May 31, 2022 OU-HET-1144

Universality in Anomaly Flow

Yutaka Hosotani

Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan

###### Abstract

Universality in anomaly flow by an Aharonov-Bohm (AB) phase $\theta_{H}$ is
shown in the flat $M^{4}\times(S^{1}/Z_{2})$ spacetime and in the Randall-
Sundrum (RS) warped space. We analyze $SU(2)$ gauge theory with doublet
fermions. With orbifold boundary conditions the $U(1)$ part of gauge symmetry
remains unbroken at $\theta_{H}=0$ and $\pi$. Chiral anomalies smoothly vary
with $\theta_{H}$ in the RS space. It is shown that anomaly coefficients
associated with this anomaly flow are expressed in terms of the values of the
wave functions of gauge fields at the UV and IR branes in the RS space. The
anomaly coefficients depend on $\theta_{H}$, the warp factor of the RS space,
and the orbifold boundary conditions for fermions, but not on the bulk mass
parameters of fermions.

##  1 Introduction

In gauge-Higgs unification (GHU), gauge symmetry is dynamically broken by an
Aharonov-Bohm (AB) phase, $\theta_{H}$, in the fifth dimension[1, 2, 3, 4, 5,
6, 7]. It has been shown recently that chiral anomalies [8, 9, 10, 11] in GHU
flow with $\theta_{H}$, that is, anomaly coefficients smoothly change with
$\theta_{H}$ in the Randall-Sundrum (RS) warped space [12]. In the GUT-
inspired $SO(5)\times U(1)_{X}\times SU(3)_{C}$ GHU models in the RS space,
chiral quarks and leptons at $\theta_{H}=0$ are transformed to vector-like
fermions at $\theta_{H}=\pi$[13]. As $\theta_{H}$ varies from 0 to $\pi$,
$SU(2)_{L}\times U(1)_{Y}\times SU(3)_{C}$ gauge symmetry is converted to
$SU(2)_{R}\times U(1)_{Y^{\prime}}\times SU(3)_{C}$ gauge symmetry. Chiral
fermions appearing as zero modes of fermion multiplets in the spinor
representation of $SO(5)$ at $\theta_{H}=0$ become massive fermions having
vector-like gauge couplings at $\theta_{H}=\pi$. The chiral anomaly induced by
each quark or lepton at $\theta_{H}=0$ smoothly changes and vanishes at
$\theta_{H}=\pi$.

In the RS space each fermion multiplet is characterized by its own
dimensionless bulk mass parameter $c$ which controls the mass and wave
function of the fermion. In the previous paper [12] it has been recognized by
numerical evaluation that the anomaly coefficients depend on $\theta_{H}$, but
not on the bulk mass parameter $c$. This fact leads to a puzzle. How can the
$\theta_{H}$-dependence of the anomaly coefficients be determined and
expressed independently of the details of the fermion field? This is the main
theme addressed in this paper. We are going to show that the anomaly
coefficients at general $\theta_{H}$ are expressed in terms of the values of
the wave functions of gauge fields at the UV and IR branes in the RS space.
The anomaly coefficients depend on $\theta_{H}$, the warp factor $z_{L}$ of
the RS space, and boundary conditions of the fermion field, but not on the
bulk mass parameter $c$. The universality of the anomaly flow is observed.

We stress that the universal behavior is highly nontrivial. In GHU in the RS
space gauge couplings of each fermion mode depend on $\theta_{H}$, $z_{L}$ and
$c$. To find the total anomaly coefficients one needs to sum all contributions
coming from triangle loop diagrams in which all possible Kaluza-Klein (KK)
excited modes of fermions are running. The universality of the anomaly flow is
established only when all contributions are taken into account.

The phenomenon of anomaly flow is different from that of anomaly inflow in
which anomalies and fermion zero modes on defects such as strings and domain
walls or on the boundary of spacetime are intertwined and related to each
other [14, 15, 16]. In orbifold gauge theory gauge couplings of fermion modes
vary with the AB phase $\theta_{H}$ in the fifth dimension, and anomalies also
vary with $\theta_{H}$. We are going to show that the $\theta_{H}$-dependence
of the anomalies is expressed by a holographic formula involving the values of
the wave functions of gauge fields.

In this paper we analyze $SU(2)$ GHU models in the flat
$M^{4}\times(S^{1}/Z_{2})$ spacetime and in the RS warped space with orbifold
boundary conditions which break $SU(2)$ to $U(1)$. The $U(1)$ gauge symmetry
survives at $\theta_{H}=0$ and $\pi$. Fermion doublet multiplets have zero
modes at $\theta_{H}=0$ or $\pi$, depending on their boundary conditions.
Chiral anomalies appear in various combinations of Kaluza-Klein (KK) modes of
gauge fields. In the flat $M^{4}\times(S^{1}/Z_{2})$ spacetime all 4D gauge
couplings are determined analytically, but the KK mass spectrum of gauge and
fermion fields exhibit level crossings as $\theta_{H}$ varies. In the RS space
there occurs no level crossing in the spectrum, and all gauge couplings
smoothly vary with $\theta_{H}$. The flat spacetime limit of the RS space
gives rise to singular behavior of the anomalies as functions of $\theta_{H}$,
reproducing the known result in the flat spacetime.

In Section 2 $SU(2)$ GHU models are introduced both in flat
$M^{4}\times(S^{1}/Z_{2})$ spacetime and in the RS space. In Section 3 chiral
anomalies are evaluated and expressed in a simple form which involves the
values of the wave functions of gauge fields at the UV and IR branes and
boundary conditions of fermion fields. In Section 4 conditions for anomaly
cancellation are derived. Section 5 is devoted to a summary and discussions.

##  2 $SU(2)$ GHU

We consider $SU(2)$ GHU in the flat $M^{4}\times(S^{1}/Z_{2})$ spacetime with
coordinate $x^{M}$ ($M=0,1,2,3,5$, $x^{5}=y$) whose action is given by

$\displaystyle I_{\rm flat}$ $\displaystyle=\int d^{4}x\int_{0}^{L}dy\,{\cal
L}_{\rm flat}~{},$ (2.1) $\displaystyle{\cal L}_{\rm flat}$
$\displaystyle=-\frac{1}{2}{\rm
Tr}\,F_{MN}F^{MN}+\overline{\Psi}\gamma^{M}D_{M}\Psi~{},$ (2.2)

where ${\cal L}_{\rm flat}(x^{\mu},y)={\cal L}_{\rm flat}(x^{\mu},y+2L)={\cal
L}_{\rm flat}(x^{\mu},-y)$. Here
$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{N}-ig_{A}[A_{M},A_{N}]$,
$A_{M}=\hbox{$\frac{1}{2}$}\sum_{a=1}^{3}A^{a}_{M}\tau^{a}$ where $\tau^{a}$’s
are Pauli matrices. We adopt the metric $\eta_{MN}={\rm diag}\,(-1,1,1,1,1)$.
$\Psi$ is an $SU(2)$ doublet and $D_{M}=\partial_{M}-ig_{A}A_{M}$.
$\overline{\Psi}=i\Psi^{\dagger}\gamma^{0}$. Orbifold boundary conditions are
given, with $(y_{0},y_{1})=(0,L)$, by

$\displaystyle\begin{pmatrix}A_{\mu}\cr A_{y}\end{pmatrix}(x,y_{j}-y)$
$\displaystyle=P_{j}\begin{pmatrix}A_{\mu}\cr-
A_{y}\end{pmatrix}(x,y_{j}+y)P_{j}^{-1}~{},$ (2.3)
$\displaystyle\Psi(x,y_{j}-y)$
$\displaystyle=\begin{cases}+P_{j}\gamma^{5}\Psi(x,y_{j}+y)&\hbox{type
1A}\cr\kern 3.0pt\cr-P_{j}\gamma^{5}\Psi(x,y_{j}+y)&\hbox{type 1B}\cr\kern
3.0pt\cr(-1)^{j}P_{j}\gamma^{5}\Psi(x,y_{j}+y)&\hbox{type 2A}\cr\kern
3.0pt\cr(-1)^{j+1}P_{j}\gamma^{5}\Psi(x,y_{j}+y)&\hbox{type 2B}\end{cases},$
(2.4) $\displaystyle P_{0}=P_{1}$ $\displaystyle=\tau^{3}~{}.$ (2.5)

The $SU(2)$ symmetry is broken to $U(1)$ by the boundary conditions (2.5).
$A^{3}_{\mu},A^{1,2}_{y}$ are parity even at both $y_{0}$ and $y_{1}$, and
have constant zero modes. The zero mode of $A^{3}_{\mu}$ is the 4D $U(1)$
gauge field, and the 4D gauge coupling is given by

$\displaystyle g_{4}=\frac{g_{A}}{\sqrt{L}}~{}.$ (2.6)

We denote the doublet field as $\Psi=(u,d)^{t}$. In type 1A (1B) $u_{R}$ and
$d_{L}$ ($u_{L}$ and $d_{R}$) are parity even at both $y_{0}$ and $y_{1}$, and
have zero modes, leading to chiral structure.

The zero modes of $A_{y}^{1,2}$ may develop nonvanishing expectation values.
Without loss of generality one may assume that $\langle A_{y}^{1}\rangle=0$.
An AB phase $\theta_{H}$ along the fifth dimension is given by

$\displaystyle P\exp\bigg{\\{}ig_{A}\int_{0}^{2L}dy\,\langle
A_{y}\rangle\bigg{\\}}=e^{i\theta_{H}\tau^{2}}=\begin{pmatrix}\cos\theta_{H}&\sin\theta_{H}\cr-\sin\theta_{H}&\cos\theta_{H}\end{pmatrix}~{},$
(2.7) $\displaystyle\theta_{H}=g_{4}L\,\langle A_{y}^{2}\rangle~{}.$ (2.8)

The AB phase $\theta_{H}$ is a physical quantity. It couples to fields,
affecting their mass spectrum. One can change the value of $\theta_{H}$ by a
gauge transformation, which also alters boundary conditions. Under a large
gauge transformation given by

$\displaystyle\tilde{A}_{M}$ $\displaystyle=\Omega
A_{M}\Omega^{-1}+\frac{i}{g_{A}}\Omega\partial_{M}\Omega^{-1}~{},~{}~{}\tilde{\Psi}=\Omega\Psi~{},$
(2.9) $\displaystyle\Omega$
$\displaystyle=\exp\Big{(}\frac{i}{2}\theta(y)\tau^{2}\Big{)}~{},~{}~{}\theta(y)=\theta_{H}\Big{(}1-\frac{y}{L}\Big{)}~{},$
(2.10)

$\tilde{\theta}_{H}=0$ and boundary condition matrices become

$\displaystyle\tilde{P}_{j}$
$\displaystyle=\Omega(y_{j}-y)P_{j}\Omega^{-1}(y_{j}+y)~{},$ (2.11)
$\displaystyle\tilde{P}_{0}$
$\displaystyle=\begin{pmatrix}\cos\theta_{H}&-\sin\theta_{H}\cr-\sin\theta_{H}&-\cos\theta_{H}\end{pmatrix},~{}~{}\tilde{P}_{1}=\tau^{3}~{}.$
(2.12)

Although the AB phase $\tilde{\theta}_{H}$ vanishes, boundary conditions
become nontrivial. Physics remains the same. This gauge is called the twisted
gauge[17, 18].

Fields in the twisted gauge satisfy free equations. KK expansions for
$\tilde{A}_{\mu}^{1},\,\tilde{A}_{\mu}^{3}$ are given by

$\displaystyle\begin{pmatrix}\tilde{A}_{\mu}^{1}(x,y)\cr\kern
5.0pt\cr\tilde{A}_{\mu}^{3}(x,y)\end{pmatrix}$
$\displaystyle=\sum_{n=-\infty}^{\infty}B_{\mu}^{(n)}(x)\frac{1}{\sqrt{\pi
R}}\begin{pmatrix}\sin\Big{[}\frac{\displaystyle ny}{\displaystyle
R}-\theta(y)\Big{]}\cr\cos\Big{[}\frac{\displaystyle ny}{\displaystyle
R}-\theta(y)\Big{]}\end{pmatrix}$ (2.13)

where $L=\pi R$. In the original gauge they become

$\displaystyle\begin{pmatrix}A_{\mu}^{1}(x,y)\cr\kern 5.0pt\cr
A_{\mu}^{3}(x,y)\end{pmatrix}$
$\displaystyle=\sum_{n=-\infty}^{\infty}B_{\mu}^{(n)}(x)\frac{1}{\sqrt{\pi
R}}\begin{pmatrix}\sin\frac{\displaystyle ny}{\displaystyle
R}\cr\cos\frac{\displaystyle ny}{\displaystyle R}\end{pmatrix}.$ (2.14)

The mass of the $B_{\mu}^{(n)}(x)$ mode is
$m_{n}(\theta_{H})=R^{-1}\big{|}n+\frac{\theta_{H}}{\pi}\big{|}$. The spectrum
is periodic in $\theta_{H}$ with period $\pi$.

Similarly the fermion field $\Psi$ in the twisted gauge

$\displaystyle\tilde{\Psi}=\begin{pmatrix}\tilde{u}\cr\tilde{d}\end{pmatrix}=\begin{pmatrix}\cos\hbox{$\frac{1}{2}$}\theta(y)&\sin\hbox{$\frac{1}{2}$}\theta(y)\cr-\sin\hbox{$\frac{1}{2}$}\theta(y)&\cos\hbox{$\frac{1}{2}$}\theta(y)\end{pmatrix}\begin{pmatrix}u\cr
d\end{pmatrix}$ (2.15)

satisfies free equations in the bulk region $0<y<L$. The KK expansion of
$\tilde{\Psi}$ in the type 1A is given by

$\displaystyle\begin{pmatrix}\tilde{u}_{R}(x,y)\cr\kern
5.0pt\cr\tilde{d}_{R}(x,y)\end{pmatrix}$
$\displaystyle=\sum_{n=-\infty}^{\infty}\psi_{R}^{(n)}(x)\frac{1}{\sqrt{\pi
R}}\begin{pmatrix}\cos\Big{[}\frac{\displaystyle ny}{\displaystyle
R}-\hbox{$\frac{1}{2}$}\theta(y)\Big{]}\cr\sin\Big{[}\frac{\displaystyle
ny}{\displaystyle R}-\hbox{$\frac{1}{2}$}\theta(y)\Big{]}\end{pmatrix},$
(2.16) $\displaystyle\begin{pmatrix}\tilde{u}_{L}(x,y)\cr\kern
5.0pt\cr\tilde{d}_{L}(x,y)\end{pmatrix}$
$\displaystyle=\sum_{n=-\infty}^{\infty}\psi_{L}^{(n)}(x)\frac{1}{\sqrt{\pi
R}}\begin{pmatrix}-\sin\Big{[}\frac{\displaystyle ny}{\displaystyle
R}-\hbox{$\frac{1}{2}$}\theta(y)\Big{]}\cr\cos\Big{[}\frac{\displaystyle
ny}{\displaystyle R}-\hbox{$\frac{1}{2}$}\theta(y)\Big{]}\end{pmatrix}.$
(2.17)

In the original gauge it becomes

$\displaystyle\underline{\hbox{type
1A}}:~{}~{}\begin{pmatrix}u_{R}(x,y)\cr\kern 5.0pt\cr d_{R}(x,y)\end{pmatrix}$
$\displaystyle=\sum_{n=-\infty}^{\infty}\psi_{R}^{(n)}(x)\frac{1}{\sqrt{\pi
R}}\begin{pmatrix}\cos\frac{\displaystyle ny}{\displaystyle
R}\cr\sin\frac{\displaystyle ny}{\displaystyle R}\end{pmatrix},$ (2.18)
$\displaystyle\begin{pmatrix}u_{L}(x,y)\cr\kern 5.0pt\cr
d_{L}(x,y)\end{pmatrix}$
$\displaystyle=\sum_{n=-\infty}^{\infty}\psi_{L}^{(n)}(x)\frac{1}{\sqrt{\pi
R}}\begin{pmatrix}-\sin\frac{\displaystyle ny}{\displaystyle
R}\cr\cos\frac{\displaystyle ny}{\displaystyle R}\end{pmatrix}.$ (2.19)

$\psi_{R}^{(n)}$ and $\psi_{L}^{(n)}$ combine to form the $\psi^{(n)}(x)$
mode, whose mass is given by
$m_{n}(\theta_{H})=R^{-1}\big{|}n+\frac{\theta_{H}}{2\pi}\big{|}$. The
spectrum is periodic in $\theta_{H}$ with period $2\pi$. The KK expansion for
type 1B is obtained by interchanging left-handed and right-handed components
in (2.19).

For $\Psi$ in type 2A the KK expansion is

$\displaystyle\underline{\hbox{type
2A}}:~{}~{}\begin{pmatrix}u_{R}(x,y)\cr\kern 5.0pt\cr d_{R}(x,y)\end{pmatrix}$
$\displaystyle=\sum_{n=-\infty}^{\infty}\psi_{R}^{(n+\frac{1}{2})}(x)\frac{1}{\sqrt{\pi
R}}\begin{pmatrix}\cos\frac{\displaystyle(n+\hbox{$\frac{1}{2}$})y}{\displaystyle
R}\cr\sin\frac{\displaystyle(n+\hbox{$\frac{1}{2}$})y}{\displaystyle
R}\end{pmatrix},$ (2.20) $\displaystyle\begin{pmatrix}u_{L}(x,y)\cr\kern
5.0pt\cr d_{L}(x,y)\end{pmatrix}$
$\displaystyle=\sum_{n=-\infty}^{\infty}\psi_{L}^{(n+\frac{1}{2})}(x)\frac{1}{\sqrt{\pi
R}}\begin{pmatrix}-\sin\frac{\displaystyle(n+\hbox{$\frac{1}{2}$})y}{\displaystyle
R}\cr\cos\frac{\displaystyle(n+\hbox{$\frac{1}{2}$})y}{\displaystyle
R}\end{pmatrix}.$ (2.21)

$\ \psi_{R}^{(n+\frac{1}{2})}$ and $\psi_{L}^{(n+\frac{1}{2})}$ combine to
form the $\psi^{(n+\frac{1}{2})}(x)$ mode, whose mass is given by
$m_{n+\frac{1}{2}}(\theta_{H})=R^{-1}\big{|}n+\hbox{$\frac{1}{2}$}+\frac{\theta_{H}}{2\pi}\big{|}$.
The KK expansion for type 2B is obtained by interchanging left-handed and
right-handed components in (2.21).

Next we examine $SU(2)$ GHU in the RS space whose metric is given by [19]

$\displaystyle ds^{2}=e^{-2\sigma(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2}$
(2.22)

where $\eta_{\mu\nu}=\mbox{diag}(-1,+1,+1,+1)$,
$\sigma(y)=\sigma(y+2L)=\sigma(-y)$ and $\sigma(y)=ky$ for $0\leq y\leq L$. It
has the same topology as $M^{4}\times(S^{1}/Z_{2})$. In the fundamental region
$0\leq y\leq L$ the metric can be written, in terms of the conformal
coordinate $z=e^{ky}$, as

$\displaystyle
ds^{2}=\frac{1}{z^{2}}\bigg{(}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+\frac{dz^{2}}{k^{2}}\bigg{)}\quad(1\leq
z\leq z_{L}=e^{kL})~{}.$ (2.23)

$z_{L}$ is called the warp factor of the RS space. The action in RS is

$\displaystyle I_{\rm RS}$ $\displaystyle=\int d^{5}x\sqrt{-\det G}\,{\cal
L}_{\rm RS}~{},$ (2.24) $\displaystyle{\cal L}_{\rm RS}$
$\displaystyle=-\frac{1}{2}{\rm Tr}\,F_{MN}F^{MN}+\overline{\Psi}{\cal
D}(c)\Psi~{},$ (2.25) $\displaystyle{\cal D}(c)$
$\displaystyle=\gamma^{A}{e_{A}}^{M}\bigg{(}D_{M}+\frac{1}{8}\omega_{MBC}[\gamma^{B},\gamma^{C}]\bigg{)}-c\,\sigma^{\prime}$
(2.26)

where $\sigma^{\prime}(y)=k$ for $0\leq y\leq L$. Note ${\cal L}_{\rm
RS}(x^{\mu},y)={\cal L}_{\rm RS}(x^{\mu},-y)={\cal L}_{\rm RS}(x^{\mu},y+2L)$.
Fields $A_{M}$ and $\Psi$ satisfy the same boundary conditions (2.5) as in the
flat spacetime. The dimensionless bulk mass parameter $c$ in ${\cal D}(c)$
controls the mass and wave function of the fermion field. The KK mass scale is
given by

$\displaystyle m_{{\rm KK}}$ $\displaystyle=\frac{\pi k}{z_{L}-1}$ (2.27)

which becomes $1/R$ in the flat spacetime limit $k\rightarrow 0$.

In the KK expansion in the $z$ coordinate, $A_{z}^{a}(x,z)=k^{-1/2}\sum
A_{z}^{a(n)}(x)h_{n}(z)$, the zero mode $A_{z}^{2(0)}$ has a wave function
$h_{0}(z)=\sqrt{2/(z_{L}^{2}-1)}\,z$. In the $y$-coordinate $A_{y}^{2(0)}$ has
a wave function $v_{0}(y)=ke^{ky}h_{0}(z)$ for $0\leq y\leq L$ and
$v_{0}(-y)=v_{0}(y)=v_{0}(y+2L)$. The AB phase $\theta_{H}$ in (2.8) becomes

$\displaystyle\theta_{H}$ $\displaystyle=\frac{\langle
A_{z}^{2(0)}\rangle}{f_{H}}~{},~{}~{}f_{H}=\frac{1}{g_{4}}\sqrt{\frac{2k}{L(z_{L}^{2}-1)}}~{}.$
(2.28)

The twisted gauge [17, 18], in which $\tilde{\theta}_{H}=0$, is related to the
original gauge by a large gauge transformation

$\displaystyle\Omega(z)$
$\displaystyle=e^{i\theta(z)\tau^{2}/2}~{},~{}~{}\theta(z)=\theta_{H}\,\frac{z_{L}^{2}-z^{2}}{z_{L}^{2}-1}~{}.$
(2.29)

In the $y$-coordinate it becomes

$\displaystyle\Omega(y)$
$\displaystyle=\exp\bigg{\\{}i\theta_{H}\sqrt{\frac{2}{z_{L}^{2}-1}}\int_{y}^{L}dy\,v_{0}(y)\cdot\frac{\tau^{2}}{2}\bigg{\\}}.$
(2.30)

In the twisted gauge $\tilde{A}_{\mu}^{1,3}(x,z)$ satisfy free equations in
$1\leq z\leq z_{L}$ and boundary conditions (2.12). The mass spectrum
$\\{m_{n}(\theta_{H})=k\lambda_{n}(\theta_{H})\\}$
($\lambda_{0}<\lambda_{1}<\lambda_{2}<\cdots$) is given by

$\displaystyle
Z_{\mu}^{(n)}:~{}SC^{\prime}(1;\lambda_{n})+\lambda_{n}\sin^{2}\theta_{H}=0$
(2.31)

where $S(z;\lambda)$ and $C(z;\lambda)$ are expressed in terms of Bessel
functions and are given by (A.5). The KK expansions in the twisted gauge in
the region $1\leq z\leq z_{L}$ are written as111Note a change in the
normalization of mode functions. $\tilde{\bf h}_{n}(z)$ in the present paper
corresponds to $\sqrt{kL}\,\tilde{\bf h}_{n}(z)$ in Ref. [12].

$\displaystyle\begin{pmatrix}\tilde{A}_{\mu}^{1}(x,z)\cr\kern
5.0pt\cr\tilde{A}_{\mu}^{3}(x,z)\end{pmatrix}$
$\displaystyle=\frac{1}{\sqrt{L}}\sum_{n=0}^{\infty}Z_{\mu}^{(n)}(x)\,\tilde{\bf
h}_{n}(z)~{},~{}~{}\tilde{\bf
h}_{n}(z)=\begin{pmatrix}\tilde{h}_{n}(z)\cr\tilde{k}_{n}(z)\end{pmatrix}$
(2.32)

where the mode functions $\tilde{\bf h}_{n}(z)$ are given in (B.6). In the
original gauge the KK expansions of $A_{\mu}^{1,3}(x,y)$ become

$\displaystyle\begin{pmatrix}A_{\mu}^{1}(x,y)\cr\kern 5.0pt\cr
A_{\mu}^{3}(x,y)\end{pmatrix}$
$\displaystyle=\frac{1}{\sqrt{L}}\sum_{n=0}^{\infty}Z_{\mu}^{(n)}(x)\,\begin{pmatrix}h_{n}(y)\cr
k_{n}(y)\end{pmatrix},$ (2.33) $\displaystyle\begin{pmatrix}h_{n}(y)\cr
k_{n}(y)\end{pmatrix}$ $\displaystyle=\begin{pmatrix}-h_{n}(-y)\cr
k_{n}(-y)\end{pmatrix}=\begin{pmatrix}h_{n}(y+2L)\cr k_{n}(y+2L)\end{pmatrix}$
(2.34)
$\displaystyle=\begin{pmatrix}\cos\theta(z)&\sin\theta(z)\cr-\sin\theta(z)&\cos\theta(z)\end{pmatrix}\begin{pmatrix}\tilde{h}_{n}(z)\cr\tilde{k}_{n}(z)\end{pmatrix}\quad{\rm
for~{}}0\leq y\leq L~{}.$ (2.35)

For a fermion field $\Psi(x,z)$ it is most convenient to express its KK
expansion for $\check{\Psi}(x,z)=z^{-2}\Psi(x,z)$. Equations of motion in the
region $1\leq z\leq z_{L}$ become

$\displaystyle-
kD_{-}(c)\,\check{\Psi}_{R}+\sigma^{\mu}\partial_{\mu}\check{\Psi}_{L}=0~{},~{}-kD_{+}(c)\,\check{\Psi}_{L}+\bar{\sigma}^{\mu}\partial_{\mu}\check{\Psi}_{R}=0~{},$
(2.36)
$\displaystyle\quad\sigma^{\mu}=(I_{2},\vec{\sigma})~{},~{}\bar{\sigma}^{\mu}=(-I_{2},\vec{\sigma})~{},~{}D_{\pm}(c)=\pm\frac{\partial}{\partial
z}+\frac{c}{z}~{}.$ (2.37)

In the presence of gauge fields $\partial_{M}$ is replaced by
$\partial_{M}-ig_{A}A_{M}$. The Neumann boundary conditions at
$z=(z_{0},z_{1})=(1,z_{L})$, corresponding to even parity, for left- and
right-handed components are given by
$D_{+}(c)\check{\Psi}_{L}\big{|}_{z_{j}}=0$ and
$D_{-}(c)\check{\Psi}_{R}\big{|}_{z_{j}}=0$.

The spectrum of the KK modes of the fermion field $\Psi$ is determined by

$\displaystyle\chi^{(n)}:~{}\begin{cases}S_{L}S_{R}(1;\lambda_{n},c)+\sin^{2}\hbox{$\frac{1}{2}$}\theta_{H}=0&\hbox{for
type 1A/B}\cr\kern 5.0pt\cr
S_{L}S_{R}(1;\lambda_{n},c)+\cos^{2}\hbox{$\frac{1}{2}$}\theta_{H}=0&\hbox{for
type 2A/B}\end{cases}$ (2.38)

where functions $S_{L/R}(z;\lambda,c)$ are given in (A.14). The spectrum is
periodic in $\theta_{H}$ with period $2\pi$. A massless mode appears at
$\theta_{H}=0$ for type 1A and 1B, whereas it appears at $\theta_{H}=\pi$ for
type 2A and 2B. There is no level crossing in the spectrum except for the case
$c=0$. The spectra of the gauge fields (2.31) and fermion fields (2.38) are
displayed in Figure 1.

Figure 1: The mass spectrum of gauge fields $Z_{\mu}^{(n)}$ and fermion fields
$\chi^{(n)}$ (type 1A) in the RS warped space is displayed. The warp factor is
$z_{L}=100$ and the bulk mass parameter of $\Psi$ is $c=0.25$. There is no
level crossing in the spectrum.

The KK expansion of the fermion field $\Psi$ in the twisted gauge in the
region $1\leq z\leq z_{L}$ is expressed as

$\displaystyle\begin{pmatrix}\tilde{\check{u}}_{R}(x,z)\cr\kern
5.0pt\cr\tilde{\check{d}}_{R}(x,z)\end{pmatrix}$
$\displaystyle=\sqrt{k}\sum_{n=0}^{\infty}\chi_{R}^{(n)}(x)\,\tilde{\bf
f}_{Rn}(z)~{},~{}\tilde{\bf
f}_{Rn}(z)=\begin{pmatrix}\tilde{f}_{Rn}(z)\cr\tilde{g}_{Rn}(z)\end{pmatrix},$
(2.39) $\displaystyle\begin{pmatrix}\tilde{\check{u}}_{L}(x,z)\cr\kern
5.0pt\cr\tilde{\check{d}}_{L}(x,z)\end{pmatrix}$
$\displaystyle=\sqrt{k}\sum_{n=0}^{\infty}\chi_{L}^{(n)}(x)\,\tilde{\bf
f}_{Ln}(z)~{},~{}\tilde{\bf
f}_{Ln}(z)=\begin{pmatrix}\tilde{f}_{Ln}(z)\cr\tilde{g}_{Ln}(z)\end{pmatrix}.$
(2.40)

The mode functions $\tilde{\bf f}_{Rn}(z)$ and $\tilde{\bf f}_{Ln}(z)$ for
type 1A are given in (B.12). In the original gauge the expansions of
$\check{u}(x,y)$ and $\check{d}(x,y)$ become

$\displaystyle\begin{pmatrix}{\check{u}}_{R}(x,y)\cr\kern
2.0pt\cr{\check{d}}_{R}(x,y)\end{pmatrix}$
$\displaystyle=\sqrt{k}\sum_{n=0}^{\infty}\chi_{R}^{(n)}(x)\begin{pmatrix}f_{Rn}(y)\cr
g_{Rn}(y)\end{pmatrix},$ (2.41)
$\displaystyle\begin{pmatrix}{\check{u}}_{L}(x,y)\cr\kern
2.0pt\cr{\check{d}}_{L}(x,y)\end{pmatrix}$
$\displaystyle=\sqrt{k}\sum_{n=0}^{\infty}\chi_{L}^{(n)}(x)\begin{pmatrix}f_{Ln}(y)\cr
g_{Ln}(y)\end{pmatrix},$ (2.42)

where

$\displaystyle\underline{\hbox{type 1A}}~{}~{}$ (2.43)
$\displaystyle\begin{pmatrix}f_{Rn}(y)\cr g_{Rn}(y)\end{pmatrix}$
$\displaystyle=\begin{pmatrix}f_{Rn}(-y)\cr-
g_{Rn}(-y)\end{pmatrix}=\begin{pmatrix}f_{Rn}(y+2L)\cr
g_{Rn}(y+2L)\end{pmatrix}$ (2.44)
$\displaystyle=\begin{pmatrix}\cos\hbox{$\frac{1}{2}$}\theta(z)&-\sin\hbox{$\frac{1}{2}$}\theta(z)\cr\kern
2.0pt\cr\sin\hbox{$\frac{1}{2}$}\theta(z)&\cos\hbox{$\frac{1}{2}$}\theta(z)\end{pmatrix}\begin{pmatrix}\tilde{f}_{Rn}(z)\cr\tilde{g}_{Rn}(z)\end{pmatrix}\quad{\rm
for~{}}0\leq y\leq L~{},$ (2.45) $\displaystyle\begin{pmatrix}f_{Ln}(y)\cr
g_{Ln}(y)\end{pmatrix}$ $\displaystyle=\begin{pmatrix}-f_{Ln}(-y)\cr
g_{Ln}(-y)\end{pmatrix}=\begin{pmatrix}f_{Ln}(y+2L)\cr
g_{Ln}(y+2L)\end{pmatrix}$ (2.46)
$\displaystyle=\begin{pmatrix}\cos\hbox{$\frac{1}{2}$}\theta(z)&-\sin\hbox{$\frac{1}{2}$}\theta(z)\cr\kern
2.0pt\cr\sin\hbox{$\frac{1}{2}$}\theta(z)&\cos\hbox{$\frac{1}{2}$}\theta(z)\end{pmatrix}\begin{pmatrix}\tilde{f}_{Ln}(z)\cr\tilde{g}_{Ln}(z)\end{pmatrix}\quad{\rm
for~{}}0\leq y\leq L~{},$ (2.47) $\displaystyle\underline{\hbox{type
2A}}~{}~{}$ (2.48) $\displaystyle\begin{pmatrix}f_{Rn}(y)\cr
g_{Rn}(y)\end{pmatrix}$ $\displaystyle=\begin{pmatrix}f_{Rn}(-y)\cr-
g_{Rn}(-y)\end{pmatrix}=\begin{pmatrix}-f_{Rn}(y+2L)\cr-
g_{Rn}(y+2L)\end{pmatrix},$ (2.49) $\displaystyle\begin{pmatrix}f_{Ln}(y)\cr
g_{Ln}(y)\end{pmatrix}$ $\displaystyle=\begin{pmatrix}-f_{Ln}(-y)\cr
g_{Ln}(-y)\end{pmatrix}=\begin{pmatrix}-f_{Ln}(y+2L)\cr-
g_{Ln}(y+2L)\end{pmatrix}.$ (2.50)

For type 1B (2B), the parity of $f_{R/Ln},g_{R/Ln}$ is reversed compared to
type 1A (2A).

##  3 Anomalies

Doublet fermions in type 1A or 1B are chiral at $\theta_{H}=0$. Massless modes
appear for right-handed $u$ and left-handed $d$ (left-handed $u$ and right-
handed $d$) for type 1A (1B). They become massive as $\theta_{H}$ varies, and
their gauge couplings become purely vector-like at $\theta_{H}=\pi$. Chiral
anomalies exist at $\theta_{H}=0$, smoothly vary as $\theta_{H}$ in the RS
space, and vanish at $\theta_{H}=\pi$. This phenomenon is called the anomaly
flow by an AB phase [12].

Chiral anomalies arise from triangular loop diagrams. Gauge couplings of
fermions have been obtained in Ref.[12]. Substituting the KK expansions (2.35)
and (2.42) into

$\displaystyle
g_{A}\int_{1}^{z_{L}}\frac{dz}{k}\,\Big{\\{}{\check{\Psi}}_{R}^{\dagger}\bar{\sigma}^{\mu}A_{\mu}{\check{\Psi}}_{R}-{\check{\Psi}}_{L}^{\dagger}\sigma^{\mu}A_{\mu}{\check{\Psi}}_{L}\Big{\\}}~{},$
(3.1)

one finds that the couplings in

$\displaystyle\frac{g_{4}}{2}\sum_{n=0}^{\infty}\sum_{\ell=0}^{\infty}\sum_{m=0}^{\infty}Z_{\mu}^{(n)}(x)\Big{\\{}t^{R}_{n\ell
m}\,\chi_{R}^{(\ell)}(x)^{\dagger}\bar{\sigma}^{\mu}\chi_{R}^{(m)}(x)+t^{L}_{n\ell
m}\,\chi_{L}^{(\ell)}(x)^{\dagger}\sigma^{\mu}\chi_{L}^{(m)}(x)\Big{\\}}$
(3.2)

are given by

$\displaystyle t^{R}_{n\ell m}$
$\displaystyle=\int_{1}^{z_{L}}dz\,\Big{\\{}\tilde{h}_{n}(z)\big{(}\tilde{f}_{R\ell}^{*}(z)\tilde{g}_{Rm}(z)+\tilde{g}_{R\ell}^{*}(z)\tilde{f}_{Rm}(z)\big{)}$
(3.3) $\displaystyle\hskip
85.35826pt+\tilde{k}_{n}(z)\big{(}\tilde{f}_{R\ell}^{*}(z)\tilde{f}_{Rm}(z)-\tilde{g}_{R\ell}^{*}(z)\tilde{g}_{Rm}(z)\big{)}\Big{\\}}$
(3.4)
$\displaystyle=\frac{k}{2}\int_{-a}^{2L-a}dy\,e^{\sigma(y)}\Big{\\{}h_{n}(y)\big{(}f_{R\ell}^{*}(y)g_{Rm}(y)+g_{R\ell}^{*}(y)f_{Rm}(y)\big{)}$
(3.5) $\displaystyle\hskip
85.35826pt+k_{n}(y)\big{(}f_{R\ell}^{*}(y)f_{Rm}(y)-g_{R\ell}^{*}(y)g_{Rm}(y)\big{)}\Big{\\}},$
(3.6) $\displaystyle t^{L}_{n\ell m}$
$\displaystyle=-\int_{1}^{z_{L}}dz\,\Big{\\{}h_{n}(z)\big{(}f_{L\ell}^{*}(z)g_{Lm}(z)+g_{L\ell}^{*}(z)f_{Lm}(z)\big{)}$
(3.7) $\displaystyle\hskip
85.35826pt+k_{n}(z)\big{(}f_{L\ell}^{*}(z)f_{Lm}(z)-g_{L\ell}^{*}(z)g_{Lm}(z)\big{)}\Big{\\}}$
(3.8)
$\displaystyle=-\frac{k}{2}\int_{-a}^{2L-a}dy\,e^{\sigma(y)}\Big{\\{}h_{n}(y)\big{(}f_{L\ell}^{*}(y)g_{Lm}(y)+g_{L\ell}^{*}(y)f_{Lm}(y)\big{)}$
(3.9) $\displaystyle\hskip
85.35826pt+k_{n}(y)\big{(}f_{L\ell}^{*}(y)f_{Lm}(y)-g_{L\ell}^{*}(y)g_{Lm}(y)\big{)}\Big{\\}}.$
(3.10)

The couplings $t^{R}_{n\ell m}$ and $t^{L}_{n\ell m}$ are gauge-invariant. In
the integral formulas in the $y$-coordinate the constant $a$ is arbitrary as
the integrands are periodic functions with period $2L$. It is convenient to
take $0<a<L$ in the following discussions. We note that the couplings
$t^{R/L}_{n\ell m}$ depend not only on $\theta_{H}$ and $z_{L}$, but also on
the bulk mass parameter $c$ of the fermion field $\Psi$.

The anomaly coefficient associated with the three legs of
$Z_{\mu_{1}}^{(n_{1})}Z_{\mu_{2}}^{(n_{2})}Z_{\mu_{3}}^{(n_{3})}$ is given by

$\displaystyle
a_{n_{1}n_{2}n_{3}}=a_{n_{1}n_{2}n_{3}}^{R}+a_{n_{1}n_{2}n_{3}}^{L}~{},$
(3.11) $\displaystyle a_{n_{1}n_{2}n_{3}}^{R}={\rm
Tr}\,T^{R}_{n_{1}}T^{R}_{n_{2}}T^{R}_{n_{3}}~{},~{}~{}(T^{R}_{n})_{m\ell}=t^{R}_{nm\ell}~{},$
(3.12) $\displaystyle a_{n_{1}n_{2}n_{3}}^{L}={\rm
Tr}\,T^{L}_{n_{1}}T^{L}_{n_{2}}T^{L}_{n_{3}}~{},~{}~{}(T^{L}_{n})_{m\ell}=t^{L}_{nm\ell}~{}.$
(3.13)

The anomaly coefficient $a_{n_{1}n_{2}n_{3}}$ depends on $\theta_{H}$,
exhibiting the anomaly flow. It has been observed by numerical evaluation in
Ref.[12] that $a_{n_{1}n_{2}n_{3}}$ does not depend on the bulk mass parameter
$c$, though $a_{n_{1}n_{2}n_{3}}^{R}$ and $a_{n_{1}n_{2}n_{3}}^{L}$ do depend
on $c$. We are going to show that $a_{n_{1}n_{2}n_{3}}(\theta_{H},z_{L})$ is
expressed in terms of the values of the wave functions $k_{n_{j}}(y)$ at $y=0$
and $y=L$.

To see it we insert the formulas for $t^{R/L}_{n\ell m}$ in (3.10) into
(3.13), and rearrange the traces.

$\displaystyle
a_{n_{1}n_{2}n_{3}}=\Big{(}\frac{k}{2}\Big{)}^{3}\int\int\int_{-a}^{2L-a}dy_{1}dy_{2}dy_{3}\,e^{\sigma(y_{1})+\sigma(y_{2})+\sigma(y_{3})}$
(3.14)
$\displaystyle\quad\times\Big{[}~{}k_{1}k_{2}k_{3}\big{\\{}A_{R}(1,2)A_{R}(2,3)A_{R}(3,1)-B_{R}(1,2)B_{R}(2,3)B_{R}(3,1)$
(3.15) $\displaystyle\hskip
62.59596pt+B_{L}(1,2)B_{L}(2,3)B_{L}(3,1)-A_{L}(1,2)A_{L}(2,3)A_{L}(3,1)\big{\\}}$
(3.16)
$\displaystyle\quad\quad+k_{1}h_{2}h_{3}\big{\\{}A_{R}(1,2)B_{R}(2,3)A_{R}(3,1)-B_{R}(1,2)A_{R}(2,3)B_{R}(3,1)$
(3.17) $\displaystyle\hskip
62.59596pt+B_{L}(1,2)A_{L}(2,3)B_{L}(3,1)-A_{L}(1,2)B_{L}(2,3)A_{L}(3,1)\big{\\}}$
(3.18)
$\displaystyle\quad\quad+h_{1}k_{2}h_{3}\big{\\{}A_{R}(1,2)A_{R}(2,3)B_{R}(3,1)-B_{R}(1,2)B_{R}(2,3)A_{R}(3,1)$
(3.19) $\displaystyle\hskip
62.59596pt+B_{L}(1,2)B_{L}(2,3)A_{L}(3,1)-A_{L}(1,2)A_{L}(2,3)B_{L}(3,1)\big{\\}}$
(3.20)
$\displaystyle\quad\quad+h_{1}h_{2}k_{3}\big{\\{}B_{R}(1,2)A_{R}(2,3)A_{R}(3,1)-A_{R}(1,2)B_{R}(2,3)B_{R}(3,1)$
(3.21) $\displaystyle\hskip
62.59596pt+A_{L}(1,2)B_{L}(2,3)B_{L}(3,1)-B_{L}(1,2)A_{L}(2,3)A_{L}(3,1)\big{\\}}~{}\Big{]}$
(3.22)

where

$\displaystyle k_{j}=k_{n_{j}}(y_{j})~{},~{}~{}h_{j}=h_{n_{j}}(y_{j})~{},$
(3.23) $\displaystyle\begin{pmatrix}A_{R/L}(j,k)\cr
B_{R/L}(j,k)\end{pmatrix}=\begin{pmatrix}A_{R/L}\cr
B_{R/L}\end{pmatrix}(y_{j},y_{k})=\sum_{n=0}^{\infty}\begin{pmatrix}f_{R/Ln}(y_{j})f_{R/Ln}^{*}(y_{k})\cr
g_{R/Ln}(y_{j})g_{R/Ln}^{*}(y_{k})\end{pmatrix}.$ (3.24)

Eqs. (2.40) and (2.42) and the orthonormality relations of the mode functions
imply that

$\displaystyle\begin{pmatrix}\check{u}_{R/L}(x,y)\cr\check{d}_{R/L}(x,y)\end{pmatrix}=\frac{k}{2}\int_{-a}^{2L-a}dy^{\prime}\,e^{\sigma(y^{\prime})}\begin{pmatrix}A_{R/L}&C_{R/L}\cr
D_{R/L}&B_{R/L}\end{pmatrix}(y,y^{\prime})\begin{pmatrix}\check{u}_{R/L}(x,y^{\prime})\cr\check{d}_{R/L}(x,y^{\prime})\end{pmatrix},$
(3.25) $\displaystyle\hskip 56.9055pt\begin{pmatrix}C_{R/L}\cr
D_{R/L}\end{pmatrix}(y,y^{\prime})=\sum_{n=0}^{\infty}\begin{pmatrix}f_{R/Ln}(y)g_{R/Ln}^{*}(y^{\prime})\cr
g_{R/Ln}(y)f_{R/Ln}^{*}(y^{\prime})\end{pmatrix}.$ (3.26)

We have made use of the relation $C_{R/L}=D_{R/L}=0$ in deriving (3.22). With
the choice of the AB phase $\theta_{H}$ in (2.28) all mode functions
$\\{f_{Rn}(y)\\}$ etc. can be taken to be real so that
$A_{R/L}(y,y^{\prime})=A_{R/L}(y^{\prime},y)$ and
$B_{R/L}(y,y^{\prime})=B_{R/L}(y^{\prime},y)$.

In addition to the relation (3.26), $A_{R/L}$ and $B_{R/L}$ must satisfy the
parity relations and boundary conditions of the mode functions. With
$(y_{0},y_{1})=(0,L)$

$\displaystyle\underline{\hbox{type 1A}}:$ (3.27)
$\displaystyle\begin{pmatrix}A_{R}\cr B_{R}\cr A_{L}\cr
B_{L}\end{pmatrix}(y_{j}-y,y^{\prime})=\begin{pmatrix}A_{R}\cr-B_{R}\cr-
A_{L}\cr B_{L}\end{pmatrix}(y_{j}+y,y^{\prime})~{},$ (3.28)
$\displaystyle\begin{pmatrix}\hat{D}_{-}(c)A_{R}(y,y^{\prime})\cr\hat{D}_{+}(c)B_{L}(y,y^{\prime})\end{pmatrix}_{y={\epsilon},L-{\epsilon}}=0~{},~{}\hat{D}_{\pm}(c)=\pm\frac{\partial}{\partial
y}+c\,k~{},$ (3.29) $\displaystyle
B_{R}(y_{j},y^{\prime})=A_{L}(y_{j},y^{\prime})=0~{},$ (3.30)
$\displaystyle\underline{\hbox{type 2A}}:$ (3.31)
$\displaystyle\begin{pmatrix}A_{R}\cr B_{R}\cr A_{L}\cr
B_{L}\end{pmatrix}(y_{j}-y,y^{\prime})=\begin{pmatrix}(-1)^{j}A_{R}\cr(-1)^{j+1}B_{R}\cr(-1)^{j+1}A_{L}\cr(-1)^{j}B_{L}\end{pmatrix}(y_{j}+y,y^{\prime})~{},$
(3.32)
$\displaystyle\begin{pmatrix}\hat{D}_{-}(c)A_{R}(y,y^{\prime})\cr\hat{D}_{+}(c)B_{L}(y,y^{\prime})\end{pmatrix}_{y={\epsilon}}=\begin{pmatrix}\hat{D}_{-}(c)B_{R}(y,y^{\prime})\cr\hat{D}_{+}(c)A_{L}(y,y^{\prime})\end{pmatrix}_{y=L-{\epsilon}}=0~{},$
(3.33) $\displaystyle
B_{R}(0,y^{\prime})=A_{R}(L,y^{\prime})=A_{L}(0,y^{\prime})=B_{L}(L,y^{\prime})=0~{}.$
(3.34)

The condition for type 1B (2B) are obtained by interchanging $R$ (right-
handed) and $L$ (left-handed) in those for type 1A (2A). For $c\not=0$, parity
even components of $A_{R/L}$ and $B_{R/L}$ functions exhibit the cusp behavior
at $y,y^{\prime}=0,\pm L,\cdots$.

It is not easy to explicitly write down $A_{R/L}(y,y^{\prime})$ and
$B_{R/L}(y,y^{\prime})$ functions for $c\not=0$ which satisfy the relations in
both (3.26) and (3.34). In the previous paper [12] it has been recognized that
the anomaly coefficient $a_{n_{1}n_{2}n_{3}}$ in (3.22) is independent of $c$.
With this observation we shall derive an analytical expression for
$a_{n_{1}n_{2}n_{3}}$ by evaluating it in the case $c=0$. We will confirm
later that numerically evaluated $a_{n_{1}n_{2}n_{3}}$ for $c\not=0$ agrees
with the analytical formula.

Fermion wave functions for $c=0$ are expressed in terms of trigonometric
functions. They are summarized in Appendix B.3. Inserting the wave functions
in (B.24) into $A_{R}(z,z^{\prime})=\sum f_{Rn}(z)f_{Rn}^{*}(z^{\prime})$, for
instance, one finds for type 1A that, for $1\leq z,z^{\prime}\leq z_{L}$,

$\displaystyle A_{R}(z,z^{\prime})^{c=0}$ (3.35)
$\displaystyle=\frac{1}{z_{L}-1}\sum_{n=-\infty}^{\infty}\cos\Big{(}n\pi\frac{z-z_{L}}{z_{L}-1}+\alpha(z)\Big{)}\cos\Big{(}n\pi\frac{z^{\prime}-z_{L}}{z_{L}-1}+\alpha(z^{\prime})\Big{)}$
(3.36)
$\displaystyle=\delta_{2(z_{L}-1)}(z-z^{\prime})\cos\big{\\{}\alpha(z)-\alpha(z^{\prime})\big{\\}}+\delta_{2(z_{L}-1)}(z+z^{\prime}-2)\cos\big{\\{}\alpha(z)+\alpha(z^{\prime})\big{\\}}$
(3.37)
$\displaystyle=\delta_{2(z_{L}-1)}(z-z^{\prime})+\delta_{2(z_{L}-1)}(z+z^{\prime}-2)~{},$
(3.38)
$\displaystyle\quad\alpha(z)=\frac{1}{2}\Big{\\{}\theta_{H}\,\frac{z-z_{L}}{z_{L}-1}+\theta(z)\Big{\\}}~{},~{}\alpha(1)=\alpha(z_{L})=0~{}.$
(3.39)

Here $\delta_{L}(x)=\sum_{n}\delta(x-nL)$. With the extension (2.50) in the
$y$-coordinate and similar manipulation one finds that

$\displaystyle\underline{\hbox{type 1A},~{}c=0}$ (3.40) $\displaystyle
A_{R}(y,y^{\prime})=B_{L}(y,y^{\prime})=\frac{e^{-\sigma(y)}}{k}\big{\\{}\delta_{2L}(y-y^{\prime})+\delta_{2L}(y+y^{\prime})\big{\\}}~{},$
(3.41) $\displaystyle
B_{R}(y,y^{\prime})=A_{L}(y,y^{\prime})=\frac{e^{-\sigma(y)}}{k}\big{\\{}\delta_{2L}(y-y^{\prime})-\delta_{2L}(y+y^{\prime})\big{\\}}~{},$
(3.42)

Formulas for type 1B are obtained by interchanging $R$ and $L$.

For fermions in type 2A, one finds for $1\leq z,z^{\prime}\leq z_{L}$ that

$\displaystyle A_{R}(z,z^{\prime})^{c=0}$ (3.43)
$\displaystyle=\frac{1}{z_{L}-1}\sum_{n=-\infty}^{\infty}\sin\Big{(}n\pi\frac{z-z_{L}}{z_{L}-1}+\beta(z)\Big{)}\sin\Big{(}n\pi\frac{z^{\prime}-z_{L}}{z_{L}-1}+\beta(z^{\prime})\Big{)}$
(3.44)
$\displaystyle=\delta_{2(z_{L}-1)}(z-z^{\prime})\cos\big{\\{}\beta(z)-\beta(z^{\prime})\big{\\}}-\delta_{2(z_{L}-1)}(z+z^{\prime}-2)\cos\big{\\{}\beta(z)+\beta(z^{\prime})\big{\\}}~{},$
(3.45)
$\displaystyle\quad\beta(z)=\frac{1}{2}\Big{\\{}(\theta_{H}+\pi)\,\frac{z-z_{L}}{z_{L}-1}+\theta(z)\Big{\\}}~{},~{}\beta(1)=-\frac{1}{2}\pi~{},~{}\beta(z_{L})=0~{}.$
(3.46)

Noting the relations in (2.50), one finds in the $y$-coordinate that

$\displaystyle\underline{\hbox{type 2A},~{}c=0}$ (3.47) $\displaystyle
A_{R}(y,y^{\prime})=B_{L}(y,y^{\prime})=\frac{e^{-\sigma(y)}}{k}\big{\\{}\hat{\delta}_{2L}(y-y^{\prime})+\hat{\delta}_{2L}(y+y^{\prime})\big{\\}}~{},$
(3.48) $\displaystyle
B_{R}(y,y^{\prime})=A_{L}(y,y^{\prime})=\frac{e^{-\sigma(y)}}{k}\big{\\{}\hat{\delta}_{2L}(y-y^{\prime})-\hat{\delta}_{2L}(y+y^{\prime})\big{\\}}~{},$
(3.49)
$\displaystyle\hat{\delta}_{2L}(y)=\delta_{4L}(y)-\delta_{4L}(y-2L)~{}.$
(3.50)

Formulas for type 2B are obtained by interchanging $R$ and $L$.

We insert the expressions (3.42) or (3.50) into (3.22). There appear products
of three delta functions in the integrand. Take $0<a<L$. Then in the
integration range $-a\leq y_{1},y_{2},y_{3}\leq 2L-a$, products of delta
functions reduce to

$\displaystyle\begin{matrix}\delta_{2L}(y_{1}-y_{2})\delta_{2L}(y_{2}-y_{3})\delta_{2L}(y_{3}+y_{1})\cr\kern
5.0pt\cr\delta_{2L}(y_{1}+y_{2})\delta_{2L}(y_{2}+y_{3})\delta_{2L}(y_{3}+y_{1})\end{matrix}~{}\bigg{\\}}$
(3.51)
$\displaystyle\quad\Rightarrow\frac{1}{2}\Big{\\{}\delta(y_{1})\delta(y_{2})\delta(y_{3})+\delta(y_{1}-L)\delta(y_{2}-L)\delta(y_{3}-L)\Big{\\}}~{},$
(3.52)
$\displaystyle\begin{matrix}\hat{\delta}_{2L}(y_{1}-y_{2})\hat{\delta}_{2L}(y_{2}-y_{3})\hat{\delta}_{2L}(y_{3}+y_{1})\cr\kern
5.0pt\cr\hat{\delta}_{2L}(y_{1}+y_{2})\hat{\delta}_{2L}(y_{2}+y_{3})\hat{\delta}_{2L}(y_{3}+y_{1})\end{matrix}~{}\bigg{\\}}$
(3.53)
$\displaystyle\quad\Rightarrow\frac{1}{2}\Big{\\{}\delta(y_{1})\delta(y_{2})\delta(y_{3})-\delta(y_{1}-L)\delta(y_{2}-L)\delta(y_{3}-L)\Big{\\}}~{}.$
(3.54)

As $h_{n}(0)=h_{n}(L)=0$, only the terms proportional to $k_{1}k_{2}k_{3}$ in
(3.22) survive. We find the formula for the anomaly coefficients;

$\displaystyle a_{n\ell m}(\theta_{H},z_{L})$
$\displaystyle=Q_{0}k_{n}(0)k_{\ell}(0)k_{m}(0)+Q_{1}k_{n}(L)k_{\ell}(L)k_{m}(L)~{},$
(3.55) $\displaystyle(Q_{0},Q_{1})$
$\displaystyle=\begin{cases}(+1,+1)&\hbox{for type 1A}\cr(-1,-1)&\hbox{for
type 1B}\cr(+1,-1)&\hbox{for type 2A}\cr(-1,+1)&\hbox{for type
2B}\end{cases}~{}.$ (3.56)

The anomaly coefficients are determined by the values of the wave functions of
the gauge fields at the UV and IR branes and the parity conditions of the
fermion fields.

The formula (3.56) is strikingly simple. The wave function $k_{n}(y)$ depends
on $\theta_{H}$ and $z_{L}$. The sum of the chiral anomalies arising from all
possible fermion KK modes are summarized in terms of $k_{n}(0)$ and
$k_{n}(L)$. The $c$-independence of those anomalies is confirmed numerically.
The anomaly coefficients $a_{n\ell m}$ given by (3.56) are compared with those
determined by first evaluating the gauge couplings $t^{R/L}_{n\ell m}$
($0\leq\ell,m\leq\ell_{0}$) in (3.10) and then taking the traces of
$(\ell_{0}+1)$-dimensional matrices in (3.13). In Figure 2 the results for
$a_{000},a_{111},a_{222}$ and $a_{012}$ are shown for type 1A fermions with
$c=0.25$, $\ell_{0}=10$ and $z_{L}=10$. One sees that the numerically
evaluated values for $c=0.25$ fall on the universal curves given by (3.56). We
have checked that the numerically evaluated values for other values of $c$
fall on the universal curves as well.

Figure 2: The anomaly coefficients $a_{000},a_{111},a_{222}$ and $a_{012}$ as
functions of $\theta_{H}$ are shown for type 1A fermions for $z_{L}=10$. Blue
curves represent the universal curves given by (3.56). Red dots represent the
values determined from the gauge couplings $t^{R/L}_{n\ell m}$
($0\leq\ell,m\leq\ell_{0}$) in (3.10) and then taking the traces of
$(\ell_{0}+1)$-dimensional matrices in (3.13) for fermions with $c=0.25$ and
$\ell_{0}=10$.

Some of $k_{n}(0;\theta_{H})$ and $k_{n}(L;\theta_{H})$ are plotted in Figure
3. Note that for $n=1,3,5,\cdots$, $|k_{n}(L;\theta_{H})|$ is much larger than
$|k_{n}(0;\theta_{H})|$ for $z_{L}\geq 10$. Massless gauge bosons
($Z_{\mu}^{(0)}$) exist at $\theta_{H}=0$ and $\pi$. $k_{0}(0;0)=k_{0}(L;0)=1$
and $k_{0}(0;\pi)=-k_{0}(L;\pi)=1$ so that $a_{000}(\theta_{H}=0)=2$ and
$a_{000}(\theta_{H}=\pi)=0$ for type 1A fermions and $a_{000}(\theta_{H}=0)=0$
and $a_{000}(\theta_{H}=\pi)=2$ for type 2A fermions. The anomaly flow is
reflected in the behavior of the wave functions of the gauge fields at $y=0$
and $L$.

Figure 3: The values of the gauge wave functions $k_{n}(y;\theta_{H})$
($n=0,1,2,3$) at $y=0$ (blue curves) and $y=L$ (red curves) for $z_{L}=10$ are
shown.

Dependence of the anomaly coefficients $a_{n\ell m}$ on fermion types has a
simple pattern. $a_{n\ell m}(\theta_{H})^{\rm{type}\,\rm{1A}}=-a_{n\ell
m}(\theta_{H})^{\rm{type}\,\rm{1B}}$ and $a_{n\ell
m}(\theta_{H})^{\rm{type}\,\rm{2A}}=-a_{n\ell
m}(\theta_{H})^{\rm{type}\,\rm{2B}}$. Further $a_{n\ell
m}(\theta_{H}+\pi)^{\rm{type}\,\rm{1A}}=a_{n\ell
m}(\theta_{H})^{\rm{type}\,\rm{2A}}$ or $a_{n\ell
m}(\theta_{H})^{\rm{type}\,\rm{2B}}$. (See Figure 4.) It follows from the
property that
$[k_{n}(0),k_{n}(L)]_{\theta_{H}+\pi}=[k_{n}(0),-k_{n}(L)]_{\theta_{H}}$ or
$[-k_{n}(0),k_{n}(L)]_{\theta_{H}}$.

Figure 4: Dependence of the anomaly coefficients $a_{000},a_{111},a_{002}$ and
$a_{012}$ on fermion types is shown for $z_{L}=10$. One sees that $a_{n\ell
m}(\theta_{H}+\pi)^{\rm{type}\,\rm{1A}}=a_{n\ell
m}(\theta_{H})^{\rm{type}\,\rm{2A}}$ or $a_{n\ell
m}(\theta_{H})^{\rm{type}\,\rm{2B}}$.

Formulas in the flat $M^{4}\times(S^{1}/Z_{2})$ spacetime simplify. With the
KK expansions (2.14), (2.19), and (2.21), the gauge couplings are written as

$\displaystyle\frac{g_{4}}{2}\sum_{n=-\infty}^{\infty}\sum_{\ell=-\infty}^{\infty}\sum_{m=-\infty}^{\infty}B_{\mu}^{(n)}(x)\Big{\\{}s^{R}_{n\ell
m}\,\psi_{R}^{(\ell)}(x)^{\dagger}\bar{\sigma}^{\mu}\psi_{R}^{(m)}(x)+s^{L}_{n\ell
m}\,\psi_{L}^{(\ell)}(x)^{\dagger}\sigma^{\mu}\psi_{L}^{(m)}(x)\Big{\\}}$
(3.57)

for type 1A and 1B fermions. For type 2A and 2B fermions $\psi_{R/L}^{(m)}(x)$
should be replaced by $\psi_{R/L}^{(m+\frac{1}{2})}(x)$. The anomaly
coefficient associated with the three legs of
$B_{\mu_{1}}^{(n_{1})}B_{\mu_{2}}^{(n_{2})}B_{\mu_{3}}^{(n_{3})}$ is given by

$\displaystyle
b_{n_{1}n_{2}n_{3}}=b_{n_{1}n_{2}n_{3}}^{R}+b_{n_{1}n_{2}n_{3}}^{L}~{},$
(3.58) $\displaystyle b_{n_{1}n_{2}n_{3}}^{R}={\rm
Tr}\,S^{R}_{n_{1}}S^{R}_{n_{2}}S^{R}_{n_{3}}~{},~{}~{}(S^{R}_{n})_{m\ell}=s^{R}_{nm\ell}~{},$
(3.59) $\displaystyle b_{n_{1}n_{2}n_{3}}^{L}={\rm
Tr}\,S^{L}_{n_{1}}S^{L}_{n_{2}}S^{L}_{n_{3}}~{},~{}~{}(S^{L}_{n})_{m\ell}=s^{L}_{nm\ell}~{}.$
(3.60)

Applying the same argument as in the case of the RS space, one finds that

$\displaystyle b_{n\ell m}$ $\displaystyle=Q_{0}k^{\rm flat}_{n}(0)k^{\rm
flat}_{\ell}(0)k^{\rm flat}_{m}(0)+Q_{1}k^{\rm flat}_{n}(L)k^{\rm
flat}_{\ell}(L)k^{\rm flat}_{m}(L)$ (3.61)

where $Q_{0},Q_{1}$ are given in (3.56). Since $k^{\rm flat}_{n}(y)=\cos(n\pi
y/L)$ from (2.14), one finds that

$\displaystyle b_{n\ell m}$ $\displaystyle=Q_{0}+(-1)^{n+\ell+m}\,Q_{1}~{},$
(3.62)

which agrees with the result in Ref.[12]. The formula (3.62) also results in
the flat spacetime limit of (3.56). In the flat spacetime the level-crossing
in the mass spectrum of gauge fields occurs at
$\theta_{H}=0,\pm\hbox{$\frac{1}{2}$}\pi,\pm\pi,\cdots$. For this reason the
flat spacetime limit of (3.56) becomes singular, as has been shown in
Ref.[12].

##  4 Anomaly cancellation

The universality of the anomaly flow, expressed in the formula (3.56), has a
profound implication in the model building, particularly in the GHU scenario.
Chiral anomalies associated with gauge currents must be cancelled for the
consistency of the theory in four dimensions [20, 21]. The fact that the
anomaly coefficients are independent of the bulk mass parameters of fermions
implies that anomaly cancellation can be achieved among various distinct
fermions in the theory. In this section we examine this problem in the $SU(2)$
model.

Let us first recall the equations following from the action $I_{\rm RS}$ in
(2.26) are, at the classical level,

$\displaystyle\frac{1}{\sqrt{-\det G}}\,\partial_{M}\big{(}\sqrt{-\det
G}\,F^{MN}\big{)}-ig_{A}[A_{M},F^{MN}]+J^{N}=0~{},$ (4.1) $\displaystyle{\cal
D}(c)\,\Psi=0~{},$ (4.2) $\displaystyle
J^{N}=J^{Na}\frac{\tau^{a}}{2}~{},~{}~{}J^{Na}=-ig_{A}\overline{\Psi}\gamma^{A}{e_{A}}^{N}\frac{\tau^{a}}{2}\Psi~{}.$
(4.3)

The current in five dimensions is covariantly conserved;

$\displaystyle\frac{1}{\sqrt{-\det G}}\,\partial_{N}\big{(}\sqrt{-\det
G}\,J^{N}\big{)}-ig_{A}[A_{N},J^{N}]=0~{}.$ (4.4)

Note that the derivative term in the fifth coordinate generates mass terms in
four dimensions when expanded in the KK modes. At the quantum level there
arises an anomaly term on the righthand side of Eq. (4.4). The four-
dimensional current $j_{(n)}^{\mu}(x)$ which couples with $Z_{\mu}^{(n)}(x)$
is

$\displaystyle j_{(n)}^{\mu}(x)$ $\displaystyle=\int_{0}^{L}dy\,\sqrt{-\det
G}\,\big{\\{}h_{n}(y)J^{\mu 1}+k_{n}(y)J^{\mu 3}\big{\\}}$ (4.5)
$\displaystyle=\frac{g_{4}}{2}\sum_{\ell=0}^{\infty}\sum_{m=0}^{\infty}\Big{\\{}t^{R}_{n\ell
m}\,\chi_{R}^{(\ell)}(x)^{\dagger}\bar{\sigma}^{\mu}\chi_{R}^{(m)}(x)+t^{L}_{n\ell
m}\,\chi_{L}^{(\ell)}(x)^{\dagger}\sigma^{\mu}\chi_{L}^{(m)}(x)\Big{\\}}.$
(4.6)

The divergence $\partial_{\mu}j_{(n)}^{\mu}$ picks up an anomalous term
$j_{(n)}^{\rm anomaly}$ given by

$\displaystyle j_{(n)}^{\rm
anomaly}=-\Big{(}\frac{g_{4}}{2}\Big{)}^{3}\sum_{\ell=0}^{\infty}\sum_{m=0}^{\infty}\frac{a_{n\ell
m}}{32\pi^{2}}\,{\epsilon}^{\mu\nu\rho\sigma}Z^{(\ell)}_{\mu\nu}Z^{(m)}_{\rho\sigma}$
(4.7)

where
$Z^{(\ell)}_{\mu\nu}=\partial_{\mu}Z_{\nu}^{(\ell)}-\partial_{\nu}Z_{\mu}^{(\ell)}$.

The conditions for the cancellation of the gauge anomalies are simple. Let the
numbers of doublet fermions of type 1A, 1B, 2A and 2B be $n_{1A}$, $n_{1B}$,
$n_{2A}$ and $n_{2B}$, respectively. It follows from (3.56) that the anomalies
are cancelled if

$\displaystyle n_{1A}=n_{1B}~{},~{}~{}n_{2A}=n_{2B}~{}.$ (4.8)

In the presence of brane fermions, namely fermions living only on the UV or IR
brane, the conditions are generalized. Suppose that there are $\hat{n}_{R}$
right-handed and $\hat{n}_{L}$ left-handed doublet brane fermions on the UV
brane at $y=0$. As the $Z_{\mu}^{(n)}$ coupling of each brane fermion is given
by $(g_{4}/2)\,k_{n}(0)$, the anomaly cancellation conditions become

$\displaystyle n_{1A}-n_{1B}+n_{2A}-n_{2B}+\hat{n}_{R}-\hat{n}_{L}=0~{},$
(4.9) $\displaystyle n_{1A}-n_{1B}-n_{2A}+n_{2B}=0~{},$ (4.10)

We stress that the conditions (4.8) and (4.10) do not depend on $\theta_{H}$
and $z_{L}$. Furthermore the conditions guarantee that not only the zero mode
anomaly $a_{000}$ but also all other anomalies $a_{n\ell m}$ are cancelled at
once.

Fermion multiplets in the triplet representation do not contribute to
anomalies in the $SU(2)$ gauge theory as is easily confirmed. The anomaly
cancellation is achieved by the condition (4.8) or (4.10), namely by the
condition for the numbers of doublet fermions with four types of orbifold
boundary conditions. It does not depend on the AB phase $\theta_{H}$, namely
the VEV of $A_{y}$. The situation is very similar to the anomaly cancellation
condition in the SM.

##  5 Summary and discussions

In this paper we have examined the anomaly flow by the AB phase $\theta_{H}$
in the $SU(2)$ gauge theory in the RS space and in the flat
$M^{4}\times(S^{1}/Z_{2})$ spacetime. The anomaly coefficients $a_{n\ell
m}(\theta_{H},z_{L})$ induced by a fermion field in the bulk smoothly changes
in $\theta_{H}$ in the RS space. Although the gauge couplings of the fermion,
$t_{n\ell m}^{R/L}(\theta_{H},z_{L},c)$, nontrivially depend on the bulk mass
parameter $c$ of the fermion, the total anomaly coefficients $a_{n\ell m}$ are
independent of $c$. We have shown that those anomaly coefficients $a_{n\ell
m}$ are expressed in terms of the values of the wave functions of the gauge
fields at the UV and IR branes. The holographic formula (3.56) manifestly
exhibits the $c$-independence. We have confirmed that the values of the
anomaly coefficients numerically evaluated directly from $t_{n\ell
m}^{R/L}(\theta_{H},z_{L},c)$ fall precisely on the curves given by (3.56). It
has been left for future investigation to find an analytic proof of the
$c$-independence of the expression (3.22).

As has been mentioned in the previous section, the universality in anomaly
flow is critically important in the construction of realistic models of
particle physics. GHU models have been proposed to unify the 4D Higgs boson
with gauge fields in the framework of gauge theory on five-dimensional
orbifolds in which the gauge hierarchy problem is naturally solved [5, 7, 22,
23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34]. In particular, $SO(5)\times
U(1)_{X}\times SU(3)_{C}$ GHU in the RS space with $\theta_{H}\sim 0.1$ and
$z_{L}=10^{5}\sim 10^{10}$ has been shown to reproduce nearly the same
phenomenology at low energies as the SM [31, 33]. As in the case of the SM,
all chiral anomalies associated with gauge currents must be cancelled.
Generalization of the argument on the universality to the group $SO(5)\times
U(1)_{X}\times SU(3)_{C}$ is necessary. Further the technology developed in
the present paper can be applied to the evaluation of anomalies of global
currents such as baryon and lepton numbers. The phenomenon of anomaly flow may
possibly be related to Chern-Simons terms in five dimensions[35, 36, 37].
These issues will be clarified in separate papers.

## Acknowledgement

This work was supported in part by Japan Society for the Promotion of Science,
Grants-in-Aid for Scientific Research, Grant No. JP19K03873.

##  Appendix A Basis functions

Wave functions of gauge fields and fermions are expressed in terms of the
following basis functions. For gauge fields we introduce

$\displaystyle C(z;\lambda)$ $\displaystyle=\frac{\pi}{2}\lambda
zz_{L}F_{1,0}(\lambda z,\lambda z_{L})~{},$ (A.1) $\displaystyle S(z;\lambda)$
$\displaystyle=-\frac{\pi}{2}\lambda zF_{1,1}(\lambda z,\lambda z_{L})~{},$
(A.2) $\displaystyle C^{\prime}(z;\lambda)$
$\displaystyle=\frac{\pi}{2}\lambda^{2}zz_{L}F_{0,0}(\lambda z,\lambda
z_{L})~{},$ (A.3) $\displaystyle S^{\prime}(z;\lambda)$
$\displaystyle=-\frac{\pi}{2}\lambda^{2}zF_{0,1}(\lambda z,\lambda z_{L})~{},$
(A.4) $\displaystyle F_{\alpha,\beta}(u,v)$ $\displaystyle\equiv
J_{\alpha}(u)Y_{\beta}(v)-Y_{\alpha}(u)J_{\beta}(v)~{},$ (A.5)

where $J_{\alpha}(u)$ and $Y_{\alpha}(u)$ are Bessel functions of the first
and second kind. They satisfy

$\displaystyle-z\frac{d}{dz}\frac{1}{z}\frac{d}{dz}\begin{pmatrix}C\cr
S\end{pmatrix}=\lambda^{2}\begin{pmatrix}C\cr S\end{pmatrix}~{},$ (A.6)
$\displaystyle
C(z_{L};\lambda)=z_{L}~{},~{}~{}C^{\prime}(z_{L};\lambda)=0~{},$ (A.7)
$\displaystyle
S(z_{L};\lambda)=0~{},~{}~{}S^{\prime}(z_{L};\lambda)=\lambda~{},$ (A.8)
$\displaystyle CS^{\prime}-SC^{\prime}=\lambda z~{}.$ (A.9)

To express wave functions of KK modes of gauge fields, we make use of

$\displaystyle\hat{S}(z;\lambda)=N_{0}(\lambda)S(z;\lambda)~{},~{}~{}\hat{C}(z;\lambda)=N_{0}(\lambda)^{-1}C(z;\lambda)~{},$
(A.10)
$\displaystyle\check{S}(z;\lambda)=N_{1}(\lambda)S(z;\lambda)~{},~{}~{}\check{C}(z;\lambda)=N_{1}(\lambda)^{-1}C(z;\lambda)~{},$
(A.11) $\displaystyle\qquad
N_{0}(\lambda)=\frac{C(1;\lambda)}{S(1;\lambda)}~{},~{}~{}N_{1}(\lambda)=\frac{C^{\prime}(1;\lambda)}{S^{\prime}(1;\lambda)}~{}.$
(A.12)

For fermion fields with a bulk mass parameter $c$, we define

$\displaystyle\begin{pmatrix}C_{L}\cr S_{L}\end{pmatrix}(z;\lambda,c)$
$\displaystyle=\pm\frac{\pi}{2}\lambda\sqrt{zz_{L}}F_{c+\frac{1}{2},c\mp\frac{1}{2}}(\lambda
z,\lambda z_{L})~{},$ (A.13) $\displaystyle\begin{pmatrix}C_{R}\cr
S_{R}\end{pmatrix}(z;\lambda,c)$
$\displaystyle=\mp\frac{\pi}{2}\lambda\sqrt{zz_{L}}F_{c-\frac{1}{2},c\pm\frac{1}{2}}(\lambda
z,\lambda z_{L})~{}.$ (A.14)

These functions satisfy

$\displaystyle D_{+}(c)\begin{pmatrix}C_{L}\cr
S_{L}\end{pmatrix}=\lambda\begin{pmatrix}S_{R}\cr C_{R}\end{pmatrix},$ (A.15)
$\displaystyle D_{-}(c)\begin{pmatrix}C_{R}\cr
S_{R}\end{pmatrix}=\lambda\begin{pmatrix}S_{L}\cr
C_{L}\end{pmatrix},~{}~{}D_{\pm}(c)=\pm\frac{d}{dz}+\frac{c}{z}~{},$ (A.16)
$\displaystyle C_{R}=C_{L}=1~{},~{}~{}S_{R}=S_{L}=0\quad{\rm
at~{}}z=z_{L}~{},$ (A.17) $\displaystyle C_{L}C_{R}-S_{L}S_{R}=1~{}.$ (A.18)

Also $C_{L}(z;\lambda,-c)=C_{R}(z;\lambda,c)$ and
$S_{L}(z;\lambda,-c)=-S_{R}(z;\lambda,c)$ hold. To express wave functions of
KK modes of fermion fields, we make use of

$\displaystyle\hat{S}_{L}(z;\lambda,c)=N_{L}(\lambda,c)S_{L}(z;\lambda,c)~{},~{}~{}\hat{C}_{L}(z;\lambda,c)=N_{R}(\lambda,c)C_{L}(z;\lambda,c)~{},$
(A.19)
$\displaystyle\hat{S}_{R}(z;\lambda,c)=N_{R}(\lambda,c)S_{R}(z;\lambda,c)~{},~{}~{}\hat{C}_{R}(z;\lambda,c)=N_{L}(\lambda,c)C_{R}(z;\lambda,c)~{},$
(A.20)
$\displaystyle\check{S}_{L}(z;\lambda,c)=N_{R}(\lambda,c)^{-1}S_{L}(z;\lambda,c)~{},~{}~{}\check{C}_{L}(z;\lambda,c)=N_{L}(\lambda,c)^{-1}C_{L}(z;\lambda,c)~{},$
(A.21)
$\displaystyle\check{S}_{R}(z;\lambda,c)=N_{L}(\lambda,c)^{-1}S_{R}(z;\lambda,c)~{},~{}~{}\check{C}_{R}(z;\lambda,c)=N_{R}(\lambda,c)^{-1}C_{R}(z;\lambda,c)~{},$
(A.22) $\displaystyle\qquad
N_{L}(\lambda,c)=\frac{C_{L}(1;\lambda,c)}{S_{L}(1;\lambda,c)}~{},~{}~{}N_{R}(\lambda,c)=\frac{C_{R}(1;\lambda,c)}{S_{R}(1;\lambda,c)}~{}.$
(A.23)

##  Appendix B Wave functions in RS

###  B.1 Gauge fields $Z_{\mu}^{(n)}$

The mode functions of the gauge fields $Z_{\mu}^{(n)}(x)$ in (2.32) are given
by

$\displaystyle\tilde{\bf h}_{0}(z)$ $\displaystyle=\bar{\bf h}_{0}^{a}(z)~{},$
(B.1) $\displaystyle\tilde{\bf h}_{2\ell-1}(z)$
$\displaystyle=(-1)^{\ell}\begin{cases}\bar{\bf h}_{2\ell-1}^{a}(z)&{\rm
for}-\frac{1}{2}\pi<\theta_{H}<\frac{1}{2}\pi\cr\bar{\bf
h}_{2\ell-1}^{b}(z)&{\rm for~{}}0<\theta_{H}<\pi\cr-\bar{\bf
h}_{2\ell-1}^{a}(z)&{\rm
for~{}}\frac{1}{2}\pi<\theta_{H}<\frac{3}{2}\pi\cr-\bar{\bf
h}_{2\ell-1}^{b}(z)&{\rm for~{}}\pi<\theta_{H}<2\pi\cr\bar{\bf
h}_{2\ell-1}^{a}(z)&{\rm
for~{}}\frac{3}{2}\pi<\theta_{H}<\frac{5}{2}\pi\end{cases}~{}~{}(\ell=1,2,3,\cdots),$
(B.2) $\displaystyle\tilde{\bf h}_{2\ell}(z)$
$\displaystyle=(-1)^{\ell}\begin{cases}\bar{\bf h}_{2\ell}^{b}(z)&{\rm
for}-\frac{1}{2}\pi<\theta_{H}<\frac{1}{2}\pi\cr-\bar{\bf
h}_{2\ell}^{a}(z)&{\rm for~{}}0<\theta_{H}<\pi\cr-\bar{\bf
h}_{2\ell}^{b}(z)&{\rm
for~{}}\frac{1}{2}\pi<\theta_{H}<\frac{3}{2}\pi\cr\bar{\bf
h}_{2\ell}^{a}(z)&{\rm for~{}}\pi<\theta_{H}<2\pi\cr\bar{\bf
h}_{2\ell}^{b}(z)&{\rm
for~{}}\frac{3}{2}\pi<\theta_{H}<\frac{5}{2}\pi\end{cases}~{}(\ell=1,2,3,\cdots),$
(B.3) $\displaystyle\bar{\bf h}_{n}^{a}(z)$
$\displaystyle=\frac{1}{\sqrt{r^{a}_{n}}}\begin{pmatrix}-s_{H}\hat{S}(z;\lambda_{n})\cr
c_{H}C(z;\lambda_{n})\end{pmatrix},~{}~{}\bar{\bf
h}_{n}^{b}(z)=\frac{1}{\sqrt{r^{b}_{n}}}\begin{pmatrix}c_{H}S(z;\lambda_{n})\cr
s_{H}\check{C}(z;\lambda_{n})\end{pmatrix},$ (B.4) $\displaystyle
s_{H}=\sin\theta_{H}~{},~{}~{}c_{H}=\cos\theta_{H}~{},$ (B.5) $\displaystyle
r_{n}=\frac{1}{kL}\int_{1}^{z_{L}}\frac{dz}{z}\big{\\{}|\hat{h}_{n}(z)|^{2}+|\hat{k}_{n}(z)|^{2}\big{\\}}\quad{\rm
for}~{}\begin{pmatrix}\hat{h}_{n}(z)\cr\hat{k}_{n}(z)\end{pmatrix}.$ (B.6)

$\hat{S}$ and $\check{C}$ are given in (A.12). In the above formulas, the two
expressions given in an overlapping $\theta_{H}$ region are the same. The
connection formulas are necessary as one of them fails to make sense at the
boundary in $\theta_{H}$.

###  B.2 Fermion fields $\chi_{R/L}^{(n)}$

The mode functions of the fermion fields $\chi_{R/L}^{(n)}(x)$ in (2.40) are
given, for type 1A and $c>0$, by

$\displaystyle\underline{\hbox{type 1A}}$ (B.7) $\displaystyle\tilde{\bf
f}_{R,2\ell}(z)$ $\displaystyle=\begin{cases}\bar{\bf f}_{R,2\ell}^{a}(z)&{\rm
for~{}}-\pi<\theta_{H}<\pi\cr\bar{\bf f}_{R,2\ell}^{b}(z)&{\rm
for~{}}0<\theta_{H}<2\pi\cr-\bar{\bf f}_{R,2\ell}^{a}(z)&{\rm
for~{}}\pi<\theta_{H}<3\pi\cr-\bar{\bf f}_{R,2\ell}^{b}(z)&{\rm
for~{}}2\pi<\theta_{H}<4\pi\cr\bar{\bf f}_{R,2\ell}^{a}(z)&{\rm
for~{}}3\pi<\theta_{H}<5\pi\cr\end{cases}~{}(\ell=0,1,2,\cdots),$ (B.8)
$\displaystyle\tilde{\bf f}_{R,2\ell-1}(z)$
$\displaystyle=\begin{cases}\bar{\bf f}_{R,2\ell-1}^{c}(z)&{\rm
for~{}}-\pi<\theta_{H}<\pi\cr\bar{\bf f}_{R,2\ell-1}^{d}(z)&{\rm
for~{}}0<\theta_{H}<2\pi\cr-\bar{\bf f}_{R,2\ell-1}^{c}(z)&{\rm
for~{}}\pi<\theta_{H}<3\pi\cr-\bar{\bf f}_{R,2\ell-1}^{d}(z)&{\rm
for~{}}2\pi<\theta_{H}<4\pi\cr\bar{\bf f}_{R,2\ell-1}^{c}(z)&{\rm
for~{}}3\pi<\theta_{H}<5\pi\cr\end{cases}~{}(\ell=1,2,3,\cdots),$ (B.9)
$\displaystyle\tilde{\bf f}_{L0}(z)$ $\displaystyle=\bar{\bf f}_{L0}^{a}(z),$
(B.10) $\displaystyle\tilde{\bf f}_{L,2\ell-1}(z)$
$\displaystyle=\begin{cases}\bar{\bf f}_{L,2\ell-1}^{a}(z)&{\rm
for~{}}-\pi<\theta_{H}<\pi\cr\bar{\bf f}_{L,2\ell-1}^{b}(z)&{\rm
for~{}}0<\theta_{H}<2\pi\cr-\bar{\bf f}_{L,2\ell-1}^{a}(z)&{\rm
for~{}}\pi<\theta_{H}<3\pi\cr-\bar{\bf f}_{L,2\ell-1}^{b}(z)&{\rm
for~{}}2\pi<\theta_{H}<4\pi\cr\bar{\bf f}_{L,2\ell-1}^{a}(z)&{\rm
for~{}}3\pi<\theta_{H}<5\pi\cr\end{cases}~{}(\ell=1,2,3,\cdots),$ (B.11)
$\displaystyle\tilde{\bf f}_{L,2\ell}(z)$ $\displaystyle=\begin{cases}\bar{\bf
f}_{L,2\ell}^{c}(z)&{\rm for~{}}-\pi<\theta_{H}<\pi\cr\bar{\bf
f}_{L,2\ell}^{d}(z)&{\rm for~{}}0<\theta_{H}<2\pi\cr-\bar{\bf
f}_{L,2\ell}^{c}(z)&{\rm for~{}}\pi<\theta_{H}<3\pi\cr-\bar{\bf
f}_{L,2\ell}^{d}(z)&{\rm for~{}}2\pi<\theta_{H}<4\pi\cr\bar{\bf
f}_{L,2\ell}^{c}(z)&{\rm
for~{}}3\pi<\theta_{H}<5\pi\cr\end{cases}~{}(\ell=1,2,3,\cdots),$ (B.12)

Here

$\displaystyle\bar{\bf
f}_{Rn}^{a}(z)=\frac{1}{\sqrt{r^{a}_{n}}}\begin{pmatrix}\bar{c}_{H}C_{R}(z;\lambda_{n},c)\cr-\bar{s}_{H}\hat{S}_{R}(z;\lambda_{n},c)\end{pmatrix},~{}~{}\bar{\bf
f}_{Rn}^{b}(z)=\frac{1}{\sqrt{r^{b}_{n}}}\begin{pmatrix}\bar{s}_{H}C_{R}(z;\lambda_{n},c)\cr\bar{c}_{H}\check{S}_{R}(z;\lambda_{n},c)\end{pmatrix},$
(B.13) $\displaystyle\bar{\bf
f}_{Rn}^{c}(z)=\frac{1}{\sqrt{r^{c}_{n}}}\begin{pmatrix}\bar{s}_{H}\hat{C}_{R}(z;\lambda_{n},c)\cr\bar{c}_{H}S_{R}(z;\lambda_{n},c)\end{pmatrix},~{}~{}\bar{\bf
f}_{Rn}^{d}(z)=\frac{1}{\sqrt{r^{d}_{n}}}\begin{pmatrix}-\bar{c}_{H}\check{C}_{R}(z;\lambda_{n},c)\cr\bar{s}_{H}S_{R}(z;\lambda_{n},c)\end{pmatrix},$
(B.14) $\displaystyle\bar{\bf
f}_{Ln}^{a}(z)=\frac{1}{\sqrt{r^{a}_{n}}}\begin{pmatrix}\bar{s}_{H}\hat{S}_{L}(z;\lambda_{n},c)\cr\bar{c}_{H}C_{L}(z;\lambda_{n},c)\end{pmatrix},~{}~{}\bar{\bf
f}_{Ln}^{b}(z)=\frac{1}{\sqrt{r^{b}_{n}}}\begin{pmatrix}-\bar{c}_{H}\check{S}_{L}(z;\lambda_{n},c)\cr\bar{s}_{H}C_{L}(z;\lambda_{n},c)\end{pmatrix},$
(B.15) $\displaystyle\bar{\bf
f}_{Ln}^{c}(z)=\frac{1}{\sqrt{r^{c}_{n}}}\begin{pmatrix}\bar{c}_{H}S_{L}(z;\lambda_{n},c)\cr-\bar{s}_{H}\hat{C}_{L}(z;\lambda_{n},c)\end{pmatrix},~{}~{}\bar{\bf
f}_{Ln}^{d}(z)=\frac{1}{\sqrt{r^{d}_{n}}}\begin{pmatrix}\bar{s}_{H}S_{L}(z;\lambda_{n},c)\cr\bar{c}_{H}\check{C}_{L}(z;\lambda_{n},c)\end{pmatrix},$
(B.16)
$\displaystyle\qquad\bar{c}_{H}=\cos\hbox{$\frac{1}{2}$}\theta_{H}~{},~{}\bar{s}_{H}=\sin\hbox{$\frac{1}{2}$}\theta_{H}~{},$
(B.17) $\displaystyle\qquad
r_{n}=\int_{1}^{z_{L}}dz\big{\\{}|\hat{f}_{n}(z)|^{2}+|\hat{g}_{n}(z)|^{2}\big{\\}}\quad{\rm
for}~{}\begin{pmatrix}\hat{f}_{n}(z)\cr\hat{g}_{n}(z)\end{pmatrix}.$ (B.18)

Functions $\hat{S}_{R/L},\check{S}_{R/L}$ etc. are defined in (A.23). In
(B.12) two expressions in an overlapping region in $\theta_{H}$ are the same.

###  B.3 Fermion fields $\chi_{R/L}^{(n)}$ for $c=0$

For $c=0$ $C_{R/L}(z;\lambda,0)$ and $S_{R/L}(z;\lambda,0)$ reduce to
trigonometric functions.

$\displaystyle\begin{pmatrix}C_{L}\cr S_{L}\end{pmatrix}(z;\lambda,0)$
$\displaystyle=\begin{pmatrix}\cos\lambda(z-z_{L})\cr\sin\lambda(z-z_{L})\end{pmatrix},$
(B.19) $\displaystyle\begin{pmatrix}C_{R}\cr S_{R}\end{pmatrix}(z;\lambda,0)$
$\displaystyle=\begin{pmatrix}\cos\lambda(z-z_{L})\cr-\sin\lambda(z-z_{L})\end{pmatrix}.$
(B.20)

The spectrum and wave functions in $1\leq z=e^{ky}\leq z_{L}$ in the original
gauge are given for type 1A by

$\displaystyle\underline{\hbox{type 1A}}:$ (B.21)
$\displaystyle\lambda_{n}=\frac{1}{z_{L}-1}\big{|}n\pi+\hbox{$\frac{1}{2}$}\theta_{H}\big{|}\quad(-\infty<n<\infty),$
(B.22) $\displaystyle\begin{pmatrix}f_{Rn}(y)\cr\kern 5.0pt\cr
g_{Rn}(y)\end{pmatrix}=\frac{1}{\sqrt{z_{L}-1}}\begin{pmatrix}\cos\Big{\\{}(n\pi+\hbox{$\frac{1}{2}$}\theta_{H})\frac{\displaystyle
z-z_{L}}{\displaystyle
z_{L}-1}+\hbox{$\frac{1}{2}$}\theta(z)\Big{\\}}\cr\sin\Big{\\{}(n\pi+\hbox{$\frac{1}{2}$}\theta_{H})\frac{\displaystyle
z-z_{L}}{\displaystyle
z_{L}-1}+\hbox{$\frac{1}{2}$}\theta(z)\Big{\\}}\end{pmatrix},$ (B.23)
$\displaystyle\begin{pmatrix}f_{Ln}(y)\cr\kern 5.0pt\cr
g_{Ln}(y)\end{pmatrix}=\frac{1}{\sqrt{z_{L}-1}}\begin{pmatrix}-\sin\Big{\\{}(n\pi+\hbox{$\frac{1}{2}$}\theta_{H})\frac{\displaystyle
z-z_{L}}{\displaystyle
z_{L}-1}+\hbox{$\frac{1}{2}$}\theta(z)\Big{\\}}\cr\cos\Big{\\{}(n\pi+\hbox{$\frac{1}{2}$}\theta_{H})\frac{\displaystyle
z-z_{L}}{\displaystyle
z_{L}-1}+\hbox{$\frac{1}{2}$}\theta(z)\Big{\\}}\end{pmatrix},$ (B.24)

and for type 2A by

$\displaystyle\underline{\hbox{type 2A}}:$ (B.25)
$\displaystyle\lambda_{n}=\frac{1}{z_{L}-1}\big{|}(n+\hbox{$\frac{1}{2}$})\pi+\hbox{$\frac{1}{2}$}\theta_{H}\big{|}\quad(-\infty<n<\infty),$
(B.26) $\displaystyle\begin{pmatrix}f_{Rn}(y)\cr\kern 5.0pt\cr
g_{Rn}(y)\end{pmatrix}=\frac{1}{\sqrt{z_{L}-1}}\begin{pmatrix}-\sin\Big{\\{}(n\pi+\hbox{$\frac{1}{2}$}\pi+\hbox{$\frac{1}{2}$}\theta_{H})\frac{\displaystyle
z-z_{L}}{\displaystyle
z_{L}-1}+\hbox{$\frac{1}{2}$}\theta(z)\Big{\\}}\cr\cos\Big{\\{}(n\pi+\hbox{$\frac{1}{2}$}\pi+\hbox{$\frac{1}{2}$}\theta_{H})\frac{\displaystyle
z-z_{L}}{\displaystyle
z_{L}-1}+\hbox{$\frac{1}{2}$}\theta(z)\Big{\\}}\end{pmatrix},$ (B.27)
$\displaystyle\begin{pmatrix}f_{Ln}(y)\cr\kern 5.0pt\cr
g_{Ln}(y)\end{pmatrix}=\frac{1}{\sqrt{z_{L}-1}}\begin{pmatrix}\cos\Big{\\{}(n\pi+\hbox{$\frac{1}{2}$}\pi+\hbox{$\frac{1}{2}$}\theta_{H})\frac{\displaystyle
z-z_{L}}{\displaystyle
z_{L}-1}+\hbox{$\frac{1}{2}$}\theta(z)\Big{\\}}\cr\sin\Big{\\{}(n\pi+\hbox{$\frac{1}{2}$}\pi+\hbox{$\frac{1}{2}$}\theta_{H})\frac{\displaystyle
z-z_{L}}{\displaystyle
z_{L}-1}+\hbox{$\frac{1}{2}$}\theta(z)\Big{\\}}\end{pmatrix}.$ (B.28)

Note that the expressions (B.24) and (B.28) reduce, up to normalization
factors, to the expressions (2.19) and (2.21) in the flat spacetime limit,
respectively. For other regions in $y$, the wave functions are defined by
(2.50).

References

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