However, a counterexample found by two students
was later presented by Debreu (1976), in which the solution $f(x)$ must not be
twice continuously differentiable. Therefore, we cannot assume that the
objective function for the consumer is twice continuously differentiable if
$g(x)$ is only continuously differentiable. Our additional property implies
the continuous differentiability for such a $g(x)$, and so our results are
applicable to consumer theory.

In Subsection 2.1, we introduce some knowledge on Banach spaces that is needed
to understand our results. In Subsection 2.2, we present the main results of
this study. Section 3 contains several remarks concerning our results.

##  2 Main Results

###  2.1 Preliminaries

In this subsection, we introduce some basic knowledge on Banach spaces. We
assume that readers know the definition of a Banach space. All of the facts
introduced in this subsection are explained and proved in standard textbooks:
see, for example, chapters 2-5 of Luenberger (1969) or chapters 5-6 of
Aliprantis and Border (2006).

Suppose that $X,Y$ are Banach spaces. Let $L(X,Y)$ be the set of all linear
and continuous functions from $X$ into $Y$. Then, $L(X,Y)$ becomes a Banach
space with respect to the following operator norm

$\|A\|=\sup\\{\|Ax\||x\in X,\ \|x\|\leq 1\\}.$

In particular, the space $X^{\prime}$ of all linear and continuous functions
from $X$ into $\mathbb{R}$ is called the dual space of $X$. Note that
$X^{\prime}$ is the same as $L(X,\mathbb{R})$, and thus it is also a Banach
space with the operator norm. For $x^{\prime}\in X^{\prime}$, we write
$\langle x,x^{\prime}\rangle$ instead of $x^{\prime}(x)$. It is easy to show
that the function $f(x,x^{\prime})=\langle x,x^{\prime}\rangle$ is continuous.

If $X=\mathbb{R}^{n}$, then $X^{\prime}=\mathbb{R}^{n}$ and $\langle
x,x^{\prime}\rangle$ is the usual inner product between two vectors
$x=(x_{1},...,x_{n})$ and $x^{\prime}=(x_{1}^{\prime},...,x_{n}^{\prime})$;
that is,

$\langle x,x^{\prime}\rangle=\sum_{i=1}^{n}x_{i}x_{i}^{\prime}.$

In this paper, $A^{T}$ denotes the transpose of an $n\times m$ matrix $A$.
Note that, if $X=\mathbb{R}^{n}$, then $\langle x,x^{\prime}\rangle$ can also
be written as $x^{T}x^{\prime}$.

Next, suppose that $X,Y$ are Banach spaces. Suppose that $U$ is an open set in
$X$ and $f:U\to Y$. If there exists an element $A\in L(X,Y)$ such that

$\lim_{h\to 0}\frac{\|f(x+h)-f(x)-Ah\|}{\|h\|}=0,$

then $f$ is said to be Fréchet differentiable at $x$, and $A$ is called the
Fréchet derivative of $f$ at $x$. It is known that if $f$ is Fréchet
differentiable at $x$, then it is continuous at $x$, and its Fréchet
derivative at $x$ is unique. In this paper, $Df(x)$ denotes the Fréchet
derivative of $f$ at $x$. If $f$ is Fréchet differentiable at any point of $U$
and $Df:x\mapsto Df(x)$ is continuous, then we say that $f$ is $C^{1}$. We
often abbreviate “Fréchet differentiable” to “differentiable” simply.

Specifically, suppose that $Y=\mathbb{R}$. Let $U$ be an open subset of the
Banach space $X$ and $f:U\to\mathbb{R}$ be $C^{1}$. Then, the mapping
$Df:x\mapsto Df(x)$ is a function from $U$ into $X^{\prime}$, and $X^{\prime}$
is a Banach space. Therefore, we can define the Fréchet derivative of $Df$ at
$x$, which is denoted by $D^{2}f(x)$. If $D^{2}f(x)$ is defined on $U$ and
$D^{2}f:x\mapsto D^{2}f(x)$ is continuous, then we say that $f$ is $C^{2}$.

Suppose that $X,Y,Z$ are Banach spaces, $U\subset X$ and $V\subset Y$ are open
sets, $f:V\to Z$ and $g:U\to Y$ are given, $g(x)\in V$, $g$ is differentiable
at $x$, and $f$ is differentiable at $g(x)$. Then, $f\circ g$ is
differentiable at $x$, and

$D(f\circ g)(x)=Df(g(x))\circ Dg(x).$

This formula is called the chain rule.

Suppose that $X$ is a linear space and $U$ is a convex subset of $X$. A
function $f:U\to\mathbb{R}$ is said to be quasi-concave if and only if for
every $x,y\in U$ and $t\in[0,1]$,

$f((1-t)x+ty)\geq\min\\{f(x),f(y)\\}.$

If, in addition,

$f((1-t)x+ty)>\min\\{f(x),f(y)\\}$

when $x\neq y$ and $0<t<1$, then we say that $f$ is strictly quasi-concave.

###  2.2 Results

Suppose that $X$ is a Banach space and $U$ is an open subset of $X$. For
$f:U\to\mathbb{R}$, we say that $f$ is $C^{1}_{*}$ if $f$ is $C^{1}$ and there
exists a pair of a $C^{1}$ function $g:U\to X^{\prime}$ and a positive
continuous function $\lambda:U\to\mathbb{R}$ such that

$Df(x)=\lambda(x)g(x).$ (1)

Clearly, if $f$ is $C^{2}$, then $f$ is $C^{1}_{*}$: choose $\lambda\equiv 1$
and $g(x)=Df(x)$. When $X$ is a Hilbert space, then the norm of $X$ is
differentiable everywhere except the zero vector. In this case, if $f$ is
$C^{1}$ and $Df(x)\neq 0$ for all $x\in U$, then $f$ is $C^{1}_{*}$ if and
only if the following function

$h(x)=\frac{1}{\|Df(x)\|}Df(x)$

is $C^{1}$. Indeed, if $h$ is $C^{1}$, then we can choose
$\lambda(x)=\|Df(x)\|$ and $g(x)=h(x)$. Conversely, if there exists a $C^{1}$
function $g$ and a continuous function $\lambda$ such that (1) holds, then
$g(x)\neq 0$ for all $x\in U$, and

$h(x)=\frac{1}{\|g(x)\|}g(x),$

which is $C^{1}$. In particular, if $X=\mathbb{R}^{n}$, then the space of all
$C^{1}_{*}$ functions whose derivatives never vanish coincides with the space
of $C^{1}$ functions $f$ such that the above $h$ is $C^{1}$.

Our first main result is as follows.

Theorem 1. Suppose that $U$ is an open and convex subset of a Banach space
$X$, $f:U\to\mathbb{R}$ is $C^{1}_{*}$, and $Df(x)\neq 0$ for all $x\in U$.
Choose a pair of a $C^{1}$ function $g$ and a positive continuous function
$\lambda$ that satisfies (1). Then, the following two claims are equivalent.

  1. 1)

$f$ is quasi-concave.

  2. 2)

$\langle w,Dg(x)w\rangle\leq 0$ for any $x\in U$ and $w\in X$ such that
$\langle w,g(x)\rangle=0$.

Proof. First, we prove a lemma.

Lemma 1(Lagrange’s multiplier rule). Suppose that $U$ is an open and convex
subset of a Banach space $X$ and $f:U\to\mathbb{R}$ is continuous and quasi-
concave. Choose any $x^{\prime}\in X^{\prime}\setminus\\{0\\}$. If $f$ is
differentiable at $x^{*}$ and $Df(x^{*})=\lambda x^{\prime}$ for some
$\lambda>0$, then $f(x^{*})\geq f(x)$ for every $x\in U$ such that $\langle
x,x^{\prime}\rangle=\langle x^{*},x^{\prime}\rangle$.

Proof of Lemma 1. Suppose not. Then, there exists $x\in U$ such that $\langle
x,x^{\prime}\rangle=\langle x^{*},x^{\prime}\rangle$ and $f(x)>f(x^{*})$.
Because $f$ is continuous and $U$ is open, there exists $y\in U$ such that
$\langle y,x^{\prime}\rangle<\langle x^{*},x^{\prime}\rangle$ and
$f(y)>f(x^{*})$. Define $y(t)=(1-t)x^{*}+ty$. Then, $y(t)\in U$ for all
$t\in[0,1]$, and by the quasi-concavity of $f$, we have that $f(y(t))\geq
f(x^{*})$ for all $t\in[0,1]$. Therefore,

$0>\langle
y-x^{*},Df(x^{*})\rangle=\left.\frac{d}{dt}f(y(t))\right|_{t=0}=\lim_{t\downarrow
0}\frac{f(y(t))-f(x^{*})}{t}\geq 0,$

which is a contradiction. This completes the proof. $\blacksquare$

Suppose that 1) holds. Choose any $x\in U$ and $w\in X$ such that $\langle
w,g(x)\rangle=0$. Define

$c(t)=f(x+tw).$

Then, there exists $\varepsilon>0$ such that $c(t)$ is defined on
$I=[-\varepsilon,\varepsilon]$. By Lagrange’s multiplier rule, we have that
$c(t)$ attains the maximum at $t=0$. Hence, $c(t)\leq c(0)$ for all
$t\in[0,\varepsilon]$. By the mean value theorem, we can obtain a sequence
$(t_{k})$ such that $t_{k}\downarrow 0$ as $k\to\infty$ and
$c^{\prime}(t_{k})\leq 0$ for all $k$. Therefore,

$\displaystyle 0\geq$
$\displaystyle~{}\limsup_{k\to\infty}\frac{c^{\prime}(t_{k})}{t_{k}}=\limsup_{k\to\infty}\frac{\langle
w,Df(x+t_{k}w)\rangle}{t_{k}}$ $\displaystyle=$
$\displaystyle~{}\limsup_{k\to\infty}\frac{\lambda(x+t_{k}w)\langle
w,g(x+t_{k}w)\rangle}{t_{k}}$ $\displaystyle\geq$
$\displaystyle~{}M\limsup_{k\to\infty}\frac{\langle
w,g(x+t_{k}w)\rangle}{t_{k}}$ $\displaystyle=$
$\displaystyle~{}M\limsup_{k\to\infty}\frac{\langle
w,g(x+t_{k}w)-g(x)\rangle}{t_{k}}=M\langle w,Dg(x)w\rangle,$

where $M=\max\\{\lambda(x+tw)|t\in I\\}>0$, which implies that 2) holds.
Therefore, 1) implies 2).

Next, suppose that 2) holds and 1) is violated. Then, there exist $x,y\in U$
and $t\in[0,1]$ such that $f((1-t)x+ty)<\min\\{f(x),f(y)\\}$. Let

$T^{*}=\arg\min\\{f((1-t)x+ty)|t\in[0,1]\\}.$

Then, $T^{*}$ is a closed subset of $[0,1]$ such that $0,1\notin T^{*}$. Let
$t^{*}=\max T^{*}$ and $z=(1-t^{*})x+t^{*}y$. Moreover, define
$x(t)=(1-t)x+ty$. Because $Df(z)\neq 0$, we have that there exists $p\in X$
such that $\langle p,Df(z)\rangle>0$. Consider the following function $u$:

$u(a,b)=f(z+a(y-x)+bp)=f(x(t^{*}+a)+bp).$

Then, $u$ is $C^{1}$ around $(0,0)$, $u(0,0)=f(z)$, and $\frac{\partial
u}{\partial b}(0,0)=\langle p,Df(z)\rangle>0$. By the implicit function
theorem, there exist $\varepsilon_{1},\varepsilon_{2}>0$, and a $C^{1}$
function
$b:[-\varepsilon_{1},\varepsilon_{1}]\to[-\varepsilon_{2},\varepsilon_{2}]$
such that $b(0)=0$ and, for each
$(a,b)\in[-\varepsilon_{1},\varepsilon_{1}]\times[-\varepsilon_{2},\varepsilon_{2}]$,
$u(a,b)=f(z)$ if and only if $b=b(a)$. Then,

$b^{\prime}(a)=-\frac{\frac{\partial u}{\partial a}(a,b(a))}{\frac{\partial
u}{\partial b}(a,b(a))}=-\frac{\langle
y-x,Df(x(t^{*}+a)+b(a)p)\rangle}{\langle p,Df(x(t^{*}+a)+b(a)p)\rangle}.$

Because of the definition of $z$ and the first-order condition, we have that

$\langle y-x,Df(z)\rangle=0,$

and thus,

$b^{\prime}(0)=-\frac{\langle y-x,Df(z)\rangle}{\langle p,Df(z)\rangle}=0.$

Define

$y(a)=x(t^{*}+a)+b(a)p,\ w(a)=b^{\prime}(a)p+(y-x),$

and

$d(a)=\langle p,Df(y(a))\rangle.$

Differentiating $u(a,b(a))=f(y(a))\equiv f(z)$ with respect to $a$, we obtain

$\langle w(a),Df(y(a))\rangle=0,$

and thus,

$\langle w(a),g(y(a))\rangle=0$

for any $a\in[-\varepsilon_{1},\varepsilon_{1}]$. By our choice of $p$,
$y(0)=z$ and $d(0)>0$, and thus, there exists $\delta>0$ such that if
$0<a<\delta$, then $d(a)>0$. For such $a>0$,

$\displaystyle 0=$ $\displaystyle~{}\liminf_{a^{\prime}\to
a}\frac{1}{a^{\prime}-a}[\langle w(a^{\prime}),Df(y(a))\rangle-\langle
w(a^{\prime}),Df(y(a))\rangle]$ $\displaystyle=$
$\displaystyle~{}\liminf_{a^{\prime}\to a}\frac{1}{a^{\prime}-a}[\langle
w(a^{\prime}),Df(y(a))\rangle-\lambda(y(a))\langle
w(a^{\prime}),g(y(a))\rangle]$ $\displaystyle=$
$\displaystyle~{}\liminf_{a^{\prime}\to a}\frac{1}{a^{\prime}-a}[\langle
w(a^{\prime})-w(a),Df(y(a))\rangle+\lambda(y(a))\langle
w(a^{\prime}),g(y(a^{\prime}))-g(y(a))\rangle]$ $\displaystyle=$
$\displaystyle~{}d(a)\times\liminf_{a^{\prime}\to
a}\frac{b^{\prime}(a^{\prime})-b^{\prime}(a)}{a^{\prime}-a}+\lambda(y(a))\langle
w(a),Dg(y(a))w(a)\rangle$ $\displaystyle\leq$
$\displaystyle~{}d(a)\times\liminf_{a^{\prime}\to
a}\frac{b^{\prime}(a^{\prime})-b^{\prime}(a)}{a^{\prime}-a}.$

Therefore,

$\liminf_{a^{\prime}\to
a}\frac{b^{\prime}(a^{\prime})-b^{\prime}(a)}{a^{\prime}-a}\geq 0,$

which implies that

$\liminf_{a^{\prime}\downarrow
a}\frac{b^{\prime}(a^{\prime})-b^{\prime}(a)}{a^{\prime}-a}\geq 0,\
\liminf_{a^{\prime}\uparrow
a}\frac{b^{\prime}(a^{\prime})-b^{\prime}(a)}{a^{\prime}-a}\geq 0.$

Fix $a\in]0,\delta[$, and define $h(s)=b^{\prime}(s)a-b^{\prime}(a)s$. Then,
$h(a)=h(0)=0$, and thus, there exists $s^{*}$ such that $0<s^{*}<a$ and $h(s)$
attains either the maximum or minimum on $[0,a]$ at $s=s^{*}$. If $h(s^{*})$
attains the maximum, then

$\displaystyle 0\geq$ $\displaystyle~{}\liminf_{s\downarrow
s^{*}}\frac{h(s)-h(s^{*})}{s-s^{*}}$ $\displaystyle=$
$\displaystyle~{}a\liminf_{s\downarrow
s^{*}}\frac{b^{\prime}(s)-b^{\prime}(s^{*})}{s-s^{*}}-b^{\prime}(a)\geq-b^{\prime}(a),$

which implies that $b^{\prime}(a)\geq 0$ for any $a\in]0,\delta[$. If
$h(s^{*})$ attains the minimum, then

$\displaystyle 0\geq$ $\displaystyle~{}\liminf_{s\uparrow
s^{*}}\frac{h(s)-h(s^{*})}{s-s^{*}}$ $\displaystyle=$
$\displaystyle~{}a\liminf_{s\uparrow
s^{*}}\frac{b^{\prime}(s)-b^{\prime}(s^{*})}{s-s^{*}}-b^{\prime}(a)\geq-b^{\prime}(a),$

which again implies that $b^{\prime}(a)\geq 0$ for any $a\in]0,\delta[$.
Because $b(0)=0$, we have that $b(a)\geq 0$ for all $a\in[0,\delta]$.

Because

$\frac{\partial u}{\partial b}(0,0)=\langle p,Df(z)\rangle>0,$

there exists a neighborhood $V$ of $(0,0)$ such that $\frac{\partial
u}{\partial b}(a,b)>0$ for all $(a,b)\in V$. If $a>0$ is sufficiently small,
then $a<\delta$ and $(a,b)\in V$ for all $b\in[0,b(a)]$. Therefore,
$\frac{\partial u}{\partial b}(a,b)>0$ for such $(a,b)$, and thus,

$f(x(t^{*}+a))\leq f(x(t^{*}+a)+b(a)p)=f(z),$

which contradicts the definition of $t^{*}$. This completes the proof.
$\blacksquare$

Regarding strict quasi-concavity, we present the following theorem.

Theorem 2. Suppose that $U$ is an open and convex subset of a Banach space $X$
and $f:U\to\mathbb{R}$ is $C^{1}_{*}$. Choose a pair of a $C^{1}$ function $g$
and a positive continuous function $\lambda$ that satisfies (1). If $\langle
w,Dg(x)w\rangle<0$ for all $x\in U$ and $w\in X$ such that $w\neq 0$ and
$\langle w,g(x)\rangle=0$, then $f$ is strictly quasi-concave.

Proof. Suppose that $f(x)$ is not strictly quasi-concave. Then, there exist
$x,y\in U$ and $t\in]0,1[$ such that $x\neq y$ and

$f((1-t)x+ty)\leq\min\\{f(x),f(y)\\}.$

Define $x(t)=(1-t)x+ty$. Then, there exists
$t^{*}\in\arg\min\\{f(x(t))|t\in[0,1]\\}$ such that $0<t^{*}<1$. Define
$z=x(t^{*})$, $w=y-x$, and

$c(t)=\langle w,Df(x(t))\rangle.$

By the first-order condition, we have that $c(t^{*})=0$, and thus $\langle
w,g(x(t^{*}))\rangle=0$. Moreover,

$\displaystyle 0>$ $\displaystyle~{}\langle w,Dg(z)w\rangle=\lim_{t\downarrow
t^{*}}\frac{\langle w,g(x(t))-g(x(t^{*}))\rangle}{t-t^{*}}$ $\displaystyle=$
$\displaystyle~{}\lim_{t\downarrow t^{*}}\frac{\langle
w,g(x(t))\rangle}{t-t^{*}}=\lim_{t\downarrow
t^{*}}\frac{1}{\lambda(x(t))}\frac{\langle w,Df(x(t))\rangle}{t-t^{*}}$
$\displaystyle\geq$ $\displaystyle~{}\limsup_{t\downarrow
t^{*}}\frac{1}{m}\frac{\langle
w,Df(x(t))\rangle}{t-t^{*}}=\frac{1}{m}\limsup_{t\downarrow
t^{*}}\frac{c(t)}{t-t^{*}},$

where $m=\min\\{\lambda(x(t))|t\in[0,1]\\}>0$. This implies that $c(t)<0$ for
all $t>t^{*}$ such that $t-t^{*}$ is sufficiently small, and thus,

$f(x(t))<f(z)$

for such $t$, which is a contradiction. This completes the proof.
$\blacksquare$

##  3 Remarks

Suppose that $X=\mathbb{R}^{n}$ and $U$ is an open and convex subset of $X$.
Let $f:U\to\mathbb{R}$ be $C^{1}$. Then, $Df(x)$ can be represented by the row
vector

$\left(\frac{\partial f}{\partial x_{1}}(x),...,\frac{\partial f}{\partial
x_{n}}(x)\right).$

That is, we can consider that $Df(x)=(\nabla f(x))^{T}$. Moreover, suppose
that $f$ is $C^{2}$. Then, $D^{2}f(x)$ can be represented by the following
Hessian matrix:

$\begin{pmatrix}\frac{\partial^{2}f}{\partial
x_{1}^{2}}(x)&...&\frac{\partial^{2}f}{\partial x_{n}\partial x_{1}}(x)\\\
\vdots&\ddots&\vdots\\\ \frac{\partial^{2}f}{\partial x_{1}\partial
x_{n}}(x)&...&\frac{\partial^{2}f}{\partial x_{n}^{2}}(x)\end{pmatrix}.$

In this context, it is known the following facts. Suppose that
$f:U\to\mathbb{R}$ is $C^{2}$. Then, the following holds.

  1. 1)

If $Df(x)\neq 0$ for all $x\in U$, then $f$ is quasi-concave if and only if
$v^{T}D^{2}f(x)v\leq 0$ for every $x\in U$ and $v\in\mathbb{R}^{n}$ such that
$Df(x)v=0$.

  2. 2)

If $Df(x)\neq 0$ and $v^{T}D^{2}f(x)v<0$ for every $x\in U$ and
$v\in\mathbb{R}^{n}$ such that $v\neq 0$ and $Df(x)v=0$, then $f$ is strictly
quasi-concave.

Fact 1) is proved by Otani (1983). Fact 2) is well known, and have been
introduced in many textbooks. If $Df(x)=0$ is admitted, then 1) is violated by
the following function

$f(x_{1},x_{2})=x_{1}^{4}.$

Indeed, we have that

$Df(x)=(4x_{1}^{3},0),D^{2}f(x)=\begin{pmatrix}12x_{1}^{2}&0\\\
0&0\end{pmatrix},$

and thus $Df(x)v=0$ if and only if $x_{1}v_{1}=0$. For such a $v$,
$v^{T}D^{2}f(x)v=0$, although $f$ is not quasi-concave.

Note that the converse of 2) is not true. Let us check this. Suppose that
$U=\\{(x_{1},x_{2})|x_{1}>0,\ x_{2}>0\\}$. Define
$f(x_{1},x_{2})=x_{1}^{3}x_{2}+x_{1}x_{2}^{3}$. If $x_{2}(x_{1})$ represents a
function whose graph consists of a level set $f(x_{1},x_{2})\equiv a$, then we
can show that the function $x_{2}(x_{1})$ is strictly convex, which implies
that $f$ is strictly quasi-concave. We also have

$Df(1,1)=(4,4),\ D^{2}f(1,1)=\begin{pmatrix}6&6\\\ 6&6\end{pmatrix},$

and thus, if $v=(1,-1)$, then $Df(1,1)v=0$ but $v^{T}D^{2}f(1,1)v=0$. This
example was found by Katzner (1968), and has an important implication in the
consumer’s optimization problem.

Actually, facts 1) and 2) are corollaries of our results. Recall that if $f$
is $C^{2}$ and $Df(x)\neq 0$ for all $x\in U$, then $f$ is $C^{1}_{*}$, and we
can use $Df(x)$ as $g(x)$. Then, 1) follows from Theorem 1, and 2) follows
from Theorem 2. In this view, our results are extensions of the above facts.

Is there an $f$ that is $C^{1}_{*}$ but is not $C^{2}$? we can answer this
question affirmatively. Let

$f(x)=\begin{cases}x_{2}&\mbox{if }x_{2}\leq 0,\\\
\frac{x_{2}}{1-x_{1}x_{2}}&\mbox{if }x_{2}>0.\end{cases}$ (2)

Then, $f$ is $C^{1}$. Later, we will show that $f$ is not $C^{2}$. Define

$\displaystyle g_{1}(x)=$ $\displaystyle~{}\begin{cases}0&\mbox{if }x_{2}\leq
0,\\\ \frac{x_{2}^{2}}{\sqrt{1+x_{2}^{4}}}&\mbox{if }x_{2}>0,\end{cases}$ (3)
$\displaystyle g_{2}(x)=$ $\displaystyle~{}\begin{cases}1&\mbox{if }x_{2}\leq
0,\\\ \frac{1}{\sqrt{1+x_{2}^{4}}}&\mbox{if }x_{2}>0,\end{cases}$ (4)
$\displaystyle\lambda(x_{1},x_{2})=$ $\displaystyle~{}\begin{cases}1&\mbox{if
}x_{2}\leq 0,\\\ \frac{\sqrt{1+x_{2}^{4}}}{(1-x_{1}x_{2})^{2}}&\mbox{if
}x_{2}>0.\end{cases}$ (5)

Then, we can easily check that $g$ is $C^{1}$, $\lambda$ is continuous, and
$g,\lambda$ satisfy (1).111The notation $Df(x)$ in (1) is changed to $\nabla
f(x)$ in this context. Thus, $f$ is $C^{1}_{*}$. Note that, on some convex
neighborhood of $0$, $f$ is quasi-concave; we can use Theorem 2 to check this
fact.

It may be worthwhile to mention a fact on the above example. The function $g$
defined by (3) and (4) was first introduced by Debreu (1976).222In this paper,
Debreu said that this example was first found by Andreu Mas-Colell and Leonard
Shapiro, and Marcel Richter informed him of this example. Debreu stated that,
for this $g$, there is no $C^{2}$ function $f$ that satisfies (1) with some
positive continuous function $\lambda$ around $0$. Actually, this claim is
correct. We now prove this fact rigorously.

First, we show that the function $f$ defined by (2) is not $C^{2}$ on any open
neighborhood of $0$. If $f$ is $C^{2}$, then

$\lambda(x)=\frac{\frac{\partial f}{\partial x_{2}}(x)}{g_{2}(x)},$

which is $C^{1}$. However, if $x_{1}\neq 0$ and $x_{2}=0$, $\lambda$ defined
by (5) is not differentiable, and thus $f$ is not $C^{2}$.

Next, suppose that there exists a $C^{2}$ function $h$ and a positive $C^{1}$
function $\mu$ defined on some open neighborhood of $0$ such that

$\nabla h(x)=\mu(x)g(x)$ (6)

for all $x$. Without loss of generality, we can assume that the domain of $h$
includes $V=[-\varepsilon,\varepsilon]^{2}$, where $0<\varepsilon<1$. Then,
$f$ is also defined on $V$. Consider the following differential equation:

$\dot{x}_{2}(x_{1})=-\frac{g_{1}(x_{1},x_{2}(x_{1}))}{g_{2}(x_{1},x_{2}(x_{1}))},\
x_{2}(0)=c,$

where $-\varepsilon<c<\varepsilon$. By the Picard-Lindelöf theorem, the
nonextendable solution of the above equation is unique. Note that, by
construction,

$\dot{x}_{2}(x_{1})=-\frac{\frac{\partial f}{\partial
x_{1}}(x_{1},x_{2}(x_{1}))}{\frac{\partial f}{\partial
x_{2}}(x_{1},x_{2}(x_{1}))}=-\frac{\frac{\partial h}{\partial
x_{1}}(x_{1},x_{2}(x_{1}))}{\frac{\partial h}{\partial
x_{2}}(x_{1},x_{2}(x_{1}))},$

and thus, the graph of $x_{2}(x_{1})$ coincides with the level sets
$f^{-1}(c)$ and $h^{-1}(h(0,c))$.333Note that $f(0,c)=c$ for all $c$. This
implies that, on $V$, $h(x)=\varphi(f(x))$, where $\varphi(c)=h(0,c)$. By
definition, we have that $\varphi$ is $C^{2}$. Moreover, by (6), we have that
$\varphi^{\prime}(c)\neq 0$, and

$\frac{\partial h}{\partial
x_{2}}(x_{1},x_{2})=\varphi^{\prime}(f(x_{1},x_{2}))\frac{\partial f}{\partial
x_{2}}(x_{1},x_{2}).$

Because $h$ and $\varphi$ are $C^{2}$, we conclude that $\frac{\partial
f}{\partial x_{2}}(x_{1},x_{2})$ is differentiable. However, we have already
stated that it is not differentiable if $x_{1}\neq 0$ and $x_{2}=0$, which is
a contradiction. Hence, such an $h$ does not exist.

There are two open problems related to our theorems. First, consider
$X=\mathbb{R}^{n}$. Suppose that $A=(a_{ij})_{i,j=1}^{n}$ is an $n\times n$
symmetric matrix and $b\in\mathbb{R}^{n}$ such that $b_{1}\neq 0$. Debreu
(1952) showed that $w^{T}Aw\leq 0$ for all $w\in\mathbb{R}^{n}$ such that
$w^{T}b=0$ if and only if for any permutation $\pi$ and $j\in\\{2,...,n\\}$,

$(-1)^{j}\begin{vmatrix}a_{i_{1}i_{1}}&...&a_{i_{1}i_{j}}&b_{i_{1}}\\\
\vdots&\ddots&\vdots&\vdots\\\ a_{i_{j}i_{1}}&...&a_{i_{j}i_{j}}&b_{i_{j}}\\\
b_{i_{1}}&...&b_{i_{j}}&0\end{vmatrix}\geq 0,$

where $i_{k}=\pi(k)$. If the pair $(A,b)$ has this property, we say that
$(A,b)$ satisfies Property N. Applying this result to fact 1), we obtain the
following result: suppose that $U\subset\mathbb{R}^{n}$ is an open and convex
set, $f;U\to\mathbb{R}$ is $C^{2}$, and $\frac{\partial f}{\partial
x_{1}}(x)\neq 0$ for all $x\in U$. Then, $f$ is quasi-concave if and only if,
for each $x\in U$, the pair $(D^{2}f(x),\nabla f(x))$ satisfies Property N. We
want to obtain the analogy of this fact. Suppose that $U\subset\mathbb{R}^{n}$
is open and convex, $f:U\to\mathbb{R}$ is $C^{1}_{*}$, and $g,\lambda$ satisfy
(1), where $g_{1}(x)\neq 0$ for all $x\in U$. Then, we expect that $f$ is
quasi-concave if and only if, for each $x\in U$, the pair $(Dg(x),g(x))$
satisfies Property N. However, $Dg(x)$ may not be symmetric, and thus Debreu’s
result cannot be applied directly. Hence, whether our conjecture is true is an
open problem.

Second, we would like to link our results to the concavity of functions.
Suppose that $U\subset X$ is open and convex, and $f:U\to\mathbb{R}$ is given.
Then, $f$ is said to be concave if and only if, for every $x,y\in U$ and
$t\in[0,1]$,

$f((1-t)x+ty)\geq(1-t)f(x)+tf(y).$

If, in addition,

$f((1-t)x+ty)>(1-t)f(x)+tf(y)$

when $x\neq y$ and $0<t<1$, then $f$ is said to be strictly concave. If
$X=\mathbb{R}^{n}$ and $f$ is $C^{2}$, then it is well known that 1) $f$ is
concave if and only if $D^{2}f(x)$ is negative semi-definite and 2) if
$D^{2}f(x)$ is negative definite, then $f$ is strictly concave. As an analogy
of these facts, we expect that for a $C^{1}_{*}$ function $f$, $f$ is concave
if and only if $Dg(x)$ is negative semi-definite, and if $Dg(x)$ is negative
definite, then $f$ is strictly concave. However, this conjecture is also
difficult to verify. This is another open problem.

## Acknowledgments

The author is grateful to Shinichi Suda and Ken Hasegawa for their kindful
comments.

## Reference

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[2] Debreu, G. (1952) “Definite and Semi-Definite Quadratic Forms.”
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[3] Debreu, G. (1972) “Smooth Preferences.” Econometrica 40, pp.603-615.

[4] Debreu, G. (1976) “Smooth Preferences, A Corrigendum.” Econometrica 44,
pp.831-832.

[5] Katzner, D. W. (1968) “A Note on the Differentiability of Consumer Demand
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[6] Luenberger, D. G. (1969) Optimization by Vector Space Methods. Wiley, New
York.

[7] Otani, K. (1983) “A Characterization of Quasi-Concave Functions.” Journal
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