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%%%%%	  Takehiro Azuma
%%%%%   first version: September 6th, 2004.
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\newcommand {\tr}{{\rm tr\,}}
\newcommand {\stwo}{{\rm S}^{2}}
\newcommand {\stwostwo}{{\rm S}^{2} \times {\rm S}^{2}}
\newcommand {\stwon}{{\rm S}_{N}^{2}}
\newcommand {\stwonn}{{\rm S_{N}}^{2} \times {\rm S_{N}}^{2}}
\newcommand {\stwolr}{{\rm S}_{N_{L}}^{2} \times {\rm S}_{N_{R}}^{2}}
\newcommand {\A}{{\tilde A}}
\newcommand {\D}{\hat {\cal D}}

\pagestyle{myheadings}
\markright{\footnotesize "Gauge Theory on Fuzzy $\stwostwo$ and Regularization on Noncommutative {\bf R}$^{4}$"
 ({\tt hep-th/0503041}), Takehiro Azuma, May. 16, 12:15 $\sim$ 13:15}
\begin{document}
%\begin{landscape}
%\setlength{\baselineskip}{12mm}
{\bf \boldmath \Large
\vspace*{-1cm}

{\color{Green}{\shadowbox{
   \begin{minipage}{23cm}
    \begin{center} 
     \vspace*{1cm}
      {\color{Blue}{\Large Gauge Theory on Fuzzy $\stwostwo$ and Regularization on 
      Noncommutative {\bf R}$^{4}$ \\
      ({\tt hep-th/0503041}) }} \\
       {\color{Blue}{\Large \it Wolfgang Behr, Frank Meyer and Harold Steinacker}} \\
    \end{center}
   \end{minipage}
}}}
\vspace*{-0.8cm}
  \begin{center}
  {\large Journal Club at KEK, presented by {\color{Magenta}{Takehiro Azuma}},\footnote{This slide is used for Takehiro Azuma's presentation at KEK. Therefore, it is not the authors {\it but the presenter Takehiro Azuma} that is responsible for any potential flaws in this slide.} }
  {\large May. 16th 2005, 12:15 $\sim$ 13:15}
 \end{center}
\vspace*{-1.2cm}
{\normalsize \tableofcontents}
\newpage

%########################### Page 2 ##########################
\section{Introduction} 
 Motivations of fuzzy manifold studies:
 \begin{itemize}
  \item{Prototype of the curved-space background in the large-$N$ reduced models.}
  \item{Dynamical generation of spacetime / gauge group.}
 \end{itemize}
 
 A plethora of works for the fuzzy sphere physics:
 
 {\color{Magenta}{\normalsize {\tt hep-th/0101102,0103192,0204256,0209057,0301055,0303120,0307007,0312241,0401038,0403242,0405096,0405277,0410263,\\ 0412303,0412312,0504217 $\cdots$} }}
 
Four-dimensional fuzzy manifolds:

 {\color{Magenta}{$\stwostwo$}}, 
 ${\rm S}^{4} = {\rm SO}(5)/{\rm U}(2)$, 
 ${\rm CP}^{2} = {\rm SU}(3)/{\rm U}(2)$, $\cdots$. \\
 
 These fuzzy manifolds are {\color{Magenta}{compact}}, and thus realized by 
 {\color{Magenta}{finite matrices}}. \\
 
 We focus on {\color{Red}{the gauge theory on fuzzy $\stwostwo$}}:
 
 {\color{Red}{$\Rightarrow$}} We obtain {\bf R}$_{\theta}^{4}$ as a scalar field.
 
\newpage
%################## Page 3 #######################################
\section{Fuzzy spaces $\stwon$ and $\stwolr$}
{\color{Blue}{\shadowbox{Definitions of fuzzy $\stwo$}}} \\
 {\color{Magenta}{$x_{i} = (x_{1}, x_{2}, x_{3})=$}} (Hermitian operators) 
{\color{Red}{ \begin{eqnarray}
 [x_i,x_j] = i \Lambda_{N} \epsilon_{ijk} x_{k}, \hspace{5mm} x_1^2 + x_2^2 + x_3^2 = R^{2}, \hspace{5mm}
 \Lambda_{N} = \frac{2R}{\sqrt{N^2 - 1}}. \nonumber
\end{eqnarray} }}
{\color{Magenta}{$\Lambda_{N}$}} = (NC parameter) = O([length]$^1$).

They are obtained from the $N$-dimensional representation of su$(2)$ {\color{Magenta}{$\lambda_i$}}:
{\color{Red}{\begin{eqnarray}
 x_{i} = \Lambda_{N} \lambda_{i}, \hspace{5mm} \Lambda_{N} = \frac{2R}{\sqrt{N^{2}-1}}, \textrm{ where }
 [\lambda_{i}, \lambda_{j}] = i \epsilon_{ijk} \lambda_{k}, \hspace{5mm} \lambda^{2}_{i} = \frac{N^2-1}{4}.
  \nonumber
\end{eqnarray} }}
 (algebra of functions $f$ on $\stwon$) $\Leftrightarrow$ (matrix algebra {\color{Magenta}{Mat$(N,{\bf C})$}}):
 {\color{Magenta}{
 \begin{eqnarray}
 \int_{\stwon} f = \frac{4 \pi R^{2}}{N} \tr f. \nonumber
 \end{eqnarray} }}
$N \to \infty$ ($R$ fixed) limit : commutative $\stwon$ is recovered.

\newpage
%######################### Page 4 #############################
 {\color{Blue}{\shadowbox{Definition of fuzzy $\stwolr$}}} \\
  {\color{Magenta}{$\lambda_{i}^{L}, \lambda_{j}^{R}$}} $\Rightarrow$ $N_L (N_R)$-dimensional representation
 of su$(2)$ algebra: 
{\color{Magenta}{\begin{eqnarray}
 [\lambda_{i}^{L}, \lambda_{j}^{L}] = i \epsilon_{ijk} \lambda_{k}^{L}, \hspace{3mm}
 [\lambda_{i}^{R}, \lambda_{j}^{R}] = i \epsilon_{ijk} \lambda_{k}^{L}, \hspace{3mm}
 [\lambda_{i}^{L}, \lambda_{j}^{R}] = 0, \textrm{ where }
  (\lambda_{i}^{L,R})^{2} = \frac{N^{2}_{L,R} - 1}{4}. \nonumber
\end{eqnarray} }}
 This representation is realized by the tensor product:
 {\color{Red}{\begin{eqnarray}
  \lambda_{i}^{L} = \lambda_{i} \otimes {\bf 1}_{N_{R}}, \hspace{5mm} 
  \lambda_{i}^{R} = {\bf 1}_{N_{L}} \otimes \lambda_{i}. \nonumber
 \end{eqnarray} }}
 (algebra of functions $f$ on $\stwolr$) = (matrix algebra {\color{Magenta}{Mat$({\cal N}, {\bf C})$}}), \\
 where {\color{Magenta}{${\cal N} = N_{L} N_{R}$}}. 
 
 Normalized coordinate function
 {\color{Magenta}{\begin{eqnarray}
  x_{i}^{L,R} = \frac{2R}{\sqrt{(N_{L,R})^{2}-1}} \lambda_{i}^{L,R}, \hspace{5mm}
  (x_{i}^{L})^{2} = (x_{i}^{R})^{2} = R^{2}. \nonumber
 \end{eqnarray} }}
 This space is {\color{Magenta}{a regularization of $\stwolr \subset {\bf R}^{6}$}}. 
 
 Normalized integral of a function {\color{Magenta}{$f \in \stwolr$}}:
 {\color{Magenta}{ \begin{eqnarray}
  \int_{\stwolr} f = \frac{V}{\cal N} \tr f, \textrm{ where we define the volume } V = 16 \pi^2 R^4. \nonumber
 \end{eqnarray} }}
\newpage
%################### Page 5 ######################
{\color{Blue}{\shadowbox{Quantum plane limit {\bf R}$_{\theta}^{4}$}}} \\
 Tangential coordinate {\color{Magenta}{$x_{1,2}$}} near the "north pole":
 
 For {\color{Magenta}{$R^{2} = N \theta /2$}}, we obtain
{\color{Magenta}{\begin{eqnarray}
 [x_{1}, x_{2}] = i \frac{2R}{N} x_{3} = i \frac{2R}{N} \sqrt{R^{2} - x_1^2 - x_2^2}
 = i \theta (1 + {\rm O}(1/N)). \nonumber
\end{eqnarray} }}
Algebra of {\color{Magenta}{the quantum plane $[x_1, x_2] = i \theta$}} ($R,N \to \infty$,
$\theta$ fixed). \\

The fuzzy $\stwolr$ case:  ($R^{2} = N_{L,R} \theta_{L,R} /2$)
{\color{Magenta}{\begin{eqnarray}
 [x_{i}^{L}, x_{j}^{L}] = i \epsilon_{ij} \theta^{L}, \hspace{5mm}
 [x_{i}^{R}, x_{j}^{R}] = i \epsilon_{ij} \theta^{R}, \hspace{5mm}
 [x_{i}^{L}, x_{j}^{R}] = 0. \nonumber
\end{eqnarray} }}
Integral of a function $f(x) \in \stwolr$:
{\color{Magenta}{\begin{eqnarray}
 \int_{\stwolr} f(x) \to 4 \pi^{2} \theta_{L} \theta_{R} \tr f(x) = \int_{{\rm R}_{\theta}^{4}} f(x). \nonumber
\end{eqnarray} }}
\newpage
%############## Page 6 ####################
\section{Gauge theory on fuzzy $\stwolr$}
 Construct {\color{Magenta}{$\stwolr$}} as "submanifold" of {\bf R}$^{6}$. 
 
 Consider a multi-matrix model with {\color{Magenta}{6 dynamical fields $B_{i}^{L,R}$}}:
 {\color{Red}{\begin{eqnarray}
  & & S = \frac{1}{g^{2}} \int_{\stwolr} \left( \frac{1}{2} F_{ia,jb} F_{ia,jb} + \varphi^{2}_{L}
  + \varphi^{2}_{R} \right), \textrm{ where } \nonumber \\
  & & \varphi_{L(R)} = \frac{1}{R^{2}} \left( B_{i}^{L(R)} B_{i}^{L(R)} - \frac{N^{2}_{L(R)}-1}{4} \right), 
  \nonumber \\
  & & F_{iL(R), jL(R)} = \frac{1}{R^{2}} \left( i [B_{i}^{L(R)}, B_{j}^{L(R)}] + \epsilon_{ijk} B_{k}^{L(R)}
    \right), \hspace{5mm}
   F_{iL,jR} = \frac{i}{R^{2}} [B_{i}^{L}, B_{j}^{R}]. \nonumber
  \end{eqnarray} }}
  Invariant under {\color{Magenta}{${\rm SU}(2)_{L} \times {\rm SU}(2)_{R}$}} and
  {\color{Magenta}{U$({\cal N})$}}. 
  
  Equation of motion 
  {\color{Blue}{\large
  \begin{eqnarray}
   \{ B_{i}^{L}, (B_{j}^{L})^{2} - \frac{N^{2}_{L}-1}{4} \} + (B_{i}^{L}
   + i \epsilon_{ijk} B_{j}^{L} B_{k}^{L} )
   + i \epsilon_{ijk} [B_{j}^{L}, (B_{k}^{L} + i \epsilon_{krs} B_{r}^{L} B_{s}^{L})]
   + [B_{j}^{R}, [B_{j}^{R}, B_{i}^{L}]] = 0.\nonumber
  \end{eqnarray} }}
  Classical solution {\color{Red}{$F = \varphi = 0$ $\Rightarrow$ $B_{i}^{L(R)} = 
  \lambda_{i}^{L(R)}$}}.
  
  Expansion around the classical solution {\color{Magenta}{$B_{i}^{a} = \lambda_{i}^{a}
  + R A_{i}^{a}$}}. 
  
  Gauge transformation of the fluctuation : {\color{Magenta}{$A_{i}^{L(R)} \to
  A_{i}^{'L(R)} = U A_{i}^{L(R)} U^{-1} + U [\lambda_{i}^{L(R)}, U^{-1}]$}}. 
  
  The field strength takes the form
 {\color{Magenta}{
 \begin{eqnarray}
  & & F_{iL(R),jL(R)} = \frac{i}{R} \left( [\lambda_{i}^{L(R)},A_{j}^{L(R)}] - [\lambda_{j}^{L(R)}, A_{i}^{L(R)}]
  + R [A_{i}^{L(R)}, A_{j}^{L(R)}] \right), \hspace{5mm} \nonumber \\
  & & F_{iL,jR} = \frac{i}{R} \left( [\lambda_{i}^{L}, A_{j}^{R}] - [\lambda_{j}^{R}, A_{i}^{R}] 
  + R [A_{i}^{L}, A_{j}^{R}] \right). \nonumber
 \end{eqnarray} }}
% \newpage
 %################# Page 7 ########################
  Commutative limit $\Rightarrow$ separate the radial/tangential degrees of freedom.
  
  $\varphi_{L(R)}$ is bounded for configurations with finite action.
 {\color{Magenta}{
 \begin{eqnarray}
  \varphi_{L(R)} = \frac{1}{R} (\lambda_{i}^{L(R)} A_{i}^{L(R)} + A_{i}^{L(R)} \lambda_{i}^{L(R)})
  + A_{i}^{L(R)} A_{i}^{L(R)} \hspace{5mm} \Rightarrow
  x_{i} A^{a}_{i} + A^{a}_{i} x_{i} = {\rm O}(\frac{\varphi}{N}). \nonumber
 \end{eqnarray} }}
 {\color{red}{$A_{i}^{a}$}} is tangential in the commutative limit. \\
 
% \item{Consider {\color{Red}{$\phi_{L(R)} = N \varphi_{L(R)}$}}, with a mass of O$(N)$. \\
%   These are decoupled from the other fields.}
% \end{enumerate}
 Standard electrodynamics on commutative $\stwolr$:
 {\color{Magenta}{
 \begin{eqnarray}
  S = \frac{1}{2g^{2}} \int_{\stwolr} F^{t}_{ia,jb} F^{t}_{ia,jb}. \nonumber
 \end{eqnarray} }}
 At the north pole, {\color{Magenta}{$\frac{i}{R} {\rm ad} \lambda_{i}^{L(R)} = - \epsilon_{ij}
 \frac{\partial}{\partial x_{j}^{L(R)}}$}}. 
 
 The field strength :{\color{Magenta}{$F^{t}_{iL,jR} = \partial_{i}^{L} A_{j}^{{\rm (cl)}R}
 - \partial_{j}^{R} A_{i}^{{\rm (cl)}L}$}}, ({\color{Blue}{$A_{i}^{{\rm (cl)}L(R)} = - \epsilon_{ij}
 A_{j}^{L(R)}$}}).
 
 \newpage
%################# Page 8 #########################
\section{Formulation based on SO$(6)$}
We cast the action on $\stwolr$ into the formulation based on {\color{Red}{SO$(6)$}}. 

Embed $\stwolr \subset {\rm R}^{6}$ (${\rm SO}(3)_{L} \times {\rm SO}(3)_{R} \subset
{\rm SO}(6)$ ):
{\color{Blue}{
\begin{eqnarray}
  & & B_{\mu} = (B_{i}^{L}, B_{j}^{R}), \hspace{5mm} \gamma_{\mu} = (\gamma_{i}^{L}, \gamma_{i}^{R}),
  \nonumber \\
  & & B^{L} = \frac{1}{2} + B_{i}^{L} \gamma_{i}^{L}, \hspace{5mm}
 B^{R} = \frac{i}{2} + B_{i}^{R} \gamma_{i}^{R}. \nonumber
\end{eqnarray} }}
 The gamma matrices' relations:
{\color{Blue}{
\begin{eqnarray}
& & (\gamma_{i}^{L})^{\dagger} = \gamma_{i}^{L}, \hspace{5mm}
 (\gamma_{i}^{R})^{\dagger} = - \gamma_{i}^{R}, \nonumber \\
& & \gamma_{i}^{L} \gamma_{j}^{L} = \delta_{ij} + i \epsilon_{ijk} \gamma_{k}^{L}, \hspace{5mm}
 \gamma_{i}^{R} \gamma_{j}^{R} = - \delta_{ij} - \epsilon_{ijk} \gamma_{k}^{R}, \hspace{5mm}
 [\gamma_{i}^{L}, \gamma_{j}^{R}] = 0. \nonumber
\end{eqnarray} }}
\newpage
%########### Page 9 #######################
{\color{Blue}{\shadowbox{(1) separate $B$ into two $4 {\cal N} \times 4 {\cal N}$ matrices}}} \\
 We break the symmetry as ${\rm SO}(6) \to {\rm SO}(3) \times {\rm SO}(3)$.
{\color{Magenta}{
\begin{eqnarray}
 & & \gamma_{i}^{L} = \sigma_{i} \otimes {\bf 1}_{2 \times 2}, \hspace{5mm}
     \gamma_{i}^{R} = {\bf 1}_{2 \times 2} \otimes i \sigma_{i}. \nonumber
\end{eqnarray} }}
For the projector {\color{Magenta}{$P = \frac{1}{2}(1 + \sigma_{i} \otimes \sigma_{i})$}},
they satisfy {\color{Magenta}{$\gamma_{i}^{R} = i P \gamma_{i}^{L} P$, $P^2 = 1$}}.

The degree of freedom reduces to {\color{Blue}{two $2 {\cal N} \times 2 {\cal N}$ matrices}}:
{\color{Blue}{
\begin{eqnarray}
 B_{L} = \underbrace{\left( B_{i}^{L} \sigma_{i} + \frac{1}{2} \right)}_{=X_{L}} 
 \otimes {\bf 1}_{2 \times 2}, \hspace{5mm}
 B_{R} = i P \left\{ \underbrace{\left( B_{i}^{R} \sigma_{i} + \frac{1}{2} \right)}_{=X_{R}}
  \otimes {\bf 1}_{2 \times 2} \right\} P. \nonumber
\end{eqnarray} }}
The action {\color{Magenta}{$S = S_{6} - 2 S_{\rm break}$}} recovers the action of $\stwolr$:
{\color{Blue}{
\begin{eqnarray}
 S_{6} &=& 2 {\rm Tr} \left( B_{L}^{2} - B_{R}^{2} - \frac{N^{2}}{2} \right)^{2}
       + 2 {\rm Tr} [B_{L}, B_{R}]^{2}, \nonumber \\
 S_{\rm break} &=& -2 {\rm Tr} \left( B_{L}^{2} - \frac{N^{2}}{4} \right)
 \left( - B_{R}^{2} - \frac{N^{2}}{4} \right). \nonumber
\end{eqnarray} }}
\newpage
%######### Page 10 ###################
{\color{Blue}{\shadowbox{(2) Embed $B$ into one $8 {\cal N} \times 8 {\cal N}$ matrices}}} \\
Construct the $8 \times 8$ gamma matrices:
{\color{Magenta}{
\begin{eqnarray}
 \Gamma^{\mu} = \left( \begin{array}{cc} 0 & \gamma^{\mu} \\ (\gamma^{\mu})^{\dagger} & 0 
 \end{array} \right), \textrm{ where }
 \{ \Gamma^{\mu}, \Gamma^{\nu} \} = 2 \delta^{\mu \nu}. \nonumber
\end{eqnarray} }}
Consider a {\color{Blue}{single Hermitean $8{\cal N} \times 8 {\cal N}$ matrix}}:
{\color{Blue}{
\begin{eqnarray}
 C &=& \Gamma^{\mu} B_{\mu} + \underbrace{(C_{0}^{L} + C_{0}^{R})}_{=C_{0}}
   = \underbrace{\left( \begin{array}{cc} 0 & B^{L} \\ B^{L} & 0 \end{array} \right)}_{=C^{L}}
   + \underbrace{\left( \begin{array}{cc} 0 & B^{R} \\ - B^{R} & 0 \end{array} \right)}_{=C^{R}},
    \textrm{ where } \nonumber \\
 C_{0}^{L} &=& - \frac{i}{2} \Gamma^{L}_{1} \Gamma^{L}_{2} \Gamma^{L}_{3}
    = \frac{1}{2} \left( \begin{array}{cc} 0 & 1 \\  1 & 0 \end{array} \right), \hspace{5mm}
 C_{0}^{R} = - \frac{i}{2} \Gamma^{R}_{1} \Gamma^{R}_{2} \Gamma^{R}_{3}
    = \frac{i}{2} \left( \begin{array}{cc} 0 & 1 \\ -1 & 0 \end{array} \right). \nonumber
\end{eqnarray} }}
 The following action is close to that of $\stwolr$:
{\color{Magenta}{
\begin{eqnarray}
 S_{6} &=& {\rm Tr} \left( (C^{2} - \frac{N^{2}}{2})^{2} \right)
   = 8 \tr (B_{\mu} B_{\mu} - \frac{N^{2}-1}{2})^{2} + 4 \tr F_{\mu \nu} F_{\mu \nu}. 
 \textrm{ where } \nonumber \\
 F_{ia,jb} &=& i [B_{ia}, B_{jb}] + \delta_{ab} \epsilon_{ijk} B_{ka}. \nonumber
\end{eqnarray} }}
But we have {\color{Blue}{$(B_{\mu} B_{\mu} - \frac{N^{2}-1}{2})^{2}$}}, instead of
{\color{Blue}{$(B_{i}^{L} B_{i}^{L} - \frac{N_{L}^{2}-1}{4})^{2}
+ (B_{i}^{R} B_{i}^{R} - \frac{N_{R}^{2}-1}{4})^{2}$}}. 

This is because its ground state should be some {\color{Blue}{S$^{5}$}}.
\newpage
%################### Page 11 #####################
We have to explicitly break the symmetry as {\color{Blue}{${\rm SO}(6) \to {\rm SO}(3) 
\times {\rm SO}(3)$}}. 

We introduce the term 
{\color{Magenta}{
\begin{eqnarray}
 S_{\rm break} = {\rm Tr} (C^{2}_{L} - \frac{N^{2}_{L}}{4} )(C^{2}_{R} - \frac{N^{2}_{R}}{4}). \nonumber
\end{eqnarray} }}
The following action recovers that of $\stwolr$:
{\color{Magenta}{
\begin{eqnarray}
 S = S_{6} - 2 S_{\rm break}
   = 8 \tr \left( (B_{iL}^{2} - \frac{N^{2}_{L}-1}{4})^{2} 
    + (B_{iR}^{2} - \frac{N^{2}_{R}-1}{4})^{2} + \frac{1}{2} F_{\mu \nu} F_{\mu \nu} \right). \nonumber
\end{eqnarray} }}
This formulation is useful in introducing fermions.
\newpage
%############ Page 12 ##################
\section{Quantization}
The quantization is straightforward by {\color{Magenta}{a path integral over
the Hermitian matrices}}:

%{\color{Red}{
%\begin{eqnarray}
% Z = \int dB_{\mu} \exp( - S[B_{\mu}] + \tr B_{\mu} J_{\mu} ). \nonumber
%\end{eqnarray} }}
No need to fix the gauge (since the gauge group U$({\cal N})$ is compact).

{\color{Magenta}{Finite path integral for any fixed ${\cal N}$}}.

(the square term of $B_{i}^{L(R)}$ further suppresses the path integral).

Perturbation around the fuzzy sphere $\to$ Gauge fixing by usual BRST prescription:
{\color{Magenta}{
\begin{eqnarray}
 S_{\rm BRST} = S + \frac{1}{\cal N} \tr \left( {\bar c}[\lambda_{\mu}, [B_{\mu}, c]] 
 - (\frac{\alpha}{2} b - [\lambda_{\mu}, B_{\mu}]) b \right). \nonumber
\end{eqnarray} }}
BRST transformation (such that $s^2 = 0$)
{\color{Blue}{
\begin{eqnarray}
 s B_{\mu} = [B_{\mu}, c], \hspace{3mm} sc = cc, \hspace{3mm} s {\bar c} = b, \hspace{3mm} sb = 0. \nonumber
\end{eqnarray} }}
\newpage
%########### Page 13 ##################
\section{Topologically nontrivial solutions on $\stwolr$}
{\color{Blue}{\shadowbox{Monopole solutions}}} 
{\color{Magenta}{
\begin{eqnarray}
 & & B_{i}^{L} = \alpha^{L} \lambda_{i}^{N - m_{L}} \otimes {\bf 1}_{N - m_{R}}, \hspace{5mm}
 B_{i}^{R} = \alpha^{R} {\bf 1}_{N-m_{L}} \otimes \lambda_{i}^{N-m_{R}}, \nonumber \\
 & & \textrm{ where } \alpha^{L,R} = 1 + \frac{m_{L,R}}{N}, \hspace{3mm} m_{L,R} \ll N. \nonumber
\end{eqnarray} }}
Associated field strength and constraint term:
{\color{Blue}{
\begin{eqnarray}
 & & F_{iL(R), jL(R)} = - \frac{m^{L(R)}}{2R^{3}} \epsilon_{ijk} x_{k}^{L(R)}, \hspace{5mm}
 F_{iL,jR} = 0, \hspace{3mm} \nonumber \\
 & & \lim_{N \to \infty} (B_{i}^{L(R)} B_{i}^{L(R)} - \frac{N^{2}-1}{4}) = 0. \nonumber
\end{eqnarray} }}
Commutative limit: {\color{Blue}{$F = - 2 \pi (m^{L} \omega^{L} + m^{R} \omega^{R})$}}, where
$\omega^{L(R)} = \frac{1}{4 \pi R^{3}} \epsilon_{ijk} x_{i}^{L(R)} dx_{j}^{L(R)} dx_{k}^{L(R)}$. 

This monopole solution is realized for {\color{Red}{matrix size ${\cal N} = (N-m_{L})(N-m_{R}) \neq N^{2}$}}. 

This mismatch can be reconciled by combining this with another type of solution.
\newpage

%########### Page 14 ##########################
{\color{Blue}{\shadowbox{Fluxon solutions}}}
{\color{Magenta}{
\begin{eqnarray}
 B_{i}^{L(R)} = {\rm diag}(d_{i,1}^{L(R)}, \cdots, d_{i,n}^{L(R)}), \textrm{ where }
 \sum_{i} d_{i,k}^{L(R)} d_{i,k}^{L(R)} = \left\{ \begin{array}{ll} 
 \frac{N^{2}-3}{4}, & \textrm{ type A}, \\
 0, & \textrm{ type B}. \end{array} \right. \nonumber
\end{eqnarray} }}
Associated field strength and constraint term:
{\color{Blue}{
\begin{eqnarray}
 & & F_{iL(R),jL(R)} = \frac{\epsilon_{ijk}}{R^{2}} {\rm diag}(d_{k,1}^{L}, \cdots,
 d_{k,n}^{L}), \hspace{5mm}
 F_{iL,jR} = 0, \nonumber \\
 & & \lim_{N \to \infty} (B_{i}^{L(R)} B_{i}^{L(R)} - \frac{N^{2}-1}{4}) 
 = \left\{ \begin{array}{ll} -\frac{1}{2}, & \textrm{type A}, \\
 - \frac{N^{2}-1}{4}, & \textrm{type B}. \end{array} \right. \nonumber
\end{eqnarray} }}
{\color{magenta}{Only the type A solution}} has a finite contribution:
{\color{Blue}{
\begin{eqnarray}
 S = \left\{ \begin{array}{ll}
 \frac{16 \pi^{2} R^{4}}{g^{2} N^{2}} \left( \frac{n}{4R^{4}} + \frac{2n}{R^{4}}
 \frac{N^{2}-3}{4} \right) \to \frac{8 \pi^{2}}{g^{2}} n, & \textrm{type A}, \\
 \frac{16 \pi^{2} R^{4}}{g^{2} N^{2}} \left( 2 \times (\frac{N^{2}-1}{4})^{2} \right) \to \infty,
 & \textrm{type B}. \end{array} \right. \nonumber
\end{eqnarray} }}
We call these type A solutions {\color{Red}{"fluxons"}}.
\newpage
%################ Page 15 #############################
{\color{Blue}{\shadowbox{Combination of monopole and fluxon solutions}}}
{\color{Magenta}{
\begin{eqnarray}
 & & B_{i}^{L} = \left( \begin{array}{cc} \alpha^{L} \lambda_{i}^{N-m_{L}} \otimes {\bf 1}_{N - m_{R}} & 0 \\
 0 & {\rm diag}(d_{i,1}^{L}, \cdots, d_{i,n}^{L}) \end{array} \right), \nonumber \\
 & & B_{i}^{R} = \left( \begin{array}{cc} \alpha^{R} {\bf 1}_{N-m_{L}} \otimes \lambda_{i}^{N-m_{R}} & 0 \\
 0 & {\rm diag}(d_{i,1}^{R}, \cdots, d_{i,n}^{R}) \end{array} \right), \textrm{ where } \nonumber \\
 & & m_{L} = - m_{R} = m, \hspace{5mm} n=m^{2}. \nonumber
\end{eqnarray} }}
%In the commutative limit {\bf R}$_{\theta}^{4}$, this becomes the {\color{Red}{"fluxon"}} solution.

The total action of this solution:
{\color{Magenta}{
\begin{eqnarray}
 S_{(m)} = \frac{4 \pi^{2}}{g^{2}} (2m^2 + 2m^2), \textrm{ as } N \to \infty. \nonumber
\end{eqnarray} }}

\newpage
%############ Page 16 ########################
\section{Gauge theory on {\bf R}$_{\theta}^{4}$ from fuzzy $\stwolr$}
The most general noncommutative {\bf R}$_{\theta}^{4}$ (for $\mu,\nu = 1,2,3,4$)
{\color{Blue}{
\begin{eqnarray}
[x_{\mu}, x_{\nu}] = i \theta_{\mu \nu}, \textrm{ where }
 \theta_{\mu \nu} = \left( \begin{array}{cccc} 0 & \theta_{12} & 0 & 0 \\ - \theta_{12} & 0 & 0 & 0 \\
 0 & 0 & 0 & \theta_{34} \\ 0 & 0 & - \theta_{34} & 0 \end{array} \right). \nonumber
 \end{eqnarray} }}.
We define the coordinates $X_{\mu}$ and $\phi$ as
{\color{Blue}{
\begin{eqnarray}
 & & X_{1,2} = \sqrt{\frac{2 \theta_{12}}{N_{L}}} B_{1,2}^{L}, \hspace{5mm}
 X_{3,4} = \sqrt{\frac{2 \theta_{34}}{N_{R}}} B_{1,2}^{R}, \nonumber \\
 & & \phi^{L(R)} = B_{3}^{L(R)} - \frac{N_{L(R)}}{2} + \frac{1}{N_{L(R)}}
 [ (B_{1}^{L(R)})^{2} + (B_{2}^{L(R)})^{2} ]. \nonumber
\end{eqnarray} }}
 Scaling limit {\color{Magenta}{$R^{2} = \frac{1}{2} N_{L} \theta_{34} = \frac{1}{2} N_{R} 
 \theta_{12} \to \infty$}}:
 
 $X$ are the {\color{Red}{covariant coordinates on the tangential {\bf R}$_{\theta}^{4}$}} as 
 $N_{L,R} \to \infty$:
 \newpage
%################## Page 17 ########################
{\large 
 \begin{eqnarray}
 \frac{1}{R^{2}}([B_{1}^{L},B^{R}_{1}]) & = & \frac{1}{\theta
  _{12}\theta _{34}}[X_{1},X_{3}], \quad\mbox{etc.,} \nonumber \\ 
\frac{1}{R^{2}}(B^{L}_{1}+i[B^{L}_{2},B^{L}_{3}]) & = & \sqrt{\frac{1}{\theta _{12}\theta _{34}R^{2}}}\;(X_{1}+i[X_{2},\phi ^{L}]-\frac{i}{2\theta _{12}}[X_{2},(X_{1})^{2}]), \nonumber \\
\frac{1}{R^{2}}(B^{L}_{2}+i[B^{L}_{3},B^{L}_{1}]) & = & \sqrt{\frac{1}{\theta _{12}\theta _{34}R^{2}}}\;(X_{2}+i[X_{1},\phi ^{L}]-\frac{i}{2\theta _{12}}[X_{1},(X_{2})^{2}]), \nonumber \\
\frac{1}{R^{2}}(B^{L}_{3}+i[B^{L}_{1},B^{L}_{2}]) & = & \frac{1}{\theta _{12}\theta _{34}}(\theta _{12}+i[X_{1},X_{2}]+\frac{\theta _{12}\theta _{34}}{R^{2}}\phi _{L}-\frac{\theta _{12}\theta _{34}^{2}}{2R^{4}}((X_{1})^{2}+(X_{2})^{2}))\, \nonumber \\
\frac{1}{R^{2}}(B^{L}_{i}B^{L}_{i}-\frac{N_{L}^{2}-1}{4}) &=&
\frac{1}{\theta _{34}}\phi ^{L}
+\frac{2}{R^{2}}((\phi^{L})^{2}+\frac{1}{4})
 -\frac{1}{\theta _{12}R^{2}}\{\phi^{L},(X_{1})^{2}+(X_{2})^{2}\} \nonumber \\
&& +\frac{1}{\theta^{2}_{12}R^{2}}((X_{1})^{2}+(X_{2})^{2})^{2}. \nonumber
 \end{eqnarray} }
The terms from the $\stwolr$ action involving $\phi^{L,R}$:
{\color{Magenta}{
\begin{eqnarray}
 \frac{1}{\theta_{34}^{2}} (\phi^{L})^{2} + \frac{1}{\theta_{12}^{2}} (\phi^{R})^{2} + {\rm O}(\frac{1}{R}).
 \nonumber
\end{eqnarray} }}
At {\color{Magenta}{$R \to \infty$}}, we obtain the action
{\color{Red}{
\begin{eqnarray}
 S = - \frac{1}{2 g^{2} \theta_{12}^{2} \theta_{34}^{2}} \int ([X_{\mu}, X_{\nu}] - i \theta_{\mu \nu})^{2}.
 \nonumber
\end{eqnarray} }}
$X_{\mu}$ can be written as {\color{Red}{$X_{\mu} = x_{\mu} + i \theta_{\mu \nu} A_{\nu}$}}.
\newpage

%################# Page 18 ##############################
{\color{Blue}{\shadowbox{U$(1)$ instantons on {\bf R}$_{\theta}^{4}$ }}}

We consider the self-dual case {\color{Magenta}{$\theta_{\mu \nu} = \frac{1}{2}
\epsilon_{\mu \nu \rho \sigma} \theta_{\rho \sigma}$}}, namely {\color{Magenta}{
$\theta_{12} = \theta_{34} (= \theta)$}}.

The action for U$(1)$ gauge theory on {\bf R}$_{\theta}^{4}$:
{\color{Blue}{
\begin{eqnarray}
 S = \frac{(2 \pi)^{2}}{2 g^{2} \theta^{2}} \tr F_{\mu \nu} F_{\mu \nu}, \textrm{ where }
 F_{\mu \nu} = i ([X_{\mu}, X_{\nu}] - i \theta_{\mu \nu}). \nonumber
\end{eqnarray} }}
Complex coordinates: {\color{Magenta}{$x_{\pm L} = x_{1} \pm i x_{2}, x_{\pm R} = x_{3} \pm i x_{4}$}}.

Commutation relation: {\color{Magenta}{$[x_{+a},x_{-b}] = 2 \theta \delta_{ab},
[x_{+a},x_{+b}] = [x_{-a}, x_{-b}] = 0$}}.

Basis of the Fock space {\color{Magenta}{${\cal H}$}} 
for these coordinates: {\color{Magenta}{$|n_1, n_2 \rangle$, 
$n_{1}, n_{2} \in {\bf N}$}}.
\begin{eqnarray*}
x_{-L}|n_{1},n_{2}\rangle=\sqrt{2\theta}\sqrt{n_{1}+1}|n_{1}+1,n_{2}\rangle, & x_{+L}|n_{1},n_{2}\rangle=\sqrt{2\theta}\sqrt{n_{1}}|n_{1}-1,n_{2}\rangle, \nonumber \\
x_{-R}|n_{1},n_{2}\rangle=\sqrt{2\theta}\sqrt{n_{2}+1}|n_{1},n_{2}+1\rangle,
&
x_{+R}|n_{1},n_{2}\rangle=\sqrt{2\theta}\sqrt{n_{2}}|n_{1},n_{2}-1\rangle\,. \nonumber
\end{eqnarray*}
Complex covariant coordinates: {\color{Blue}{$X_{\pm L} = X_{1} \pm i X_{2}$,
$X_{\pm R} = X_{3} \pm i X_{4}$}}. 

The action is rewritten as
{\color{Magenta}{
\begin{eqnarray}
 S = \frac{\pi^{2}}{g^{2} \theta^{2}} \tr \left( \sum_{a} F_{+a,-a} F_{+a,-a}
 - \sum_{a,b} F_{+a,+b} F_{-a,-b} \right), \textrm{ where }
 F_{a \alpha,b \beta} = [X_{\alpha a}, X_{\beta b}] - 2 \theta \epsilon_{\alpha \beta}
 \delta_{ab}. \nonumber
\end{eqnarray} }}
Equation of motion: {\color{Magenta}{$\sum_{a, \alpha}[X_{\alpha,a}, (F_{\alpha a,\beta b})^{\dagger}]=0$}}.
\newpage
%############ Page 19 ########################
Finite-dimensional subvector space {\color{Magenta}{$V_{n} \subset {\cal H}$}}:
{\color{Blue}{
\begin{eqnarray}
 V_{n} = \{ | i_{k}, j_{k} \rangle; k=1,2, \cdots, n \}. \nonumber
\end{eqnarray} }}
Solutions of the equation of motion:
{\color{Blue}{
\begin{eqnarray}
 X^{(n)}_{+L(R)} = S x_{+L(R)} S^{\dagger} + \sum_{k=1}^{n} \gamma_{k}^{L(R)}
 |i_{k}, j_{k} \rangle \langle i_{k}, j_{k} |. \nonumber
\end{eqnarray} }}
\begin{itemize}
\item{$\gamma_{k}^{L,R} \in {\bf C}$: position of fluxons}
\item{{\color{Magenta}{$S = \sum_{k=1}^{\infty} |i_{k+n},j_{k+n} \rangle \langle 
i_{k}, j_{k} |$}}, with {\color{Magenta}{$S^{\dagger} S = 1$, $S S^{\dagger} =
1 - \underbrace{\sum_{k=1}^{n} |i_{k}, j_{k} \rangle \langle i_{k}, j_{k} |}_{=P_{V_{n}}}$}}. \\
{\color{Magenta}{$P_{V_{n}}$}}: projection operator onto $V_{n}$.
}
\end{itemize}
\begin{eqnarray}
 F_{\mu \nu} = P_{V_{n}} \theta_{\mu \nu} \to S[X_{\pm,a}^{(n)}] = \frac{8 \pi^{2}}{g^{2}}
 \tr (P_{V_{n}}) = \frac{8 \pi^{2} n}{g^{2}}. \nonumber
\end{eqnarray}
This is {\color{Red}{the U$(1)$-instanton solutions}} on R$_{\theta}^{4}$.
\newpage
%############ Page 20 ########################
\section{Fermions}
{\color{Blue}{\shadowbox{Commutative Dirac operator on $\stwostwo$}}} 
{\color{Red}{
\begin{eqnarray}
 D_{4} = \Gamma^{\mu} J_{\mu} + 2 C_{0}. \nonumber
\end{eqnarray} }}
The 8-component spinor {\color{Magenta}{$\Psi_{8}$}} is split into
{\color{Magenta}{two independent 4-component Dirac spinors}}. 

Operators that anticommute with $D_{4}$:
\begin{itemize}
\item{{\color{Blue}{6-dimensional chirality operator:
$\Gamma = i \Gamma_{1}^{L} \Gamma_{2}^{L} \Gamma_{3}^{L} \Gamma_{1}^{R} \Gamma_{2}^{R} \Gamma_{3}^{R}
 = \left( \begin{array}{cc} -1 & 0 \\ 0 & 1 \end{array} \right)$}}. \\
This satisfies {\color{Magenta}{$\{ D_{4}, \Gamma \} = 0$, $\Gamma^{\dagger} = \Gamma$ and 
$\Gamma^{2} = 1$}}. \\
The 8-component spinor is split as {\color{Magenta}{$\Psi_{8} = \left( \begin{array}{c}
\psi_{\alpha} \\ {\bar \psi}_{\bar \beta} \end{array} \right)$}}.
}
\item{Operators {\color{Blue}{$\chi_{L(R)} = \Gamma^{i L(R)} x_{iL(R)}$}}, which preserves
{\color{Magenta}{${\rm SO}(3) \times {\rm SO}(3) \subset {\rm SO}(6)$}}. \\
They satisfy {\color{Magenta}{$\{ D_4, \chi_{L(R)} \} = \{ \chi_{L}, \chi_{R} \} =0$, 
$\chi_{L(R)}^{2} = 1$}}.}
\end{itemize}
\newpage
%############## Page 21 #################
We introduce a projector {\color{Red}{$P_{\pm} = \frac{1}{2} (1 \pm i \chi_{L} \chi_{R})$}}.
{\color{Magenta}{
\begin{eqnarray}
 P_{\pm}^{2} = P_{\pm}, \hspace{3mm} P_{+} + P_{-} = 1, \hspace{3mm} P_{+} P_{-} = 0, \hspace{3mm}
 [P_{\pm}, D_{4}] = [P_{\pm}, \Gamma] = 0. \nonumber
\end{eqnarray} }}
The projected space is {\color{Blue}{preserved by $D_{4}$ and $\Gamma$}}.
{\color{Blue}{
\begin{eqnarray}
 \textrm{spinor Lagrangian : } S_{\rm f} = \Psi_{8}^{\dagger} D_{4} \Psi_{8} = \Psi_{+}^{\dagger} D_{4} \Psi_{+} 
 + \Psi_{-}^{\dagger} D_{4} \Psi_{-}, \textrm{ where } \Psi_{\pm} = P_{\pm} \Psi_{8}. \nonumber
\end{eqnarray} }}
Two ways to obtain a 4-component Dirac spinor:
\begin{itemize}
\item{Impose the constraint {\color{Blue}{$P_{+} \Psi_{8} = \Psi_{8}$}}.}
\item{Add a mass term 
{\color{Blue}{$M_{-} \Psi^{\dagger}_{8} P_{-} \Psi_{8}$}}, with $M_{-} \to \infty$.}
\end{itemize}

At the north pole {\color{Magenta}{$x_{L} = x_{R} = ^{t}(1,0,0)$}},
{\color{Blue}{
\begin{eqnarray}
 P_{+} = \frac{1}{2} \left\{ 1 + i \left( \begin{array}{cc}
 - \gamma_{1}^{L} \gamma_{1}^{R} & 0 \\ 0 & \gamma_{1}^{L} \gamma_{1}^{R} 
 \end{array} \right) \right\} = \frac{1}{2} (1 + \sigma_{3} \otimes 
 \sigma_{3} \otimes \sigma_{3})
 = {\rm diag}(1,0,0,1,0,1,1,0). \nonumber
\end{eqnarray} }}
$P_{\pm}$ actually projects onto a 4-dimensional subspace exactly.
\newpage
%############## Page 22 #############
{\color{Blue}{\shadowbox{Gauged fuzzy Dirac and chirality operators}}}

Fuzzy analogs of {\color{Magenta}{$D_{4}$}} and {\color{Magenta}{$\chi$}}.

Natural fuzzy spinor action : {\color{Red}{$S_{F} = \Psi^{\dagger} C \Psi$}} ($\Psi$ = $8 {\cal N}
\times {\cal N}$ matrix).

$C$ can be split into {\color{Magenta}{fuzzy Dirac operator ${\hat D}$}} and operator {\color{Magenta}{
${\hat \chi}$}}:
{\color{Blue}{
\begin{eqnarray}
 {\hat \chi} \Psi = \sqrt{\frac{2}{N^{2}}} (\Gamma^{\mu} \Psi \lambda_{\mu} - C_{0} \Psi),
 \textrm{ such that } {\hat \chi}^{2} = 1. \nonumber
\end{eqnarray} }}
Fuzzy Dirac operator:
{\color{Blue}{
\begin{eqnarray}
 {\hat D} = C - \sqrt{\frac{N^{2}}{2}} {\hat \chi}
 = \Gamma^{\mu} \D_{\mu} + 2 C_{0}, \textrm{ where }
 \D_{\mu} \Psi = \underbrace{[\lambda_{\mu}, \Psi]}_{={\hat J}_{\mu} \Psi} + A_{\mu} \Psi. \nonumber
\end{eqnarray} }}
$\D_{\mu}$ is a covariant derivative operator, i.e. {\color{Magenta}{$U \D_{\mu} \psi
= \D'_{\mu} U \psi$}} 

(where $D'_{\mu} \Psi = [\lambda_{\mu}, \Psi] + \underbrace{UA_{\mu} U^{-1}}_{=A'_{\mu}}$)

${\hat D}$ satisfies {\color{Magenta}{$\{ {\hat D}, \Gamma \} = 0$}}, however
{\color{Magenta}{$\{ {\hat D}, \chi \} \neq 0$}}:
\begin{eqnarray}
 \{ {\hat D}, {\hat \chi} \} = 
%- \sqrt{\frac{2}{N^{2}-1}}
% \left( 2 (\lambda_{\mu} + A_{\mu}) {\hat J} - 2 A_{\mu} \lambda_{\mu}
% + \{ \Gamma^{\mu}, C_{0} \} \D_{\mu} + 2 \right)=
 {\rm O}(\frac{1}{N}). \nonumber
\end{eqnarray} 
\newpage
%################### Page 23 #############################
{\color{Magenta}{
\begin{eqnarray}
 {\hat D}^{2} \psi = ( \Sigma^{\mu \nu} F_{\mu \nu} + \underbrace{ \D_{\mu} \D_{\mu}
 + \{ \Gamma^{\mu}, C_{0} \} \D_{\mu} }_{={\hat \Box}} + 2 ) \psi. \nonumber
\end{eqnarray} }}
This corresponds to the {\color{Magenta}{${\hat D}^{2}$}} on the curved space. 

Projectors on the fuzzy spinors:
{\color{Red}{
\begin{eqnarray}
 {\hat \chi}_{L(R)} \Psi = \frac{2}{N} (\Gamma^{iL(R)} \Psi \lambda_{iL(R)}
 + C_{0}^{L(R)} ). \nonumber
\end{eqnarray} }}
We define the projection operators {\color{Red}{${\hat P}_{\pm} = \frac{1}{2}
(1 \pm i {\hat \chi}_{L} {\hat \chi}_{R} )$}}.
{\color{Magenta}{
\begin{eqnarray}
 & & {\hat \chi}_{L(R)}^{2} = 1, \hspace{3mm} \{ {\hat \chi}_{L}, {\hat \chi}_{R} \} = 0, \hspace{3mm}
 ({\hat \chi}_{L} {\hat \chi}_{R})^{2} = 1, \nonumber \\
 & & [ {\hat D}^{2}, \Gamma ] = [ \Sigma^{\mu \nu} F_{\mu \nu}, \Gamma ]
 = [ {\hat D}^{2}, {\hat P}_{\pm}] = [\Sigma^{\mu \nu} F_{\mu \nu}, {\hat P}_{\pm} ] = 0.
 \nonumber
\end{eqnarray} }}
The projector no longer commutes with the fuzzy Dirac operator:
{\color{Magenta}{$[{\hat D}, {\hat \chi}_{L} {\hat \chi}_{R}] \neq 0$}}.

We have to add a mass term to reduce the degrees of freedom:
{\color{Magenta}{$M_{-} \Psi^{\dagger}_{8} {\hat P}_{-} \Psi_{8}$}}.

The complete action for a Dirac fermion on fuzzy $\stwostwo$:
{\color{Red}{
\begin{eqnarray}
 S_{\rm Dirac} = \int \left( \Psi^{\dagger}_{8} ({\hat D}+m) \Psi_{8}
  + M_{-} \Psi^{\dagger}_{8} {\hat P}_{-} \Psi_{8} \right). \nonumber
\end{eqnarray} }}
\newpage
%################### Page 24 ##########################
\section{Summary}
In this paper, we have studied the following:
\begin{itemize}
 \item{Gauge theory on fuzzy $\stwostwo$ as a multi-matrix model.}
 \item{Alternative formulations using "collective matrices" based on SO$(6)$.}
 \item{Quantization by a finite path integral.}
 \item{The monopole and fluxon solution}
 \item{The quantum field theory in the flat noncommutative plane {\bf R}$_{\theta}^{4}$.}
 \item{Fermionic term and the chiral Dirac operator.}
\end{itemize}
}
 \end{document}