OU-HET 599 February 2008

Fiber Bundles and Matrix Models

Takaaki Ishii*** e-mail address<EMAIL_ADDRESS>Goro
Ishiki††† e-mail address<EMAIL_ADDRESS>Shinji
Shimasaki‡‡‡ e-mail address<EMAIL_ADDRESS>and Asato
Tsuchiya§§§ e-mail address<EMAIL_ADDRESS>address
after April 2008 : Department of Physics, Shizuoka University, 836 Ohya,
Suruga-ku, Shizuoka 422-8529, Japan

Department of Physics, Graduate School of Science
Osaka University, Toyonaka, Osaka 560-0043, Japan

We investigate relationship between a gauge theory on a principal bundle and
that on its base space. In the case where the principal bundle is itself a
group manifold, we also study relations of those gauge theories with a matrix
model obtained by dimensionally reducing them to zero dimensions. First, we
develop the dimensional reduction of Yang-Mills (YM) on the total space to YM-
higgs on the base space for a general principal bundle. Second, we show a
relationship that YM on an $SU(2)$ bundle is equivalent to the theory around a
certain background of YM-higgs on its base space. This is an extension of our
previous work [29], in which the same relationship concerning a $U(1)$ bundle
is shown. We apply these results to the case of $SU(n+1)$ as the total space.
By dimensionally reducing YM on $SU(n+1)$, we obtain YM-higgs on
$SU(n+1)/SU(n)\simeq S^{2n+1}$ and on $SU(n+1)/(SU(n)\times U(1))\simeq
CP^{n}$ and a matrix model. We show that the theory around each monopole
vacuum of YM-higgs on $CP^{n}$ is equivalent to the theory around a certain
vacuum of the matrix model in the commutative limit. By combining this with
the relationship concerning a $U(1)$ bundle, we realize YM-higgs on
$SU(n+1)/SU(n)\simeq S^{2n+1}$ in the matrix model. We see that the
relationship concerning a $U(1)$ bundle can be interpreted as Buscher’s
T-duality.

###### Contents

  1. 1 Introduction and conclusion
  2. 2 Typical relationships
  3. 3 Dimensional reduction on a principal bundle
  4. 4 Extension of the matrix T-duality
    1. 4.1 Nontrivial vacua and transformation between patches
    2. 4.2 $G=U(1),SU(2)$
    3. 4.3 Example: $S^{7}\rightarrow S^{4}$
  5. 5 Gauge theories on $SU(n+1)(/H)$ and matrix model
    1. 5.1 Dimensional reduction of YM theory on a group manifold
    2. 5.2 Dimensional reduction of YM theory on $SU(n+1)$
    3. 5.3 Relations among gauge theories on $SU(n+1)/H$
  6. 6 Interpretation as Buscher’s T-duality
  7. A Spherical harmonics
  8. B Derivation of (4.22)
  9. C Group manifold and coset space
  10. D Metrics of $SU(3)$, $S^{5}$ and $CP^{2}$
  11. E Fuzzy $CP^{n}$

##  1 Introduction and conclusion

Emergence of space-time is one of the key concepts in matrix models as
nonperturbative definition of superstring [1, 2, 3]. This phenomenon was first
observed in the relationship between a gauge theory and a matrix model. This
is the so-called large N reduction [4]. It states that a large $N$ planar
gauge theory is equivalent to the matrix model that is its dimensional
reduction to zero dimensions unless the $U(1)^{D}$ symmetry is broken, where
$D$ denotes the dimensionality of the original gauge theory. However, the
$U(1)^{D}$ symmetry is in general spontaneously broken for $D>2$. There are
two improved versions of the large $N$ reduced model that preserve the
$U(1)^{D}$ symmetry. One is the quenched reduced model [5, 6, 7, 8]. The other
is the twisted reduced model [9], which was later rediscovered in the context
of the noncommutative field theories [10]. The T-duality for D-brane effective
theories [11], which we call the matrix T-duality in this paper, share the
same idea with the large N reduced model. The statement of the matrix
T-duality is that $U(N)$ Yang-Mills (YM) on $R^{p}\times S^{1}$ is equivalent
to $U(N\times\infty)$ YM-higgs on $R^{p}$ which is a dimensional reduction of
$U(N\times\infty)$ YM on $R^{p}\times S^{1}$ if a periodicity (orbifolding)
condition is imposed. Also, deconstruction [12] and supersymmetric lattice
gauge theories inspired by it [13] are analogs of the matrix T-duality. The
above developments are all concerning gauge theories on flat space-time. It is
important to understand how gauge theories on curved space-time are realized
in matrix models or gauge theories in lower dimensions, because it would lead
us to gain some insights into how curved space-time is realized in matrix
models as nonperturbative definition of superstring. Note that an interesting
approach to the description of curved spacetime by matrices was proposed in
[14].

In [15], Takayama and three of the present authors found relationships among
the $SU(2|4)$ symmetric theories. Here the $SU(2|4)$ symmetric theories
include ${\cal N}=4$ super Yang Mills (SYM) on $R\times S^{3}/Z_{k}$, 2+1 SYM
on $R\times S^{2}$ [16] and the plane wave matrix model (PWMM) [17]. These
theories are related by dimensional reductions and possess common features:
mass gap, discrete spectrum and many discrete vacua. From the gravity duals of
those vacua proposed in [18], the following relations between these theories
are suggested: the theory around each vacuum of 2+1 SYM on $R\times S^{2}$ is
equivalent to the theory around a certain vacuum of PWMM, and the theory
around each vacuum of ${\cal N}=4$ SYM on $R\times S^{3}/Z_{k}$ is equivalent
to the theory around a certain vacuum of 2+1 SYM on $R\times S^{2}$ with the
periodicity imposed. Combining these two equivalences, we can say that the
theory around each vacuum of ${\cal N}=4$ SYM on $R\times S^{3}/Z_{k}$ is
realized in PWMM. In [15], these equivalences were shown directly on the gauge
theory side. The results in [15] not only serve as a nontrivial check of the
gauge/gravity correspondence for the $SU(2|4)$ theories, but they are also
interesting in the following aspects. Much work has been already done on the
realization of the gauge theories on the fuzzy sphere [19, 20, 21, 22] by
matrix models [23] and on the monopoles on the fuzzy sphere [24, 25, 26, 28,
27]. Note that the realization of the fuzzy sphere by matrix models can be
viewed as an extension of the twisted reduced model to curved space. Here in
the relation between 2+1 SYM on $R\times S^{2}$ and PWMM, it was manifestly
shown that the continuum limit of concentric fuzzy spheres correspond to multi
monopoles. The relation between ${\cal N}=4$ SYM on $R\times S^{3}/Z_{k}$ and
2+1 SYM on $R\times S^{2}$ can be regarded as an extension of the matrix
T-duality to that on a nontrivial $U(1)$ bundle, $S^{3}/Z_{k}$, whose base
space is $S^{2}$. Furthermore, in [29], we generalized the matrix T-duality to
that on an arbitrary $U(1)$ bundle. As an application of these results, in
[30], Ohta and the present authors investigated relationships among Chern-
Simons theory on a $U(1)$ bundle over a Riemann surface, BF theory with a mass
term on the Riemann surface, which is equivalent to two-dimensional Yang-Mills
on the Riemann surface, and a matrix model. It was discussed that the former
two (topological) field theories associated with topological strings can be
realized in the matrix model. The results in [15] also suggests an interesting
possibility of a nonperturbative formulation of ${\cal N}=4$ SYM on $R\times
S^{3}$ by PWMM, which would lead to a nonperturbative test of the AdS/CFT
correspondence.

This paper is aimed at further investigation of the above developments
concerning the large $N$ reduction and the matrix T-duality on curved space.
First, we develop a dimensional reduction of YM on the total space to YM-higgs
on the base space for a general principal bundle. This also enables us to
dimensionally reduce YM on a group manifold to a matrix model. Second, as an
extension of the work [29], in the case in which the fiber is $SU(2)$, we show
that YM on the total space is equivalent to a certain vacuum111Throughout this
paper, we consider gauge theories on manifolds with the Euclidean signature.
Here ‘vacuum’ represents a configuration that gives the global minimum of the
classical action. of YM-higgs on the base space with the periodicity imposed.
This enables us to realize YM on an $SU(2)^{k}\times U(1)^{l}$ bundle in YM-
higgs on its base space. We apply the above results to the case of $SU(n+1)$
as the total space. $SU(n+1)$ is viewed as $SU(n)$ bundle over
$SU(n+1)/SU(n)\simeq S^{2n+1}$ or $SU(n)\times U(1)$ bundle over
$SU(n+1)/(SU(n)\times U(1))\simeq CP^{n}$, and $SU(n+1)/SU(n)\simeq S^{2n+1}$
is viewed as $U(1)$ bundle over $CP^{n}$. By the dimensional reduction, we
obtain YM-higgs on $S^{2n+1}$ and $CP^{n}$ and a matrix model. We find the
commutative (continuum) limit of gauge theory on fuzzy $CP^{n}$ [28, 31, 32,
33, 34, 35] realized in the matrix model coincides with YM-higgs on $CP^{n}$.
Namely, we show that the theory around each monopole vacuum of YM-higgs on
$CP^{n}$ is equivalent to the theory around a certain vacuum of the matrix
model. By combing this with the extended matrix T-duality, we realize YM-higgs
on $SU(n+1)/SU(n)\simeq S^{2n+1}$ in the matrix model. We also show that the
extended matrix T-duality of the $U(1)$ case developed in [29] can be
interpreted as Buscher’s T-duality [36].

In the remainder of this section, we describe the organization of the present
paper, providing our results in detail, and finally describe some outlook.
From the same reasoning as the case of the $SU(2|4)$ symmetric theories, the
following relationships among YM on $S^{3}$, YM-higgs on $S^{2}$ and a matrix
model hold. These theories are related to each other by dimensional
reductions. The theory around each vacuum of YM-higgs on $S^{2}$ is equivalent
to the theory around a certain vacuum of the matrix model. YM on $S^{3}$ is
equivalent to the theory around a certain vacuum of YM-higgs on $S^{2}$ with
the periodicity imposed. Eventually, YM on $S^{3}$ is realized in the matrix
model. It can be said that our results in this paper are extension of these
relationships. In section 2, we show these relationships in order to
illustrate our basic ideas.

In section 3, we develop a dimensional reduction on a general principal fiber
bundle. We start with YM on the total space, dimensionally reduce the fiber
directions and obtain a YM-higgs on the base space.

In section 4, we examine a relationship between YM on the total space and YM-
higgs on the base space obtained in section 3. In section 4.1, we first
examine the transformations of the fields from a local patch to another local
patch in YM-higgs on the base space. In section 4.2, using the observation in
section 4.1, we show that when the fiber is $U(1)$ or $SU(2)$, YM on the total
space is equivalent to the theory around a certain vacuum of YM-higgs on the
base space with the periodicity imposed. This vacuum is given by multimonopole
configuration on the base space. We already found the $U(1)$ case of this
equivalence in [29]. In the $SU(2)$ case, we also use the result in section 2
that YM on $S^{3}$ is realized in the matrix model. As a generalization, we
realize YM on $SU(2)^{k}\times U(1)^{l}$ bundle in YM-higgs on its base space.
In section 4.3, as an example, we consider $S^{7}$ which is an $SU(2)$ bundle
over $S^{4}$. In Fig. 1, we summarize our results in sections 3 and 4.

Figure 1: Matrix T-duality for $G=U(1)$, $SU(2)$

In section 5, we examine a series of $SU(n+1)$ symmetric theories. Fig. 2
summarizes our findings in section 5 and their relation to other sections. The
case of $n=1$ is nothing but the example discussed in section 2. In this case,
YM on $SU(2)$ is the same as YM-higgs on $S^{3}$ because $SU(2)\simeq S^{3}$.
In section 5.1, as a special case of section 3, we consider a dimensional
reduction of YM on a group manifold $\tilde{G}$ to a coset space $\tilde{G}/H$
where $H$ is a subgroup of $\tilde{G}$. Namely, we view $\tilde{G}$ as an $H$
bundle over $\tilde{G}/H$. By dimensionally reducing the Killing vectors on
$\tilde{G}$ to those on $\tilde{G}/H$, we obtain a theory on $\tilde{G}/H$
expressed in terms of the Killing vectors. Then, we show that this theory on
$\tilde{G}/H$ is rewritten into YM-higgs on $\tilde{G}/H$ obtained in section
3. In section 5.2, we apply the results in section 5.1 to the case of
$\tilde{G}=SU(n+1)$ and obtain a series of theories in Fig. 2 which possess
$SU(n+1)$ symmetry. If we take $SU(n)$ as $H$, we obtain YM-higgs on
$S^{2n+1}$. Note that the isometry of this $S^{2n+1}$ is not $SO(2n+2)$ but
$SU(n+1)$. For $n\geq 2$, it is different from the ordinary $S^{2n+1}$ but
homeomorphic to the ordinary one, and is called a squashed $S^{2n+1}$. If we
take $SU(n)\times U(1)$ as $H$, we obtain YM-higgs on $CP^{n}$. Finally if we
take $SU(n+1)$ itself as $H$, we obtain a matrix model whose action is shown
in Fig. 2, where $f_{ABC}$ is the structure constant of the $SU(n+1)$ Lie
algebra. As indicated in Fig. 2, these dimensional reductions can also be
performed step by step: we obtain YM-higgs on $CP^{n}$ from YM-higgs on
$S^{2n+1}$ and the matrix model from YM-higgs on $CP^{n}$. In the case of
$n=2$, as an application of the result in section 4.2, we see that YM on
$SU(3)$ is equivalent to the theory around a vacuum of YM-higgs on $S^{5}$
with the periodicity imposed ((i) in Fig. 2). Since $S^{2n+1}$ can be viewed
as a $U(1)$ bundle over $CP^{n}$, in section 5.3, we show as an application of
the results in section 4.2 that the theory around each vacuum of YM-higgs on
$S^{2n+1}$ is equivalent to the theory around a vacuum of YM-higgs on $CP^{n}$
with the periodicity imposed ((ii) in Fig. 2). In section 5.3, we show that
the theory around each abelian monopole vacuum of YM-higgs on $CP^{n}$ is
equivalent to a certain vacuum of the matrix model ((iii) in Fig. 2).
Combining these results, we also show that the theory around the trivial
vacuum of YM-higgs on $S^{2n+1}$ is realized in the matrix model ((iv) in Fig.
2). YM on $SU(3)$ is realized in YM-higgs on $CP^{2}$ ((v) in Fig. 2).
Finally, we make a comment: it follows from the result in section 4 that YM on
$SU(n+1)$ is realized in YM-higgs on $SU(n+1)/(SU(2)^{k}\times U(1)^{l})$.

Figure 2: A series of theories studied in section 5.

In section 6, we discuss how the extended matrix T-dulaity found in [29] and
reviewed in section 4.2 is interpreted as Buscher’s T-duality. In appendices
A-D, we describe some details.

It is an open problem whether YM on $SU(n+1)$ with $n\geq 2$ is realized in
the matrix model. Presumably, we need to construct noncommutative counterparts
of non-Abelian monopoles of YM-higgs on $S^{2n+1}$ or $CP^{n}$ in the matrix
model. Realization of YM on $SU(n+1)$ in the matrix model should enable us to
extend the matrix T-duality to the case of $G=SU(n+1)$. Of course, the matrix
T-duality for a general $G$ should still be investigated. It is important to
see whether the matrix T-duality in the $SU(2)$ case is associated with the
nonabelian T-duality discussed within the nonlinear sigma models [37]. It is
also relevant to identify the commutative limit of the matrix model consisting
of the square of the commutators and the generalized Myers term with the
$SU(n+1)$ structure constant which has been examined in [33, 38] and find its
higher-dimensional origin. Analysis in this paper is classical. Whether the
relationships among the gauge theories we found hold quantum mechanically is a
nontrivial and important problem. It should be noted that in the quantum
correspondence no orbifolding condition is needed in the matrix T-duality as
far as the planar limit is concerned. This is nothing but the large $N$
reduction and enables us to make the size of matrices become finite and play a
role of the ultraviolet cutoff. In particular, we expect to give a
nonperturbative definition of ${\cal N}=4$ SYM on $R\times S^{3}$ in the
planar limit in terms of PWMM [39].

##  2 Typical relationships

In this section, to illustrate our ideas, we describe relationships among YM
on $S^{3}$, YM-higgs on $S^{2}$ and a matrix model. These relationships are
essentially the same as those among the $SU(2|4)$ symmetric theories found in
[15].

We consider $S^{3}$ with radius $2/\mu$ and regard it as the $U(1)$ ($S^{1}$)
Hopf bundle on $S^{2}$ with radius $1/\mu$. $S^{3}$ with radius $2/\mu$ is
defined by

$\displaystyle\\{(w_{1},w_{2})\in
C^{2}\>|\>|w_{1}|^{2}+|w_{2}|^{2}=4/\mu^{2}\\}.$ (2.1)

The Hopf map $\pi:\>S^{3}\rightarrow CP^{1}\;(S^{2})$ is defined by

$\displaystyle(w_{1},w_{2})\rightarrow[(w_{1},w_{2})]\equiv\\{\lambda(w_{1},w_{2})|\lambda\in
C\backslash\\{0\\}\\}.$ (2.2)

Two patches are introduced on $CP^{1}$: the patch I $(w_{1}\neq 0)$ and the
patch II $(w_{2}\neq 0)$. On the patch I the local trivialization is given by

$\displaystyle(w_{1},w_{2})\rightarrow\left(\frac{w_{2}}{w_{1}},\frac{w_{1}}{|w_{1}|}\right)\in\mbox{(patch
I)}\times U(1),$ (2.3)

while on the patch II the local trivialization is given by

$\displaystyle(w_{1},w_{2})\rightarrow\left(\frac{w_{1}}{w_{2}},\frac{w_{2}}{|w_{2}|}\right)\in\mbox{(patch
II)}\times U(1).$ (2.4)

The equation (2.1) is solved as

$\displaystyle
w_{1}=\frac{2}{\mu}\cos\frac{\theta}{2}\>e^{i\sigma_{1}},\;\;\;w_{2}=\frac{2}{\mu}\sin\frac{\theta}{2}\>e^{i\sigma_{2}},$
(2.5)

where $0\leq\theta\leq\pi$ and $0\leq\sigma_{1},\>\sigma_{2}<2\pi$. We put

$\displaystyle\varphi=\sigma_{1}-\sigma_{2},\;\;\;\psi=\sigma_{1}+\sigma_{2},$
(2.6)

and can change the ranges of $\varphi$ and $\psi$ to $0\leq\varphi<2\pi$ and
$0\leq\psi<4\pi$. The periodicity is expressed as

$\displaystyle(\theta,\varphi,\psi)\sim(\theta,\varphi+2\pi,\psi+2\pi)\sim(\theta,\varphi,\psi+4\pi).$
(2.7)

From the local trivializations (2.3) and (2.4), one can see that $\theta$ and
$\varphi$ are regarded as the angular coordinates of the base space $S^{2}$
through the stereographic projection. The patch I corresponds to
$0\leq\theta<\pi$, while the patch II corresponds to $0<\theta\leq\pi$. The
metric of $S^{3}$ is given as follows:

$\displaystyle ds_{S^{3}}^{2}$ $\displaystyle=|dw_{1}|^{2}+|dw_{2}|^{2}$
$\displaystyle=\frac{1}{\mu^{2}}(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}+(d\psi+\cos\theta d\varphi)^{2}).$ (2.8)

In the remainder of this section, the upper sign is taken in the patch I and
the lower sign in the patch II. From (2.3), (2.4) and (2.8), one sees that the
fiber $S^{1}$ is parameterized by $y=\frac{1}{\mu}(\psi\pm\varphi)$ and its
radius is given by $2/\mu$. The connection 1-form is given by

$\displaystyle\omega=\frac{\mu}{2}\left(dy+\frac{1}{\mu}(\cos\theta\mp
1)d\varphi\right).$ (2.9)

The connection 1-form provides the vertical-horizontal decomposition by
determining the inverse of the dreibein $E_{A}^{M}$ through
$\omega(E_{\alpha}^{M})=0$, $E_{3}^{\mu}=0$ and $E_{3}^{y}=1$, where
$A=1,2,3$, $\alpha=1,2$, $M=\theta,\varphi,y$ and $\mu=\theta,\varphi$. The
inverse of the dreibein is determined as

$\displaystyle
E_{1}^{\theta}=\mu,\;\;\;E_{2}^{\varphi}=\frac{\mu}{\sin\theta},$
$\displaystyle E_{2}^{y}=\mu\frac{\cos\theta\mp
1}{\sin\theta},\;\;\;E_{3}^{y}=1,$ $\displaystyle\mbox{others}=0.$ (2.10)

The dreibein are given by

$\displaystyle
E^{1}_{\theta}=e^{1}_{\theta}=\frac{1}{\mu},\;\;\;E^{2}_{\varphi}=e^{2}_{\varphi}=\frac{1}{\mu}\sin\theta,$
$\displaystyle E^{3}_{\varphi}=\frac{1}{\mu}(\cos\theta\mp
1),\;\;\;E^{3}_{y}=1,$ $\displaystyle\mbox{others}=0,$ (2.11)

where $e_{\mu}^{\alpha}$ are the zweibein of $S^{2}$.

We start with YM on $S^{3}$

$\displaystyle
S_{S^{3}}=\frac{1}{4g_{S^{3}}^{2}}\int\frac{d\Omega_{3}}{(\mu/2)^{3}}\mbox{tr}(F_{AB}F_{AB}).$
(2.12)

The vertical-horizontal decomposition tells us how to relate the gauge field
on $S^{3}$ to the gauge field and the higgs field on $S^{2}$:

$\displaystyle A_{\alpha}=a_{\alpha},$ $\displaystyle A_{3}=\phi.$ (2.13)

Or equivalently

$\displaystyle A_{\theta}=a_{\theta},$ $\displaystyle
A_{\varphi}=a_{\varphi}+\frac{1}{\mu}(\cos\theta\mp 1)\phi,$ $\displaystyle
A_{y}=\phi.$ (2.14)

In (2.13) and (2.14), in order to make a dimensional reduction, we assume that
the both sides are independent of $y$. Then, substituting (2.13) into (2.12)
yields a YM-higgs on $S^{2}$,

$\displaystyle
S_{S^{2}}=\frac{1}{g_{S^{2}}^{2}}\int\frac{d\Omega_{2}}{\mu^{2}}{\rm
tr}\left(\frac{1}{2}(f_{12}+\mu\phi)^{2}+\frac{1}{2}(D_{\alpha}\phi)^{2}\right),$
(2.15)

where $g_{S^{2}}^{2}=\frac{\mu}{4\pi}g_{S^{3}}^{2}$. It is convenient for us
to rewrite (2.15) using the three-dimensional flat space notation. We define a
three-dimensional vector field in terms of $a_{\alpha}$ and $\phi$ [16]:

$\displaystyle\vec{X}=\phi\vec{e}_{r}+a_{1}\vec{e}_{\varphi}-a_{2}\vec{e}_{\theta},$
(2.16)

where $\vec{e}_{r}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta)$
and
$\vec{e}_{\theta}=\frac{\partial\vec{e}_{r}}{\partial\theta},\;\;\vec{e}_{\varphi}=\frac{1}{\sin\theta}\frac{\partial\vec{e}_{r}}{\partial\varphi}$.
We also introduce the angular momentum operator in three-dimensional flat
space,

$\displaystyle\vec{L}^{(0)}=-i\vec{e}_{\phi}\partial_{\theta}+i\frac{1}{\sin\theta}\vec{e}_{\theta}\partial_{\phi}.$
(2.17)

Then, (2.15) is rewritten as

$\displaystyle
S_{S^{2}}=\frac{1}{g_{S^{2}}^{2}}\int\frac{d\Omega_{2}}{\mu^{2}}\frac{1}{2}\mbox{tr}\left(\mu
X_{A}+i\mu\epsilon_{ABC}L^{(0)}_{B}X_{C}+\frac{i}{2}\epsilon_{ABC}[X_{B},X_{C}]\right)^{2}.$
(2.18)

By dropping all the derivatives, we dimensionally reduce (2.18) to zero
dimensions to obtain a matrix model:

$\displaystyle S_{mm}=\frac{1}{g_{mm}^{2}}\frac{1}{2}\mbox{tr}\left(\mu
X_{A}+\frac{i}{2}\epsilon_{ABC}[X_{B},X_{C}]\right)^{2},$ (2.19)

where $g_{mm}^{2}=\frac{\mu^{2}}{4\pi}g_{S^{2}}^{2}$. The cross term in the
above action is nothing but the Myers term [40]. It was first found in [41]
that (2.19) is obtained from (2.12) through the dimensional reduction.

We can obtain (2.18) and (2.19) directly from (2.12) in the following way. We
parameterize the gauge field on $S^{3}$ as $A=X_{A}E^{A}$ [18], where $E^{A}$
is the right invariant 1-form defined in appendix A. Then, by using the
Maurer-Cartan equation (A.4), we evaluate the curvature 2-form as

$\displaystyle F$ $\displaystyle=dA+iA\wedge A$
$\displaystyle=\frac{1}{2}\epsilon_{ABC}\left(i\mu\epsilon_{CDE}{\cal
L}_{D}X_{E}+\mu X_{C}+i\epsilon_{CDE}X_{D}X_{E}\right)E^{A}\wedge E^{B},$
(2.20)

where ${\cal L}_{A}$ are the Killing vector dual to $E^{A}$, the explicit form
of which is given in (A.7). Noting that ${\cal L}_{A}$ reduces to
$L^{(0)}_{A}$ when $X_{A}$ is independent of $y$, one can easily see that
(2.12) is dimensionally reduced to (2.18). Moreover, if we assume that $X_{A}$
is independent of all coordinates, we obtain the matrix model (2.19) directly
from (2.12).

The theories (2.15) and (2.19) possess many nontrivial vacua. Let us see how
those vacua are described. First, the vacuum configurations of (2.15) with the
gauge group $U(M)$ are determined by

$\displaystyle f_{12}+\mu\phi=0,$ $\displaystyle D_{\alpha}\phi=0.$ (2.21)

In the gauge in which $\phi$ is diagonal, (2.21) is solved as

$\displaystyle\hat{a}_{1}=0,$ $\displaystyle\hat{a}_{2}=\frac{\cos\theta\mp
1}{\sin\theta}\hat{\phi},$
$\displaystyle\hat{\phi}=\frac{\mu}{2}\mbox{diag}(\cdots,\underbrace{n_{s-1},\cdots,n_{s-1}}_{N_{s-1}},\underbrace{n_{s},\cdots,n_{s}}_{N_{s}},\underbrace{n_{s+1},\cdots,n_{s+1}}_{N_{s+1}},\cdots),$
(2.22)

where the gauge field takes the configurations of Dirac’s monopoles, so that
$n_{s}$ must be integers due to Dirac’s quantization condition. Note also that
$\sum_{s}N_{s}=M$. Thus the vacua of YM-higgs on $S^{2}$ are classified by the
monopole charges $n_{s}/2$ and their degeneracies $N_{s}$. Next, the vacuum
configurations of (2.19) with the gauge group $U(\hat{M})$ are determined
by222There is a solution to the equations of motion of the matrix model
(2.19), $X_{A}=\frac{\mu}{2}L_{A}$, which does not satisfy (2.23). It turns
out that the theory around this solution is unstable.

$\displaystyle[X_{A},X_{B}]=i\mu\epsilon_{ABC}X_{C}.$ (2.23)

(2.23) is solved as

$\displaystyle\hat{X}_{A}=\mu L_{A},$ (2.24)

where $L_{A}$ are the representation matrices of the $SU(2)$ generators which
are in general reducible, and are decomposed into irreducible representations:

$\displaystyle
L_{A}=\begin{pmatrix}\rotatebox[origin={tl}]{-35.0}{$\cdots\;\;\;\overbrace{\rotatebox[origin={c}]{35.0}{$L_{A}^{[j_{s-1}]}$}\;\cdots\;\rotatebox[origin={c}]{35.0}{$L_{A}^{[j_{s-1}]}$}}^{\rotatebox{35.0}{$N_{s-1}$}}\;\;\;\overbrace{\rotatebox[origin={c}]{35.0}{$L_{A}^{[j_{s}]}$}\;\cdots\;\rotatebox[origin={c}]{35.0}{$L_{A}^{[j_{s}]}$}}^{\rotatebox{35.0}{$N_{s}$}}\;\;\;\overbrace{\rotatebox[origin={c}]{35.0}{$L_{A}^{[j_{s+1}]}$}\;\cdots\;\rotatebox[origin={c}]{35.0}{$L_{A}^{[j_{s+1}]}$}}^{\rotatebox{35.0}{$N_{s+1}$}}\;\;\;\cdots$}\end{pmatrix},$
(2.25)

where $L_{A}^{[j]}$ are the spin $j$ representation matrices of $SU(2)$ and
$\sum_{s}N_{s}(2j_{s}+1)=\hat{M}$. The vacua of the matrix model are
classified by the $SU(2)$ representations $[j_{s}]$ and their degeneracies
$N_{s}$. (2.25) represents concentric fuzzy spheres with different radii.

In the remainder of this section, we show relationships among the theories
(2.12), (2.15) and (2.19). First, we show that the theory around the vacuum
(2.22) of YM-higgs on $S^{2}$ is equivalent to the theory around the vacuum
(2.24) of the matrix model if one puts $2j_{s}+1=N_{0}+n_{s}$ and takes the
$N_{0}\rightarrow\infty$ limit with $g_{mm}^{2}/N_{0}$ fixed to
$g_{S^{2}}^{2}\mu^{2}/4\pi$. We decompose the fields into the background
corresponding (2.22) and the fluctuation as
$X_{A}^{(s,t)}\rightarrow\hat{X}_{A}^{(s,t)}+X_{A}^{(s,t)}$, where $(s,t)$
label the (off-diagonal) blocks. Then, (2.18) is expanded around (2.22) as

$\displaystyle S_{S^{2}}$
$\displaystyle=\frac{1}{g_{S^{2}}^{2}}\int\frac{d\Omega_{2}}{\mu^{2}}\frac{1}{2}\sum_{s,t}\mbox{tr}\left[\left(\mu
X_{A}^{(s,t)}+i\mu\epsilon_{ABC}L^{(q_{st})}_{B}X_{C}^{(s,t)}+\frac{i}{2}\epsilon_{ABC}[X_{B},X_{C}]^{(s,t)}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\qquad\times\left(\mu
X_{A}^{(t,s)}+i\mu\epsilon_{ADE}L^{(q_{ts})}_{D}X_{E}^{(t,s)}+\frac{i}{2}\epsilon_{ADE}[X_{D},X_{E}]^{(t,s)}\right)\right],$
(2.26)

where $q_{st}=(n_{s}-n_{t})/2$. $\vec{L}^{(q)}$ is the angular momentum
operator in the presence of a monopole with the magnetic charge $q$ at the
origin, which takes the form [42]

$\displaystyle\vec{L}^{(q)}=\vec{L}^{(0)}-q\frac{\cos\theta\mp
1}{\sin\theta}\vec{e}_{\theta}-q\vec{e}_{r}.$ (2.27)

We make a harmonic expansion of (2.26) by expanding the fluctuation in terms
of the monopole vector spherical harmonics $\tilde{Y}_{JmqA}^{\rho}$ defined
in appendix A as

$\displaystyle X_{A}^{(s,t)}=\sum_{\rho=0,\pm
1}\sum_{\tilde{Q}\geq|q_{st}|}\sum_{m=-Q}^{Q}X_{Jm\rho}^{(s,t)}\tilde{Y}_{JmqA}^{\rho},$
(2.28)

where $Q=J+\frac{(1+\rho)\rho}{2}$ and $\tilde{Q}=J-\frac{(1-\rho)\rho}{2}$.
Substituting (2.28) into (2.26) yields

$\displaystyle S_{S^{2}}$
$\displaystyle=\frac{4\pi}{g_{S^{2}}^{2}\mu^{2}}\mbox{tr}\left[\frac{\mu^{2}}{2}\sum_{s,t}\rho^{2}(J+1)^{2}X_{Jm\rho}^{(s,t)\dagger}X_{Jm\rho}^{(s,t)}\right.$
$\displaystyle+i\mu\sum_{s,t,u}\rho_{1}(J_{1}+1){\cal
E}_{J_{1}m_{1}q_{st}\rho_{1}\;J_{2}m_{2}q_{tu}\rho_{2}\;J_{3}m_{3}q_{us}\rho_{3}}X_{J_{1}m_{1}\rho_{1}}^{(s,t)}X_{J_{2}m_{2}\rho_{2}}^{(t,u)}X_{J_{3}m_{3}\rho_{3}}^{(u,s)}$
$\displaystyle-\frac{1}{2}\sum_{s,t,u,v}(-1)^{m-q_{su}+1}{\cal
E}_{J-mq_{us}\rho\;J_{1}m_{1}q_{st}\rho_{1}\;J_{2}m_{2}q_{tu}\rho_{2}}{\cal
E}_{Jmq_{su}\rho\;J_{3}m_{3}q_{uv}\rho_{3}\;J_{4}m_{4}q_{vs}\rho_{4}}$
$\displaystyle\left.\qquad\qquad\qquad\times
X_{J_{1}m_{1}\rho_{1}}^{(s,t)}X_{J_{2}m_{2}\rho_{2}}^{(t,u)}X_{J_{3}m_{3}\rho_{3}}^{(u,v)}X_{J_{4}m_{4}\rho_{4}}^{(v,s)}\right],$
(2.29)

where ${\cal
E}_{J_{1}m_{1}q_{st}\rho_{1}\;J_{2}m_{2}q_{tu}\rho_{2}\;J_{3}m_{3}q_{us}\rho_{3}}$
is defined in (A.42) and we have used (A.39). Similarly we decompose the
matrices into the background given by (2.24) and the fluctuation as
$X_{i}\rightarrow\hat{X}_{i}+X_{i}$ and obtain the theory around (2.24):

$\displaystyle S_{mm}$
$\displaystyle=\frac{1}{g_{mm}^{2}}\frac{1}{2}\sum_{s,t}\mbox{tr}\left[\left(\mu
X_{A}^{(s,t)}+i\mu\epsilon_{ABC}L_{B}\circ
X_{C}^{(s,t)}+\frac{i}{2}\epsilon_{ABC}[X_{B},X_{C}]^{(s,t)}\right)\right.$
$\displaystyle\left.\qquad\qquad\qquad\qquad\times\left(\mu
X_{A}^{(t,s)}+i\mu\epsilon_{ADE}L_{D}\circ
X_{E}^{(t,s)}+\frac{i}{2}\epsilon_{ADE}[X_{D},X_{E}]^{(t,s)}\right)\right],$
(2.30)

where $L_{A}\circ$ is defined by

$\displaystyle L_{A}\circ
X_{B}^{(s,t)}=L_{A}^{[j_{s}]}X_{B}^{(s,t)}-X_{B}^{(s,t)}L_{A}^{[j_{t}]}.$
(2.31)

We make a harmonic expansion for (2.30) by expanding the fluctuation in terms
of the fuzzy vector spherical harmonics $\hat{Y}_{Jm(j_{s}j_{t})A}^{\rho}$
defined in appendix A as

$\displaystyle X_{A}^{(s,t)}=\sum_{\rho=0,\pm
1}\sum_{\tilde{Q}\geq|j_{s}-j_{t}|}^{j_{s}+j_{t}}\sum_{m=-Q}^{Q}X_{Jm\rho}^{(s,t)}\otimes\hat{Y}_{Jm(j_{s}j_{t})A}^{\rho}.$
(2.32)

Since $j_{s}+j_{t}=N_{0}+\frac{n_{s}+n_{t}}{2}-1$, $N_{0}$ plays a role of the
ultraviolet cutoff. Note also that $j_{s}-j_{t}=(n_{s}-n_{t})/2=q_{st}$.
Substituting (2.32) into (2.30) yields

$\displaystyle S_{mm}$
$\displaystyle=\frac{N_{0}}{g_{mm}^{2}}\mbox{tr}\left[\frac{\mu^{2}}{2}\sum_{s,t}\rho^{2}(J+1)^{2}X_{Jm\rho}^{(s,t)\dagger}X_{Jm\rho}^{(s,t)}\right.$
$\displaystyle+i\mu\sum_{s,t,u}\rho_{1}(J_{1}+1)\hat{{\cal
E}}_{J_{1}m_{1}(j_{s}j_{t})\rho_{1}\;J_{2}m_{2}(j_{t}j_{u})\rho_{2}\;J_{3}m_{3}(j_{u}j_{s})\rho_{3}}X_{J_{1}m_{1}\rho_{1}}^{(s,t)}X_{J_{2}m_{2}\rho_{2}}^{(t,u)}X_{J_{3}m_{3}\rho_{3}}^{(u,s)}$
$\displaystyle-\frac{1}{2}\sum_{s,t,u,v}(-1)^{m-q_{su}+1}\hat{{\cal
E}}_{J-m(j_{u}j_{s})\rho\;J_{1}m_{1}(j_{s}j_{t})\rho_{1}\;J_{2}m_{2}(j_{t}j_{u})\rho_{2}}\hat{{\cal
E}}_{Jm(j_{s}j_{u})\rho\;J_{3}m_{3}(j_{u}j_{v})\rho_{3}\;J_{4}m_{4}(j_{v}j_{s})\rho_{4}}$
$\displaystyle\left.\qquad\qquad\qquad\times
X_{J_{1}m_{1}\rho_{1}}^{(s,t)}X_{J_{2}m_{2}\rho_{2}}^{(t,u)}X_{J_{3}m_{3}\rho_{3}}^{(u,v)}X_{J_{4}m_{4}\rho_{4}}^{(v,s)}\right],$
(2.33)

where $\hat{{\cal
E}}_{J_{1}m_{1}(j_{s}j_{t})\rho_{1}\;J_{2}m_{2}(j_{t}j_{u})\rho_{2}\;J_{3}m_{3}(j_{u}j_{s})\rho_{3}}$
is defined in (A.42) and we have used (A.39). In the $N_{0}\rightarrow\infty$
limit, the ultraviolet cutoff goes to infinity and $\hat{{\cal
E}}_{J_{1}m_{1}(j_{s}j_{t})\rho_{1}\;J_{2}m_{2}(j_{t}j_{u})\rho_{2}\;J_{3}m_{3}(j_{u}j_{s})\rho_{3}}$
reduces to ${\cal
E}_{J_{1}m_{1}q_{st}\rho_{1}\;J_{2}m_{2}q_{tu}\rho_{2}\;J_{3}m_{3}q_{us}\rho_{3}}$
as shown in appendix A. Namely, this limit corresponds to the commutative
(continuum) limit of the fuzzy spheres. Hence, in the limit in which
$N_{0}\rightarrow\infty$ and $g_{mm}\rightarrow\infty$ such that
$g_{mm}^{2}/N_{0}=g_{S^{2}}^{2}\mu^{2}/4\pi$, (2.33) agrees with (2.29). We
have proven our statement.

Next, we show that the theory around a certain vacuum of $U(M=N\times\infty)$
YM-higgs on $S^{2}$ with a periodicity condition imposed is equivalent to
$U(N)$ YM on $S^{3}$. This is an extension of the matrix T-duality to a
nontrivial fiber bundle. The vacuum of YM-higgs on $S^{2}$ we take is given by
(2.22) with $s$ running from $-\infty$ to $\infty$, $n_{s}=s$ and $N_{s}=N$.
$4\pi g_{S^{2}}^{2}/\mu$ is identified with the coupling constant on $S^{3}$,
$g_{S^{3}}^{2}$. We decompose the fields on $S^{2}$ into the background and
the fluctuation,

$\displaystyle a_{\alpha}\rightarrow\hat{a}_{\alpha}+a_{\alpha},$
$\displaystyle\phi\rightarrow\hat{\phi}+\phi,$ (2.34)

and impose the periodicity (orbifolding) condition on the fluctuation,

$\displaystyle a_{\alpha}^{(s+1,t+1)}=a_{\alpha}^{(s,t)}\equiv
a_{\alpha}^{(s-t)},$
$\displaystyle\phi^{(s+1,t+1)}=\phi^{(s,t)}\equiv\phi^{(s-t)}.$ (2.35)

The fluctuations are gauge-transformed from the patch I to the patch II as
[29]

$\displaystyle{a^{\prime}}_{\\!\alpha}^{(s-t)}=e^{-i(s-t)\phi}a_{\alpha}^{(s-t)},$
$\displaystyle{\phi^{\prime}}^{(s-t)}=e^{-i(s-t)\phi}\phi^{(s-t)}.$ (2.36)

We make the Fourier transformation for the fluctuations on each patch to
construct the gauge field on the total space from the fields on the base
space:

$\displaystyle A_{\alpha}(\theta,\varphi,\psi)$
$\displaystyle=\sum_{w}a_{\alpha}^{(w)}(\theta,\varphi)e^{-i\frac{\mu}{2}wy},$
$\displaystyle A_{y}(\theta,\varphi,\psi)$
$\displaystyle=\sum_{w}\phi^{(w)}(\theta,\varphi)e^{-i\frac{\mu}{2}wy}.$
(2.37)

We see from (2.36) that the lefthand sides of (2.37) are indeed independent of
the patches. We substitute (2.37) into (2.26) and divide an overall factor
$\sum_{s}$ to extract a single period. Then, we obtain $U(N)$ YM on $S^{3}$.
The details of this calculation are given as a special case of (4.13) and
(4.14).

Finally, combining the above two statements, we see that the theory around
(2.24) of the matrix model where $s$ runs from $-\infty$ to $\infty$,
$2j_{s}+1=N_{0}+s$ is equivalent to $U(N)$ YM on $S^{3}$ if the
$N_{0}\rightarrow\infty$ limit is taken with $g_{mm}^{2}/N_{0}$ fixed to
$\frac{g_{S^{3}}^{2}\mu^{3}}{16\pi^{2}}$, the periodicity condition is imposed
on the fluctuation on $S^{2}$ and the overall factor $\Sigma_{s}$ is divided.
In this way, $S^{3}$ is realized in terms of the three matrices
$X_{1},\;X_{2},\;X_{3}$.

In sections 3-5, we generalize the results in this section. We set $\mu=1$ and
set all other dimensionful parameters to a certain constant value.

##  3 Dimensional reduction on a principal bundle

In this section, we provide the dimensional reduction of YM on a principal $G$
bundle to its base space. The case of principal $U(1)$ bundles was already
given in [29]. Here we consider the case where $G$ is nonabelian.

First, we give a metric and a vielbein of a fiber bundle on which pure YM is
defined. We consider a principal $G$-bundle $P$ on a manifold $M$. The base
space $M$ has a covering $\cal S$, and the total space has a covering
$\\{\pi^{-1}(U)|U\in{\cal S}\\}$. $\pi^{-1}(U)$ is diffeomorphic to $U\times
G$ by the local trivialization. Thus it is parameterized by
$z^{M}=(x^{\mu},y^{m})\;(\mu=1,\cdots,\dim M\;;\;m=1,\cdots,\dim G)$, where
$x^{\mu}$ parameterize the local patch $U$ and $y^{m}$ parameterize an element
of $G$. We assume that the connection of $P$ is expressed as

$\displaystyle\omega=g^{-1}(y)b(x)g(y)-i\>g^{-1}(y)dg(y).$ (3.1)

where $g(y)\in G$, $b(x)=b_{\mu}^{a}(x)T^{a}dx^{\mu}$ and $T^{a}$ are the
generators of the Lie group $G$.

The transition functions of a principal bundle act on fibers by left
multiplication. If there is overlap between $U$ and $U^{\prime}$, the relation
between fiber coodinates, $g(y)$ on $U$ and $g(y^{\prime})$ on $U^{\prime}$,
is given by

$\displaystyle g(y^{\prime})=k(x)\>g(y)$ (3.2)

where $k(x)\in G$. In the overlapping region $U\cap U^{\prime}$, $b(x)$ must
transform as

$\displaystyle
b^{\prime}(x^{\prime})=k(x)\>b(x)\>k^{-1}(x)+i\>dk(x)k^{-1}(x).$ (3.3)

Indeed, by using (3.3), we can show

$\displaystyle\omega=g(y)^{-1}\>b(x)\>g(y)-i\>g(y)^{-1}dg(y)=g(y^{\prime})^{-1}\>b^{\prime}(x^{\prime})\>g(y^{\prime})-i\>g(y^{\prime})^{-1}dg(y^{\prime}).$
(3.4)

We assume that the total space is endowed with a metric that has the fibered
structure determined by the connection (3.1) and the isometry. As shown in
[43], such metric can be locally expressed as333Throughout of this paper, we
use the following normalizations for the traces:
$\mbox{Tr}(T^{a}T^{b})=\frac{1}{2}\delta_{ab}$ for the structure group of the
fiber bundle and $\mbox{tr}(T^{a}T^{b})=\delta_{ab}$ for the gauge group.

$\displaystyle ds^{2}$ $\displaystyle=G_{MN}dz^{M}dz^{N}$
$\displaystyle=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+2{\rm Tr}\omega^{2}$
$\displaystyle=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+\\{e_{m}^{a}(y)dy^{m}-b^{a}_{\mu}(x)dx^{\mu}\\}^{2}.$
(3.5)

Here $g_{\mu\nu}$ is a metric on the base space and $e_{m}^{a}(y)\;(a=\dim
M+1,\cdots,\dim P)$ are the components of the right invariant Maurer-Cartan
1-form of $G$, which is defined by

$\displaystyle dg(y)g(y)^{-1}=-ie^{a}_{m}(y)T^{a}dy^{m}.$ (3.6)

We have assumed that the coefficient of the second term in (3.5) is just
$\delta_{ab}$ so that the resultant dimensionally reduced theory is simple,
although it is allowed to take $y$ independent function $\xi_{ab}(x)$. The
Maurer-Cartan 1-form satisfies the Maurer-Cartan equation

$\displaystyle de^{a}-\frac{1}{2}f^{abc}e^{b}\wedge e^{c}=0,$ (3.7)

where $f^{abc}$ is the structure constant of the Lie algebra of $G$, and is
regarded as the vielbein of the Cartan-Killing metric on $G$ defined by

$\displaystyle h_{mn}(y)dy^{m}dy^{n}$ $\displaystyle\equiv-2{\rm
Tr}\left(dgg^{-1}\right)^{2}$
$\displaystyle=e^{a}_{m}(y)e^{a}_{n}(y)dy^{m}dy^{n}.$ (3.8)

Note that $e^{a}_{m}(y)$ and $b(x)$ in the metric (3.5) are defined locally on
$U$ and must be transformed from $U$ to $U^{\prime}$: the transformation of
$e^{a}_{m}(y)$ is determined by (3.2) and an equality

$\displaystyle
dg(y^{\prime})g(y^{\prime})^{-1}=-ie^{a}_{m}(y^{\prime})T^{a}dy^{\prime m},$
(3.9)

while the transformation of $b(x)$ is given in (3.3). By introducing a
vielbein on the base space, $e^{\alpha}_{\mu}(x)\;(\alpha=1,\cdots,\dim M)$,
one can write a vielbein and its inverse on the total space as follows:

$\displaystyle E^{A}_{\;\;M}(z)=\begin{pmatrix}e^{\alpha}_{\mu}(x)&0\\\
-b^{a}_{\mu}(x)&e^{a}_{m}(y)\end{pmatrix},\quad
E^{M}_{\;\;A}(z)=\begin{pmatrix}e^{\mu}_{\alpha}(x)&0\\\
e^{m}_{a}(y)b^{a}_{\alpha}(x)&e^{m}_{a}(y)\end{pmatrix},$ (3.10)

where $e^{\mu}_{\alpha}$ and $e^{m}_{a}$ are the inverse of $e^{\alpha}_{\mu}$
and $e^{a}_{m}$, respectively, and $b^{a}_{\alpha}(x)\equiv
e^{\mu}_{\alpha}(x)b^{a}_{\mu}(x)$. The local Lorentz frame defined by (3.10)
gives the vertical-horizontal decomposition of vectors and 1-forms on the
total space. Namely, $\alpha=1,\cdots,\dim M$ correspond to the directions to
those of the base space and $a=\dim M+1,\cdots,\dim P$ correspond of the fiber
space. Again, we remark that these expressions are defined locally on $U$.
From (3.3) and (3.9), we can obtain relationships of the vielbeins between on
$U$ and on $U^{\prime}$ as

$\displaystyle E^{\prime\alpha}=E^{\alpha},$ $\displaystyle E^{\prime a}={\rm
Ad}(k)^{ab}E^{b}.$ (3.11)

where ${\rm Ad}(k)$ is the adjoint representation of $k(x)$. (2.8) is a
counterpart of (3.5), and (2.10) and (2.11) are a counterpart of (3.10).

We next consider a gauge theory on the total space and make a dimensional
reduction of the fiber direction to obtain a gauge theory on the base space.
We start with $U(N)$ YM on the total space:

$\displaystyle S_{P}=\frac{1}{g^{2}_{P}}\int d^{D}z\sqrt{G}\>{\rm
tr}\left(\frac{1}{4}F_{MN}F^{MN}\right).$ (3.12)

where $D=\dim P$ and
$F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}+i[A_{M},A_{N}]$. In order to make
the reduction, we perform the vertical-horizontal decompostion for the gauge
field $A_{M}(z)$ and the derivatives $\partial_{M}$ according to (3.10). The
gauge field is decomposed as

$\displaystyle A_{M}(z)=A_{\alpha}(z)E^{\alpha}_{M}(x)+A_{a}(z)E^{a}_{M}(z).$
(3.13)

After the reduction, horizontal components $A_{\alpha}$ and vertical
components $A_{a}$ of the gauge field will be naturally identified with the
gauge field and the higgs fields on the base space, respectively. The field
strength in the local Lorentz frame is rewritten as follows:

$\displaystyle
F_{\alpha\beta}=\nabla^{(M)}_{\alpha}A_{\beta}-\nabla^{(M)}_{\beta}A_{\alpha}+i[A_{\alpha},A_{\beta}]-b^{a}_{\alpha\beta}A_{a}+ib^{a}_{\alpha}{\cal
L}_{a}A_{\beta}-ib^{a}_{\beta}{\cal L}_{a}A_{\alpha},$ $\displaystyle
F_{\alpha
a}=e^{\mu}_{\alpha}\partial_{\mu}A_{a}+i\>[A_{\alpha},A_{b}]-f^{abc}b^{b}_{\alpha}A_{c}-i{\cal
L}_{a}A_{\alpha}+ib^{b}_{\alpha}{\cal L}_{b}A_{a},$ $\displaystyle
F_{ab}=f^{abc}A_{c}+i\>[A_{a},A_{b}]+i{\cal L}_{a}A_{b}-i{\cal L}_{b}A_{a}.$
(3.14)

Here we have defined the following quantities:

$\displaystyle b^{a}_{\alpha\beta}$ $\displaystyle\equiv
e^{\mu}_{\alpha}e^{\nu}_{\beta}\\{\partial_{\mu}b^{a}_{\nu}-\partial_{\nu}b^{a}_{\mu}-f^{abc}b^{b}_{\mu}b^{c}_{\nu}\\},$
$\displaystyle\nabla^{(M)}_{\alpha}A_{\beta}$ $\displaystyle\equiv
e^{\mu}_{\alpha}\left(\partial_{\mu}A_{\beta}+\omega_{\mu\>\beta}^{\quad\gamma}A_{\gamma}\right),$
$\displaystyle{\cal L}_{a}$ $\displaystyle\equiv-ie^{m}_{a}\partial_{m},$
(3.15)

where $\omega$ is the spin connection on the base space defined by
$e^{\alpha}_{\mu}$ and ${\cal L}_{a}$ are the right invariant Killing vectors
on the total space, which represent the isometry. Note that our calculations
have been performed on $U$ so far. When it is performed on $U^{\prime}$, the
quantities on $U^{\prime}$ must be used. The transformation of
$b^{a}_{\alpha}(x)$ between on $U$ and on $U^{\prime}$ is given by (3.3), so
that that of $b^{a}_{\alpha\beta}(x)$ is given by

$\displaystyle b^{\prime a}_{\alpha\beta}(x^{\prime})={\rm
Ad}(k)^{ab}b^{b}_{\alpha\beta}(x).$ (3.16)

The gauge field with the local Lorentz index must be transformed according to
(3.11) as

$\displaystyle A^{\prime}_{\alpha}=A_{\alpha},$ $\displaystyle
A^{\prime}_{a}={\rm Ad}(k)^{ab}A_{b}.$ (3.17)

In order to make the dimensional reduction, we relate the fields on the total
space to those on the base space as

$\displaystyle A_{\alpha}$ $\displaystyle=a_{\alpha},$ $\displaystyle A_{a}$
$\displaystyle=\phi_{a},$ (3.18)

where $a_{\alpha}$ are the gauge field in the local Lorentz frame and
$\phi_{a}$ are higgs fields on the base space. We assume the both sides in
(3.18) are independent of $y^{m}$. Using subscript of curved space, we can
write (3.18) equivalently as

$\displaystyle A_{\mu}$ $\displaystyle=a_{\mu}-b^{a}_{\mu}\phi_{a},$
$\displaystyle A_{m}$ $\displaystyle=e^{a}_{m}\phi_{a}.$ (3.19)

Here (3.18) and (3.19) are a generalization of (2.13) and (2.14),
respectively. Substituting (3.14) and (3.18) into (3.12) and using
$\sqrt{G}=\sqrt{g}\sqrt{h}$, we obtain YM-higgs on the base space:

$\displaystyle S_{M}$ $\displaystyle=\frac{1}{g_{M}^{2}}\int
d^{d}x\sqrt{g}\>{\rm
tr}\bigg{\\{}\frac{1}{4}\left(f_{\alpha\beta}-b^{a}_{\alpha\beta}\phi_{a}\right)^{2}+\frac{1}{2}\left(\nabla^{(M)}_{\alpha}\phi_{a}+i\>[a_{\alpha},\phi_{a}]-f^{abc}b^{b}_{\alpha}\phi_{c}\right)^{2}$
$\displaystyle\qquad+\frac{1}{4}\left(f^{abc}\phi_{c}+i\>[\phi_{a},\phi_{b}]\right)^{2}\bigg{\\}},$
(3.20)

where $g_{M}^{2}=(\int
dy\sqrt{h})^{-1}g_{P}^{2}=\frac{1}{\text{Vol}(G)}g_{P}^{2}$, $d=\dim M$ and
$f_{\alpha\beta}=\nabla^{(M)}_{\alpha}a_{\beta}-\nabla^{(M)}_{\beta}a_{\alpha}+i[a_{\alpha},a_{\beta}]$.
Note that the connection in the fiber bundle can generate nontrivial mass
terms of the higgs fields. This is reminiscent of the flux compactification in
string theory.

##  4 Extension of the matrix T-duality

In this section, we extend the matrix T-duality on nontrivial $U(1)$ bundles
developed in [29] to that on nontrivial $SU(2)$ bundles.

###  4.1 Nontrivial vacua and transformation between patches

As in the example in section 2, the theory on the base space (3.20) has
monopolelike vacua, which are in general patch-dependent if the principal
bundle we consider is nontrivial. Here we describe the vacua and their patch-
dependence. We also consider how the fields of the theory are transformed from
a patch to another. We examine, in particular, the transformation properties
of fluctuations around the vacua.

It is seen from (3.20) that the condition for the vacua is given by

$\displaystyle f_{\alpha\beta}-b^{a}_{\alpha\beta}\phi_{a}=0,$
$\displaystyle\nabla^{(M)}_{\alpha}\phi_{a}+i\>[a_{\alpha},\phi_{a}]-f^{abc}b^{b}_{\alpha}\phi_{c}=0,$
$\displaystyle f^{abc}\phi_{c}+i\>[\phi_{a},\phi_{b}]=0.$ (4.1)

They are satisfied by the following configurations:

$\displaystyle\hat{a}_{\alpha}(x)=b^{a}_{\alpha}(x)\hat{\phi}_{a}=b^{a}_{\alpha}(x)L_{a},$
$\displaystyle\hat{\phi}_{a}=L_{a},$ (4.2)

where $L_{a}$ are the generators of the Lie algebra of $G$ satisfying
$[L_{a},L_{b}]=if^{abc}L^{c}$ and generally reducible. Note that as mentioned
in section 3.1, $b^{a}(x)$ are generally patch-dependent quantities. The vacua
are, therefore, also patch-dependent. From (3.3) and (4.2), we can read off
the transformation properties for the vacua between patches:

$\displaystyle\hat{a}^{\prime}(x)=K(x)\hat{a}(x)K(x)^{-1}+i\>dK(x)K(x)^{-1},$
$\displaystyle\hat{\phi}^{\prime}_{a}={\rm
Ad}(k(x))_{ab}K(x)\hat{\phi}_{b}K(x)^{-1}=\hat{\phi}_{a},$ (4.3)

where $K(x)$ is obtained by replacing $T^{a}$ in $k(x)$ in (3.2) by
$\hat{\phi}_{a}=L_{a}$. Note that this is the gauge transformation by $K(x)$
except for the rotation of $\hat{\phi}_{a}$ by ${\rm Ad}(k(x))$, which comes
from (3.17).

Let us consider the theory around the vacua (4.2) and decompose the fields
into the backgrounds and fluctuations:

$\displaystyle a_{\alpha}(x)=\hat{a}_{\alpha}(x)+\tilde{a}_{\alpha}(x),$
$\displaystyle\phi_{a}(x)=\hat{\phi}_{a}+\tilde{\phi}_{a}(x).$ (4.4)

The fluctuations are transformed between patches as

$\displaystyle\tilde{a}^{\prime}_{\alpha}(x)=K(x)\tilde{a}_{\alpha}(x)K(x)^{-1},$
$\displaystyle\tilde{\phi}^{\prime}_{a}(x)={\rm
Ad}(k(x))_{ab}K(x)\tilde{\phi}_{b}(x)K(x)^{-1}.$ (4.5)

One can easily see that the action (3.20) is indeed invariant under the
transformation (4.3) and (4.5).

###  4.2 $G=U(1),SU(2)$

In this subsection, we consider the case in which the fiber is $U(1)$ or
$SU(2)$. In the case of $G=U(1)$, the matrix T-duality indeed works as shown
in [29] and its typical example was given in section 2. We extend the matrix
T-duality to the case of $G=SU(2)$ by applying the fact described in section 2
that YM on $S^{3}$ is realized in the matrix model.

First, we review the matrix T-duality in the case of $G=U(1)$, which is a
generalization of the relationship between YM on $S^{3}$ and YM-higgs on
$S^{2}$ in section 2. In this case, the metric (3.5) reduces to the following
form:

$\displaystyle ds^{2}=g_{\mu\nu}(x)dx^{\mu}dx^{\nu}+(dy-
b_{\mu}(x)dx^{\mu})^{2},$ (4.6)

where $y$ represents the fiber direction and $0\leq y<2\pi$. We put $\dim
M=d$. (2.8) indeed takes the form of (4.6). YM-higgs on the base space
obtained from YM on the total space is given as the $U(1)$ case of (3.20):

$\displaystyle S_{M}$ $\displaystyle=\frac{1}{g_{M}^{2}}\int
d^{d}x\sqrt{g}\>{\rm
tr}\bigg{\\{}\frac{1}{4}\left(f_{\alpha\beta}-b_{\alpha\beta}\phi\right)^{2}+\frac{1}{2}\left(\nabla^{(M)}_{\alpha}\phi+i\>[a_{\alpha},\phi]\right)^{2}\bigg{\\}}.$
(4.7)

(2.15) is a special case of (4.7). We show that we obtain the $U(N)$ YM on the
total space from the $U(N\times\infty)$ YM-higgs on the base space through the
following procedure: we choose a certain background of the $U(N\times\infty)$
YM-higgs on the base space, expand the theory around the background and impose
a periodicity condition.

Note, first, that a general background of (4.7) is given by

$\displaystyle\hat{a}_{\alpha}=b_{\alpha}\hat{\phi},$
$\displaystyle\hat{\phi}=-\>\mbox{diag}(\cdots,\underbrace{n_{s-1},\cdots,n_{s-1}}_{N_{s-1}},\underbrace{n_{s},\cdots,n_{s}}_{N_{s}},\underbrace{n_{s+1},\cdots,n_{s+1}}_{N_{s+1}},\cdots),$
(4.8)

which is a counterpart of (2.22). We decompose the fields into the backgrounds
and the fluctuations as

$\displaystyle a_{\alpha}$
$\displaystyle\rightarrow\hat{a}_{\alpha}+a_{\alpha},$ $\displaystyle\phi$
$\displaystyle\rightarrow\hat{\phi}+\phi.$ (4.9)

In particular, we take the following background: $s$ running from $-\infty$ to
$\infty$, $n_{s}=s$ and $N_{s}=N$. We label the (off-diagonal) blocks by
$(s,t)$ and impose the periodicity (orbifolding) condition on the fluctuations
as in (2.35):

$\displaystyle a_{\alpha}^{(s+1,t+1)}=a_{\alpha}^{(s,t)}\equiv
a^{(s-t)}_{\alpha},$
$\displaystyle\phi^{(s+1,t+1)}=\phi^{(s,t)}\equiv\phi^{(s-t)}.$ (4.10)

The fluctuations are gauge-transformed from $U$ to $U^{\prime}$ as

$\displaystyle{a^{\prime}}_{\alpha}^{(s-t)}$
$\displaystyle=e^{-i(s-t)v(x)}a^{(s-t)}_{\alpha},$
$\displaystyle\phi^{\prime(s-t)}$ $\displaystyle=e^{-i(s-t)v(x)}\phi^{(s-t)},$
(4.11)

where $e^{-iv}$ is a transition function;
$e^{-iy^{\prime}}=e^{-iv(x)}e^{-iy}$. (2.36) is a special case of (4.11). We
make the Fourier transformation for the fluctuations on each patch to
construct the gauge field on the total space:

$\displaystyle A_{\alpha}(x,y)$
$\displaystyle=\sum_{w}a^{(w)}_{\alpha}(x)e^{-iwy},$ $\displaystyle
A_{d+1}(x,y)$ $\displaystyle=\sum_{w}\phi^{(w)}(x)e^{-iwy}.$ (4.12)

We can see from (4.11) that the lefthand sides in the above equations are
indeed invariant under the transformation between patches. Using (4.9) and
(4.12), we can rewrite each term in (4.7) as

$\displaystyle\left(f_{\alpha\beta}-b_{\alpha\beta}\phi\right)^{(s,t)}$
$\displaystyle\rightarrow\left(\nabla^{(M)}_{\alpha}a_{\beta}-\nabla^{(M)}_{\beta}a_{\alpha}+i[\hat{a}_{\alpha},a_{\beta}]+i[a_{\alpha},\hat{a}_{\beta}]+i[a_{\alpha},a_{\beta}]-b_{\alpha\beta}\phi\right)^{(s,t)}$
$\displaystyle=\left(\nabla^{(M)}_{\alpha}a_{\beta}^{(s-t)}-\nabla^{(M)}_{\beta}a_{\alpha}^{(s-t)}+i[a_{\alpha},a_{\beta}]^{(s-t)}-i(s-t)b_{\alpha}a_{\beta}^{(s-t)}+i(s-t)b_{\beta}a_{\alpha}^{(s-t)}-b_{\alpha\beta}\phi^{(s-t)}\right)$
$\displaystyle=\frac{1}{2\pi}\int
dy\left(\nabla^{(M)}_{\alpha}A_{\beta}-\nabla^{(M)}_{\beta}A_{\alpha}+i[A_{\alpha},A_{\beta}]-b_{\alpha\beta}A_{d+1}+b_{\alpha}\partial_{y}A_{\beta}-b_{\beta}\partial_{y}A_{\alpha}\right)\>e^{i(s-t)y}$
$\displaystyle=\frac{1}{2\pi}\int dyF_{\alpha\beta}\>e^{i(s-t)y},$
$\displaystyle\left(\nabla^{(M)}_{\alpha}\phi+i\>[a_{\alpha},\phi]\right)^{(s,t)}$
$\displaystyle\rightarrow\left(\nabla^{(M)}_{\alpha}\phi+i\>[\hat{a}_{\alpha},\phi]+i\>[a_{\alpha},\hat{\phi}]+i\>[a_{\alpha},\phi]\right)^{(s,t)}$
$\displaystyle=\nabla^{(M)}_{\alpha}\phi^{(s-t)}+i\>[a_{\alpha},\phi]^{(s-t)}-i\>(s-t)b_{\alpha}\phi^{(s-t)}+i\>(s-t)a_{\alpha}^{(s-t)}$
$\displaystyle=\frac{1}{2\pi}\int
dy\left(\nabla^{(M)}_{\alpha}A_{d+1}+i\>[A_{\alpha},A_{d+1}]-\partial_{y}A_{\alpha}+b_{\alpha}\partial_{y}A_{d+1}\right)\>e^{i(s-t)y}$
$\displaystyle=\frac{1}{2\pi}\int dyF_{\alpha(d+1)}\>e^{i(s-t)y}.$ (4.13)

Then (4.7) becomes

$\displaystyle S_{M}$ $\displaystyle=\frac{1}{g_{M}^{2}}\int
d^{d}x\sqrt{g}\>{\rm
tr}\biggl{\\{}\frac{1}{4}\left(f_{\alpha\beta}-b_{\alpha\beta}\phi\right)^{2}+\frac{1}{2}\left(\nabla^{(M)}_{\alpha}\phi_{a}+i\>[a_{\alpha},\phi]\right)^{2}\biggr{\\}}$
$\displaystyle=\frac{1}{g_{M}^{2}}\int d^{d}x\sqrt{g}\>{\rm
tr}\Biggl{[}\sum_{s,t}\biggl{\\{}\frac{1}{4}\left(f_{\alpha\beta}-b_{\alpha\beta}\phi\right)^{(s,t)}\left(f_{\alpha\beta}-b_{\alpha\beta}\phi\right)^{(t,s)}$
$\displaystyle\hskip
142.26378pt+\frac{1}{2}\left(\nabla^{(M)}_{\alpha}\phi+i\>[a_{\alpha},\phi]\right)^{(s,t)}\left(\nabla^{(M)}_{\alpha}\phi+i\>[a_{\alpha},\phi]\right)^{(t,s)}\biggr{\\}}\Biggr{]}$
$\displaystyle\rightarrow\frac{1}{g_{M}^{2}}\frac{1}{2\pi}\sum_{w}\int
d^{D}z\sqrt{G}\>\frac{1}{4}{\rm tr}\left(F_{AB}F_{AB}\right).$ (4.14)

By dividing an overall factor $\sum_{w}$ in the last line in (4.14) to extract
a single period, we obtain Yang-Mills theory on the total space.

Next we consider the case where fiber is $SU(2)$. In this case, YM-higgs on
the base space takes the form

$\displaystyle S_{M}$ $\displaystyle=\frac{1}{g_{M}^{2}}\int
d^{d}x\sqrt{g}\>{\rm
tr}\bigg{\\{}\frac{1}{4}\left(f_{\alpha\beta}-b^{a}_{\alpha\beta}\phi_{a}\right)^{2}+\frac{1}{2}\left(\nabla^{(M)}_{\alpha}\phi_{a}+i\>[a_{\alpha},\phi_{a}]-\epsilon^{abc}b^{b}_{\alpha}\phi_{c}\right)^{2}$
$\displaystyle\qquad\qquad+\frac{1}{4}\left(\epsilon^{abc}\phi_{c}+i\>[\phi_{a},\phi_{b}]\right)^{2}\bigg{\\}}.$
(4.15)

We show that we can obtain the $U(N)$ YM on the total space of a nontrivial
$SU(2)$-bundle from the YM with three higgs on its base space in a way similar
to the case of $G=U(1)$.

The vacuum of YM-higgs is given by (4.2) with $L_{a}$ satisfying the $SU(2)$
algebra, $[L_{a},L_{b}]=i\>\epsilon_{abc}L_{c}$, and $L_{a}$ generically take
a reducible representation (2.25). We expand the fields around this
background,

$\displaystyle a_{\alpha}(x)$
$\displaystyle\rightarrow\hat{a}_{\alpha}(x)+a_{\alpha}(x),$
$\displaystyle\phi_{a}(x)$
$\displaystyle\rightarrow\hat{\phi}_{a}+\phi_{a}(x).$ (4.16)

We label the (off-diagonal) blocks of the fluctuations by $(s,t)$, which is
$(N_{s}(2j_{s}+1))\times(N_{t}(2j_{t}+1))$ matrix, and expand them by the
fuzzy spherical harmonics:

$\displaystyle a_{\alpha}^{(s,t)}(x)$
$\displaystyle=\sum_{J=|j_{s}-j_{t}|}^{j_{s}+j_{t}}\sum_{m=-J}^{J}a_{\alpha,Jm}^{(s,t)}(x)\otimes\hat{Y}_{Jm(j_{s}j_{t})},$
$\displaystyle\phi_{a}^{(s,t)}(x)$
$\displaystyle=\sum_{J=|j_{s}-j_{t}|}^{j_{s}+j_{t}}\sum_{m=-J}^{J}\phi_{a,Jm}^{(s,t)}(x)\otimes\hat{Y}_{Jm(j_{s}j_{t})}.$
(4.17)

We verify from (4.5), (4.17) and (A.26) that the modes are gauge-transformed
from $U$ to $U^{\prime}$ as

$\displaystyle a^{\prime(s,t)}_{\alpha,Jm}(x)=\sum_{m^{\prime}}\langle
Jm|k^{[J]}|Jm^{\prime}\rangle a_{\alpha,Jm^{\prime}}^{(s,t)}(x),$
$\displaystyle\phi^{\prime(s,t)}_{a,Jm}(x)=\sum_{m^{\prime}}{\rm
Ad}(k)_{ab}\langle
Jm|k^{[J]}|Jm^{\prime}\rangle\phi_{b,Jm^{\prime}}^{(s,t)}(x),$ (4.18)

where $k^{[J]}$ is the spin $J$ representation of $SU(2)$ for $k(x)$.

In what follows, we assume that as a background we set $2j_{s}+1=N_{0}+s$ with
$s$ running from $-T$ to $T$ in (2.25) and take the limit of
$N_{0}\rightarrow\infty$ and $T\rightarrow\infty$ in order. For the modes, we
impose the periodicity condition:

$\displaystyle a_{\alpha,Jm}^{(s+1,t+1)}=a_{\alpha,Jm}^{(s,t)}\equiv
a_{\alpha,Jm}^{(q_{st})},$
$\displaystyle\phi_{a,Jm}^{(s+1,t+1)}=\phi_{a,Jm}^{(s,t)}\equiv\phi_{a,Jm}^{(q_{st})},$
(4.19)

where $q_{st}\equiv\frac{s-t}{2}$. By using these modes and the spherical
harmonics on $S^{3}$, we make Fourier transformation on each patch to
construct the gauge field on the total space:

$\displaystyle
A_{\alpha}(z)=\sum_{Jm\tilde{m}}a_{\alpha,Jm}^{(\tilde{m})}(x)Y_{Jm\tilde{m}}(y),$
$\displaystyle
A_{a}(z)=\sum_{Jm\tilde{m}}\phi_{a,Jm}^{(\tilde{m})}(x)Y_{Jm\tilde{m}}(y).$
(4.20)

Its inverse is

$\displaystyle
a_{\alpha,Jm}^{(\tilde{m})}(x)=\int\frac{d\Omega_{3}}{2\pi^{2}}A_{\alpha}(z)Y_{Jm\tilde{m}}^{\dagger}(y),$
$\displaystyle\phi_{a,Jm}^{(\tilde{m})}(x)=\int\frac{d\Omega_{3}}{2\pi^{2}}A_{a}(z)Y_{Jm\tilde{m}}^{\dagger}(y).$
(4.21)

From (4.18) and (A.8), it is verified that the lefthand sides in (4.20) are
indeed transformed between patches as the gauge field on the total space
(3.17).

Using (4.17) and (4.21), we can obtain the following equalities:

$\displaystyle[L_{a},a_{\alpha}(x)]^{(s,t)}$
$\displaystyle=\int\frac{d\Omega_{3}}{2\pi^{2}}\left({\cal
L}_{a}A_{\alpha}(z)\right)Y_{Jpq_{st}}^{\dagger}\otimes\hat{Y}_{Jp(j_{s}j_{t})},$
$\displaystyle[\phi_{a},\phi_{b}]^{(s,t)}$
$\displaystyle=\int\frac{d\Omega_{3}}{2\pi^{2}}[A_{a}(z),A_{b}(z)]Y_{Jmq_{st}}^{\dagger}(y)\otimes\hat{Y}_{Jm(j_{s}j_{t})}.$
(4.22)

The derivation of the above equalities is given in appendix B. Substituting
these into (4.15), we obtain

$\displaystyle S_{M}$ $\displaystyle=\frac{1}{g_{M}^{2}}\int
d^{d}x\sqrt{g}\>{\rm
tr}\bigg{\\{}\frac{1}{4}\left(f_{\alpha\beta}-b^{a}_{\alpha\beta}\phi_{a}\right)^{2}+\frac{1}{2}\left(\nabla^{(M)}_{\alpha}\phi_{a}+i\>[a_{\alpha},\phi_{a}]-\epsilon^{abc}b^{b}_{\alpha}\phi_{c}\right)^{2}$
$\displaystyle\qquad\qquad+\frac{1}{4}\left(\epsilon^{abc}\phi_{c}+i\>[\phi_{a},\phi_{b}]\right)^{2}\bigg{\\}}$
$\displaystyle=\frac{1}{g_{M}^{2}}\int d^{d}x\sqrt{g}\>{\rm
tr}\Biggl{[}\sum_{s,t}\biggl{\\{}\frac{1}{4}\left(f_{\alpha\beta}-b^{a}_{\alpha\beta}\phi_{a}\right)^{(s,t)}\left(f_{\alpha\beta}-b^{a}_{\alpha\beta}\phi_{a}\right)^{(t,s)}$
$\displaystyle\qquad\qquad+\frac{1}{2}\left(\nabla^{(M)}_{\alpha}\phi_{a}+i\>[a_{\alpha},\phi_{a}]-\epsilon^{abc}b^{b}_{\alpha}\phi_{c}\right)^{(s,t)}\left(\nabla^{(M)}_{\alpha}\phi_{a}+i\>[a_{\alpha},\phi_{a}]-\epsilon^{abc}b^{b}_{\alpha}\phi_{c}\right)^{(t,s)}$
$\displaystyle\qquad\qquad+\frac{1}{4}\left(\epsilon^{abc}\phi_{c}+i\>[\phi_{a},\phi_{b}]\right)^{(s,t)}\left(\epsilon^{abc}\phi_{c}+i\>[\phi_{a},\phi_{b}]\right)^{(t,s)}\biggr{\\}}\Biggr{]}$
$\displaystyle\rightarrow\frac{1}{g_{M}^{2}}\frac{N_{0}}{2\pi^{2}}\sum_{w}\int
d^{D}z\sqrt{G}$ $\displaystyle\qquad\qquad\times{\rm
tr}\bigg{\\{}\frac{1}{4}\left(\nabla^{(M)}_{\alpha}A_{\beta}-\nabla^{(M)}_{\beta}A_{\alpha}+i[A_{\alpha},A_{\beta}]-b^{a}_{\alpha\beta}A_{a}+ib^{a}_{\alpha}{\cal
L}_{a}A_{\beta}-ib^{a}_{\beta}{\cal L}_{a}A_{\alpha}\right)^{2}$
$\displaystyle\qquad\qquad+\frac{1}{2}\left(\nabla^{(M)}_{\alpha}A_{a}+i\>[A_{\alpha},A_{b}]-f^{abc}b^{b}_{\alpha}A_{c}-i{\cal
L}_{a}A_{\alpha}+ib^{b}_{\alpha}{\cal L}_{b}A_{a}\right)^{2}$
$\displaystyle\qquad\qquad+\frac{1}{4}\left(f^{abc}A_{c}+i\>[A_{a},A_{b}]+i{\cal
L}_{a}A_{b}-i{\cal L}_{b}A_{a}\right)^{2}\bigg{\\}}$
$\displaystyle=\frac{1}{g_{M}^{2}}\frac{N_{0}}{2\pi^{2}}\sum_{w}\int
d^{d+1}z\sqrt{G}\mbox{tr}\left(\frac{1}{4}F_{AB}F_{AB}\right)$ (4.23)

By dividing an overall factor $\sum_{w}$ in the last line in (4.23) to extract
a single period, we obtain Yang-Mills theory on the total space.

We can easily extend the above matrix T-duality to the case in which the fiber
is $SU(2)^{k}\times U(1)^{l}$. As an example, we consider an $SU(2)\times
U(1)$ bundle, $P$. Let $a,b,c$ in (3.20) run $0,1,2,3$ such that ‘$0$’
corresponds to the $U(1)$ direction and ‘$1,2,3$’ correspond to the $SU(2)$
direction. We assign $i,j,k$ to the $SU(2)$ direction. We can consider YM-
higgs on the $U(1)$ bundle on $M$, $M^{\prime}$, which is obtained by making
the dimensional reduction of the $SU(2)$ fiber direction for YM on the
$SU(2)\times U(1)$ bundle. We realize the theory around an $SU(2)$
multimonopole background of YM-higgs on $M^{\prime}$ by taking the following
background in YM-higgs on $M$ (3.20) and imposing the periodicity condition to
the fluctuations:

$\displaystyle\hat{\phi}_{0}=-\frac{1}{R}\mbox{diag}(\cdots,t-1,t,t+1,\cdots)\otimes
1_{\hat{M}}+b_{0}^{i}\hat{\phi}_{i},$
$\displaystyle\hat{\phi}_{i}=1_{\infty}\otimes(L_{i}\;\mbox{in (\ref{matrix
background})}),$ $\displaystyle\hat{a}_{\alpha}=b_{\alpha}^{a}\hat{\phi}_{a},$
(4.24)

where $b_{\alpha}^{0}$ represents the $U(1)$ monopole and $b_{\alpha}^{i}$
represents the SU(2) monopole. $R$ is a certain constant depending on the
fiber structure. By setting $2j_{s}+1=N_{0}+s$ with $s$ running from $-T$ to
$T$, taking the limit of $N_{0}\rightarrow\infty$ and $T\rightarrow\infty$ in
order and imposing the periodicity condition to the fluctuations again, we
realize YM on $P$ in YM-higgs on $M$. In a similar way, we can realize YM on
an $SU(2)^{k}\times U(1)^{l}$ in YM-higgs on its base space.

###  4.3 Example: $S^{7}\rightarrow S^{4}$

We present an example of our findings in the previous subsection: we consider
$S^{7}$ with radius $2$ and regard it as $SU(2)\cong S^{3}$ Hopf bundle on
$S^{4}$ with radius $1$.

In order to describe $S^{7}$ as $SU(2)$ bundle on $S^{4}$, it is convenient to
introduce the quaternion $H$ (see for example [44, 45, 46]). The quaternion
algebra is defined by

$\displaystyle\bm{i}^{2}=\bm{j}^{2}=\bm{k}^{2}=-1,\quad\bm{i}\bm{j}=-\bm{j}\bm{i}=\bm{k},$
(4.25)
$\displaystyle\bm{j}\bm{k}=-\bm{k}\bm{j}=\bm{i},\quad\bm{k}\bm{i}=-\bm{i}\bm{k}=\bm{j}.$
(4.26)

An arbitrary element of $H$ is written as

$\displaystyle q=a+b\bm{i}+c\bm{j}+d\bm{k}.$ (4.27)

where $a,b,c,d\in R$. Its conjugation $q^{*}$ is defined by

$\displaystyle q^{*}\equiv a-b\bm{i}-c\bm{j}-d\bm{k}.$ (4.28)

The absolute value is given by

$\displaystyle|q|\equiv\sqrt{q^{*}q}=\sqrt{a^{2}+b^{2}+c^{2}+d^{2}}\geq 0.$
(4.29)

$S^{7}$ with radius $2$ is expressed by using quaternions as follows:

$\displaystyle\\{(q_{1},q_{2})\in H^{2}||q_{1}|^{2}+|q_{2}|^{2}=4\\}.$ (4.30)

The Hopf map $\pi:S^{7}\rightarrow S^{4}$ is defined by

$\displaystyle\pi:(q_{1},q_{2})\rightarrow[(q_{1},q_{2})]\equiv\\{(q_{1},q_{2})q|q\in
H\backslash\\{0\\}\\}.$ (4.31)

In order to introduce local coordinates one needs to divide $S^{4}$ in two
patches: $U_{1}$ ($q_{1}\neq 0$) and $U_{2}$ ($q_{2}\neq 0$). The local
trivialization is given on each patch by

$\displaystyle\pi^{-1}(U_{1})\ni(q_{1},q_{2})$
$\displaystyle\rightarrow(q_{2}q_{1}^{-1},q_{1}|q_{1}|^{-1})\in U_{1}\times
SU(2),$ $\displaystyle\pi^{-1}(U_{2})\ni(q_{1},q_{2})$
$\displaystyle\rightarrow(q_{1}q_{2}^{-1},q_{2}|q_{2}|^{-1})\in U_{2}\times
SU(2).$ (4.32)

We parameterize $(q_{1},q_{2})$ by using a matrix representation of
quaternions as

$\displaystyle q_{1}$ $\displaystyle=2\cos\frac{\chi}{2}\;\lambda,$
$\displaystyle q_{2}$ $\displaystyle=2\sin\frac{\chi}{2}\;\kappa\lambda.$
(4.33)

where $\kappa,\>\lambda\in SU(2)$ are defined by using Pauli matrices
$\sigma^{a}\>(a=1,2,3)$ as

$\displaystyle\kappa$
$\displaystyle=e^{i\eta\frac{\sigma^{3}}{2}}e^{i\xi\frac{\sigma^{2}}{2}}e^{i\zeta\frac{\sigma^{3}}{2}},$
$\displaystyle\lambda$
$\displaystyle=e^{-i\psi\frac{\sigma^{3}}{2}}e^{-i\theta\frac{\sigma^{2}}{2}}e^{-i\phi\frac{\sigma^{3}}{2}}.$
(4.34)

The ranges of variables in the above equations are

$\displaystyle 0\leq\chi\leq\pi,$ $\displaystyle
0\leq\xi\leq\pi,\;0\leq\eta<2\pi,\;0\leq\zeta<4\pi,$ $\displaystyle
0\leq\theta\leq\pi,\;0\leq\phi<2\pi,\;0\leq\psi<4\pi.$ (4.35)

In particular, $|\lambda|^{2}=\det\lambda=1$ and $|\kappa|^{2}=\det\kappa=1$
hold. One can easily see from (4.32) and (4.33) that on $U_{1}$ the fiber
space $SU(2)$ is described by $\lambda$ while on $U_{2}$ that is described by
$\lambda^{\prime}\equiv\kappa\lambda$. In the following, we restrict ourselves
to the region $U_{1}$. We denote sets of coordinates as
$x^{\mu}=(\chi,\xi,\eta,\zeta)=(\chi,x^{\bar{\mu}})$ and
$y^{m}=(\theta,\phi,\psi)$. $x^{\mu}$ are coordinates of $S^{4}$,
$x^{\bar{\mu}}$ are those of $S^{3}$ inside of $S^{4}$ and $y^{m}$ are those
of $SU(2)$ of fiber. In order to describe a metric of $S^{7}$ explicitly, we
introduce the Maurer-Cartan 1-forms for $\kappa$ and $\lambda$

$\displaystyle\kappa(\bar{x})^{\dagger}d\kappa(\bar{x})$
$\displaystyle=i\bar{e}^{a}_{\bar{\mu}}(\bar{x})\frac{\sigma^{a}}{2}dx^{\bar{\mu}},$
$\displaystyle d\lambda(y)\lambda(y)^{\dagger}$
$\displaystyle=-i\>e^{a}_{m}(y)\frac{\sigma^{a}}{2}dy^{m},$ (4.36)

where $\bar{x}$ represents the set of $\\{x^{\bar{\mu}}\\}$. Then we define
the metric of $S^{7}$ as

$\displaystyle ds_{S^{7}}^{2}$ $\displaystyle=\det(dq_{1})+\det(dq_{2}),$
(4.37)

which is evaluated as

$\displaystyle
ds_{S^{7}}^{2}=\Bigl{(}d\chi^{2}+\frac{1}{4}\sin^{2}\chi\;\bar{e}^{a}_{\bar{\mu}}(\bar{x})\bar{e}^{a}_{\bar{\nu}}(\bar{x})dx^{\bar{\mu}}dx^{\bar{\nu}}\Bigr{)}+\Bigl{(}e^{a}_{m}(y)dy^{m}-\sin^{2}\frac{\chi}{2}\>\bar{e}^{a}_{\bar{\mu}}(\bar{x})dx^{\bar{\mu}}\Bigr{)}^{2}.$
(4.38)

In the above expression, the first term represents the metric of the base
space $S^{4}$ and the second one represents that of the fiber space $SU(2)$
locally. Note that $\frac{1}{4}\bar{e}^{a}_{\bar{\mu}}\bar{e}^{a}_{\bar{\nu}}$
and $\frac{1}{4}e^{a}_{m}e^{a}_{n}$ are a metric of $S^{3}$ with radius $1$.
From (4.38) one can read off the vielbein on $S^{4}$ and the local connections
of the fiber bundle as

$\displaystyle e^{\alpha}_{\;\;\mu}(x)$ $\displaystyle=\begin{pmatrix}1&0\\\
0&\frac{1}{2}\sin\chi\>\bar{e}^{a}_{\;\;\bar{\mu}}(\bar{x})\end{pmatrix},\qquad
e^{\mu}_{\;\;\alpha}(x)=\begin{pmatrix}1&0\\\
0&\frac{2}{\sin\chi}\>\bar{e}^{\bar{\mu}}_{\;\;a}(\bar{x})\end{pmatrix},$
$\displaystyle b^{a}_{\chi}(x)=0,\quad
b^{a}_{\bar{\mu}}(x)=\tan\frac{\chi}{2}\>e^{a}_{\bar{\mu}}(x),$ $\displaystyle
b^{a}_{\chi\bar{\nu}}(x)=e^{a}_{\bar{\nu}}(x),\quad
b^{a}_{\bar{\mu}\bar{\nu}}(x)=f^{abc}e^{b}_{\bar{\mu}}(x)e^{c}_{\bar{\nu}}(x).$
(4.39)

As noted before, when we move to the other region, $U_{2}$, we must change
$\lambda$ to $\lambda^{\prime}\equiv\kappa\lambda$. Then, one can easily find
that the local connections change to

$\displaystyle{b^{\prime}}^{a}_{\chi}(x)=0,\quad{b^{\prime}}^{a}_{\bar{\mu}}(x)=-\cot\frac{\chi}{2}\>{\rm
Ad}(\kappa)^{ab}e^{b}_{\bar{\mu}}(x),$ $\displaystyle
b^{a}_{\chi\bar{\nu}}(x)={\rm Ad}(\kappa)^{ab}b^{b}_{\chi\bar{\nu}}(x),\quad
b^{a}_{\bar{\mu}\bar{\nu}}(x)={\rm
Ad}(\kappa)^{ab}b^{b}_{\bar{\mu}\bar{\nu}}(x).$ (4.40)

This transformation property is consistent with (3.3). The vacua of (4.15) are
given by (4.2), (2.25), (4.39) and (4.40) on each patch. $b^{a}_{\mu}$ and
${b^{\prime}}^{a}_{\mu}$ are known as the gauge field of the Yang monopole
[47].

By applying the arguments in the previous subsection, we can show that YM on
$S^{7}$ is equivalent to the theory around the multi Yang monopole background
of YM-higgs on $S^{4}$ with the periodicity imposed.

##  5 Gauge theories on $SU(n+1)(/H)$ and matrix model

In this section, we reveal various relations among gauge theories on $SU(n+1)$
and $SU(n+1)/H$, where $H$ is $SU(n)$ or $SU(n)\times U(1)$ or $SU(n+1)$ which
is a subgroup of $SU(n+1)$. Note that $SU(n+1)/SU(n)\simeq S^{2n+1}$ and
$SU(n+1)/(SU(n)\times U(1))\simeq CP^{n}$ and for $H=SU(n+1)$ the
corresponding gauge theory reduces to a matrix model. First, we develop a
general formalism of a dimensional reduction by which one can obtain YM-higgs
on $\tilde{G}/H$ from YM on $\tilde{G}$, where $\tilde{G}$ is an arbitrary
group manifold. Applying this formalism to the case of $\tilde{G}=SU(n+1)$, we
obtain YM-higgs on $S^{2n+1}$ and on $CP^{n}$ and the matrix model. Next, by
using the facts explained in appendix E, we show that the YM-higgs on $CP^{n}$
in the most general $U(1)$ monopole background is obtained by taking the
commutative limit of the theory around a certain background of the matrix
model. We have found the correct form of the YM-higgs type action of such
theory on $CP^{n}$. Third, by using the extended matrix T-duality of the
$U(1)$ case reviewed in section 4, we show that YM-higgs on $S^{2n+1}$ is
equivalent to the theory around a certain background of YM-higgs on $CP^{n}$
with the orbifolding condition imposed. Combining these two facts, we also
show that YM-higgs on $S^{2n+1}$ is realized as the theory around an
appropriate background of the matrix model with the orbifolding condition
imposed. Finally, by using the results in section 4, we show that YM on
$SU(n+1)$ is realized in YM-higgs on $SU(N+1)/(SU(2)^{k}\times U(1)^{l})$. In
particular, it follows that YM on $SU(3)$ is realized in YM-higgs on $S^{5}$
and on $CP^{2}$.

###  5.1 Dimensional reduction of YM theory on a group manifold

In this subsection, we restrict ourselves to the case in which the total space
$P$ is itself a group manifold $\tilde{G}$. In this case, we can take the
Maurer-Cartan basis and rewrite the YM action on $\tilde{G}$ in such a way
that the relation between YM on the total space and YM-higgs on the base space
becomes more manifest. In terms of this expression of the YM action, we can
easily perform the dimensional reduction to obtain the YM-higgs theory on a
coset space $\tilde{G}/H$, where $H$ is a subgroup of $\tilde{G}$. Some
conventions on the group manifold $\tilde{G}$ and the coset space
$\tilde{G}/H$ are summarized in appendix C.

Let us consider pure YM on $\tilde{G}$. In the Maurer-Cartan basis, the gauge
potential is written as $A=X_{A}E^{A}$ where $E^{A}$ are the right invariant
1-forms on $\tilde{G}$ which are defined in (C.2). In this basis, the field
strength is written as

$\displaystyle F$ $\displaystyle=$ $\displaystyle dA+iA\wedge A$ (5.1)
$\displaystyle=$ $\displaystyle\frac{1}{2}\left(f_{ABC}X_{C}+i{\cal
L}_{A}X_{B}-i{\cal L}_{B}X_{A}+i[X_{A},X_{B}]\right)E^{A}\wedge E^{B},$

where we have used the Maurer-Cartan equation (C.3) and ${\cal L_{A}}$ are the
right invariant Killing vectors on $\tilde{G}$ which are defined in (C.13).
This is a counterpart of (2.20). Then, the original YM action on $\tilde{G}$
is rewritten as follows:

$\displaystyle\frac{1}{g_{\tilde{G}}^{2}}\int{\rm
tr}\left(\frac{1}{2}F\wedge*F\right)=\frac{1}{g_{\tilde{G}}^{2}}\int
d^{D}z\sqrt{G}\;{\rm tr}\left\\{\frac{1}{4}\left(f_{ABC}X_{C}+i{\cal
L}_{A}X_{B}-i{\cal L}_{B}X_{A}+i[X_{A},X_{B}]\right)^{2}\right\\},$ (5.2)

where $D={\rm dim}(\tilde{G})$, $G=\det G_{MN}$ and $G_{MN}$ is the metric on
$\tilde{G}$. Note that the gauge transformation in this basis is given by

$\displaystyle X_{A}\rightarrow UX_{A}U^{-1}-{\cal L}_{A}U\>U^{-1}.$ (5.3)

As explained in appendix C, if one drops the derivatives along the fiber
direction in ${\cal L}_{A}$, these operators are reduced to the $L_{A}$ which
are the Killing vectors on $\tilde{G}/H$ defined in (C.16). By dropping the
derivatives along the fiber direction in ${\cal L}_{A}$ in (5.2), therefore,
we can obtain the theory on $\tilde{G}/H$,

$\displaystyle\frac{1}{g^{2}_{\tilde{G}}}\int{\rm
tr}\left(\frac{1}{2}F\wedge*F\right)\rightarrow\frac{1}{g^{2}_{\tilde{G}/H}}\int
d^{d}x\sqrt{g}\;{\rm
tr}\left\\{\frac{1}{4}\left(f_{ABC}X_{C}+iL_{A}X_{B}-iL_{B}X_{A}+i[X_{A},X_{B}]\right)^{2}\right\\},$
(5.4)

where $g^{2}_{\tilde{G}/H}=g^{2}_{\tilde{G}}/{\rm Vol}(H)$,
$d=\dim{\tilde{G}/H}$, $g=\det g_{\mu\nu}$ and $g_{\mu\nu}$ is the metric on
$\tilde{G}/H$. This is a counterpart of (2.18).

The action (5.4) is also rewritten into the YM-higgs form which was obtained
in section 3. The relation between the fields $X_{A}$ and the gauge and higgs
fields on $\tilde{G}/H$ is given as follows. We introduce the orthogonal
vectors to $L_{A}$ as

$N_{A}^{a}={\rm Ad}(L(x))_{A}{}^{a},$ (5.5)

where $L(x)$ is a representative element of $\tilde{G}/H$ which is defined in
(C.1), and ${\rm Ad}$ represents the adjoint action: $gT^{A}g^{-1}=T^{B}{\rm
Ad}(g)_{BA}$. One can show the orthonormality conditions,

$L_{A}^{\mu}L_{A}^{\nu}=-g^{\mu\nu},\;\;N_{A}^{a}N_{A}^{b}=\delta^{ab},\;\;L_{A}^{\mu}N_{A}^{a}=0,$
(5.6)

where $g^{\mu\nu}$ is the inverse of the metric on $\tilde{G}/H$. Furthermore,
the following equalities hold:

$\displaystyle
L_{A}^{\mu}\partial_{\mu}N_{B}^{a}-L_{B}^{\mu}\partial_{\mu}N_{A}^{a}=-2iL_{A}^{\mu}L_{B}^{\nu}b_{\mu\nu}^{a}-f_{abc}(L_{A}^{\mu}N_{B}^{b}-L_{B}^{\mu}N_{A}^{b})b_{\mu}^{c},$
$\displaystyle
f_{ABC}N_{C}^{a}-f_{abc}N_{A}^{b}N_{B}^{c}+L_{A}^{\mu}L_{B}^{\nu}b_{\mu\nu}^{a}=0.$
(5.7)

We decompose $X_{A}$ into the gauge and higgs fields in terms of $L_{A}^{\mu}$
and $N_{A}^{a}$ as follows [33]:

$X_{A}=iL_{A}^{\mu}a_{\mu}+N_{A}^{a}\phi_{a}.$ (5.8)

This is a generalization of (2.16). Then, each term in the action (5.4) is
rewritten as

$\displaystyle f_{ABC}X_{C}$ $\displaystyle=$ $\displaystyle
if_{ABC}L_{C}^{\mu}a_{\mu}-L_{A}^{\mu}L_{B}^{\nu}b_{\mu\nu}^{a}\phi_{a}+f_{abc}N_{A}^{b}N_{B}^{c}\phi_{a},$
$\displaystyle iL_{A}X_{B}-iL_{B}X_{A}$ $\displaystyle=$ $\displaystyle-
if_{ABC}L_{C}^{\mu}a_{\mu}-L_{A}^{\mu}L_{B}^{\nu}(\partial_{\mu}a_{\nu}-\partial_{\nu}a_{\mu}-2b_{\mu\nu}^{a}\phi_{a})$
$\displaystyle+i(L_{A}^{\mu}N_{B}^{a}-L_{B}^{\mu}N_{A}^{a})(\partial_{\mu}\phi_{a}-f_{abc}b_{\mu}^{b}\phi_{c}),$
$\displaystyle i[X_{A},X_{B}]$ $\displaystyle=$ $\displaystyle-
iL_{A}^{\mu}L_{B}^{\nu}[a_{\mu},a_{\nu}]-(L_{A}^{\mu}N_{B}^{a}-L_{B}^{\mu}N_{A}^{a})[a_{\mu},\phi_{a}]+iN_{A}^{a}N_{B}^{b}[\phi_{b},\phi_{a}],$
(5.9)

where we have used (5.7). By substituting these equations into the action
(5.4) and using (5.6), we indeed obtain the YM-higgs type action (3.20),

$\displaystyle S_{\tilde{G}/H}=\frac{1}{g^{2}_{\tilde{G}/H}}\int
d^{d}x\sqrt{g}\;{\rm
tr}\Biggl{\\{}\frac{1}{4}\left(f_{\mu\nu}-b_{\mu\nu}^{a}\phi_{a}\right)^{2}+\frac{1}{2}\left(D_{\mu}\phi_{a}-f_{abc}b_{\mu}^{b}\phi_{c}\right)^{2}$
$\displaystyle+\frac{1}{4}\left(f_{abc}\phi_{c}+i[\phi_{a},\phi_{b}]\right)^{2}\Biggr{\\}}.$
(5.10)

Finally, we consider the case in which $P=\tilde{G}$ and the base manifold is
just a point. This is the special case of the above dimensional reduction in
which $H$ equals $\tilde{G}$ itself. In this case, the theory on the base
space is given by a zero-dimensional matrix model. Dropping all the
derivatives in (5.2), we can easily make a dimensional reduction to the matrix
model:

$\displaystyle\frac{1}{g^{2}_{\tilde{G}}}\int{\rm
tr}\left(\frac{1}{2}F\wedge*F\right)\rightarrow\frac{1}{g^{2}_{mm}}\;{\rm
tr}\left\\{\frac{1}{4}\left(f_{ABC}X_{C}+i[X_{A},X_{B}]\right)^{2}\right\\},$
(5.11)

where $g^{2}_{mm}=g^{2}_{\tilde{G}}/{\rm Vol}(\tilde{G})$. This is a
counterpart of (2.19). Of course, we can obtain the matrix model (5.11) also
from the theory (5.4) on $\tilde{G}/H$ by dropping the derivatives $L_{A}$. If
we regard the original YM on $\tilde{G}$ as YM on a principal $\tilde{G}$
bundle over a point, we obtain (5.11) as a special case of (3.20).

###  5.2 Dimensional reduction of YM theory on $SU(n+1)$

In this subsection, we derive the YM-higgs on $S^{2n+1}$ and on $CP^{n}$ by
applying the dimensional reduction discussed in the previous subsection. We
also derive the 0-dimensional matrix model in which the YM-higgs on $S^{2n+1}$
and on $CP^{n}$ will be realized.

Let us consider the group manifold $SU(n+1)$. We can apply the dimensional
reduction developed in section 5.1 to the case of $P=\tilde{G}=SU(n+1)$ and
obtain a theory on a coset space $SU(n+1)/H$, where $H$ is a subgroup of
$SU(n+1)$. We begin with pure YM on the group manifold $SU(n+1)$ in the
Maurer-Cartan basis,

$S_{SU(n+1)}=\frac{1}{g^{2}_{SU(n+1)}}\int d^{n(n+2)}z\sqrt{G}\;{\rm
tr}\left\\{\frac{1}{4}\left(f_{ABC}X_{C}+i{\cal L}_{A}X_{B}-i{\cal
L}_{B}X_{A}+i[X_{A},X_{B}]\right)^{2}\right\\},$ (5.12)

where $f_{ABC}$ is the structure constant of $SU(n+1)$, $G=\det G_{MN}$ and
$G_{MN}$ is the Cartan-Killing metric on $\tilde{G}$ which is defined in
(C.10).

Let us consider the dimensional reduction of the above theory to a theory on
$\tilde{G}/H$. If we take $H$ to be $SU(n)$, the coset space is given by
$SU(n+1)/SU(n)\simeq S^{2n+1}$. By applying the dimensional reduction (5.4) to
YM on $SU(n+1)$, therefore, we obtain the YM-higgs theory on $S^{2n+1}$,

$S_{S^{2n+1}}=\frac{1}{g^{2}_{S^{2n+1}}}\int
d^{2n+1}\tilde{x}\sqrt{\tilde{g}}\;{\rm
tr}\left\\{\frac{1}{4}\left(f_{ABC}X_{C}+i\tilde{L}_{A}X_{B}-i\tilde{L}_{B}X_{A}+i[X_{A},X_{B}]\right)^{2}\right\\},$
(5.13)

where $\tilde{g}$ represents the determinant of the metric on $S^{2n+1}$, and
$\tilde{L}_{A}$’s are the Killing vectors on $S^{2n+1}$. Note that $S^{2n+1}$
that we consider here possesses only $SU(n+1)$ isometry which is smaller than
$SO(2n+2)$. In fact, this is not the ordinary round sphere but a squashed
sphere. In the case of $n=2$, the metric of this squashed $S^{5}$ is
explicitly given in appendix D.

Next, we consider the case of $H=SU(n)\times U(1)$. In this case, the coset
space is $SU(n+1)/(SU(n)\times U(1))\simeq CP^{n}$. Then, we can obtain the
theory on $CP^{n}$ from YM on $SU(n+1)$ through the dimensional reduction,

$S_{CP^{n}}=\frac{1}{g^{2}_{CP^{n}}}\int d^{2n}x\sqrt{g}\;{\rm
tr}\left\\{\frac{1}{4}\left(f_{ABC}X_{C}+iL_{A}X_{B}-iL_{B}X_{A}+i[X_{A},X_{B}]\right)^{2}\right\\}.$
(5.14)

As in the case of $S^{2n+1}$, $g=\det g_{\mu\nu}$, $g_{\mu\nu}$ and $L_{A}$
represent the metric and the Killing vectors on $CP^{n}$ respectively. The
theory (5.14) can be obtained also from the theory (5.13) by dropping the
derivative along the extra $U(1)$ fiber direction. We can also rewrite (5.13)
and (5.14) into the YM-higgs type actions as in (5.10) by using the relation
(5.8). For example, (5.14) is rewritten into (5.10) with $\mu,\nu=1,\cdots,2n$
and $a,b,c=0,\cdots,n^{2}-1$. Here, $a,b,c$ are indices of $SU(n)\times U(1)$
and $a=0$ corresponds to the $U(1)$ direction.

Finally, we consider the case in which $H$ is $SU(n+1)$ itself. In this case,
the coset space is just a point. Then, we obtain the following matrix model by
using (5.11):

$S_{mm}=\frac{1}{g^{2}_{mm}}\;{\rm
tr}\left\\{\frac{1}{4}\left(f_{ABC}X_{C}+i[X_{A},X_{B}]\right)^{2}\right\\}.$
(5.15)

This theory is used to realize the theories (5.13) and (5.14) in the next
subsection. For $n=1$, the dimensional reductions in this subsection are
equivalent to those in section 2.

###  5.3 Relations among gauge theories on $SU(n+1)/H$

In this subsection, we show that the theory (5.14) in a monopole background
can be realized by taking the commutative limit of the theory around a
nontrivial background of (5.15). Combining this construction and the matrix
T-duality, we also show that the theory (5.13) on $S^{2n+1}$ can be realized
as the theory around a certain background of the matrix model with the
orbifolding condition imposed. Furthermore, we apply the extended matrix
T-duality developed in section 4 to YM-higgs on $SU(n+1)/(SU(2)^{k}\times
U(1)^{l})$ and show that YM on $SU(n+1)$ is equivalent to the theory around a
certain vacuum of YM-higgs on $SU(n+1)/(SU(2)^{k}\times U(1)^{l})$ with the
periodicity condition imposed. For $n=2$, we obtain YM on $SU(3)$ from YM-
higgs on $S^{5}$ and on $CP^{2}$ through the extended matrix T-duality.

First, we review nontrivial backgrounds of the theory (5.14) on $CP^{n}$ and
the matrix model (5.15). The theory on $CP^{n}$ has many nontrivial monopole
vacua. In particular, we focus on the $U(1)$ monopole background. Recall that
we have $n^{2}$ higgs fields $\phi_{a}$. In the $U(1)$ monopole background,
only the higgs field along the $U(1)$ direction $\phi_{0}$ acquires its
nonzero vacuum expectation value. In the gauge where $\phi_{0}$ is diagonal,
the vacuum configurations of the $U(1)$ monopole with the gauge group $U(M)$
are given by

$\displaystyle\hat{a}_{\mu}$ $\displaystyle=b_{\mu}^{0}\hat{\phi}_{0},$
$\displaystyle\hat{\phi}_{0}$ $\displaystyle=-\frac{1}{\sqrt{2n(n+1)}}{\rm
diag}(\cdots,\underbrace{n_{s-1},\cdots,n_{s-1}}_{N_{s-1}},\underbrace{n_{s},\cdots,n_{s}}_{N_{s}},\underbrace{n_{s+1},\cdots,n_{s+1}}_{N_{s+1}},\cdots),$
$\displaystyle\hat{\phi}_{a}$ $\displaystyle=0,\;\;({\rm for}\;\;a\neq 0).$
(5.16)

Here, $\sum_{s}N_{s}=M$ and $n_{s}$ must be integers due to Dirac’s
quantization condition. Because of (5.8), the vacuum configurations of $X_{A}$
are equivalently given by

$\hat{X}_{A}=-\frac{iL_{A}^{\mu}b_{\mu}^{0}+N_{A}^{0}}{\sqrt{2n(n+1)}}\;{\rm
diag}(\cdots,\underbrace{n_{s-1},\cdots,n_{s-1}}_{N_{s-1}},\underbrace{n_{s},\cdots,n_{s}}_{N_{s}},\underbrace{n_{s+1},\cdots,n_{s+1}}_{N_{s+1}},\cdots).$
(5.17)

The theory around the background (5.17) is obtained by expanding each block of
the fields in (5.14) as
$X_{A}^{(s,t)}\rightarrow\hat{X}_{A}^{(s,t)}+X_{A}^{(s,t)}$. Then, the
following action is obtained,

$\displaystyle\frac{1}{g^{2}_{CP^{n}}}\int d^{2n}x\sqrt{g}\sum_{s,t}\;{\rm
tr}\Big{\\{}\frac{1}{4}$
$\displaystyle\left(f_{ABC}X^{(s,t)}_{C}+iL_{A}^{(q_{st})}X_{B}^{(s,t)}-iL_{B}^{(q_{st})}X_{A}^{(s,t)}+i[X_{A},X_{B}]^{(s,t)}\right)$
$\displaystyle\times$
$\displaystyle\left(f_{ABD}X^{(t,s)}_{D}+iL_{A}^{(q_{ts})}X_{B}^{(t,s)}-iL_{B}^{(q_{ts})}X_{A}^{(t,s)}+i[X_{A},X_{B}]^{(t,s)}\right)\Big{\\}},$
(5.18)

where $q_{st}=\frac{n_{s}-n_{t}}{2}$ and $L_{A}^{(q)}$ are the angular
momentum operators in the presence of a monopole with the magnetic charge $q$,
which take the form

$L_{A}^{(q)}=L_{A}+\frac{2q}{\sqrt{2n(n+1)}}(iL_{A}^{\mu}b_{\mu}^{0}+N_{A}^{0}).$
(5.19)

These operators are the generalization of (2.27) in the case of $S^{2}$.

The vacua of the theory (5.15) are determined by

$[X_{A},X_{B}]=if_{ABC}X_{C}.$ (5.20)

In addition to the trivial solution $X_{A}=0$, there are nontrivial solutions
which are given by the representation matrices of the $SU(n+1)$ generators,

$\hat{X}_{A}=\hat{L}_{A}.$ (5.21)

$\hat{L}_{A}$ are generally in a reducible representation. In order to
construct a theory on $CP^{n}$ in a $U(1)$ monopole background, we consider
the following representation:

$\displaystyle\hat{L}_{A}=\begin{pmatrix}\rotatebox[origin={tl}]{-35.0}{$\cdots\;\;\;\overbrace{\rotatebox[origin={c}]{35.0}{$\hat{L}_{A}^{(s-1)}$}\;\cdots\;\rotatebox[origin={c}]{35.0}{$\hat{L}_{A}^{(s-1)}$}}^{\rotatebox{35.0}{$N_{s-1}$}}\;\;\;\overbrace{\rotatebox[origin={c}]{35.0}{$\hat{L}_{A}^{(s)}$}\;\cdots\;\rotatebox[origin={c}]{35.0}{$\hat{L}_{A}^{(s)}$}}^{\rotatebox{35.0}{$N_{s}$}}\;\;\;\overbrace{\rotatebox[origin={c}]{35.0}{$\hat{L}_{A}^{(s+1)}$}\;\cdots\;\rotatebox[origin={c}]{35.0}{$\hat{L}_{A}^{(s+1)}$}}^{\rotatebox{35.0}{$N_{s+1}$}}\;\;\;\cdots$}\end{pmatrix}.$
(5.22)

Here $\hat{L}_{A}^{(s)}$ are the abbreviations of
$\hat{L}_{A}^{[\Lambda_{s},0,\cdots,0]}$ which are the generators of $SU(n+1)$
in the irreducible representation specified by the Dynkin index of $SU(n+1)$,
$[\Lambda_{s},0,\cdots,0]$. We consider the matrix model (5.15) around the
background (5.22) by expanding the each block of the fields around the
background :$X_{A}^{(s,t)}\rightarrow\hat{X}_{A}^{(s,t)}+X_{A}^{(s,t)}$. Then,
the action takes the following form:

$\displaystyle S_{mm}=\frac{1}{g^{2}_{mm}}\sum_{s,t}\;{\rm
tr}\Big{\\{}\frac{1}{4}$
$\displaystyle\left(f_{ABC}X_{C}^{(s,t)}+i\hat{L}_{A}\circ
X_{B}^{(s,t)}-i\hat{L}_{B}\circ X_{A}^{(s,t)}+i[X_{A},X_{B}]^{(s,t)}\right)$
$\displaystyle\times$
$\displaystyle\left(f_{ABD}X_{D}^{(t,s)}+i\hat{L}_{A}\circ
X_{B}^{(t,s)}-i\hat{L}_{B}\circ
X_{A}^{(t,s)}+i[X_{A},X_{B}]^{(t,s)}\right)\Big{\\}}.$ (5.23)

$\hat{L}_{A}\circ$ are defined as

$\hat{L}_{A}\circ
X_{B}^{(s,t)}\equiv\hat{L}_{A}^{(s)}X_{B}^{(s,t)}-X_{B}^{(s,t)}\hat{L}_{A}^{(t)}.$
(5.24)

We show in the following that the theory (5.23) is equivalent to the theory
(5.18) if we put $\Lambda_{s}=N_{0}+n_{s}$ and take $N_{0}\rightarrow\infty$
limit. In order to show this equivalence, we make a harmonic expansion [31,
32, 35]. As explained in appendix E, the $(s,t)$ blocks $X_{A}^{(s,t)}$ in the
matrix model are expanded by the basis of rectangular matrices (E.21) as

$X_{A}^{(s,t)}=\sum_{J=|q_{st}|}^{(\Lambda_{s}+\Lambda_{t})/2}X_{A}^{(s,t)}{}^{{\bm{\beta}}_{J+q_{st}}}{}_{{\bm{\alpha}}_{J-q_{st}}}\otimes\hat{Y}^{(q_{st})}{}_{{\bm{\beta}}_{J+q_{st}}}{}^{{\bm{\alpha}}_{J-q_{st}}}.$
(5.25)

Then, the diagonal coherent map allows us to map the $(s,t)$ blocks to local
sections of the monopole bundle on $CP^{n}$ with the charge $q_{st}$,

$\displaystyle X_{A}^{(s,t)}$
$\displaystyle=\sum_{J=q_{st}}^{\infty}X_{A}^{(q_{st})}{}^{{\bm{\beta}}_{J+q_{st}}}{}_{{\bm{\alpha}}_{J-q_{st}}}\otimes\hat{Y}^{(q_{st})}{}_{{\bm{\beta}}_{J+q_{st}}}{}^{{\bm{\alpha}}_{J-q_{st}}}$
$\displaystyle\rightarrow\sum_{J=q_{st}}^{\infty}X_{A}^{(q_{st})}{}^{{\bm{\beta}}_{J+q_{st}}}{}_{{\bm{\alpha}}_{J-q_{st}}}\tilde{Y}^{(q_{st})}{}_{{\bm{\beta}}_{J+q_{st}}}{}^{{\bm{\alpha}}_{J-q_{st}}}(w,\bar{w})=X_{A}^{CP(s,t)}(w,\bar{w}),$
(5.26)

where we have taken the commutative limit $N_{0}\rightarrow\infty$ and
$\tilde{Y}^{(q_{st})}{}_{{\bm{\beta}}_{J+q_{st}}}{}^{{\bm{\alpha}}_{J-q_{st}}}$
are the basis of local sections of the $U(1)$ monopole bundle on $CP^{n}$
which are defined in (E.26). Note that we have put the superscript $CP$ on the
quantity in the righthand side of the above equation in order to emphasis that
the $X_{A}^{CP(s,t)}$ are the fields on $CP^{n}$ appearing in (5.18).
Similarly, $\hat{L}_{A}\circ$ is mapped to $L_{A}^{(q)}$ as shown in
(E.31)444In [32], (5.26) and (5.27) are proven to the quadratic order in the
fields for all $q$ and to all order for $q=0$. In this paper, we assume that
these are also valid to all order for all $q$.:

$\hat{L}_{A}\circ X^{(s,t)}_{B}\rightarrow
L_{A}^{(q)}X^{CP(s,t)}_{B}(w,\bar{w}).$ (5.27)

Using (5.26) and (5.27), we find that the matrix model (5.23) is equivalent to
the theory (5.18) on $CP^{n}$ in the commutative limit
$N_{0}\rightarrow\infty$.

Next, we show that the theory around a certain vacuum of $U(M=N\times\infty)$
YM-higgs on $CP^{n}$ with a periodicity condition imposed is equivalent to
$U(N)$ YM-higgs on $S^{2n+1}$. This statement is nothing but the matrix
T-duality. As explained in section 4, therefore, we consider the appropriate
vacuum which is given by (5.17) (or equivalently (5.16)) with $s$ running from
$-\infty$ to $\infty$, $n_{s}=s$ and $N_{s}=N$. We expand the fields on
$CP^{n}$ around the background as

$\displaystyle X_{A}\rightarrow\hat{X}_{A}+X_{A},$ (5.28)

and impose the periodicity (orbifolding) condition on the fluctuation,

$\displaystyle X_{A}^{(s+1,t+1)}=X_{A}^{(s,t)}\equiv X_{A}^{(s-t)}.$ (5.29)

Then, we define the gauge and higgs fields on $S^{2n+1}$ by the Fourier
transforms of the fluctuations on each local coordinate patch:

$\displaystyle X_{A}^{S}=\sum_{u}X_{A}^{CP(u)}e^{-iuy},$ (5.30)

where $y$ is a coordinate which parameterizes the fiber ($U(1)$) direction and
satisfies $0\leq\tau\leq 2\pi$. Here, the superscripts $S$ and $CP$ indicate
that $X_{A}^{S}$ and $X_{A}^{CP(w)}$ are the fields on $S^{2n+1}$ and $CP^{n}$
respectively. We substitute (5.30) into (5.18) and divide an overall factor
$\sum_{s}$ to extract a single period. Then, we obtain $U(N)$ YM-higgs on
$S^{2n+1}$ written in the basis of $X_{A}$ (5.13).

Combining the above matrix T-duality and the construction of (5.18) in terms
of the matrix model, we find that the theory around (5.22) of the matrix
model, where $s$ runs from $-\infty$ to $\infty$ and $\Lambda_{s}=N_{0}+s$, is
equivalent to $U(N)$ YM-higgs on $S^{2n+1}$ if we take the limit
$N_{0}\rightarrow\infty$, impose the periodicity condition on the
fluctuations, and finally divide the overall factor $\sum_{s}$.

Finally, it is straightforward to apply the extended matrix T-duality to
$SU(2)^{k}\times U(1)^{l}$ bundle on $SU(n+1)/(SU(2)^{k}\times U(1)^{l})$ and
show that YM on $SU(n+1)$ is equivalent to the theory around a certain vacuum
of YM-higgs on $SU(n+1)/(SU(2)^{k}\times U(1)^{l})$ with the periodicity
condition imposed.

##  6 Interpretation as Buscher’s T-duality

In this section, let us see that the extended matrix T-duality of the $U(1)$
case, which was obtained in [29] and reviewed in section 4.2, is actually
interpreted as the T-duality in Buscher’s sense. We put $\mbox{dim}M=p$. For
$G=U(1)$, as in (4.6), the metric of the total space is given by

$\displaystyle ds^{2}=G_{MN}dz^{M}dz^{N}=g_{\mu\nu}dx^{\mu}dx^{\nu}+(dy-
b_{\mu}dx^{\mu})^{2},$ (6.1)

where $M,N=1,\cdots,p+1$ and $\mu,\nu=1,\cdots,p$. We assume that the other
fields such as the antisymmetric fields and the dilaton field are trivial.
Then, YM on the total space is viewed as the low energy effective theory for
the D$p$-branes wrapped on the total space555Here we ignore the transverse
directions.. We make the T-duality transformation for the fiber direction to
obtain a new geometry [36]:

$\displaystyle{ds^{\prime}}^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2},$
$\displaystyle{B^{\prime}}_{\mu\nu}=0,\;\;\;{B^{\prime}}_{\mu y}=-b_{\mu}.$
(6.2)

The D$p$-branes should be transformed to the D$(p-1)$-branes wrapped on the
base space. The D$(p-1)$-brane effective action on the new geometry (6.2) is
given by

$\displaystyle S_{p-1}=\tau_{p-1}\int d^{p}\sigma
e^{-\Phi}\sqrt{\det(\tilde{G}_{ab}+\tilde{B}_{ab}+2\pi\alpha^{\prime}F_{ab})},$
(6.3)

where $\sigma^{a}\;(a=1,\cdots,p)$ parameterize the world volume of the
D$(p-1)$-brane, and $\tilde{G}_{ab}$ and $\tilde{B}_{ab}$ are the pullback of
(6.2) on the world volume which is defined through the embedding of world
volume $z^{M}(\sigma)$ as

$\displaystyle\tilde{G}_{ab}=\frac{\partial
z^{M}}{\partial\sigma^{a}}\frac{\partial
z^{N}}{\partial\sigma^{b}}{G^{\prime}}_{MN},$
$\displaystyle\tilde{B}_{ab}=\frac{\partial
z^{M}}{\partial\sigma^{a}}\frac{\partial
z^{N}}{\partial\sigma^{b}}{B^{\prime}}_{MN}.$ (6.4)

In the static gauge $x^{\mu}(\sigma)=\sigma^{\mu}$ and
$z^{y}(\sigma)=2\pi\alpha^{\prime}\phi$, (6.3) reduces to

$\displaystyle S_{p-1}=\tau_{p-1}\int
d^{p}x\sqrt{\det(g_{\mu\nu}+(2\pi\alpha^{\prime})^{2}\partial_{\mu}\phi\partial_{\nu}\phi+2\pi\alpha^{\prime}(F_{\mu\nu}+\partial_{\mu}\phi
b_{\nu}-\partial_{\nu}\phi b_{\mu}))}.$ (6.5)

Up to ${\cal O}({\alpha^{\prime}}^{3})$, this equals

$\displaystyle\frac{1}{g_{YM}^{2}}\int
d^{p}x\sqrt{g}\left(\frac{1}{4}(F_{\alpha\beta}+\nabla_{\alpha}\phi
b_{\beta}-\nabla_{\beta}\phi
b_{\alpha})^{2}+\frac{1}{2}(\nabla_{\alpha}\phi)^{2}\right),$ (6.6)

where $g_{YM}^{2}=\frac{1}{4\pi\alpha^{2}\tau_{p-1}}$. If we redefine the
gauge field as $a_{\alpha}\rightarrow a_{\alpha}+b_{\alpha}\phi$ and make non-
abelianization, we obtain from (6.6)

$\displaystyle\frac{1}{g_{YM}^{2}}\int
d^{p}x\sqrt{g}\mbox{tr}\left(\frac{1}{4}(F_{\alpha\beta}-b_{\alpha\beta}\phi)^{2}+\frac{1}{2}(D_{\alpha}\phi)^{2}\right),$
(6.7)

which indeed agrees with (3.20) with $G=U(1)$.

## Acknowledgements

We would like to thank T. Higashi for discussions. The work of G.I. is
supported in part by the JSPS Research Fellowship for Young Scientists. The
work of A.T. is supported in part by Grant-in-Aid for Scientific Research (No.
19540294) from the Ministry of Education, Culture, Sports, Science and
Technology.

##  Appendix A Spherical harmonics

In this appendix, we review the spherical harmonics on $S^{3}$, the monopole
harmonics on $S^{2}$ [42] and the fuzzy spherical harmonics [24, 49, 15]. For
more details, see [48, 29] and references therein.

### A.1 Spherical harmonics on $S^{3}$

We regard $S^{3}$ as the $SU(2)$ group manifold. We parameterize an element of
$SU(2)$ in terms of the Euler angles as

$g=e^{-i\varphi J_{3}}e^{-i\theta J_{2}}e^{-i\psi J_{3}},$ (A.1)

where $J_{A}$ satisfy $[J_{A},J_{B}]=i\epsilon_{ABC}J_{C}$ and
$0\leq\theta\leq\pi$, $0\leq\varphi<2\pi$, $0\leq\psi<4\pi$. The isometry of
$S^{3}$ is $SO(4)=SU(2)\times SU(2)$, and these two $SU(2)$’s act on $g$ from
left and right, respectively. We construct the right invariant 1-forms,

$dgg^{-1}=-i\mu E^{A}J_{A},$ (A.2)

where the radius of $S^{3}$ is $2/\mu$. They are explicitly given by

$\displaystyle E^{1}=\frac{1}{\mu}(-\sin\varphi d\theta+\sin\theta\cos\varphi
d\psi),$ $\displaystyle E^{2}=\frac{1}{\mu}(\cos\varphi
d\theta+\sin\theta\sin\varphi d\psi),$ $\displaystyle
E^{3}=\frac{1}{\mu}(d\varphi+\cos\theta d\psi),$ (A.3)

and satisfy the Maure-Cartan equation

$dE^{A}-\frac{\mu}{2}\epsilon_{ABC}E^{B}\wedge E^{C}=0.$ (A.4)

The metric is constructed from $E^{A}$ as

$ds^{2}=E^{A}E^{A}=\frac{1}{\mu^{2}}\left(d\theta^{2}+\sin^{2}\theta
d\varphi^{2}+(d\psi+\cos d\varphi)^{2}\right).$ (A.5)

The Killing vectors dual to $E^{A}$ are given by

${\cal{L}}_{A}=-\frac{i}{\mu}E^{M}_{A}\partial_{M},$ (A.6)

where $E^{M}_{A}$ are inverse of $E^{A}_{M}$. The explicit form of the Killing
vectors are

$\displaystyle{\cal{L}}_{1}=-i\left(-\sin\varphi\partial_{\theta}-\cot\theta\cos\varphi\partial_{\varphi}+\frac{\cos\varphi}{\sin\theta}\partial_{\psi}\right),$
$\displaystyle{\cal{L}}_{2}=-i\left(\cos\varphi\partial_{\theta}-\cot\theta\sin\varphi\partial_{\varphi}+\frac{\sin\varphi}{\sin\theta}\partial_{\psi}\right),$
$\displaystyle{\cal{L}}_{3}=-i\partial_{\varphi}.$ (A.7)

Because of the Maure-Cartan equation (A.4), the Killing vectors satisfy the
SU(2) algebra, $[{\cal{L}}_{A},{\cal{L}}_{B}]=i\epsilon_{ABC}{\cal{L}}_{C}$.

The scalar spherical harmonics on $S^{3}$ are given by

$Y_{Jm\tilde{m}}(\Omega_{3})=(-1)^{J-\tilde{m}}\sqrt{2J+1}\langle
J-\tilde{m}|g^{-1}|Jm\rangle.$ (A.8)

These spherical harmonics form the basis of SU(2) algebra generated by
${\cal{L}}_{A}$’s.

$\displaystyle{\cal{L}}^{2}Y_{Jm\tilde{m}}$ $\displaystyle=$ $\displaystyle
J(J+1)Y_{Jm\tilde{m}},$ $\displaystyle{\cal{L}}_{\pm}Y_{Jm\tilde{m}}$
$\displaystyle=$ $\displaystyle\sqrt{(J\mp m)(J\pm m+1)}Y_{Jm\pm 1\tilde{m}},$
$\displaystyle{\cal{L}}_{3}Y_{Jm\tilde{m}}$ $\displaystyle=$ $\displaystyle
mY_{Jm\tilde{m}}.$ (A.9)

The complex conjugates of the spherical harmonics are evaluated as

$\left(Y_{Jm\tilde{m}}\right)^{*}=(-1)^{m-\tilde{m}}Y_{J-m-\tilde{m}}.$ (A.10)

The spherical harmonics also satisfy the orthonormality condition

$\int\frac{d\Omega_{3}}{2\pi^{2}}\left(Y_{Jm\tilde{m}}\right)^{*}Y_{J^{\prime}m^{\prime}\tilde{m}^{\prime}}=\delta_{JJ^{\prime}}\delta_{mm^{\prime}}\delta_{\tilde{m}\tilde{m}^{\prime}}.$
(A.11)

The integral of the product of three spherical harmonics is given as follows:

$\displaystyle{\cal{C}}^{J_{1}m_{1}\tilde{m}_{1}}_{J_{2}m_{2}\tilde{m}_{2}J_{3}m_{3}\tilde{m}_{3}}$
$\displaystyle\equiv$
$\displaystyle\int\frac{d\Omega_{3}}{2\pi^{2}}\left(Y_{J_{1}m_{1}\tilde{m}_{1}}\right)^{*}Y_{J_{2}m_{2}\tilde{m}_{2}}Y_{J_{3}m_{3}\tilde{m}_{3}}$
(A.12) $\displaystyle=$
$\displaystyle\sqrt{\frac{(2J_{2}+1)(2J_{3}+1)}{2J_{1}+1}}C^{J_{1}m_{1}}_{J_{2}m_{2}J_{3}m_{3}}C^{J_{1}\tilde{m}_{1}}_{J_{2}\tilde{m}_{2}J_{3}\tilde{m}_{3}},$

where $C^{J_{1}m_{1}}_{J_{2}m_{2}J_{3}m_{3}}$ is the Clebsch-Gordan
coefficient of $SU(2)$. Finally, the spherical harmonics satisfy the
completeness condition,

$\displaystyle\sum_{Jm\tilde{m}}\left(Y_{Jm\tilde{m}}\right)^{*}(\Omega_{3})Y_{Jm\tilde{m}}(\Omega_{3}^{\prime})=2\pi^{2}\delta(\Omega_{3}-\Omega_{3}^{\prime}),$
(A.13)

where

$\delta(\Omega_{3})=\frac{8}{\sin\theta}\delta(\theta)\delta(\varphi)\delta(\psi).$
(A.14)

### A.2 Monopole spherical harmonics on $S^{2}$

We adopt the following metric for $S^{2}$:

$ds^{2}=\frac{1}{\mu^{2}}(d\theta^{2}+\sin^{2}\theta d\varphi^{2}).$ (A.15)

We define two local patches on $S^{2}$ to describe nontrivial $U(1)$ bundles
over $S^{2}$: the patch I is specified by $0\leq\theta<\pi$ and the patch II
is specified by $0<\theta\leq\pi$. In the following expressions, the upper
sign is taken in the patch I and the lower sign in the patch II.

The angular momentum operator in the presence of a monopole with magnetic
charge $q$ at the origin takes the form

$\displaystyle
L_{1}^{(q)}=i(\sin\varphi\partial_{\theta}+\cot\theta\cos\varphi\partial_{\varphi})-q\frac{1\mp\cos\theta}{\sin\theta}\cos\varphi,$
$\displaystyle
L_{2}^{(q)}=i(-\cos\varphi\partial_{\theta}+\cot\theta\sin\varphi\partial_{\varphi})-q\frac{1\mp\cos\theta}{\sin\theta}\sin\varphi,$
$\displaystyle L_{3}^{(q)}=-i\partial_{\varphi}\mp q,$ (A.16)

where $q$ is quantized as $q=0,\pm\frac{1}{2},\pm 1,\pm\frac{3}{2},\cdots$.
These operators act on the local sections on $S^{2}$ and satisfy the $SU(2)$
algebra $[L_{A}^{(q)},L_{B}^{(q)}]=i\epsilon_{ABC}L_{C}^{(q)}$. Note that when
$q=0$, these operators are reduced to the ordinary angular momentum operators
on $S^{2}$ (or $R^{3}$). if we regard $S^{3}$ as a $U(1)$ bundle over $S^{2}$,
and parameterize the fiber direction by $y=\psi\pm\varphi$, the above
expression (A.16) can be obtained by making a replacement in (A.7):
$\partial_{y}\rightarrow-iq$.

The monopole spherical harmonics are the basis of local sections on $S^{2}$
and also form the basis of the $SU(2)$ algebra generated by $L_{A}^{(q)}$. The
monopole scalar spherical harmonics are given by

$\tilde{Y}_{Jmq}(\Omega_{2})=(-1)^{J-q}\sqrt{2J+1}\langle J-q|e^{i\theta
J_{2}}|Jm\rangle e^{i(\pm q+m)\varphi}.$ (A.17)

Here $J=|q|,|q|+1,|q+2|,\cdots$, $m=-J,-J+1,\cdots,J-1,J$. The existence of
the lower bound of the angular momentum $J$ is due to the fact that the
magnetic field produced by the monopole also has nonzero angular momentum.
Note that the monopole harmonics with $q=0$ do not transform on the overlap of
two patches. They correspond to global sections (functions) on $S^{2}$ which
are expressed by the ordinary spherical harmonics on $S^{2}$. The action of
$L^{(q)}_{A}$ on the monopole spherical harmonics is given by

$\displaystyle L^{(q)2}\tilde{Y}_{Jmq}$ $\displaystyle=$ $\displaystyle
J(J+1)\tilde{Y}_{Jmq},$ $\displaystyle L^{(q)}_{\pm}\tilde{Y}_{Jmq}$
$\displaystyle=$ $\displaystyle\sqrt{(J\mp m)(J\pm m+1)}\tilde{Y}_{Jm\pm 1q},$
$\displaystyle L^{(q)}_{3}\tilde{Y}_{Jmq}$ $\displaystyle=$ $\displaystyle
m\tilde{Y}_{Jmq}.$ (A.18)

The complex conjugates of the monopole spherical harmonics are evaluated as

$\left(\tilde{Y}_{Jmq}\right)^{*}=(-1)^{m-q}\tilde{Y}_{J-m-q}.$ (A.19)

The monopole spherical harmonics are orthonormal to each other,

$\int\frac{d\Omega_{2}}{4\pi}\left(\tilde{Y}_{Jmq}\right)^{*}\tilde{Y}_{J^{\prime}m^{\prime}q}=\delta_{JJ^{\prime}}\delta_{mm^{\prime}}.$
(A.20)

The integral of three monopole spherical harmonics is equal to the
corresponding integral (A.12) on $S^{3}$ with the identification
$\tilde{m}=q$,

$\displaystyle\int\frac{d\Omega_{2}}{4\pi}\left(\tilde{Y}_{J_{1}m_{1}q_{1}}\right)^{*}\tilde{Y}_{J_{2}m_{2}q_{2}}\tilde{Y}_{J_{3}m_{3}q_{3}}$
$\displaystyle=$
$\displaystyle\sqrt{\frac{(2J_{2}+1)(2J_{3}+1)}{2J_{1}+1}}C^{J_{1}m_{1}}_{J_{2}m_{2}J_{3}m_{3}}C^{J_{1}q_{1}}_{J_{2}q_{2}J_{3}q_{3}}$
(A.21) $\displaystyle=$
$\displaystyle{\cal{C}}^{J_{1}m_{1}q_{1}}_{J_{2}m_{2}q_{2}J_{3}m_{3}q_{3}},$

where the monopole charges must be conserved in the lefthand side of the above
equation as $q_{1}+q_{2}+q_{3}=0$. Note that the monopole spherical harmonics
are expressed in terms of the spherical harmonics on $S^{3}$:

$\displaystyle\tilde{Y}_{Jmq}(\Omega_{2})$ $\displaystyle=$ $\displaystyle
e^{iq(\psi\pm\varphi)}Y_{Jmq}(\Omega_{3}),$ $\displaystyle
L^{(q)}_{A}\tilde{Y}_{Jmq}(\Omega_{2})$ $\displaystyle=$ $\displaystyle
e^{iq(\psi\pm\varphi)}{\cal L}_{A}Y_{Jmq}(\Omega_{3}).$ (A.22)

(A.21) and (A.22) represent a map between the local sections on $S^{2}$ and
the Kaluza-Klein modes on $S^{3}$.

### A.3 Fuzzy spherical harmonics

Let us consider $(2j+1)\times(2j^{\prime}+1)$ rectangular complex matrices.
Such matrices are generally expressed as

$M=\sum_{r,r^{\prime}}M_{rr^{\prime}}|jr\rangle\langle j^{\prime}r^{\prime}|.$
(A.23)

We can define linear maps $\hat{L}_{A}\circ$, which map the set of
$(2j+1)\times(2j^{\prime}+1)$ rectangular complex matrices to itself, by their
operation on the basis:

$\hat{L}_{A}\circ|jr\rangle\langle
j^{\prime}r^{\prime}|\equiv\hat{L}_{A}^{[j]}|jr\rangle\langle
j^{\prime}r^{\prime}|-|jr\rangle\langle
j^{\prime}r^{\prime}|\hat{L}_{A}^{[j^{\prime}]},$ (A.24)

where $\hat{L}_{A}^{[j]}$ are the spin $j$ representation matrices of the
$SU(2)$ generators. $\hat{L}_{A}\circ$ satisfy the $SU(2)$ algebra
$[\hat{L}_{A}\circ,\hat{L}_{B}\circ]=i\epsilon_{ABC}\hat{L}_{C}\circ$.

We make a change of a basis of the rectangular matrices from the above basis
$\\{|jr\rangle\langle j^{\prime}r^{\prime}|\\}$ to the new basis which is
called the fuzzy spherical harmonics:

$\hat{Y}_{Jm(jj^{\prime})}=\sqrt{N_{0}}\sum_{r,r^{\prime}}(-1)^{-j+r^{\prime}}C^{Jm}_{jr\;j^{\prime}-r^{\prime}}|jr\rangle\langle
j^{\prime}r^{\prime}|,$ (A.25)

where $N_{0}$ is a positive integer which will be specified below. For a fixed
$J$ the fuzzy spherical harmonics also form a basis of the spin $J$
irreducible representation of $SU(2)$ which is generated by
$\hat{L}_{A}\circ$,

$\displaystyle(\hat{L}_{A}\circ)^{2}\hat{Y}_{Jm(jj^{\prime})}$
$\displaystyle=$ $\displaystyle J(J+1)\hat{Y}_{Jm(jj^{\prime})},$
$\displaystyle\hat{L}_{\pm}\circ\hat{Y}_{Jm(jj^{\prime})}$ $\displaystyle=$
$\displaystyle\sqrt{(J\mp m)(J\pm m+1)}\hat{Y}_{Jm\pm 1(jj^{\prime})},$
$\displaystyle\hat{L}_{3}\circ\hat{Y}_{Jm(jj^{\prime})}$ $\displaystyle=$
$\displaystyle m\hat{Y}_{Jm(jj^{\prime})}.$ (A.26)

The hermitian conjugates of the fuzzy spherical harmonics are evaluated as

$\left(\hat{Y}_{Jm(jj^{\prime})}\right)^{\dagger}=(-1)^{m-(j-j^{\prime})}\hat{Y}_{J-m(j^{\prime}j)}.$
(A.27)

The fuzzy spherical harmonics satisfy the orthonormality condition under the
following normalized trace:

$\frac{1}{N_{0}}{\rm
tr}\left\\{\left(\hat{Y}_{Jm(jj^{\prime})}\right)^{\dagger}\hat{Y}_{J^{\prime}m^{\prime}(jj^{\prime})}\right\\}=\delta_{JJ^{\prime}}\delta_{mm^{\prime}},$
(A.28)

where ${\rm tr}$ stands for the trace over
$(2j^{\prime}+1)\times(2j^{\prime}+1)$ matrices. The trace of three fuzzy
spherical harmonics is given by

$\displaystyle\hat{C}^{J_{1}m_{1}(jj^{\prime\prime})}_{J_{2}m_{2}(jj^{\prime})J_{3}m_{3}(j^{\prime}j^{\prime\prime})}$
$\displaystyle\equiv\frac{1}{N_{0}}{\rm
tr}\left\\{\left(\hat{Y}_{J_{1}m_{1}(jj^{\prime\prime})}\right)^{\dagger}\hat{Y}_{J_{2}m_{2}(jj^{\prime})}\hat{Y}_{J_{3}m_{3}(j^{\prime}j^{\prime\prime})}\right\\}$
$\displaystyle=(-1)^{J_{1}+j+j^{\prime\prime}}\sqrt{N_{0}(2J_{2}+1)(2J_{3}+1)}C^{J_{1}m_{1}}_{J_{2}m_{2}J_{3}m_{3}}\left\\{\begin{array}[]{ccc}J_{1}&J_{2}&J_{3}\\\
j^{\prime}&j^{\prime\prime}&j\end{array}\right\\},$ (A.31)

where the last factor of the above equation is the $6-j$ symbol.

In order to reveal relationships among the fuzzy spherical harmonics, the
monopole harmonics on $S^{2}$ and the spherical harmonics on $S^{3}$, we
introduce the following parameterization for $j$, $j^{\prime}$ and
$j^{\prime\prime}$,

$2j+1=N_{0}+\zeta,\;\;2j^{\prime}+1=N_{0}+\zeta^{\prime},\;\;2j^{\prime\prime}+1=N_{0}+\zeta^{\prime\prime}.$
(A.32)

$\zeta$, $\zeta^{\prime}$ and $\zeta^{\prime\prime}$ are integers which are
grater than $-N_{0}$. Then, in the limit $N_{0}\rightarrow\infty$, one can
show that

$\hat{C}^{J_{1}m_{1}(jj^{\prime\prime})}_{J_{2}m_{2}(jj^{\prime})J_{3}m_{3}(j^{\prime}j^{\prime\prime})}\rightarrow{\cal{C}}^{J_{1}m_{1}q_{1}}_{J_{2}m_{2}q_{2}J_{3}m_{3}q_{3}}$
(A.33)

with the identification $j-j^{\prime\prime}=q_{1}$, $j-j^{\prime}=q_{2}$ and
$j^{\prime}-j^{\prime\prime}=q_{3}$. This relation can be proved by using the
following asymptotic form of the $6-j$ symbols. If $R\gg 1$, one obtains [50]

$\left\\{\begin{array}[]{ccc}a&b&c\\\ d+R&e+R&f+R\\\
\end{array}\right\\}\approx\frac{(-1)^{a+b+c+2(d+e+f+R)}}{\sqrt{2R}}\left(\begin{array}[]{ccc}a&b&c\\\
e-f&f-d&d-e\\\ \end{array}\right),$ (A.34)

where the $3-j$ symbol is related to the Clebsch-Gordan coefficient as

$\left(\begin{array}[]{ccc}J_{1}&J_{2}&J_{3}\\\ m_{1}&m_{2}&m_{3}\\\
\end{array}\right)=(-1)^{J_{3}+m_{3}+2J_{1}}\frac{1}{\sqrt{2J_{3}+1}}C^{J_{3}m_{3}}_{J_{1}-m_{1}J_{2}-m_{2}}.$
(A.35)

The relation (A.33) implies that the fuzzy spherical harmonics
$\hat{Y}_{Jm(jj^{\prime})}$ give a matrix regularization of the monopole
harmonics $\tilde{Y}_{Jmq}$ through the following correspondence:

$\displaystyle j-j^{\prime}$ $\displaystyle\leftrightarrow$ $\displaystyle q,$
$\displaystyle\hat{L}_{A}\circ$ $\displaystyle\leftrightarrow$ $\displaystyle
L_{A}^{(q)},$ $\displaystyle\frac{1}{N_{0}}{\rm tr}$
$\displaystyle\leftrightarrow$ $\displaystyle\int\frac{d\Omega_{2}}{4\pi}.$
(A.36)

Furthermore, combining the above correspondence and the relations (A.21) and
(A.22), we can also map the fuzzy spherical harmonics to the spherical
harmonics on $S^{3}$.

### A.4 Vector spherical harmonics

We introduce vector spherical harmonics for three different types of the
spherical harmonics that we have defined above. The vector spherical harmonics
are given by

$\displaystyle Y^{\rho}_{Jm\tilde{m}A}(\Omega_{3})$
$\displaystyle=i^{\rho}\sum_{n,p}U_{An}C^{Qm}_{\tilde{Q}p\;1n}Y_{\tilde{Q}p\tilde{m}}(\Omega_{3}),$
$\displaystyle\tilde{Y}^{\rho}_{JmqA}(\Omega_{2})$
$\displaystyle=i^{\rho}\sum_{n,p}U_{An}C^{Qm}_{\tilde{Q}p\;1n}\tilde{Y}_{\tilde{Q}pq}(\Omega_{2}),$
$\displaystyle\hat{Y}^{\rho}_{Jm(jj^{\prime})A}$
$\displaystyle=i^{\rho}\sum_{n,p}U_{An}C^{Qm}_{\tilde{Q}p\;1n}\hat{Y}_{\tilde{Q}p(jj^{\prime})},$
(A.37)

where $\rho=-1,0,1$ and $Q=J+\delta_{\rho 1}$,
$\tilde{Q}=J+\delta_{\rho\;-1}$. These spherical harmonics transform as the
vector representations under $SU(2)$ rotation. The unitary matrix $U$ is given
by

$U=\left(\begin{array}[]{ccc}-1&0&1\\\ -i&0&-i\\\ 0&\sqrt{2}&0\\\
\end{array}\right).$ (A.38)

The vector spherical harmonics satisfy

$\displaystyle\frac{1}{\mu}\epsilon_{ABC}\nabla_{B}Y^{\rho}_{Jm\tilde{m}C}=i\epsilon_{ABC}{\cal
L}_{B}Y^{\rho}_{Jm\tilde{m}C}+Y^{\rho}_{Jm\tilde{m}A}=\rho(J+1)Y^{\rho}_{Jm\tilde{m}A},$
$\displaystyle
i\epsilon_{ABC}L_{B}^{(q)}\tilde{Y}^{\rho}_{JmqC}+\tilde{Y}^{\rho}_{JmqA}=\rho(J+1)\tilde{Y}^{\rho}_{JmqA},$
$\displaystyle
i\epsilon_{ABC}\hat{L}_{B}\circ\hat{Y}^{\rho}_{Jm(jj^{\prime})C}+\hat{Y}^{\rho}_{Jm(jj^{\prime})A}=\rho(J+1)\hat{Y}^{\rho}_{Jm(jj^{\prime})A}.$
(A.39)

The complex (hermitian) conjugates of these vector harmonics are evaluated as

$\displaystyle(Y^{\rho}_{Jm\tilde{m}A})^{*}$
$\displaystyle=(-1)^{m-\tilde{m}+1}Y^{\rho}_{J-m-\tilde{m}A},$
$\displaystyle(\tilde{Y}^{\rho}_{JmqA})^{*}$
$\displaystyle=(-1)^{m-q+1}\tilde{Y}^{\rho}_{J-m-qA},$
$\displaystyle(\hat{Y}^{\rho}_{Jm(jj^{\prime})A})^{\dagger}$
$\displaystyle=(-1)^{m-(j-j^{\prime})+1}\hat{Y}^{\rho}_{J-m(j^{\prime}j)A}.$
(A.40)

The orthonormal relations are

$\displaystyle\int\frac{d\Omega_{3}}{2\pi^{2}}(Y^{\rho}_{Jm\tilde{m}A})^{*}Y^{\rho^{\prime}}_{J^{\prime}m^{\prime}\tilde{m}^{\prime}A}$
$\displaystyle=\delta_{JJ^{\prime}}\delta_{mm^{\prime}}\delta_{\tilde{m}\tilde{m}^{\prime}}\delta_{\rho\rho^{\prime}},$
$\displaystyle\int\frac{d\Omega_{2}}{4\pi}(\tilde{Y}^{\rho}_{JmqA})^{*}\tilde{Y}^{\rho^{\prime}}_{J^{\prime}m^{\prime}qA}$
$\displaystyle=\delta_{JJ^{\prime}}\delta_{mm^{\prime}}\delta_{\rho\rho^{\prime}},$
$\displaystyle\frac{1}{N_{0}}{\rm
tr}\left((\hat{Y}^{\rho}_{Jm(jj^{\prime})A})^{\dagger}\hat{Y}^{\rho^{\prime}}_{J^{\prime}m^{\prime}(j^{\prime}j)A}\right)$
$\displaystyle=\delta_{JJ^{\prime}}\delta_{mm^{\prime}}\delta_{\rho\rho^{\prime}}.$
(A.41)

Finally, the integrals (or trace) of three vector harmonics are given by

$\displaystyle\int\frac{d\Omega_{3}}{2\pi^{2}}\epsilon_{ABC}Y^{\rho_{1}}_{J_{1}m_{1}\tilde{m}_{1}A}Y^{\rho_{2}}_{J_{2}m_{2}\tilde{m}_{2}B}Y^{\rho_{3}}_{J_{3}m_{3}\tilde{m}_{3}C}$
$\displaystyle={\cal
E}_{J_{1}m_{1}\tilde{m}_{1}\rho_{1}J_{2}m_{2}\tilde{m}_{2}\rho_{2}J_{3}m_{3}\tilde{m}_{3}\rho_{3}},$
$\displaystyle\int\frac{d\Omega_{2}}{4\pi}\epsilon_{ABC}\tilde{Y}^{\rho_{1}}_{J_{1}m_{1}q_{1}A}\tilde{Y}^{\rho_{2}}_{J_{2}m_{2}q_{2}B}\tilde{Y}^{\rho_{3}}_{J_{3}m_{3}q_{3}C}$
$\displaystyle={\cal
E}_{J_{1}m_{1}q_{1}\rho_{1}J_{2}m_{2}q_{2}\rho_{2}J_{3}m_{3}q_{3}\rho_{3}},$
$\displaystyle\epsilon_{ABC}\frac{1}{N_{0}}{\rm
tr}\left(\hat{Y}^{\rho_{1}}_{J_{1}m_{1}(jj^{\prime})A}\hat{Y}^{\rho_{2}}_{J_{2}m_{2}(j^{\prime}j^{\prime\prime})B}\hat{Y}^{\rho_{3}}_{J_{3}m_{3}(j^{\prime\prime}j)C}\right)$
$\displaystyle=\hat{{\cal
E}}_{J_{1}m_{1}(jj^{\prime})\rho_{1}J_{2}m_{2}(j^{\prime}j^{\prime\prime})\rho_{2}J_{3}m_{3}(j^{\prime\prime}j)\rho_{3}},$
(A.42)

where the monopole charges must be conserved in the lefthand side of the
second equality as $q_{1}+q_{2}+q_{3}=0$ and ${\cal E}$, $\hat{\cal E}$ are
given by

$\displaystyle{\cal
E}_{J_{1}m_{1}\tilde{m}_{1}\rho_{1}J_{2}m_{2}\tilde{m}_{2}\rho_{2}J_{3}m_{3}\tilde{m}_{3}\rho_{3}}$
$\displaystyle=\sqrt{6(2J_{1}+1)(2J_{1}+2\rho_{1}^{2}+1)(2J_{2}+1)(2J_{2}+2\rho_{2}^{2}+1)(2J_{3}+1)(2J_{3}+2\rho_{3}^{2}+1)}$
$\displaystyle\;\;\;\times(-1)^{-\frac{\rho_{1}+\rho_{2}+\rho_{3}+1}{2}}\left\\{\begin{array}[]{ccc}Q_{1}&\tilde{Q}_{1}&1\\\
Q_{2}&\tilde{Q}_{2}&1\\\
Q_{3}&\tilde{Q}_{3}&1\end{array}\right\\}\left(\begin{array}[]{ccc}Q_{1}&Q_{2}&Q_{3}\\\
m_{1}&m_{2}&m_{3}\end{array}\right)\left(\begin{array}[]{ccc}\tilde{Q}_{1}&\tilde{Q}_{2}&\tilde{Q}_{3}\\\
\tilde{m}_{1}&\tilde{m}_{2}&\tilde{m}_{3}\end{array}\right),$ (A.50)
$\displaystyle\hat{{\cal
E}}_{J_{1}m_{1}(jj^{\prime})\rho_{1}J_{2}m_{2}(j^{\prime}j^{\prime\prime})\rho_{2}J_{3}m_{3}(j^{\prime\prime}j)\rho_{3}}$
$\displaystyle=\sqrt{6N_{0}(2J_{1}+1)(2J_{1}+2\rho_{1}^{2}+1)(2J_{2}+1)(2J_{2}+2\rho_{2}^{2}+1)(2J_{3}+1)(2J_{3}+2\rho_{3}^{2}+1)}$
$\displaystyle\;\;\;\times(-1)^{-\frac{\rho_{1}+\rho_{2}+\rho_{3}+1}{2}-\tilde{Q}_{1}-\tilde{Q}_{2}-\tilde{Q}_{3}+2j+2j^{\prime}+2j^{\prime\prime}}\left\\{\begin{array}[]{ccc}Q_{1}&\tilde{Q}_{1}&1\\\
Q_{2}&\tilde{Q}_{2}&1\\\
Q_{3}&\tilde{Q}_{3}&1\end{array}\right\\}\left(\begin{array}[]{ccc}Q_{1}&Q_{2}&Q_{3}\\\
m_{1}&m_{2}&m_{3}\end{array}\right)\left\\{\begin{array}[]{ccc}\tilde{Q}_{1}&\tilde{Q}_{2}&\tilde{Q}_{3}\\\
j^{\prime\prime}&j&j^{\prime}\end{array}\right\\}.$ (A.58)

As in (A.33), we can show

$\hat{{\cal
E}}_{J_{1}m_{1}(jj^{\prime})\rho_{1}J_{2}m_{2}(j^{\prime}j^{\prime\prime})\rho_{2}J_{3}m_{3}(j^{\prime\prime}j)\rho_{3}}\rightarrow{\cal
E}_{J_{1}m_{1}\tilde{m}_{1}\rho_{1}J_{2}m_{2}\tilde{m}_{2}\rho_{2}J_{3}m_{3}\tilde{m}_{3}\rho_{3}},$
(A.59)

in the limit $N_{0}\rightarrow\infty$ with $j-j^{\prime}=q_{1}$,
$j^{\prime}-j^{\prime\prime}=q_{2}$ and $j^{\prime\prime}-j=q_{3}$ fixed.

##  Appendix B Derivation of (4.22)

In this appendix, we give the derivation of (4.22) in some detail.

$\displaystyle[L_{a},a_{\alpha}(x)]^{(s,t)}$
$\displaystyle=a_{\alpha,Jm}^{(q_{st})}(x)\otimes
L_{a}\circ\hat{Y}_{Jm(j_{s}j_{t})}$
$\displaystyle=\int\frac{d\Omega_{3}}{2\pi^{2}}A_{\alpha}(z)Y_{Jmq_{st}}^{\dagger}(y)\otimes
L_{a}\circ\hat{Y}_{Jm(j_{s}j_{t})}$
$\displaystyle=\int\frac{d\Omega_{3}}{2\pi^{2}}A_{\alpha}(z){\cal
L}_{a}Y_{Jmq_{st}}^{\dagger}\otimes\hat{Y}_{Jm(j_{s}j_{t})}$
$\displaystyle=\int\frac{d\Omega_{3}}{2\pi^{2}}\left({\cal
L}_{a}A_{\alpha}(z)\right)Y_{Jmq_{st}}^{\dagger}\otimes\hat{Y}_{Jm(j_{s}j_{t})},$

where we have used (A.9) and (A.26).

$\displaystyle[\phi_{a},\phi_{b}]^{(s,t)}$
$\displaystyle=\sum_{u}\left(\phi_{a,Jm}^{(q_{su})}\phi_{b,J^{\prime}m^{\prime}}^{(q_{ut})}-\phi_{b,Jm}^{(q_{su})}\phi_{a,J^{\prime}m^{\prime}}^{(q_{ut})}\right)\otimes\hat{Y}_{Jm(j_{s}j_{u})}\hat{Y}_{J^{\prime}m^{\prime}(j_{u}j_{t})}$
$\displaystyle=\sum_{u}\left(\phi_{a,Jm}^{(q_{su})}\phi_{b,J^{\prime}m^{\prime}}^{(q_{ut})}-\phi_{b,Jm}^{(q_{su})}\phi_{a,J^{\prime}m^{\prime}}^{(q_{ut})}\right)\otimes\hat{{\cal
C}}^{J^{\prime\prime}m^{\prime\prime}(j_{s}j_{t})}_{Jm(j_{s}j_{u})\;J^{\prime}m^{\prime}(j_{u}j_{t})}\hat{Y}_{J^{\prime\prime}m^{\prime\prime}(j_{s}j_{t})}$
$\displaystyle=\sum_{u,v}\int\frac{d\Omega_{3}}{2\pi^{2}}\frac{d\Omega_{3}^{\prime}}{2\pi^{2}}\left\\{A_{a}(z)A_{b}(z^{\prime})-A_{b}(z)A_{a}(z^{\prime})\right\\}Y_{Jmq_{su}}^{\dagger}(y)Y_{J^{\prime}m^{\prime}q_{vt}}^{\dagger}(y^{\prime})$
$\displaystyle\qquad\qquad\times\int\frac{d\Omega_{3}^{\prime\prime}}{2\pi^{2}}Y_{J^{\prime\prime}m^{\prime\prime}q_{st}}^{\dagger}(y^{\prime\prime})Y_{Jmq_{su}}(y^{\prime\prime})Y_{J^{\prime}m^{\prime}q_{vt}}(y^{\prime\prime})\otimes\hat{Y}_{J^{\prime\prime}m^{\prime\prime}(j_{s}j_{t})}$
$\displaystyle=\int\frac{d\Omega_{3}}{2\pi^{2}}[A_{a}(z),A_{b}(z)]Y_{Jmq_{st}}^{\dagger}(y)\otimes\hat{Y}_{Jm(j_{s}j_{t})}.$
(B.2)

In the third and fourth lines of the righthand side, we have used (A.33), the
charge conservation $\tilde{m}^{\prime\prime}=\tilde{m}+\tilde{m}^{\prime}$ of
${\cal
C}^{J^{\prime\prime}m^{\prime\prime}\tilde{m}^{\prime\prime}}_{Jm\tilde{m}\;J^{\prime}m^{\prime}\tilde{m}^{\prime}}$
and (A.12), so that we have added the new summation over $v$ additionally and
replaced $q_{ut}$ by $q_{vt}$. Then, we can regard the summation $\sum_{u,v}$
as $\sum_{q_{su},q_{vt}}$, and the last equality holds due to (A.13).

##  Appendix C Group manifold and coset space

In this appendix, we describe some conventions on the group manifold
$\tilde{G}$ and the coset space $\tilde{G}/H$ which we follow in this paper.

We parameterize an element of $\tilde{G}$ as

$g(z)=L(x)h(y),$ (C.1)

where $L(x)\in\tilde{G}/H$, $h(y)\in H$, the coordinates $z^{M}$, $x^{\mu}$
and $y^{m}$ parameterize $\tilde{G}$, $\tilde{G}/H$ and $H$ respectively and
$z^{M}$ are decomposed into $(x^{\mu},y^{i})$. We can construct the right and
left invariant 1-forms on $\tilde{G}$ as

$dgg^{-1}=-iE^{A}_{R}T^{A},\;\;\;g^{-1}dg=iE_{L}^{A}T^{A},$ (C.2)

where $A=1,\cdots,{\rm dim}\tilde{G}$ and $T^{A}$ represent the generators of
$\tilde{G}$ which satisfy the Lie algebra of $\tilde{G}$,
$[T^{A},T^{B}]=if_{ABC}T^{C}$. We decompose $T^{A}$ into $(T^{\alpha},T^{a})$
where $\alpha=1,\cdots,{\rm dim}\tilde{G}/H,\;a={\rm
dim}\tilde{G}/H+1,\cdots,{\rm dim}\tilde{G}$, and we assume that $T^{a}$
satisfy the Lie algebra of $H$ which is a subalgebra of $\tilde{G}$,
$[T^{a},T^{b}]=if_{abc}T^{c}$. The both of $E^{A}_{R}$ and $E^{A}_{L}$ satisfy
the Maurer-Cartan equation,

$dE^{A}_{R}-\frac{1}{2}f_{ABC}E^{B}_{R}\wedge
E^{C}_{R}=0,\;\;\;dE^{A}_{L}-\frac{1}{2}f_{ABC}E^{B}_{L}\wedge E^{C}_{L}=0,$
(C.3)

We also introduce the right and left invariant 1-form for $L(x)\in\tilde{G}/H$
and $h(y)\in H$ as follows:

$\displaystyle
dLL^{-1}=-i(e_{R})^{A}_{\mu}(x)T^{A}dx^{\mu},\;\;\;L^{-1}dL=i(e_{L})^{A}_{\mu}(x)T^{A}dx^{\mu},$
$\displaystyle
dhh^{-1}=-i(\tilde{e}_{R})^{a}_{m}(y)T^{a}dy^{m},\;\;\;h^{-1}dh=i(\tilde{e}_{L})^{a}_{m}(y)T^{a}dy^{m}.$
(C.4)

Then, we can write down the components of $E^{A}_{R}$ and $E_{L}^{A}$
explicitly:

$\displaystyle(E_{R})^{A}{}_{M}=\left(\begin{array}[]{cc}(e_{R})^{\alpha}_{\mu}&{\rm
Ad}(L)^{\alpha}{}_{b}(\tilde{e}_{R})^{b}_{m}\\\ (e_{R})^{a}_{\mu}&{\rm
Ad}(L)^{a}{}_{b}(\tilde{e}_{R})^{b}_{m}\end{array}\right),\;\;\;(E_{L})^{A}{}_{M}=\left(\begin{array}[]{cc}(e_{L})^{\beta}_{\mu}{\rm
Ad}(h)_{\beta\alpha}&0\\\ (e_{L})^{\beta}_{\mu}{\rm Ad}(h)_{\beta
a}&(\tilde{e}_{L})^{a}_{m}\end{array}\right),$ (C.9)

where ${\rm Ad}$ is defined as the adjoint action $gT^{A}g^{-1}=T^{B}{\rm
Ad}(g)_{BA}$. The Cartan-Killing metric on $\tilde{G}$ is defined as

$ds^{2}=G_{MN}dz^{M}dz^{N}=-2{\rm Tr}(dgg^{-1}dgg^{-1}).$ (C.10)

In terms of the components (C.9), the above metric is written as

$ds^{2}=(e_{L})^{\alpha}_{\mu}(e_{L})^{\alpha}_{\nu}dx^{\mu}dx^{\nu}+\\{(\tilde{e}_{R})^{a}_{m}dy^{m}-(e_{L})^{a}_{\mu}dx^{\mu}\\}^{2}.$
(C.11)

We can regard the group manifold $\tilde{G}$ as the principal $H$ bundle on
$\tilde{G}/H$. By comparing (C.11) and (3.5), therefore, we can make the
following identifications:

$(e_{L})^{\alpha}_{\mu}(e_{L})^{\alpha}_{\nu}=g_{\mu\nu},\;(e_{L})^{a}_{\mu}=b_{\mu}^{a},\;(\tilde{e}_{R})^{a}_{m}(\tilde{e}_{R})^{a}_{n}=h_{mn},$
(C.12)

where $g_{\mu\nu}$ and $h_{mn}$ are the metrics on $\tilde{G}/H$ and $H$,
respectively, and $b_{\mu}^{a}$ are the local connection 1-forms of the
principal $H$ bundle. Namely, we can regard $(e_{L})^{\alpha}_{\mu}$ and
$(\tilde{e}_{R})^{a}_{m}$ as the vielbein on $\tilde{G}/H$ and $H$,
respectively. The metric (C.10) is invariant under the right and left actions
of $\tilde{G}$. The corresponding right and left invariant Killing vectors on
$\tilde{G}$ are defined in terms of the inverse of $E^{A}$ as

${\cal L}^{R}_{A}=-i(E_{R})_{A}^{M}\partial_{M},\;\;\;{\cal
L}^{L}_{A}=-i(E_{L})_{A}^{M}\partial_{M}.$ (C.13)

By using (C.3), we can show that ${\cal L}_{A}^{R}$ and ${\cal L}_{A}^{L}$
satisfy the Lie algebra of $\tilde{G}\times\tilde{G}$,

$[{\cal L}_{A}^{R},{\cal L}_{B}^{R}]=if_{ABC}{\cal L}_{C}^{R},\;\;\;[{\cal
L}_{A}^{L},{\cal L}_{B}^{L}]=if_{ABC}{\cal L}_{C}^{L},\;\;\;[{\cal
L}_{A}^{R},{\cal L}_{B}^{L}]=0,$ (C.14)

and they also satisfy the Killing vector equations,

$\nabla_{M}{\cal L}_{AN}+\nabla_{N}{\cal L}_{AM}=0,$ (C.15)

where $\nabla_{M}$ are the covariant derivative on $\tilde{G}$ and ${\cal
L}_{AM}=G_{MN}{\cal L}_{A}^{N}$. We also define the following operators:

$L_{A}=-i(E_{R})_{A}^{\mu}\partial_{\mu}.$ (C.16)

One can show that $L_{A}$ do not depend on $y^{m}$ and they satisfy
$[L_{A},L_{B}]=if_{ABC}L_{C}$ by using (C.14). Furthermore, we can show that

$\nabla^{(\tilde{G}/H)}_{\mu}L_{A\nu}+\nabla^{(\tilde{G}/H)}_{\nu}L_{A\mu}=0,$
(C.17)

where $L_{A\mu}=g_{\mu\nu}L_{A}^{\nu}$. Namely, $L_{A}$ are the Killing
vectors on the coset space $\tilde{G}/H$.

##  Appendix D Metrics of $SU(3)$, $S^{5}$ and $CP^{2}$

In this appendix, for concreteness, we give an explicit form of the metrics of
$SU(3)$, $SU(3)/SU(2)\simeq S^{5}$ and $SU(3)/(SU(2)\times U(1))\simeq CP^{2}$
[51]. We parameterize an element $g$ of $SU(3)$ as

$\displaystyle g=L(\chi,\theta,\varphi,\psi)Z(\tau)V(a,b,c),$ (D.1)

where

$\displaystyle
L(\chi,\theta,\varphi,\psi)=e^{i\varphi\lambda_{3}}e^{i\theta\lambda_{2}}e^{i\psi\lambda_{3}}e^{2i\chi\lambda_{5}},$
$\displaystyle Z(\tau)=e^{-i\sqrt{3}(\tau-2\pi)\lambda_{8}},$ $\displaystyle
V(a,b,c)=e^{-ia\lambda_{3}}e^{-ib\lambda_{2}}e^{-ic\lambda_{3}},$ (D.2)

and $0\leq\chi\leq\frac{\pi}{2}$, $0\leq\theta\leq\pi$, $0\leq\varphi<2\pi$,
$0\leq\psi<4\pi$, $0\leq\tau<2\pi$, $0\leq a<2\pi$, $0\leq b\leq\pi$ and
$0\leq c<4\pi$. $\lambda_{1},\cdots,\lambda_{8}$ are the Gell-Mann matrices
and satisfy $\mbox{Tr}(\lambda_{a}\lambda_{b})=\frac{1}{2}\delta_{ab}$. The
metric of $SU(3)$ is given by

$\displaystyle dS_{SU(3)}^{2}=$
$\displaystyle-\frac{1}{2}\mbox{Tr}(dgg^{-1}dgg^{-1})$ $\displaystyle=$
$\displaystyle
d\chi^{2}+\frac{1}{4}\sin^{2}\chi\left\\{d\theta^{2}+\sin^{2}\theta
d\varphi^{2}+\cos^{2}\chi(d\psi+\cos\theta d\varphi)^{2}\right\\}$
$\displaystyle+\frac{3}{4}\left\\{d\tau+\frac{1}{2}\sin^{2}\chi(d\psi+\cos\theta
d\varphi)\right\\}^{2}$
$\displaystyle+\frac{1}{4}\left\\{e^{1}+\cos\chi(\sin\psi
d\theta-\sin\theta\cos\psi d\varphi)\right\\}^{2}$
$\displaystyle+\frac{1}{4}\left\\{e^{2}-\cos\chi(\cos\psi
d\theta+\sin\theta\sin\psi d\varphi)\right\\}^{2}$
$\displaystyle+\frac{1}{4}\left\\{e^{3}-\frac{1}{2}(1+\cos^{2}\chi)(d\psi+\cos\theta
d\varphi)\right\\}^{2},$ (D.3)

where

$\displaystyle e^{1}=-\sin adb+\cos a\sin bdc,$ $\displaystyle e^{2}=\cos
adb+\sin a\sin bdc,$ $\displaystyle e^{3}=da+\cos bdc,$ (D.4)

which are the right invariant 1-form of $SU(2)$. $SU(3)$ is an $SU(2)\times
U(1)$ bundle over $CP^{2}$. The second line in the righthand side of (D.3) is
the Fubini-Study metric of $CP^{2}$. The third line represents the $U(1)$
fiber structure while the fourth, fifth and sixth lines represent the $SU(2)$
fiber structure. $SU(3)$ is also viewed as an $SU(2)$ bundle over $S^{5}\simeq
SU(3)/SU(2)$. The second and third lines together correspond to the metric of
$S^{5}\simeq SU(3)/SU(2)$. $S^{5}\simeq SU(3)/SU(2)$ is viewed as a $U(1)$
bundle over $CP^{2}$. The metric of the ordinary unit $S^{5}$ is given by the
sum of the second and third lines with the factor $3/4$ in the third line
replaced by $1/4$.

##  Appendix E Fuzzy $CP^{n}$

In this appendix, we give a brief review of a construction of fuzzy $CP^{n}$
[28, 31, 32, 33, 34, 35].

### E.1 Functions on fuzzy $CP^{n}$

Fuzzy $CP^{n}$ is a well-known example of noncommutative space which is given
by the quantization of coadjoint orbit of $SU(n+1)$ in terms of a certain
matrix algebra acting on an appropriate representation space $V$. We can
determine this matrix algebra and the representation space $V$ by matching the
spectrum of functions on $CP^{n}$ and that on fuzzy $CP^{n}$.

In order to consider the spectrum of functions on $CP^{n}$, We regard $CP^{n}$
as a coadjoint orbit in the Lie algebra of $SU(n+1)$.

$CP^{n}=\\{\;gtg^{-1}\;|\;g\in SU(n+1)\\}\simeq SU(n+1)/(SU(n)\times U(1)),$
(E.1)

where $t$ is an element of the $SU(n+1)$ Lie algebra such that the stabilizer
of $t$ is given by $SU(n)\times U(1)$. For example, for the case of $CP^{2}$,
we can take $t$ to be $\lambda_{8}$ which is invariant under $SU(2)\times
U(1)$ adjoint action generated by $\lambda_{1},\lambda_{2},\lambda_{3}$ and
$\lambda_{8}$. Functions on $CP^{n}$ should be invariant under the action of
$SU(n)\times U(1)$. Then, the space of functions on $CP^{n}$ is given by a
direct sum of the representation spaces of $SU(n+1)$ which contain
$SU(n)\times U(1)$ invariant states:

$C^{\infty}(CP^{n})=\bigoplus_{J=0}^{\infty}V_{[J,0,\cdots,0,J]}$ (E.2)

where we denote $[J,0,\cdots,0,J]$ as the Dynkin index of $SU(n+1)$, and
$V_{[J,0,\cdots,0,J]}$ represents the corresponding irreducible representation
space of the $SU(n+1)$ Lie algebra. One can show that $V_{[J,0,\cdots,0,J]}$
are the only spaces which contain the $SU(n)\times U(1)$ singlets.

The space of functions on fuzzy $CP^{n}$ is obtained by introducing a cutoff
$\Lambda$ in (E.2) as

$\bigoplus_{J=0}^{\Lambda}V_{[J,0,\cdots,0,J]}=V_{[\Lambda,0,\cdots,0]}\otimes
V_{[\Lambda,0,\cdots,0]}^{*}.$ (E.3)

By definition, it is obvious that the above spectrum on fuzzy $CP^{n}$ tends
to the spectrum (E.2) on $CP^{n}$ in the commutative limit
$\Lambda\rightarrow\infty$. Note that the righthand side of the above equation
can be viewed as a space of matrices. From this viewpoint, we make an
identification $V=V_{[\Lambda,0,\cdots,0]}$ and regard functions on fuzzy
$CP^{n}$ as matrices acting on the vector space $V$. In particular, the
coordinates on fuzzy $CP^{n}$ are identified with

$\hat{\xi}_{A}=\hat{L}_{A}^{[\Lambda,0,\cdots,0]},$ (E.4)

which are the generators of $SU(n+1)$ in the irreducible representation
specified by the Dynkin index $[\Lambda,0,\cdots,0]$. These coordinates on
fuzzy $CP^{n}$ are actually reduced to the coordinates on $CP^{n}$ in the
commutative limit through a map which will be defined in the last part of this
section.

### E.2 Derivatives on fuzzy $CP^{n}$

In order to construct differential operators on fuzzy $CP^{n}$, let us recall
the simplest case of fuzzy $CP^{1}\simeq S^{2}$. In this case, we established
the differential operators on fuzzy $S^{2}$ in appendix A. As shown in (A.36),
the adjoint action of the $SU(2)$ generators is reduced to the action of the
Killing vectors on $S^{2}$ in the commutative limit. We can generalize this
fact into the case of fuzzy $CP^{n}$ with $n\geq 2$. The adjoint action of the
$SU(n+1)$ generators on the space of square matrices (E.3),
$[\hat{L}_{A}^{[\Lambda,0,\cdots,0]},\;\cdot\;]$, is mapped into the action of
the Killing vectors on the space of functions on $CP^{n}$ in the commutative
limit.

### E.3 $U(1)$ monopoles on fuzzy $CP^{n}$

Topologically nontrivial field configurations including $U(1)$ monopoles can
be realized on fuzzy $CP^{n}$. If we consider rectangular matrices in addition
to the square matrices (E.3), the concept of fiber bundles naturally arises.
Let us again consider the case of fuzzy $CP^{1}$. We have shown in appendix A
that the basis of $(2j+1)\times(2j^{\prime}+1)$ rectangular matrices,
$\hat{Y}_{Jm(jj^{\prime})}$, are mapped into local sections of the $U(1)$
fiber bundle on $S^{2}$. In this correspondence, The difference $j-j^{\prime}$
is identified with the monopole charge $q$ of the $U(1)$ bundle. This fact is
also generalized into the case of $CP^{n}$ with $n\geq 2$. For the case of
$CP^{n}$, we consider a space of rectangular matrices,

$V_{[\Lambda+q,0,\cdots,0,]}\otimes V_{[\Lambda-q,0,\cdots,0,]}^{*}.$ (E.5)

Here, the charge $q$ is a half integer and we take $\Lambda\pm q$ to be
integers. When $q=0$, $\Lambda$ is an integer and this is the case of square
matrices (E.3). We can show that elements of (E.5) are mapped into local
sections of $U(1)$ fiber bundle on $CP^{n}$ with the monopole charge $q$.
Furthermore, we can extend the action of the differential operators
$[\hat{L}_{A}^{[\Lambda,0,\cdots,0]},\;\cdot\;]$ discussed above to the action
on rectangular matrices as follows.

$\hat{L}_{A}\circ\hat{M}_{q}=\hat{L}_{A}^{[\Lambda+q,0,\cdots,0]}\hat{M}_{q}-\hat{M}_{q}\hat{L}_{A}^{[\Lambda-q,0,\cdots,0]},$
(E.6)

where $\hat{M}_{q}$ is an element of (E.5). When $q=0$, $\hat{M}_{0}$ is just
a square matrix and $\hat{L}_{A}\circ$ are nothing but the commutators
$[\hat{L}_{A}^{[\Lambda,0,\cdots,0]},\;\cdot\;]$. The operators
$\hat{L}_{A}\circ$ map the space (E.5) to itself and they are reduced to the
angular momentum operators in the presence of a $U(1)$ monopole with the
magnetic charge $q$ in the commutative limit. We will show these facts in the
following subsections.

### E.4 Fock space representation

In order to construct a map between matrices and functions on $CP^{n}$, we
introduce the Fock space representation developed in [35]. Let
$a^{\dagger}_{\alpha}$, $\alpha=1,2,\cdots,n+1$ be a set of creation operators
and $a^{\alpha}$ be a set of annihilation operators which annihilate the Fock
vacuum $|0\rangle$. They satisfy the Heisenberg commutation relations.

$[a^{\alpha},a^{\beta}]=[a^{\dagger}_{\alpha},a^{\dagger}_{\beta}]=0,\;\;\;[a^{\alpha},a^{\dagger}_{\beta}]=\delta_{\beta}^{\alpha}.$
(E.7)

By acting the creation operators on the vacuum state $|0\rangle$, we can
construct the entire Fock space ${\cal F}$ which is spanned by

$|p_{1},p_{2},\cdots,p_{n+1}\rangle=\frac{1}{\sqrt{p_{1}!p_{2}!\cdots
p_{n+1}!}}(a_{1}^{\dagger})^{p_{1}}(a_{2}^{\dagger})^{p_{2}}\cdots(a_{n+1}^{\dagger})^{p_{n+1}}|0\rangle.$
(E.8)

In terms of the operators (E.7), we can construct elements of the Lie algebra
of $SU(n+1)$,

$\hat{L}_{A}=a^{\dagger}_{\alpha}(T_{A})^{\alpha}{}_{\beta}a^{\beta},$ (E.9)

where $T_{A}=\hat{L}_{A}^{[1,0,\cdots,0]}$ represent the generators of
$SU(n+1)$ in the fundamental representation. We also define the number
operator which commutes with all the operators in (E.9).

$\hat{N}=a^{\dagger}_{\alpha}a^{\alpha}.$ (E.10)

The operators (E.9) and (E.10) act on the Fock space ${\cal F}$, and satisfy

$[\hat{L}_{A},\hat{L}_{B}]=if_{ABC}\hat{L}_{C},\;\;\;[\hat{L}_{A},\hat{N}]=0.$
(E.11)

We can decompose the Fock space ${\cal F}$ into the eigenspaces of $\hat{N}$
as

${\cal F}=\bigoplus_{p=0}^{\infty}V_{[p,0,\cdots,0]},$ (E.12)

where $p$ represent an eigenvalue of $\hat{N}$. The basis of each eigenspace
$V_{[p,0,\cdots,0]}$ is formed by

$|{\bm{\alpha}}_{p}\rangle=|\alpha_{1},\alpha_{2},\cdots,\alpha_{p}\rangle=\frac{1}{\sqrt{p!}}a_{\alpha_{1}}^{\dagger}a_{\alpha_{2}}^{\dagger}\cdots
a_{\alpha_{p}}^{\dagger}|0\rangle,$ (E.13)

where ${\bm{\alpha}}_{p}$ is an abbreviation of a set of $p$ indices,
$(\alpha_{1},\alpha_{2},\cdots,\alpha_{p})$.

Let us consider square matrices which are elements of (E.3). These matrices
are generally written as

$\hat{M}=\hat{M}^{{\bm{\alpha}}_{\Lambda}}{}_{{\bm{\beta}}_{\Lambda}}|{\bm{\alpha}}_{\Lambda}\rangle\langle{\bm{\beta}}_{\Lambda}|.$
(E.14)

We define a new basis of these matrices to see the correspondence with the
spectrum of functions on $CP^{n}$, (E.2):

$\hat{Y}_{{\bm{\beta}}_{J}}{}^{{\bm{\alpha}}_{J}}=N_{\Lambda J}^{n}{\cal
P}_{{\bm{\beta}}_{J},{\bm{\tau}}_{J}}{}^{{\bm{\alpha}}_{J},{\bm{\sigma}}_{J}}|{\bm{\sigma}}_{J},{\bm{\gamma}}_{\Lambda-J}\rangle\langle{\bm{\tau}}_{J},{\bm{\gamma}}_{\Lambda-J}|,$
(E.15)

where $\Lambda-J$ indices ${\bm{\gamma}}_{\Lambda-J}$ are contracted and
${\cal
P}_{{\bm{\beta}}_{J},{\bm{\tau}}_{J}}{}^{{\bm{\alpha}}_{J},{\bm{\sigma}}_{J}}$
is the projection operator onto the representation space
$V_{[J,0,\cdots,0,J]}$ which appeared in the decomposition (E.3), that is, it
removes all traces between ${\bm{\alpha}}_{J}$ and ${\bm{\beta}}_{J}$. For
example,

$\displaystyle\hat{Y}=N_{\Lambda
0}^{n}{\bm{1}},\;\;\;\hat{Y}_{\beta}{}^{\alpha}=N_{\Lambda
1}^{n}\left(|\beta,{\bm{\gamma}}_{\Lambda-1}\rangle\langle\alpha,{\bm{\gamma}}_{\Lambda-1}|-\frac{1}{2}\delta_{\beta}^{\alpha}{\bm{1}}\right).$
(E.16)

Hence, $\hat{Y}_{{\bm{\beta}}_{J}}{}^{{\bm{\alpha}}_{J}}$ belong to the
representation $V_{[J,0,\cdots,0,J]}$ and they are mapped to the corresponding
spherical harmonics on $CP^{n}$ in the commutative limit which are elements of
(E.2). $N_{\Lambda J}^{n}$ is an appropriate normalization constant which is
determined by the following orthonormality of the basis,

${\rm
tr}\left((\hat{Y}_{{\bm{\beta}}_{J}}{}^{{\bm{\alpha}}_{J}})^{\dagger}\hat{Y}_{{\bm{\tau}}_{J^{\prime}}}{}^{{\bm{\sigma}}_{J^{\prime}}}\right)=\delta_{JJ^{\prime}}{\cal
P}_{{\bm{\alpha}}_{J},{\bm{\tau}}_{J}}{}^{{\bm{\beta}}_{J},{\bm{\sigma}}_{J}}.$
(E.17)

In the case of $n=1$, $\hat{Y}_{{\bm{\beta}}_{J}}{}^{{\bm{\alpha}}_{J}}$ are
essentially the same as the fuzzy spherical harmonics which are defined in
(A.25). The action of differential operators on fuzzy $CP^{n}$ is given by the
adjoint action of operators in (E.9). Then, one can evaluate the eigenvalues
of the Laplacian as follows:

$[\hat{L}_{A},[\hat{L}_{A},\hat{Y}_{{\bm{\beta}}_{J}}{}^{{\bm{\alpha}}_{J}}]]=J(J+n)\hat{Y}_{{\bm{\beta}}_{J}}{}^{{\bm{\alpha}}_{J}}.$
(E.18)

The above spectrum completely matches the spectrum of functions on $CP^{n}$ up
to the cutoff $\Lambda$.

In terms of the Fock space representation, we can also express rectangular
matrices which are elements of (E.5). Those rectangular matrices are generally
expressed as

$\hat{M}_{q}=(\hat{M}_{q})^{{\bm{\alpha}}_{\Lambda+q}}{}_{{\bm{\beta}}_{\Lambda-q}}|{\bm{\alpha}}_{\Lambda+q}\rangle\langle{\bm{\beta}}_{\Lambda-q}|.$
(E.19)

These matrices are expanded by a similar basis to (E.15). Note that the direct
product representation $(\ref{rectangular matrices})$ is decomposed as

$\bigoplus_{J=|q|}^{\Lambda}V_{[J+q,0,\cdots,0,J-q]}.$ (E.20)

For each representation space in (E.20) with fixed $J$, we can use the
following basis:

$\hat{Y}^{(q)}{}_{{\bm{\beta}}_{J+q}}{}^{{\bm{\alpha}}_{J-q}}=N_{\Lambda
Jq}^{n}{\cal
P}_{{\bm{\beta}}_{J+q},{\bm{\tau}}_{J-q}}{}^{{\bm{\alpha}}_{J-q},{\bm{\sigma}}_{J+q}}|{\bm{\sigma}}_{J+q},{\bm{\gamma}}_{\Lambda-J}\rangle\langle{\bm{\tau}}_{J-q},{\bm{\gamma}}_{\Lambda-J}|.$
(E.21)

As in the case of square matrices, ${\cal
P}_{{\bm{\beta}}_{J+q},{\bm{\tau}}_{J-q}}{}^{{\bm{\alpha}}_{J-q},{\bm{\sigma}}_{J+q}}$
is a projection operator onto the space (E.20) with fixed $J$ and $N_{\Lambda
Jq}^{n}$ is a normalization constant which is determined by

${\rm
tr}\left((\hat{Y}^{(q)}{}_{{\bm{\beta}}_{J+q}}{}^{{\bm{\alpha}}_{J-q}})^{\dagger}\hat{Y}^{(q)}{}_{{\bm{\tau}}_{J^{\prime}+q}}{}^{{\bm{\sigma}}_{J^{\prime}-q}}\right)=\delta_{JJ^{\prime}}{\cal
P}_{{\bm{\tau}}_{J+q},{\bm{\alpha}}_{J-q}}{}^{{\bm{\sigma}}_{J-q},{\bm{\beta}}_{J+q}}.$
(E.22)

When $q=0$, $\hat{Y}^{(0)}{}_{{\bm{\beta}}_{J}}{}^{{\bm{\alpha}}_{J}}$ are
identical with the square matrices (E.15). The action of differential