\hfill \parbox{4cm}{\normalsize %UT-***\\ hep-th/0206012}\\ \vspace{2cm} D-geometric structure of orbifolds \vspace{2cm}
{\normalsize\em Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan}
\normalsize We study D-branes on abelian orbifolds ${C}^d/{Z}_N$ for $d=2, 3$. The toric data describing the D-brane vacuum moduli space, which represents the geometry probed by D-branes, has certain redundancy compared with the classical geometric description of the orbifolds. We show that the redundancy has a simple combinatorial structure and find analytic expressions for degrees of the redundancy. For $d=2$ the structure of the redundancy has a connection with representations of $SU(N)$ Lie algebra, which provides a new correspondence between geometry and representation theory. We also prove that non-geometric phases do not appear in the K$\ddot{a}$hler moduli space for $d=2$.
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\title{
\hfill
\parbox{4cm}{\normalsize
%UT-***\\
hep-th/0206012}\\
\vspace{2cm}
D-geometric structure of orbifolds
\vspace{2cm}}
\author{Tomomi Muto\thanks{e-mail address:\tt 
tmuto@ms.u-tokyo.ac.jp}
\vspace{1.0cm}\\
{\normalsize\em Graduate School of Mathematical Science,
University of Tokyo}\\
{\normalsize\em Komaba 3-8-1, Meguro-ku, Tokyo 153-8914, Japan}}
\date{\normalsize}
\maketitle

\vspace{1.0cm}


\begin{abstract}
\normalsize
We study D-branes on abelian orbifolds ${\bf C}^d/{\bf Z}_N$ for $d=2, 3$.
The toric data describing the D-brane vacuum moduli space,
which represents the geometry probed by D-branes,
has certain redundancy compared with the classical geometric description of the orbifolds.
We show that the redundancy has a simple combinatorial structure
and find analytic expressions for degrees of the redundancy.
For $d=2$ the structure of the redundancy has a connection with representations of $SU(N)$ Lie algebra,
which provides a new correspondence between geometry and representation theory.
We also prove that non-geometric phases do not appear in the K$\ddot{\rm a}$hler moduli space for $d=2$.
\end{abstract}






\newpage
\section{Introduction}

%It has been found that 
D-geometry, the geometry as seen by D-branes,
has qualitatively different features from that probed by fundamental strings.
One of the remarkable features of the D-geometry is that
non-geometric phases, in which the orbifold singularity is resolved,
are projected out from the K$\ddot{\rm a}$hler moduli space \cite{DGM, Muto}
in contrast to the analysis based on fundamental strings \cite{AGM, Witten}.
By inspecting the calculation in \cite{DGM},
one can see that the projection of the non-geometric phases stems from certain redundancy
of lattice vectors describing the toric data of the orbifold:
the redundancy of the  lattice vectors implies  redundancy of coordinates describing the toric variety,
and elimination of the redundant variables leads to the resolution of the orbifold singularity.
%For example, toric data of the orbifold ${\bf C}^3/{\bf Z}_3$ analyzed in \cite{DGM}
%three lattice vectors degenerate to a point corresponding to the exceptional divisor.
%The precise meaning of the redundancy will be clarified later.
Therefore it is important to clarify the structure of the redundancy to understand
the nature of D-geometry.
The redundancy, however, occurs as a result of a combinatorial algorithm
based on toric geometry, and even the degree of the redundancy has not been known except for 
the models for which explicit calculation was carried out\footnote{The meaning of the redundancy
for the orbifold ${\bf C}^3/{\bf Z}_2 \times {\bf Z}_2$ is discussed in \cite{MT}.}.
For example, the orbifolds of the form
${\bf C}^3/{\bf Z}_N$, ($(x,y,z) \equiv  (\omega x,\omega y,\omega^{-2}z)$ with $\omega^N=1$, $N$:odd)
have been analyzed only for $N \leq 11$ \cite{Muto}.
Total number $n_D(N)$ of lattice vectors in the toric data which describes the D-geometry of the orbifold
are shown in table \ref{table:number}.
For reference we also show the total number $n_C(N)$ of lattice vectors
in the classical geometric description of the orbifold.
As one easily see, $n_C(2N+1)=N+3$,
while the analytic expression for the number $n_D(N)$ has not been known. 
\begin{table}[h]
\begin{center}
\label{table:number}
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$N$&3&5&7&9&11 \\ \hline
$n_C(N)$&4&5&6&7&8 \\ \hline
$n_D(N)$&6&13&31&78&201 \\ \hline
\end{tabular}
\caption{$n_D(N)$ is the total number of lattice vectors
necessary to describe the D-geometry the orbifold ${\bf C}^3/{\bf Z}_N$.
$n_C(N)$ is the total number of lattice vectors
necessary to describe the orbifold ${\bf C}^3/{\bf Z}_N$ in classical geometry.}
\end{center}
\end{table}

The purpose of the present paper is to clarify the structure of the redundancy
by re-examining the toric data carefully.
In section 2 we investigate two-dimensional abalian orbifold ${\bf C}^2/{\bf Z}_3$
with ${\bf Z}_3 \in SU(2)$ to illustrate the idea of our analysis.
We find that the structure of the redundancy is related to representations of $SU(3)$
Lie algebra.
It implies that there is a correspondence between exceptional divisors
of the resolution of the orbifold ${\bf C}^2/{\bf Z}_3$
and representations of the $SU(3)$ Lie algebra.
We discuss the meaning of the elimination of the redundant variables from the viewpoint of representation theory.
In section 3, we generalize the analysis to the orbifold ${\bf C}^2/{\bf Z}_N$
and find a relation between the D-geometry of ${\bf C}^2/{\bf Z}_N$ and $SU(N)$ Lie algebra.
We also show that non-geometric phases are projected out for any $N$.
In section 4, we study the three-dimensional abelian orbifold mentioned above.
We show that there is a simple combinatrial structure similar to the two-dimensional case
and derive the analytic expression for the number $n_D(N)$.

After completion of this work, a paper \cite{FFHH} appeared which has overlap with ours.















\section{D-geometric structure of ${\bf C}^2/{\bf Z}_3$}
\reseteqnum


In this section we study D-branes on the orbifold ${\bf C}^2/{\bf Z}_3$
where the action of $g \in {\bf Z}_3$ on $(x,y) \in {\bf C}^2$ is given by
\begin{equation}
(x, y) \rightarrow (\omega_3 x, \omega_3^{-1}y), \quad \omega_3={\rm exp}(2\pi i/3).
\end{equation}
Field contents of the worldvolume gauge theory of the D-brane
is encoded in the quiver diagram \cite{DM}.
The quiver diagram of this model is depicted in Figure \ref{fig:quiverC2Z3}.
Each node represents a factor $U(1)$ of the gauge group and each arrow represents a complex
scalar field.
Note that the gauge group is $U(1)^2$ since the diagonal $U(1)$ of $U(1)^3$ is trivial. 
\begin{figure}[htp]
\begin{center}
\begin{picture}(200,100)
%\put(0,0){\framebox(200,100)}
\put(60,60){$x_1$}
\put(76,44){$y_1$}
\put(97,8){$x_2$}
\put(97,28){$y_2$}
\put(129,60){$x_3$}
\put(113,44){$y_3$}
\put(40,20){\circle{10}}
\put(160,20){\circle{10}}
\put(100,80){\circle{10}}
\put(50,17){\vector(1,0){100}}
\put(150,23){\vector(-1,0){100}}
\put(50,25){\vector(1,1){45}}
\put(90,75){\vector(-1,-1){45}}
\put(105,70){\vector(1,-1){45}}
\put(155,30){\vector(-1,1){45}}
\end{picture}
\caption{The quiver diagram for ${\bf C}^2/{\bf Z}_3$.}
\label{fig:quiverC2Z3}
\end{center}
\end{figure}

The classical moduli space $\cal M$ of the quiver gauge theory,
which is interpreted as the space probed by the D-brane,
is obtained from the space ${\bf C}^6$ of complex scalars $(x_i,y_i)$ 
by imposing the F-flatness conditions
\begin{equation}
x_1 y_1=x_2 y_2=x_3 y_3,
\label{eq:F-flat3}
\end{equation}
and D-flatness conditions
\begin{eqnarray}
&&|x_1|^2-|y_1|^2-|x_2|^2+|y_2|^2 = \zeta_1,\nonumber \\
&&|x_2|^2-|y_2|^2-|x_3|^2+|y_3|^2 = \zeta_2,
\label{eq:D-flat3}
\end{eqnarray}
and further dividing by the gauge group $U(1)^2$.
The Fayet-Iliopoulos parameters $\zeta_i$ coming from twisted sector of closed strings
parameterize the K$\ddot{\rm a}$hler moduli space of the orbiofld.

We calculate the space $\cal M$
following the procedure given in \cite{DGM}.
We first consider the space $\cal W$ on which only the F-flatness constraints (\ref{eq:F-flat3})
are imposed,
\begin{equation}
{\cal W}=\{(x_i,y_i) \in {\bf C}^6 | x_1 y_1=x_2 y_2=x_3 y_3 \}.
\end{equation}
As explained in \cite{DGM},
$\cal W$ can be expressed as a holomophic quotient of the form
\begin{equation}
({\bf C}^k-F)/({\bf C}^*)^{k-4},
\label{eq:holomorphic}
\end{equation}
and furthermore it is realized as a vacuum moduli space of
a certain two-dimensional $N=(2,2)$ supersymmetric $U(1)^{k-4}$ gauged linear sigma model \cite{Witten}.
Homogeneous coordinates $p_i$ of ${\bf C}^k$ is interpreted as scalar components
of chiral superfields $P_i$ in the gauge theory,
and the ${\bf C}^*$ quotient is realized by the combined operation of imposing D-flatness constraint
and then dividing by the $U(1)$ gauge symmetry.
%Thus the procedure to realize the space $\cal W$ as a vacuum moduli space of a gauge theory is
%considered as a conversion of the F-flatness conditions to D-flatness conditions.
The procedure for realizing $\cal W$ as a vacuum moduli space of a gauge theory
is carried out by a combinatorial algorithm based on toric geometry.
The first step of the algorithm is to write the solution of the F-flatness constraints as
\begin{eqnarray}
&&x_1=u_1, \quad y_1=u_2, \quad y_2=u_3, \quad y_3=u_4, \nonumber \\
&&x_2=u_1 u_2 u_3^{-1}, \quad x_3=u_1 u_2 u_4^{-1}
\end{eqnarray}
by introducing four complex variables $u_1, \cdots, u_4$.
(Note that the dimensions of $\cal W$ is four.)
The form of the solution
%\begin{equation}
%x_i = \prod_j u_j^{K_{ij}}. 
%\end{equation}
is specified by the matrix
\begin{equation}
K=\left(
\begin{array}{lcccc}
&u_1&u_2&u_3&u_4\\
m_1 \, &1&0&0&0\\
m_2 \, &0&1&0&0\\
m_3 \, &0&0&1&0\\
m_4 \, &0&0&0&1\\
m_5 \, &1&1&-1&0\\
m_6 \, &1&1&0&-1\\
\end{array}
\right).
\end{equation}
The row vectors $m_i$ of the matrix $K$
%are vectors in the lattice $M={\bf Z}^4$ and 
define the edges of a cone $\hat \sigma$ in ${\bf R}^4$.
The next step is to calculate the dual cone of $\hat \sigma$ defined by
\begin{equation}
\sigma =\{n \in {\bf R}^4 | m \cdot n \geq 0, \forall m \in \hat \sigma\}.
\end{equation}
Calculation of $\sigma$ is a problem of integer programming,
and in the present case
the cone $\sigma$ is generated by eight vectors $n_i$ in the lattice ${\bf Z}^4$.
They are represented by column vectors in the matrix,
\begin{equation}
T=\left(
\begin{array}{lcccccccc}
&n_1&n_2&n_3&n_4&n_5&n_6&n_7&n_8\\
u_1&1&0&1&1&0&0&1&0\\
u_2&0&1&0&0&1&1&0&1\\
u_3&0&0&1&0&1&0&1&1\\
u_4&0&0&0&1&0&1&1&1
\label{eq:cone}
\end{array}
\right).
\end{equation}
Chiral multiplets $P_i$ in the gauge theory realization
corresponds to each vector $n_i$,
so the number $k$ in (\ref{eq:holomorphic}) is equal to 8.
D-flatness conditions imposed on the variables $p_i$ are
determined from linear relations among the vectors $n_i$.
The four linear relations
\begin{eqnarray}
&&n_1-n_2-n_3+n_5=0, \nonumber \\
&&n_1-n_3-n_4+n_7=0, \nonumber \\
&&n_1-n_4-n_2+n_6=0, \\
&&n_4-n_6-n_7+n_8=0, \nonumber
\end{eqnarray}
imply four D-flatness conditions
in the gauge theory realization,
\begin{eqnarray}
&&|p_1|^2-|p_2|^2-|p_3|^2+|p_5|^2=0, \nonumber \\
&&|p_1|^2-|p_3|^2-|p_4|^2+|p_7|^2=0, \nonumber \\
&&|p_1|^2-|p_4|^2-|p_2|^2+|p_6|^2=0, \label{eq:FD} \\
&&|p_4|^2-|p_6|^2-|p_7|^2+|p_8|^2=0. \nonumber
\end{eqnarray}
Thus $\cal W$ is obtained from the space ${\bf C}^8$ with coordinates $(p_1,\cdots,p_8)$
by imposing the four D-flatness conditions (\ref{eq:FD}) and dividing by the $U(1)^4$ gauge symmetry.

To combine the D-flatness conditions ($\ref{eq:D-flat3}$)
existing from the beginning,
we must represent the equations ($\ref{eq:D-flat3}$) in terms of the variables $p_i$.
Since the relation between the coordinates
$(v_1,\cdots,v_6)=(x_1,y_1,y_2,y_3,x_2,x_3)$ and
$(p_1,\cdots,p_8)$ is given by
\begin{equation}
v_i=\prod_j p_j^{m_i \cdot n_j},
\end{equation}
the D-flatness conditions are rewritten as
\begin{eqnarray}
&&|p_2|^2-|p_3|^2=\zeta_1,\nonumber \\
&&|p_3|^2-|p_4|^2=\zeta_2. 
\label{eq:DD}
\end{eqnarray}
Thus the moduli space $\cal M$ is described by the six D-flatness conditions
(\ref{eq:FD}) and (\ref{eq:DD}).
For later use we make some rearrangements,
\begin{eqnarray}
&&|p_1|^2-|p_3|^2-|p_4|^2+|p_7|^2=0, \nonumber \\
&&|p_4|^2-|p_6|^2-|p_7|^2+|p_8|^2=0, \nonumber \\
&&|p_2|^2-|p_3|^2=\zeta_1, \nonumber \\
&&|p_3|^2-|p_4|^2=\zeta_2, \label{eq:D} \\
&&|p_5|^2-|p_6|^2=\zeta_2, \nonumber \\
&&|p_6|^2-|p_7|^2=\zeta_1. \nonumber
\end{eqnarray}



The analysis of $\cal M$ described by (\ref{eq:D}) depends on the values of $(\zeta_1,\zeta_2)$.
We first examine in the region $\zeta_1>0$ and $\zeta_2>0$,
where we obtain inequalities $|p_2|^2>|p_3|^2>|p_4|^2$ and $|p_5|^2>|p_6|^2>|p_7|^2$
from the last four equations of (\ref{eq:D}).
Hence the four variables $p_2$, $p_3$, $p_5$ and $p_6$ are not zero in this region.
In such a situation, the four variables can be eliminated by using the last four D-flatness conditions of (\ref{eq:D})
and the corresponding $U(1)^4$ gauge symmetry.
Thus we obtain two D-flatness conditions on four variables $p_1$, $p_4$, $p_7$ and $p_8$,
\begin{eqnarray}
&&|p_1|^2-2|p_4|^2+|p_7|^2=\zeta_2, \nonumber \\
&&|p_4|^2-2|p_7|^2+|p_8|^2=\zeta_1.
\end{eqnarray}
Since we are considering in the region $\zeta_1>0$ and $\zeta_2>0$,
the right hand side of the equations are positive,
which implies that $\cal M$ is the resolution of the orbifold ${\bf C}^2/{\bf Z}_3$.
The topology of $\cal M$ is represented by the toric diagram in Figure \ref{fig:toric3}.
In the figure, the vector $n_i$ corresponding to the variable $p_i$ is attached to each vertex.
\begin{figure}[htp]
\begin{center}
\begin{picture}(250,30)
%\put(0,0){\framebox(250,30)}
\put(45,5){$n_1$}
\put(95,5){$n_4$}
\put(145,5){$n_7$}
\put(195,5){$n_8$}
\put(50,20){\circle*{8}}
\put(100,20){\circle*{8}}
\put(150,20){\circle*{8}}
\put(200,20){\circle*{8}}
\thicklines
\put(50,20){\line(1,0){150}}
\end{picture}
\caption{Toric diagram representing $\cal M$ in the region $\zeta_1>0$ and $\zeta_2>0$.}
\label{fig:toric3}
\end{center}
\end{figure}



Next we examine (\ref{eq:D}) in the region $\zeta_1<0$ and $\zeta_1+\zeta_2>0$ as another example.
Then we obtain inequalities $|p_3|^2>|p_2|^2>|p_4|^2$ and $|p_5|^2>|p_7|^2>|p_6|^2$
from the last four equations of (\ref{eq:D}),
and hence the four variables $p_2$, $p_3$, $p_5$ and $p_7$ are not zero.
In such a situation, the four variables are redundant.
%and eliminated by using the last four D-flatness conditions and the corresponding $U(1)^4$ gauge symmetry.
Thus we obtain two D-flatness conditions on four variables $p_1$, $p_4$, $p_6$ and $p_8$,
\begin{eqnarray}
&&|p_1|^2-2|p_4|^2+|p_6|^2=\zeta_1+\zeta_2, \nonumber \\
&&|p_4|^2-2|p_6|^2+|p_8|^2=-\zeta_1.
\end{eqnarray}
Although the variable $p_7$ is replaced by $p_6$ compared with the first case,
the form of the equations are the same
and furthermore the right hand side of the two D-flatness equations remain to be positive
since we are now considering in the region $\zeta_1<0$ and $\zeta_1+\zeta_2>0$.
Hence $\cal M$ is the resolution of the orbifold ${\bf C}^2/{\bf Z}_3$ also in this region.
These two examples show that the exchange of variables from $p_7$ to $p_6$
is the key to the fact that
$\cal M$ remains to be the resolution of the orbifold ${\bf C}^2/{\bf Z}_3$:
if such an exchange of variables does not occur,
% that enables us 
$\cal M$ becomes singular after crossing the line $\zeta_1=0$.
%Of course such an exchange of variables is possible because there is some redundancy
%of the variables.

Similar analysis shows that the equations (\ref{eq:D}) describe
the resolution of the orbifold ${\bf C}^2/{\bf Z}_3$
independent of the region in the $(\zeta_1,\zeta_2)$ space
by choosing appropriate set of variables,
thus non-geometric phases are projected out from the the K$\ddot{\rm a}$hler moduli space 
of the orbifold\footnote{On
codimension one loci in the $\zeta$-space,
the singularity does not completely resolved at the classical level.}.
In Figure \ref{fig:phase}
we summarize toric diagrams representing $\cal W$ and lattice vectors $n_i$ corresponding to the
variables $p_i$ chosen in each region in the $\zeta$-space.
\begin{figure}[htp]
\begin{center}
\begin{picture}(200,200)
%\put(0,0){\framebox(200,200)}
\put(120,162){\line(1,0){60}}
\put(120,162){\circle*{3}}
\put(140,162){\circle*{3}}
\put(160,162){\circle*{3}}
\put(180,162){\circle*{3}}
\put(115,150){$n_1$}
\put(135,150){$n_4$}
\put(155,150){$n_7$}
\put(175,150){$n_8$}
\put(30,192){\line(1,0){60}}
\put(30,192){\circle*{3}}
\put(50,192){\circle*{3}}
\put(70,192){\circle*{3}}
\put(90,192){\circle*{3}}
\put(25,180){$n_1$}
\put(45,180){$n_4$}
\put(65,180){$n_6$}
\put(85,180){$n_8$}
\put(10,122){\line(1,0){60}}
\put(10,122){\circle*{3}}
\put(30,122){\circle*{3}}
\put(50,122){\circle*{3}}
\put(70,122){\circle*{3}}
\put(05,110){$n_1$}
\put(25,110){$n_2$}
\put(45,110){$n_6$}
\put(65,110){$n_8$}
\put(20,62){\line(1,0){60}}
\put(20,62){\circle*{3}}
\put(40,62){\circle*{3}}
\put(60,62){\circle*{3}}
\put(80,62){\circle*{3}}
\put(15,50){$n_1$}
\put(35,50){$n_2$}
\put(55,50){$n_5$}
\put(75,50){$n_8$}
\put(110,22){\line(1,0){60}}
\put(110,22){\circle*{3}}
\put(130,22){\circle*{3}}
\put(150,22){\circle*{3}}
\put(170,22){\circle*{3}}
\put(105,10){$n_1$}
\put(125,10){$n_3$}
\put(145,10){$n_5$}
\put(165,10){$n_8$}
\put(130,92){\line(1,0){60}}
\put(130,92){\circle*{3}}
\put(150,92){\circle*{3}}
\put(170,92){\circle*{3}}
\put(190,92){\circle*{3}}
\put(125,80){$n_1$}
\put(145,80){$n_3$}
\put(165,80){$n_7$}
\put(185,80){$n_8$}
\thicklines
\put(0,100){\vector(1,0){200}}
\put(100,0){\vector(0,1){200}}
\put(0,200){\line(1,-1){200}}
\put(190,110){$\zeta_1$}
\put(107,190){$\zeta_2$}
\end{picture}
\caption{Phase structure for ${\bf C}^2/{\bf Z}_3$.}
\label{fig:phase}
\end{center}
\end{figure}

As shown in the Figure, the vector $n_1$ ($n_8$) corresponds to the left (right) vertex
independent of the values of $(\zeta_1,\zeta_2)$.
On the other hand, the three vectors $n_2,n_3,n_4$ ($n_5,n_6,n_7$)
correspond to the second vertex from the left (right).
One of the three vectors is chosen in each region in the $\zeta$-space and
the other two are redundant.

To understand the meaning of the redundancy,
let us look closely at the vectors $n_i$ in ($\ref{eq:cone}$).
We first note that every vector $n_i$ satisfies the following condition
\begin{equation}
n_i \cdot \rho_0 =1, \quad {\rm for} \quad \rho_0=(1,1,0,0),
\end{equation}
which corresponds to the fact that the space ${\cal M}$ satisfies Calabi-Yau condition.
Thus $n_i$ lies on a three-dimensional hyperplane,
and it is enough to consider three-dimensional vectors $\hat n_i$ obtained by omitting
the first entry of $n_i$.
The vectors $\hat n_i$ corresponding to each vertex of the toric diagram
are given in Figure \ref{fig:toricC2Z3}.
\begin{figure}[htp]
\begin{center}
\begin{picture}(400,100)
%\put(0,0){\framebox(400,100)}
\put(20,35){$\hat n_1=(0,0,0)$}
\put(120,50){$\hat n_2=(1,0,0)$}
\put(120,35){$\hat n_3=(0,1,0)$}
\put(120,20){$\hat n_4=(0,0,1)$}
\put(220,50){$\hat n_5=(1,1,0)$}
\put(220,35){$\hat n_6=(1,0,1)$}
\put(220,20){$\hat n_7=(0,1,1)$}
\put(320,35){$\hat n_8=(1,1,1)$}
\put(50,80){\circle*{8}}
\put(150,80){\circle*{8}}
\put(250,80){\circle*{8}}
\put(350,80){\circle*{8}}
\thicklines
\put(50,80){\line(1,0){300}}
\end{picture}
\caption{Toric diagram for ${\bf C}^2/{\bf Z}_3$.}
\label{fig:toricC2Z3}
\end{center}
\end{figure}

Structure of the redundancy is clear from the form of the vectors;
the three entries of each vector are equal to 0 or 1,
and vectors whose $k$ components are equal to 1
correspond to the $k+1$-th vertex from the left.
Hence the multiplicity of the vectors associated to the $k+1$-th vertex from the left
is given by $(^3_k)$.

Now we would like to show that the above structure has correspondence
with representations of the $SU(3)$ Lie algebra.
Let us consider the eigenvalues $\nu_i$ of the vectors $\hat n_i$
for the generators of the Cartan subalgebra of $SU(3)$,
\begin{equation}
\frac{1}{2}
\left(
\begin{array}{ccc}
1&0&0 \\
0&-1&0 \\
0&0&0
\end{array}
\right),\quad
\frac{1}{2\sqrt 3}
\left(
\begin{array}{ccc}
1&0&0 \\
0&1&0 \\
0&0&-2
\end{array}
\right).
\end{equation}
We obtain the following results,
\begin{eqnarray}
&&\nu_1=(0,0),\\
&&\nu_2=\left(\frac{1}{2},\frac{1}{2\sqrt 3}\right),
\nu_3=\left(-\frac{1}{2},\frac{1}{2\sqrt 3}\right),
\nu_4=\left(0,-\frac{1}{\sqrt 3}\right),\\
&&\nu_5=\left(0,\frac{1}{\sqrt 3}\right),
\nu_6=\left(\frac{1}{2},-\frac{1}{2\sqrt 3}\right),\nu_7=\left(-\frac{1}{2},-\frac{1}{2\sqrt 3}\right),\\
&&\nu_8=(0,0),
\end{eqnarray}
and they are depicted in Figure \ref{fig:weight}.
\begin{figure}[htp]
\begin{center}
\begin{picture}(200,200)
%\put(0,0){\framebox(200,200)}
\put(30,60){\dashbox{4}(140,80)}
\put(100,100){\circle*{8}}
\put(170,140){\circle*{8}}
\put(30,140){\circle*{8}}
\put(100,20){\circle*{8}}
\put(100,180){\circle*{8}}
\put(170,60){\circle*{8}}
\put(30,60){\circle*{8}}
\put(110,90){$\nu_1$}
\put(180,140){$\nu_2$}
\put(10,140){$\nu_3$}
\put(110,20){$\nu_4$}
\put(110,180){$\nu_5$}
\put(180,60){$\nu_6$}
\put(10,60){$\nu_7$}
\put(80,90){$\nu_8$}
\put(160,90){$\frac{1}{2}$}
\put(10,90){$-\frac{1}{2}$}
\put(70,20){$-\frac{1}{\sqrt 3}$}
\put(80,180){$\frac{1}{\sqrt 3}$}
\put(70,50){$-\frac{1}{2\sqrt 3}$}
\put(80,130){$\frac{1}{2\sqrt 3}$}
\thicklines
\put(0,100){\vector(1,0){200}}
\put(100,0){\vector(0,1){200}}
\end{picture}
\caption{Eigenvalues of the vectors $\hat n_i$.}
\label{fig:weight}
\end{center}
\end{figure}
The figure implies that the three vectors $\nu_2$, $\nu_3$ and $\nu_4$ form the weights of the representation
${\bf 3}$ of $SU(3)$
and the three vectors $\nu_5$, $\nu_6$ and $\nu_7$ form the weights of the representation
$\bar {\bf 3}$ of SU(3).
Thus the three vectors $n_2,n_3,n_4$ ($n_5,n_6,n_7$)
associated to the second vertex from the left (right) in the toric diagram
correspond to the representation ${\bf 3}$ ($\bar {\bf 3}$) of $SU(3)$.
Furthermore it is reasonable to consider the vector $n_1$ ($n_8$)
associated to the left (right) vertex in the toric diagram
corresponds to the representation
${\bf 1}$ ($\bar {\bf 1}$) of $SU(3)$,
where ${\bf 1}$ is the trivial representation.
Thus antisymmetric representations of $SU(3)$ corresponds to each vertex in the toric diagram
as depicted in Figure \ref{fig:representationC2Z3}.
The degrees of the redundancy (the multiplicity of the vectors)
coincide with the dimensions of the representations.
\begin{figure}[htp]
\begin{center}
\begin{picture}(400,60)
%\put(0,0){\framebox(400,60)}
\put(47,27){1}
\put(145,25){\framebox(10,10)}
\put(245,15){\framebox(10,10)}
\put(245,25){\framebox(10,10)}
\put(345,5){\framebox(10,10)}
\put(345,15){\framebox(10,10)}
\put(345,25){\framebox(10,10)}
\put(50,50){\circle*{8}}
\put(150,50){\circle*{8}}
\put(250,50){\circle*{8}}
\put(350,50){\circle*{8}}
\thicklines
\put(50,50){\line(1,0){300}}
\end{picture}
\caption{Correspondence between representations of $SU(3)$
and the toric diagram for ${\bf C}^2/{\bf Z}_3$.
The Dynkin diagrams indicate the representations.}
\label{fig:representationC2Z3}
\end{center}
\end{figure}

Since the vertices except for the both ends of the diagram represent exceptional divisors,
the graph implies that there is a correspondence between exceptional divisors
of the resolution of the orbifold ${\bf C}^2/{\bf Z}_3$
and antisymmetric representations of the Lie algebra $SU(3)$.
Note that the McKay correspondence \cite{McKay}
gives the connection between exceptional divisors of the resolution of ${\bf C}^2/\Gamma$
and irreducible representations of $\Gamma$.
The above correspondence gives a new kind of connection between geometry and representation theory.




Finally we would like to discuss positivity of roots of the $SU(3)$ Lie algebra.
It gives a simple explanation on the choice of four variables $p_i$ in each
region in the $\zeta$-space.
In identifying the degeneracy of the lattice vectors $n_i$ with representations of 
$SU(3)$,
$\nu_i-\nu_j$ is identified with a root of the $SU(3)$ Lie algebra.
However, the positivity of roots is irrelevant in the discussion.
Now we define the positivity of the roots as
\begin{eqnarray}
&&sign(\nu_2-\nu_3)=sign (\zeta_1), \nonumber \\
&&sign(\nu_3-\nu_4)=sign (\zeta_2), \\
&&sign(\nu_2-\nu_4)=sign (\zeta_1+\zeta_2).\nonumber
\end{eqnarray}
The definition comes from the D-flatness conditions
\begin{eqnarray}
&&|p_2|^2-|p_3|^2=\zeta_1, \nonumber \\
&&|p_3|^2-|p_4|^2=\zeta_2.
\end{eqnarray}
The conditions lead to the relations
\begin{eqnarray}
&&sign(\nu_5-\nu_6)=sign (\zeta_2) \nonumber \\
&&sign(\nu_6-\nu_7)=sign (\zeta_1)\\
&&sign(\nu_5-\nu_7)=sign (\zeta_1+\zeta_2),\nonumber
\end{eqnarray}
which corresponds to the D-flatness conditions
\begin{eqnarray}
&&|p_5|^2-|p_6|^2=\zeta_2, \nonumber \\
&&|p_6|^2-|p_7|^2=\zeta_1.
\end{eqnarray}

By definition, positivity depend on the values of $(\zeta_1,\zeta_2)$.
In the region $\zeta_1>0$ and $\zeta_2>0$, for example,
we obtain inequalities
\begin{eqnarray}
&&\nu_2>\nu_3>\nu_4, \nonumber \\
&&\nu_5>\nu_6>\nu_7,
\end{eqnarray}
They mean that $\nu_4$ and $\nu_7$ are lowest weight states
of the representations ${\bf 3}$ and $\bar {\bf 3}$ respectively.
Thus the coordinate $p_i$ which remains after elimination of the redundant variables
corresponds to the lowest weight state in each representation.
This property also holds for any region in the $\zeta$-space.








\section{D-geometric structure of ${\bf C}^2/{\bf Z}_N$}
\reseteqnum

The discussion in the last section is generalized to the orbifold ${\bf C}^2/{\bf Z}_N$
where the action of $g \in {\bf Z}_N$ on $(x,y) \in {\bf C}^2$ is given by
\begin{equation}
(x, y) \rightarrow (\omega_N x, \omega_N^{-1}y), \quad \omega_N={\rm exp}(2\pi i/N).
\end{equation}
The quiver diagram of this model is depicted in Figure \ref{fig:quiverC2ZN}.
\begin{figure}[htp]
\begin{center}
\begin{picture}(400,100)
%\put(0,0){\framebox(400,100)}
\put(125,65){$x_1$}
\put(145,50){$y_1$}
\put(117,8){$x_2$}
\put(117,28){$y_2$}
\put(270,8){$x_{N-1}$}
\put(270,28){$y_{N-1}$}
\put(263,65){$x_N$}
\put(245,50){$y_N$}
\put(80,20){\circle{10}}
\put(160,20){\circle{10}}
\put(240,20){\circle{10}}
\put(320,20){\circle{10}}
\put(200,90){\circle{10}}
\put(90,17){\vector(1,0){60}}
\put(150,23){\vector(-1,0){60}}
\put(250,17){\vector(1,0){60}}
\put(310,23){\vector(-1,0){60}}
\put(90,30){\vector(2,1){100}}
\put(185,85){\vector(-2,-1){100}}
\put(210,80){\vector(2,-1){100}}
\put(315,35){\vector(-2,1){100}}
\put(188,20){\line(1,0){4}}
\put(198,20){\line(1,0){4}}
\put(208,20){\line(1,0){4}}
\end{picture}
\caption{Quiver diagram for ${\bf C}^2/{\bf Z}_N$.}
\label{fig:quiverC2ZN}
\end{center}
\end{figure}
The classical moduli space $\cal M$ of the corresponding quiver gauge theory is
obtained from the space ${\bf C}^{2N}$ of variables $(x_i,y_i)$ 
by imposing the F-flatness conditions
\begin{equation}
x_1 y_1=x_2 y_2 = \cdots =x_N y_N,
\label{eq:F-flatN}
\end{equation}
and D-flatness conditions
\begin{equation}
|x_i|^2-|y_i|^2-|x_{i+1}|^2+|y_{i+1}|^2 = \zeta_i, \quad (i=1,2,\cdots,N-1),
\label{eq:D-flatN}
\end{equation}
and further dividing by the gauge group $U(1)^{N-1}$.

We first consider the space 
$\cal W$ imposed only the F-flatness constraints (\ref{eq:F-flatN}).
The solution of the constraints (\ref{eq:F-flatN}) of the form
\begin{eqnarray}
&&x_1=u_1, \quad y_1=u_2, \quad y_2=u_3, \quad \cdots, \quad y_N=u_{N+1}, \nonumber \\
&&x_2=u_1 u_2 u_3^{-1}, \quad x_3=u_1 u_2 u_4^{-1}, \quad \cdots, \quad x_N=u_1 u_2 u_{N+1}^{-1},
\end{eqnarray}
is encoded in the rows of the matrix $K$,
\begin{equation}
K=\left(
\begin{array}{lcccccc}
&u_1&u_2&u_3&u_4&\cdots&u_{N+1}\\
m_1&1&0&0&0&\cdots&0\\
m_2&0&1&0&0&\cdots&0\\
m_3&0&0&1&0&\cdots&0\\
m_4&0&0&0&1&\cdots&0\\
&&&\vdots&&&\\
m_{N+1}&0&0&0&0&\cdots&1\\
m_{N+2}&1&1&-1&0&\cdots&0\\
m_{N+3}&1&1&0&-1&\cdots&0\\
&&&\vdots&&&\\
m_{2N}&1&1&0&0&\cdots&-1
\end{array}
\right).
\end{equation}
The $N+1$-dimensional vectors $m_i$ form a cone $\hat \sigma$ in ${\bf R}^{N+1}$.
To represent the space ${\cal W}$ as a holomorphic quotient,
we consider the dual cone $\sigma$ defined by
\begin{equation}
\sigma =\{n \in {\bf R}^{N+1} | m \cdot n \geq 0, \forall m \in \hat \sigma\}.
\label{eq:dual}
\end{equation}
After some calculations, we obtain the cone $\sigma$ generated by
$2^N$ vectors $n_i$ $(i=1,\cdots,2^N)$ of the form
\begin{equation}
(1,0,*,*,\cdots,*) \quad {\rm or} \quad (0,1,*,*,\cdots,*),
\label{eq:generators}
\end{equation}
where each of the last $N-1$ entries
denoted by $*$ is equal to 0 or 1.
Since the $N+1$-dimensional vector $n_i$ satisfies the relation
$\rho_0 \cdot n_i=1$ for $\rho_0=(1,1,0,\cdots,0)$,
$n_i$ lies on a $N$-dimensional hyperplane,
so we consider $N$-dimensional vectors $\hat n_i$ obtained by omitting
the first entry of $n_i$.
As one can see from (\ref{eq:generators}),
every entry of $\hat n_i$ is equal to 0 or 1,
hence the $2^N$ vectors $\hat n_i$ are labeled by the elements of ${\bf Z}_2^N$;
in other words,
$\hat n_i$ corresponds to a vertex of the $N$-dimensional cube with volume one.

For later use, we define degree $d(\hat n_i)$ of a vector $\hat n_i$ by $d(\hat n_i) =\hat \rho_1 \cdot \hat n_i$
for an $N$-dimensional vector $\hat \rho_1=(1,1,\cdots,1)$.
By definition, the degree $d(\hat n_i)$ is the number of components of $\hat n_i$
whose entry is 1,
and $d(\hat n_i)$ takes values from 0 to $N$.
Thus the vectors ${\hat n_i}$ are classified according to their degrees.
We denote the set of vectors with degree $k$ as $F_k$,
in which there are $(^N_{\, k} )$ elements.
%Now to reflect the structure of ${\bf Z}_2^N$,
We also introduce $N$-dimensional vectors $e_i$ $(i=1,\cdots,N)$ defined by
\begin{equation}
e_i=(0,\cdots,0,1,0,\cdots,0)
\end{equation}
where the $i$-th entry is equal to 1.
The vectors in $F_k$ are written in the form
\begin{equation}
\hat n_{(a_1,\cdots,a_k)}=e_{a_1}+e_{a_2}+\cdots+e_{a_k}
\end{equation}
where $(a_1,a_2,\cdots,a_N)$ is a permutation of $(1,2,\cdots,N)$.
For $k=0$, we define $\hat n_{(\phi)}=(0,\cdots,0)$.
Note that the vectors $n_i$ and the corresponding complex scalars $p_i$
are also labeled by the same indices.

Under these definitions,
$2^N-N-1$ linear relations among the $2^N$ vectors $n_i$ are
written in the form
\begin{equation}
n_{(a_1,\cdots,a_k)}-n_{(a_1,\cdots,a_k,a_{k+1})}
-n_{(a_1,\cdots,a_k,a_{k+2})}+n_{(a_1,\cdots,a_k,a_{k+1},a_{k+2})}=0,
\label{eq:relation}
\end{equation}
where $k=0,1,\cdots,N-2$.
%To realize the space $\cal W$ as a vacuum moduli space of a gauge theory,
%we introduce a complex scalar $p_{(a_1,\cdots,a_k)}$
%to each vector $n_{(a_1,\cdots,a_k)}$.
%The linear relations (\ref{eq:relation})
These relations imply $2^N-N-1$ D-flatness conditions,
\begin{equation}
|p_{(a_1,\cdots,a_k)}|^2-|p_{(a_1,\cdots,a_k,a_{k+1})}|^2
-|p_{(a_1,\cdots,a_k,a_{k+2})}|^2+|p_{(a_1,\cdots,a_k,a_{k+1},a_{k+2})}|^2=0.
\label{eq:FDN}
\end{equation}
These are the D-flatness conditions converted from the F-flatness conditions (\ref{eq:F-flatN}).
On the other hand,
the D-flatness conditions (\ref{eq:D-flatN}) existing from the beginning are rewritten as
\begin{equation}
|p_{(i)}|^2-|p_{(i+1)}|^2=\zeta_i
\label{eq:DDN}
\end{equation}
where $i=1,2,\cdots,N-1$.
Thus the vacuum moduli space $\cal M$ is described by the $2^N-2$ D-flatness conditons (\ref{eq:FDN}) and (\ref{eq:DDN}).

Once the Fayet-Iliopoulos parameters $(\zeta_1,\cdots,\zeta_{N-1})$ are fixed,
relative size of $|p_{(i)}|^2$ are determined by (\ref{eq:DDN}),
and one obtains the following inequalities\footnote{We
ignore codimension one hyperplanes in the parameter space.},
\begin{equation}
|p_{(L_1)}|^2 < |p_{(L_2)}|^2 < \cdots < |p_{(L_N)}|^2.
\label{eq:ordering}
\end{equation}
Here $(L_1,L_2,\cdots,L_N)$ is a certain permutation of $(1,2,\cdots,N)$.
%Combining with the equations $(\ref{eq:FDN})$,
%we obtain inequalities among the size $|p_{(a_1,\cdots,a_k)}|^2$ of variables.

Next we rewrite the $2^N-2$ D-flatness conditions
by taking the ordering (\ref{eq:ordering}) into account\footnote{The rearrangement
of the D-flatness conditions is similar to that
considered in \cite{BGLP}.}.
They are classified into two types.
The first type of equations are written as
\begin{equation}
|p_{(L_1,\cdots,L_k)}|^2-|p_{(L_1,\cdots,L_k,L_{k+1})}|^2
-|p_{(L_1,\cdots,L_k,L_{k+2})}|^2+|p_{(L_1,\cdots,L_k,L_{k+1},L_{k+2})}|^2=0
\label{eq:D-flat-k}
\end{equation}
where $k=0,1,\cdots,N-2$.
The second type of equations are written as
\begin{equation}
|p_{(L_1,\cdots,L_i)}|^2-|p_{(a_1,\cdots,a_i)}|^2=\zeta_{(L_1,\cdots,L_i)-(a_1,\cdots,a_i)}
\label{eq:D-flat-i}
\end{equation}
where $i=1,2,\cdots,N-1$, and
$(a_1,\cdots,a_N)$ is a permutation of $(1,\cdots,N)$
which gives $p_{(a_1,\cdots,a_i)}$ other than $p_{(L_1,\cdots,L_i)}$.
Note that the number of independent equations in the second type is
\begin{equation}
\sum_{i=1}^{N-1} \left( (^N_{\, k})-1 \right) = 2^N-N-1.
\end{equation}
The parameter in the right hand side of the equation (\ref{eq:D-flat-i}) is defined by
\begin{equation}
\zeta_{(L_1,\cdots,L_i)-(a_1,\cdots,a_i)}
=\sum_{j=1}^i \zeta_{(L_j)-(a_j)},
\end{equation}
where $\zeta_{(L_i)-(a_i)}$ is a linear combination of $\zeta_i$ defined by
\begin{equation}
\zeta_{(L_j)-(a_j)}
=\left\{
\begin{array}{ll}
\zeta_{L_j}+\zeta_{L_j+1}+\cdots +\zeta_{a_j-1},
& ({\rm if} \; L_j<a_j)\\
-(\zeta_{a_j}+\zeta_{a_j+1}+\cdots +\zeta_{L_j-1}),
& ({\rm if} \; a_j<L_j)\\
0. & ({\rm if} \; L_j=a_j)\\
\end{array}
\right.
\end{equation}
By the definition of the permutation $(L_1,\cdots,L_N)$,
the parameter $\zeta_{(L_1,\cdots,L_i)-(a_1,\cdots,a_j)}$ turns out to be negative.
This implies that $|p_{(a_1,\cdots,a_i)}|^2$ is not zero,
and the variable $p_{(a_1,\cdots,a_i)}$ can be eliminated
by (\ref{eq:D-flat-i}) and the corresponding $U(1)$ gauge symmetry.
After the elimination of the redundant variables,
we obtain $N-1$ D-flatness conditions on
$N+1$ variables $p_{(L_1,\cdots,L_i)}$ $(i=0,\cdots,N)$,
\begin{equation}
|p_{(L_1,\cdots,L_k)}|^2-2|p_{(L_1,\cdots,L_k,L_{k+1})}|^2
+|p_{(L_1,\cdots,L_k,L_{k+1},L_{k+2})}|^2=-\zeta_{(L_{k+1})-(L_{k+2})},
\label{eq:DN}
\end{equation}
where $k=0,1,\cdots,N-2$.
By the definition of $(L_1,\cdots,L_N)$, the right hand side of 
(\ref{eq:DN}) is positive.
This implies that $\cal M$ is the resolution of the orbifold ${\bf C}^2/{\bf Z}_N$.
Thus the singularity of the orbifold is resolved in every region in the $(\zeta_1,\cdots,\zeta_{N-1})$
space (except for codimension one loci)
and non-geometric phases are projected out.


Finally we would like to discuss correspondence between geometry of ${\bf C}^2/{\bf Z}_N$
and representation theory.
Similar argument to the $N=3$ case
shows that the $(^N_{\, k})$ vectors in $F_k$ corresponds to the
$k$-th vertex from the left in the toric diagram
and they form an antisymmetric representation of $SU(N)$ Lie algebra
as shown in Figure \ref{fig:representationC2ZN}.
\begin{figure}[htp]
\begin{center}
\begin{picture}(400,120)
%\put(0,0){\framebox(400,120)}
\put(47,77){1}
\put(95,75){\framebox(10,10)}
\put(145,75){\framebox(10,10)}
\put(145,65){\framebox(10,10)}
\put(295,75){\framebox(10,10)}
\put(295,65){\framebox(10,10)}
\put(295,35){\framebox(10,10)}
\put(308,60){\oval(7,50)[r]}
\put(315,57){$N$-1}
\put(345,75){\framebox(10,10)}
\put(345,65){\framebox(10,10)}
\put(345,25){\framebox(10,10)}
\put(358,55){\oval(7,60)[r]}
\put(365,52){$N$}
\put(50,100){\circle*{8}}
\put(100,100){\circle*{8}}
\put(150,100){\circle*{8}}
\put(300,100){\circle*{8}}
\put(350,100){\circle*{8}}
\put(200,100){\line(1,0){10}}
\put(220,100){\line(1,0){10}}
\put(240,100){\line(1,0){10}}
\put(300,50){\line(0,1){2}}
\put(300,54){\line(0,1){2}}
\put(300,58){\line(0,1){2}}
\put(350,45){\line(0,1){2}}
\put(350,49){\line(0,1){2}}
\put(350,53){\line(0,1){2}}
\thicklines
\put(50,100){\line(1,0){130}}
\put(270,100){\line(1,0){80}}
\end{picture}
\caption{Correspondence between representations of $SU(N)$
and the toric diagram for ${\bf C}^2/{\bf Z}_N$.}
\label{fig:representationC2ZN}
\end{center}
\end{figure}
By defining positivity of roots of $SU(N)$ Lie algebra in a similar manner to the last section,
we find out that the variables $p_{(L_1,\cdots,L_i)}$ remaining after the elimination of the redundant variables
correspond to the lowest weight state of each antisymmetric representation of $SU(N)$.

























\section{Three-dimensional orbifolds}
\reseteqnum
In this section we consider D-branes  on the orbifold ${\bf C}^3/{\bf Z}_N$
where the action of $g \in {\bf Z}_N$ on $(x,y,z) \in {\bf C}^3$ is given by
\begin{equation}
(x, y, z) \rightarrow (\omega_N x, \omega_N y,\omega_N^{-2}z),
\quad \omega_N={\rm exp}(2\pi i/N).
\label{eq:ZN}
\end{equation}
By a similar analysis to the two-dimensional case,
we obtain toric data of the vacuum moduli space of the quiver gauge theory.
As an example, we consider the orbifold ${\bf C}^3/{\bf Z}_7$.
F-flatness conditions are represented by a cone $\hat \sigma$ in ${\bf R}^9$ generated
by 21 vectors $m_i$,
and its dual cone $\sigma$ turns out to be generated by 31 vectors $n_i$.
(The form of the vectors is given in the Appendix.)
After combining D-flatness conditions,
we obtain the toric diagram depicted in Figure \ref{fig:toricC3Z7}.
In the toric diagram
7 vectors $n_4,\cdots,n_{10}$ correspond to the left vertex inside the diagram,
14 vectors $n_{11},\cdots,n_{24}$ correspond to the middle one
and 7 vectors $n_{25},\cdots,n_{31}$ correspond to the right one.
\begin{figure}[htp]
\begin{center}
\begin{picture}(400,120)
%\put(0,0){\framebox(400,120)}
\put(260,110){$n_3$}
\put(310,10){$n_1$}
\put(80,60){$n_2$}
\put(80,32){$\{n_4,\cdots,n_{10}\}$}
\put(140,18){$\{n_{11},\cdots,n_{24}\}$}
\put(200,4){$\{n_{25},\cdots,n_{31}\}$}
\put(125,38){\vector(1,1){18}}
\put(180,24){\vector(1,2){16}}
\put(235,10){\vector(1,3){14}}
\put(100,60){\circle*{8}}
\put(150,60){\circle*{8}}
\put(200,60){\circle*{8}}
\put(250,60){\circle*{8}}
\put(250,110){\circle*{8}}
\put(300,10){\circle*{8}}
\thicklines
\put(100,60){\line(1,0){150}}
\put(100,60){\line(3,1){150}}
\put(100,60){\line(4,-1){200}}
\put(150,60){\line(2,1){100}}
\put(150,60){\line(3,-1){150}}
\put(200,60){\line(1,1){50}}
\put(200,60){\line(2,-1){100}}
\put(250,60){\line(0,1){50}}
\put(250,60){\line(1,-1){50}}
\put(250,110){\line(1,-2){50}}
\end{picture}
\caption{Toric diagram for ${\bf C}^3/{\bf Z}_7$.}
\label{fig:toricC3Z7}
\end{center}
\end{figure}

The structure of the multiplicity can be analyzed from the explicit form of the
nine-dimensional vectors $n_i$.
However, not all the components of the vector are necessary
to describe the structure of the multiplicity as in the two-dimensional case.
In this case it is enough to consider only seven components of $n_i$, which we will denote by $\hat n_i$.
The 7 vectors corresponding to the left vertex inside the toric diagram are given by the columns of the
matrix,
\begin{equation}
\left(
\begin{array}{ccccccc}
\hat n_4&\hat n_5&\hat n_6&\hat n_7&\hat n_8&\hat n_9&\hat n_{10}\\
1&0&0&0&0&0&0\\
0&1&0&0&0&0&0\\
0&0&1&0&0&0&0\\
0&0&0&1&0&0&0\\
0&0&0&0&1&0&0\\
0&0&0&0&0&1&0\\
0&0&0&0&0&0&1
\end{array}
\right),
\label{eq:left}
\end{equation}
the 14 vectors corresponding to the middle vertex inside the toric diagram
are given by the columns of the matrix,
\begin{equation}
\left(
\begin{array}{cccccccccccccc}
\hat n_{11}&\hat n_{12}&\hat n_{13}&\hat n_{14}&\hat n_{15}&\hat n_{16}&\hat n_{17}&
\hat n_{18}&\hat n_{19}&\hat n_{20}&\hat n_{21}&\hat n_{22}&\hat n_{23}&\hat n_{24}\\
1&0&0&0&0&1&0&1&0&0&0&1&0&0\\
0&1&0&0&0&0&1&0&1&0&0&0&1&0\\
1&0&1&0&0&0&0&0&0&1&0&0&0&1\\
0&1&0&1&0&0&0&1&0&0&1&0&0&0\\
0&0&1&0&1&0&0&0&1&0&0&1&0&0\\
0&0&0&1&0&1&0&0&0&1&0&0&1&0\\
0&0&0&0&1&0&1&0&0&0&1&0&0&1
\end{array}
\right),
\label{eq:middle}
\end{equation}
and the 7 vectors corresponding to the right vertex inside the toric diagram
are given by the columns of the matrix,
\begin{equation}
\left(
\begin{array}{ccccccc}
\hat n_{25}&\hat n_{26}&\hat n_{27}&\hat n_{28}&\hat n_{29}&\hat n_{30}&\hat n_{31}\\
1&0&0&1&0&1&0\\
0&1&0&0&1&0&1\\
1&0&1&0&0&1&0\\
0&1&0&1&0&0&1\\
1&0&1&0&1&0&0\\
0&1&0&1&0&1&0\\
0&0&1&0&1&0&1
\end{array}
\right).
\label{eq:right}
\end{equation}

The structure of these vectors can be represented diagrammatically
as in Figure \ref{fig:graph}.
\begin{figure}[htdp]
\begin{center}
\leavevmode
\epsfysize=50mm
\epsfbox{graph.eps}
\caption{Diagrams representing the structure of $\hat n_i$ for ${\bf C}^3/{\bf Z}_7$.
The black circles represent positions where the entries are equal to 1.
$(a)$ represents the vectors $\{ \hat n_4,\cdots,\hat n_{10} \}$ corresponding to the
left vertex inside the toric diagram.
$(b)$ and $(c)$ represent the vectors $\{ \hat n_{11},\cdots,\hat n_{24} \}$
corresponding to the middle vertex inside the toric diagram.
$(d)$ represents vectors $\{ \hat n_{25},\cdots,\hat n_{31} \}$ corresponding to
the right vertex inside the toric diagram.}
\label{fig:graph}
\end{center}
\end{figure}
The seven small circles in the graph correspond to the seven components of each
vector $\hat n_i$
and the black circles represent the positions where the entries are equal to 1.
Each diagram represents seven vectors by performing ${\bf Z}_7$ rotations.
Thus the vectors on the $k$-th vertex from the left inside the toric diagram
are obtained by  choosing $k$ positions from seven
with the constraints that two neighboring positions must not be chosen.

This structure also holds for the orbifold ${\bf C}^3/{\bf Z}_N$
defined by (\ref{eq:ZN}),
and the total number $n_D(N)$ of the vectors $n_i$ necessary to describe the D-geometry
of the orbifold is given by
\begin{equation}
n_D(N)=\sum_{k=1}^{[N/2]} \frac{N}{N-k}
\left(
\begin{array}{c}
N-k \\
k
\end{array}
\right)+3.
\end{equation}
Note that this equation also holds when $N$ is even.
If we define
\begin{equation}
f(N)=n_D(N)-2,
\end{equation}
$f(N)$ satisfies the recursion relation similar to the Fibonacci sequence,
\begin{equation}
f(N)=f(N-1)+f(N-2)
\end{equation}
with the conditions $f(1)=1$ and $f(2)=3$.
The solution of the recursion relation is given by
\begin{equation}
f(N)=\left(\frac{1+\sqrt 5}{2}\right)^N +\left(\frac{1-\sqrt 5}{2}\right)^N,
\end{equation}
and hence analytic expression of $n_D(N)$ is
\begin{equation}
n_D(N)=\left(\frac{1+\sqrt 5}{2}\right)^N +\left(\frac{1-\sqrt 5}{2}\right)^N+2.
\end{equation}
It reproduces the numbers given in the Table in the Introduction.









\vskip 1cm
\centerline{\large\bf Acknowledgements}

I would like to thank T. Tani for valuable discussions.



\appendix

\section{Toric data for the orbifold ${\bf C}^3/{\bf Z}_7$.}
\reseteqnum


\begin{equation}
K=\left(
\begin{array}{cccccccccc}
m_1&1&0&0&0&0&0&0&0&0\\
m_2&0&1&0&0&0&0&0&0&0\\
m_3&0&0&1&0&0&0&0&0&0\\
m_4&0&0&0&1&0&0&0&0&0\\
m_5&0&0&0&0&1&0&0&0&0\\
m_6&0&0&0&0&0&1&0&0&0\\
m_7&0&0&0&0&0&0&1&0&0\\
m_8&0&0&0&0&0&0&0&1&0\\
m_9&0&0&0&0&0&0&0&0&1\\
m_{10}&1&0&1&0&-1&0&0&0&0\\
m_{11}&1&0&1&1&-1&-1&0&0&0\\
m_{12}&1&0&1&1&0&-1&-1&0&0\\
m_{13}&1&0&1&1&0&0&-1&-1&0\\
m_{14}&1&0&1&1&0&0&0&-1&-1\\
m_{15}&1&0&0&1&0&0&0&0&-1\\
m_{16}&0&1&1&0&0&0&0&0&-1\\
m_{17}&0&1&0&1&0&0&0&0&-1\\
m_{18}&0&1&0&0&1&0&0&0&-1\\
m_{19}&0&1&0&0&0&1&0&0&-1\\
m_{20}&0&1&0&0&0&0&1&0&-1\\
m_{21}&0&1&0&0&0&0&0&1&-1
\end{array}
\right)
\end{equation}


\[
T=\left(
\begin{array}{ccccccccccccccccc}
n_1&n_2&n_3&n_4&n_5&n_6&n_7&n_8&n_9&n_{10}&
n_{11}&n_{12}&n_{13}&n_{14}&n_{15}&n_{16}&n_{17}\\
0&1&0&0&0&1&1&1&1&1&0&0&1&1&1&0&0\\
0&0&1&0&0&0&0&0&0&1&0&0&0&0&1&0&1\\
1&0&0&1&0&0&0&0&0&0&1&0&0&0&0&1&0\\
1&0&0&0&1&0&0&0&0&0&0&1&0&0&0&0&1\\
1&0&0&0&0&1&0&0&0&0&1&0&1&0&0&0&0\\
1&0&0&0&0&0&1&0&0&0&0&1&0&1&0&0&0\\
1&0&0&0&0&0&0&1&0&0&0&0&1&0&1&0&0\\
1&0&0&0&0&0&0&0&1&0&0&0&0&1&0&1&0\\
1&0&0&0&0&0&0&0&0&1&0&0&0&0&1&0&1
\end{array}
\right.
\nonumber
\]

\begin{equation}
\left.
\begin{array}{ccccccccccccccc}
n_{18}&n_{19}&n_{20}&n_{21}&n_{22}&n_{23}&n_{24}&
n_{25}&n_{26}&n_{27}&n_{28}&n_{29}&n_{30}&n_{31}\\
0&0&1&1&0&0&1&0&0&1&0&0&0&0\\
0&0&0&1&0&0&1&0&0&1&0&1&0&1\\
1&0&0&0&1&0&0&1&0&0&1&0&1&0\\
0&1&0&0&0&1&0&0&1&0&0&1&0&1\\
0&0&1&0&0&0&1&1&0&1&0&0&1&0\\
1&0&0&1&0&0&0&0&1&0&1&0&0&1\\
0&1&0&0&1&0&0&1&0&1&0&1&0&0\\
0&0&1&0&0&1&0&0&1&0&1&0&1&0\\
0&0&0&1&0&0&1&0&0&1&0&1&0&1
\end{array}
\right)
\end{equation}





\begin{thebibliography}{99}


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\bibitem{FFHH}
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\bibitem{BGLP}
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Nucl.\ Phys.\ B {\bf 566} (2000) 599
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\end{thebibliography}



\end{document}



hep-th0206012.extracted_bib
\begin{thebibliography}{99}


\bibitem{DGM}
M.~R.~Douglas, B.~R.~Greene and D.~R.~Morrison,
``Orbifold resolution by D-branes,''
Nucl.\ Phys.\ B {\bf 506} (1997) 84
[hep-th/9704151].

\bibitem{Muto}
T.~Muto,
``D-branes on orbifolds and topology change,''
Nucl.\ Phys.\ B {\bf 521} (1998) 183
[hep-th/9711090].


\bibitem{AGM}
P.~S.~Aspinwall, B.~R.~Greene and D.~R.~Morrison,
``Calabi-Yau moduli space, mirror manifolds and spacetime 
topology  change in string theory,''
Nucl.\ Phys.\ B {\bf 416} (1994) 414
[hep-th/9309097].


\bibitem{Witten}
E.~Witten,
``Phases of N = 2 theories in two dimensions,''
Nucl.\ Phys.\ B {\bf 403} (1993) 159
[hep-th/9301042].



\bibitem{MT}
T.~Muto and T.~Tani,
``Stability of quiver representations and topology change,''
JHEP {\bf 0108} (2001) 008
[hep-th/0107217].




\bibitem{FFHH}
B.~Feng, S.~Franco, and A.~Hanany and Y.~H.~He,
``Symmetries of Toric Duality,''
hep-th/0205144.


\bibitem{DM}
M.~R.~Douglas and G.~Moore,
``D-branes, Quivers, and ALE Instantons,''
hep-th/9603167.

\bibitem{McKay}
J.~McKay,
``Graphs, Singularities, and Finite Groups,''
Proc.\ Symp.\ Pure\ Math. {\bf 37} (1980) 183.



\bibitem{BGLP}
C.~Beasley, B.~R.~Greene, C.~I.~Lazaroiu and M.~R.~Plesser,
``D3-branes on partial resolutions of abelian quotient 
singularities of  Calabi-Yau threefolds,''
Nucl.\ Phys.\ B {\bf 566} (2000) 599
[hep-th/9907186].


\end{thebibliography}