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%\pubinfo{Vol. 101, No. 4, April 1999}  %Editorial Office use
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%KUNS-1325\\ HE(TH)~97/04\\ hep-th/9702083}
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\markboth{%     %running head for odd-page (authors' name)
H. Kataoka and M. Shimojo
}{%             %running head for even-page (`short' title)
$SU(3)\times SU(2)\times U(1)$ Chiral Models 
from Intersecting D4-/D5-Branes
}


\title{%        %You can use \\ for explicit line-break
{\bf $ SU(3)\times SU(2)\times U(1)$} Chiral Models from 
Intersecting D4-/D5-Branes
}
%\subtitle{This is a Subtitle}    %use this when you want a subtitle

\author{%       %Use \sc for the family name

Hironobu {\sc Kataoka}$^1$\footnote{E-mail: kataoka@yukawa.kyoto-u.ac.jp} 
and 
Masafumi {\sc Shimojo}$^2$\footnote{E-mail: shimo0@ei.fukui-nct.ac.jp} 
}

\inst{%         %Affiliation, neglected when [addenda] or [errata]
$^1$ Department of Physics, Hyogo University of Education,
Yashiro-cho, Hyogo 673-1494, Japan,
\\
$^{2}$Department of Electronics and Information Technology, 
Fukui National College of Technology, Sabae 916-8507, Japan
\\
}

%\publishedin{%      %Write this ONLY in cases of addenda and errata
%Prog.~Theor.~Phys.\ {\bf XX} (19YY), page}

\recdate{%      %Editorial Office will fill in this.
% \today
December 28, 2001
}

\abst{%       %this abstract is neglected when [addenda] or [errata]
% Write your ABSTRACT here.
We clarify RR tadpole cancellation conditions for intersecting 
D4-/D5-branes. We determine all of the D4-brane models that have D=4 
three-generation chiral fermions with 
$SU(3)\times SU(2)\times U(1)^n$ symmetries. 
For the D5-brane case, we present a solution to the conditions 
that gives exactly the matter content of the standard model with 
$U(1)$ anomalies.
}

\begin{document}

\maketitle

%\section{Introduction}
% Start your paper from here.


%\section{Intersecting D4/5-branes}
Intersecting D$p$-branes are useful tools to obtain chiral 
fermions existing on their intersections.\cite{rf:AF}  
Although configurations of type IIB D6-branes wrapped on ${\bf T}^6$ 
give rise to identically the matter content of the standard 
model,$^{\ref{rf:IM})}$ the D6-brane models do not solve the 
fine-tuning problem caused by its breakdown of supersymmetry. 
Contrastingly, this difficulty is avoided by  
using D4-/D5-branes wrapped on ${\bf T}^{2d}\times {\bf R}^{6-2d}/Z_N$ 
through reduction of the string scale with a very large 
volume of the $Z_N$ orbifold. 
Intersecting D4-/D5-branes in both oriented and 
unoriented theories, we construct D=4 chiral models 
whose matter content transforms as $(3,2)_{\frac{1}{6}}$+
$(\overline{3},1)_{-\frac{2}{3}}$+ $(\overline{3},1)_{\frac{1}{3}}$ 
under $SU(3)\times SU(2)\times U(1)_Y$. 

We start with the definition of the spacetime geometry in the 
oriented theory. The compactified six dimensional space is 
the product of $d$ rectangular two-tori 
${\bf T}_I^2\ (1 \leq I\leq d\leq 2)$ and a ${\bf R}^{6-2d}/Z_N$ orbifold. 
The space is parameterized by the complex coordinates 
$Y_I=X_{2I+2}+iX_{2I+3}$ with radii $R_{2I+2}$ and $R_{2I+3}$ for $I=1,2,3$. 

Let us consider $K$ different stacks, each of which 
is a set of $N_a$ coincident D4-/D5-branes. They are labeled by the index 
$a$$(a=1,\cdots,K)$ and wrapped around a 1-cycle denoted by 
$(n_a^{(I)}, m_a^{(I)})$ on each of the $d$ two-tori, 
where $I=1$ for D4-branes and $I=1,2$ for D5-branes. 
When the $a$-th stack and the $b$-th stack intersect, 
we refer to their intersections as an $a$-$b$ sector. 
The intersection number of the $a$-$b$ sector is given by 
\begin{equation}
I_{ab}=\prod_{I=1}^dI_{ab}^{(I)}=
\prod_{I=1}^d(n_a^{(I)}m_b^{(I)}-n_b^{(I)}m_a^{(I)}).
\label{iab}
\end{equation}

The $Z_N$ action is realized by powers of the twist generator 
represented by a shift vector $v$ in the space of the $SO(6-2d)$ Cartan 
subalgebra. In the configurations with intersecting D4-branes, 
modular invariance conditions for one-loop amplitudes 
restrict the form of the twist vector 
$v=(v_1, v_2, v_3)$ to $(0,1,b_2)/N$, where $b_2$ is an odd integer. 
For the case of D5-branes, $v=(0,0,2/N)$. 

Since $N_a$ coincident D-branes have $U(N_a)$ gauge factors in 
their world-volume, open strings stretched between the $a$-th stack and 
the $b$-th stack give rise to states in the $a$-$b$ sector 
that belong to bifundamental representations of $U(N_a)\times U(N_b)$. 
The gauge degrees of freedom are 
expressed by the Chan-Paton (CP) factors as $|ij\rangle $, where $i$ and $j$ 
correspond to a D-brane in the $a$-th and the $b$-th stack, respectively. 
We express the embedding of the $Z_N$ action $\theta^k$  
using the unitary matrix $\gamma_k$.
We diagonalize $\gamma_{k}$ so that
\begin{equation}
\gamma_k={\rm diag}
(\zeta^{l_1}{\bf 1}_{N_{1}},
\zeta^{l_2}{\bf 1}_{N_{2}},
\cdots,\zeta^{l_K}{\bf 1}_{N_K}),
\end{equation}
where $\zeta=e^{2\pi i\frac{1}{N}}$ and $l_a$ are integers.
Then the $N_a$ D-branes in the $a$-th stack have the same CP phase, 
$\zeta^{l_a}$. 

The RR tadpole cancellation conditions$^{\ref{rf:AF})}$\cite{rf:blumen} are 
\begin{eqnarray}
\sum_{a=1}^KN_a\prod_{I=1}^dn_a^{(I)} =   
\sum_{a=1}^KN_a\prod_{I=1}^dm_a^{(I)} & = & 0, \label{rrtnor1}\\
\sum_{k=1}^{N-1}\left(\prod_{J=d+1}^3|{\rm sin}\pi k v_J|\right)
\left(\sum_{a=1}^K (\prod_{I=1}^dn_a^{(I)}){\rm Tr}_a\gamma_k\right)^2 
& = & 0,\label{rrtnor2} \\
\sum_{k=1}^{N-1}\left(\prod_{J=d+1}^3|{\rm sin}\pi k v_J|\right)
\left(\sum_{a=1}^K (\prod_{I=1}^dm_a^{(I)}){\rm Tr}_a\gamma_k\right)^2 
& = & 0, \label{rrtnor3}
\end{eqnarray}
where ${\rm Tr}_a$ represents the trace over the CP factor on 
D-branes that belong to the $a$-th stack.
For each shift vector $v$ of $Z_N$, we can obtain solutions 
for 1-cycles $(n_a^{(I)},m_a^{(I)})$ of the $a$-th stack and CP phases 
$\zeta^{l_a}$. The solution produces an open string spectrum 
that is invariant under the $Z_N$ action.\cite{rf:doug}

For the D4-brane case, left-handed 
fermions in the $a$-$b$ sector are given in bifundamental 
representations under $U(N_a)\times U(N_b)$ as 
$^{\ref{rf:AF})}$
\begin{equation}
I_{ab}\{(N_a^i,\overline{N_b}^{i-\frac{1-b_2}{2}})
+(N_a^i,\overline{N_b}^{i+\frac{1-b_2}{2}})
+(\overline{N_a}^i,N_b^{i-\frac{1+b_2}{2}})
+(\overline{N_a}^i,N_b^{i+\frac{1+b_2}{2}})\},
\end{equation}
where the symbol $N_a^i$ expresses the meaning that this is the 
representation $N_a$ of $U(N_a)$ and the CP phase 
of the $a$-$th$ stack is $\zeta ^i$.
A negative value of $I_{ab}$ indicates a positive multiplicity of 
the field that transforms as the complex conjugate 
of each bifundamental representation. 
To obtain the three (3,2) representations, we require $N_1=3$ and $N_2=2$ 
stacks and the intersection number $I_{12}=3$. 
The two types of three-generation right-handed quarks require  
$N_3=1$ and $N_4=1$ stacks and intersection numbers 
$I_{13}=\pm 3$ and $I_{14}=\pm 3$.  One more stack $N_5=1$ is 
necessary for the consistency conditions (\ref{rrtnor1})-- 
(\ref{rrtnor3}) to be satisfied. We have investigated all of 
these configurations and found two solutions, given  
in Table I. 
\begin{table}
\caption{D4-brane 1-cycles and CP phases giving rise to three generations of 
quarks.}
\begin{center}
\begin{tabular}{|l|c|c|c|c|c|c|c|}
\hline 
\hline
 & \raisebox{-2.5ex}{$Z_N$} & & \multicolumn{5}{c|}{cycle} \\
 & & & \multicolumn{5}{c|}{CP phase} \\  
\cline{4-8}
\raisebox{3.5ex}{number} & \raisebox{2.5ex}{orbifold} & \raisebox{3.5ex}{$b_2$} &
$N_1=3$ & $N_2=2$ & $N_3=1$ & $N_4=1$ & $N_5=1$ \\
\hline
%%%%D4-1
 & & & $(1,0)$ & $(n_2,3)$ & $(n_3,-3)$ & $(-2n_2-n_3,-3)$ & $(-3,0)$ \\
 \raisebox{1.5ex}{D4-1} & \raisebox{1.5ex}{$N\geq 3$} 
 & \raisebox{1.5ex}{$b_2\neq -1$} &  1
 & $\zeta ^\frac{b_2-1}{2}$ & $\zeta^{\frac{b_2-1}{2}}$ &
 $\zeta^{\frac{b_2-1}{2}}$ & $1$ \\
\hline
%%%%D4-1'
% & & $b_2=\pm\frac{N}{2}$ & $(1,0)$ & $(n_2,3)$ & $(n_3,-3)$ & $(-2n_2-n_3,-3)$%   & $(3,0)$ \\
% \raisebox{1.5ex}{D4-$1^{\prime}$} & \raisebox{1.5ex}{$Z_{2n}$} & 
%  $(\zeta^{b_2}=-1)$ & 1 & $\zeta ^\frac{b_2-1}{2}$ 
%  & $\zeta^{\frac{b_2-1}{2}}$ 
%  & $\zeta^{\frac{b_2-1}{2}}$ & $-1$  \\ 
% \hline
%%%%d4-2
 & & & $(1,0)$ & $(n_2,3)$ & $(n_3,-3)$ & $(-2n_2-n_3,-3)$ & $(-3,0)$ \\
\raisebox{1.5ex}{D4-2} & \raisebox{1.5ex}{$N\geq 3$} 
& \raisebox{1.5ex}{$b_2=-1$} &  1
 & $\zeta ^\frac{b_2-1}{2}$ & $\zeta^{\frac{b_2-1}{2}}$ &
 $\zeta^{\frac{b_2-1}{2}}$ & $1$ \\
 \hline
\end{tabular}
\vspace{10pt}
\end{center}
\end{table}

Since the gauge symmetry $U(1)$ of the stack $N_a=1$ with 1-cycle $(n_a,0)$ 
extends to $U(1)^{|n_a|}$,\cite{rf:AF}
the gauge symmetry of the models is $U(3)\times U(2)\times 
U(1)^5=SU(3)\times SU(2)\times U(1)^7$. The fermion spectrum 
under the symmetry given by D4-1 in  
Table I is as follows:   
\begin{equation}
\begin{array}{l}
3(3,2)_{(1,-1,0,0,0^3)}+3(\bar{3},1)_{(-1,0,1,0,0^3)}
+3(\bar{3},1)_{(-1,0,0,1,0^3)}  \\
+3(1,2)_{(0,1,0,0,\underline{-1,0,0})}
+3(1,1)_{(0,0,-1,0,\underline{1,0,0})}+3(1,1)_{(0,0,0,-1,\underline{1,0,0})},
\end{array}
\label{spd4-1}
\end{equation}
where the underlines indicate permutation of indices. 
Computing the mixed anomalies which need to be cancelled by the 
Green-Schwartz mechanism, we find six non-anomalous $U(1)$ linear 
combinations. They include the suitable hypercharge
\begin{equation}
Q_Y=-\frac{1}{6}Q_2+\frac{1}{3}Q_3-\frac{2}{3}Q_4+\frac{1}{3}(Q_5^{(1)}+Q_5^{(2)})
-\frac{2}{3}Q_5^{(3)},
\label{hyd41}
\end{equation} 
where $Q_a$ is the generator of the $a$-th $U(1)$. 
With this hypercharge, the fermion spectrum is 
\begin{equation}
\begin{array}{l}
3(3,2)_{\frac{1}{6}}+3(\overline{3},1)_{\frac{1}{3}}+
3(\overline{3},1)_{-\frac{2}{3}}+6(1,2)_{-\frac{1}{2}}
+3(1,2)_{\frac{1}{2}}\\
\ \ \ +6(1,1)_1+9(1,1)_0+3(1,1)_{-1}.
\end{array}
\label{spd4-1y}
\end{equation} 
When we set $n_2+n_3=1$, D4-2 in Table I gives the following fermion spectrum:
\begin{equation}
\begin{array}{l}
3(3,2)_{(1,-1,0,0,0^3)}+3(\bar{3},1)_{(-1,0,1,0,0^3)}+
3(\bar{3},1)_{(-1,0,0,1,0^3)} \\
+3(1,2)_{(0,1,0,0,\underline{-1,0,0})}
+3(1,1)_{(0,0,-1,0,\underline{1,0,0})}+3(1,1)_{(0,0,0,-1,\underline{1,0,0})}
\\
+6(1,2)_{(0,1,-1,0,0^3)}+6(1,2)_{(0,-1,0,1,0^3)}+12(1,1)_{(0,0,1,-1,0^3)}. \\
\end{array}
\label{spd4-2}
\end{equation}
We choose the hypercharge from 
anomaly-free realizations of $U(1)$ as 
\begin{equation}
Q_Y=\frac{1}{3}Q_1+\frac{1}{6}Q_2+\frac{2}{3}Q_3-\frac{1}{3}Q_4
-\frac{1}{3}(Q_5^{(1)}+Q_5^{(2)}+Q_5^{(3)}).
\label{hyd42}
\end{equation} 
With this hypercharge, the spectrum (\ref{spd4-2}) become  
\begin{equation}
\begin{array}{l}
3(3,2)_{\frac{1}{6}}+3(\overline{3},1)_{\frac{1}{3}}+3(\overline{3},1)_{-\frac{2}{3}}
+12(1,2)_{-\frac{1}{2}}+9(1,2)_{\frac{1}{2}}\\
\ \ \ +12(1,1)_1+9(1,1)_0+9(1,1)_{-1}.
\end{array}
\label{spd4-2y}
\end{equation}
When we set $n_2+n_3=-1$, we get the same spectrum (\ref{spd4-2y}) 
by exchanging $Q_3$ and $Q_4$ in the hypercharge (\ref{hyd42}).
For the case $n_2+n_3\neq \pm 1$, we have not been able to obtain the 
solution with the correct hypercharge $Q_Y$.

When the $a$-th stack and $b$-th stack have the same CP phase, 
there are $I_{ab}$ tachyon fields in the $a$-$b$ sector.\cite{rf:AF} 
In these models, the 2-3 and 2-4 sectors give rise to 
tachyon fields that transform as the $SU(2)$ doublet. 
This indicates that the configurations of branes that have $U(2)$ and $U(1)$ 
symmetries are unstable. This instability suggests a stringy Higgs mechanism of 
electroweak symmetry breaking.$^{\ref{rf:AF})}$    

For the D5-brane case, the fact that there are 
many wrapping numbers $(n_a^{(I)}$, $m_a^{(I)})$ with $(I=1,2)$ makes it 
difficult to obtain general solutions.  
For this reason, in this paper, we content ourselves with a single example. 
We will return to this problem 
at the end of the investigation of the unoriented theory.
% ---------------------------------------------------

We now discuss the unoriented theory. We introduce an orientifold group 
written as $Z_N+\Omega RZ_N$, where $\Omega$ is the world sheet parity. 
$R$ transforms $Y_I$ to $\overline{Y_I}$ for 
$1\leq I\leq d$ and $Y_I$ to $-Y_I$ for 
$d+1\leq I \leq 3$. The space-time is expressed as 
$$\frac{{\cal M}_4\times \prod_{I=1}^d{\bf T_I^2} 
\times (\prod_{I=1+d}^3 {\bf T_I^2}/Z_N)}{\Omega R}.$$
We express the $\Omega R$ action on the CP factors using 
unitary matrices $\gamma_{\Omega R}$. 
Since $\Omega R \theta^k (\Omega R)^{-1}$ $=\theta ^{2k}$,
we have
\begin{equation}
\gamma _{\Omega R k}=\gamma _k\gamma_{\Omega R}=
\pm\gamma_{2k}\gamma_{\Omega R k}^{T}.
\label{g2kgor}
\end{equation}
This leads to 
$\gamma_{\Omega R} =  \pm \gamma_{\Omega R}^T$.
In the following equations, the upper and lower sign correspond 
to the symmetric and antisymmetric $\gamma_{\Omega R}$, respectively.
 
In addition to the $a$-$b$ sector, there are intersections of 
the $a$-th stack and mirror branes of the $b$-th stack, which we refer 
to as $a$-$b*$ sector. The intersection number  
$I_{ab*}$ is obtained by changing  
$m_b^{(I)}$ to $-m_b^{(I)}$ in the expression (\ref{iab}).

The $Z_N$ action with even $N$ contains an order 2 element, so that
the orientifold group contains an element 
that does not vary $Y_I(I\geq d+1)$. 
Then there will be D8-branes in the type IIA D4-brane theory. 
Since the concept of intersecting D-branes involves use of the same 
dimensional D-branes, we restrict ourselves to the case that 
the order $N$ of $Z_N$ is odd. 

Tadpole divergences come from Klein bottle and M\"obius strip 
amplitudes as well as cylinder amplitudes.\cite{rf:blumen}
The RR tadpole cancellation conditions common to configurations 
with intersecting D4-branes and configurations with D5-branes 
are given by 
\begin{eqnarray}
\sum_{a=1}^KN_a\prod_{I=1}^dn_a^{(I)} & = &  \pm 16, \label{tcNn} \\  
\sum_{a=1}^KN_a\prod_{I=1}^dm_a^{(I)} & = & 0 \label{nmama}, \\
\sum_{a=1}^K\left(\prod_{I=1}^dm_a^{(I)}\right){\rm Tr}_a\gamma_k & = & 0 
\ \ \ {\rm for}\ k=1,\cdots ,N-1.
\label{nmatr}
\end{eqnarray}
There is one more condition on the product of 
the cycles $n_a^{(I)}$ and the trace of $\gamma_k$. 
% The equation (\ref{g2kgor}) simplifies the last condition 
When we consider a $Z_3$ orbifold, it is given by 
\begin{equation}
\sum_{a=1}^Kn_a^{(1)}{\rm Tr}_a\gamma_k=\pm 4 \ \ \ {\rm for}\ k=1,2,
\label{ntrg}
\end{equation}
for the D4-brane case and
\begin{equation}
\sum_{a=1}^Kn_a^{(1)}n_a^{(2)}{\rm Tr}_a\gamma_k=\mp 8 
\ \ \ {\rm for}\ k=1,2,
\label{nntrg}
\end{equation}
for the D5-brane case. For other $Z_N$ models with odd $N$, the condition 
is expressed by a linear summation of $\prod_{J=d+1}^3{\rm sin}\pi kv_J$ 
and $\prod_{J=d+1}^3{\rm sin}2\pi kv_J$ over $k=1,\cdots, N-1$ similar to 
(\ref{rrtnor2}), and we could not find any solution to 
1-cycles and $\gamma$ matrices. 
% ---------------------------------------------------

In order to satisfy the conditions (\ref{tcNn}) and 
(\ref{ntrg}) for unoriented D4-brane configurations 
giving a standard-like model, 
we must introduce many $U(1)$ stacks or 
an $N_2=2$ stack that has a 1-cycle $n_2^{(1)}\neq 1$.
Some of these $U(1)$ stacks and the $N_1=3$ stack always have 
intersections that lead to non-standard 
$(3,1)$ or $(\overline{3},1)$
representations under $SU(3)\times SU(2)$. 
The 1-cycles $n_2^{(1)}\neq 1$ in the 
unoriented theory produce fields that transform as a three-dimensional 
representation under $U(2)$.\cite{rf:IM} Thus we were not able to 
obtain the matter content of the standard model.

For models with intersecting D5-branes,
left-handed fermions in the $a$-$b$ sector and $a$-$b*$ sector
are given by$^{\ref{rf:AF})}$ $^{\ref{rf:IM})}$
\begin{equation} 
I_{ab}((N_a^i,\overline{N_b}^{i+1})+(\overline{N_a}^i,N_b^{i-1})),
\end{equation}
\begin{equation}
I_{ab*}((N_a^i,N_b^{-i-1})+(\overline{N_a}^i,\overline{N_b}^{-i+1})),
\label{d5iab*}
\end{equation}
under $U(N_a)\times U(N_b)$.

The bifundamental representations of (\ref{d5iab*}) for the $a$-$a*$ 
sector change to $N_a^i\wedge N_a^{-i-1}$ and $\overline{N_a}^i\wedge
\overline{N_a}^{-i+1}$ under $U(N_a)$.
On-orientifold intersections of the $a$-$a*$ sector
give $4m_a^{(1)}m_a^{(2)}$ fermions in the representation of 
either an antisymmetric or symmetric tensor, depending on whether 
$\gamma_{\Omega R}$ is antisymmetric or symmetric. Off-orientifold intersections of the 
$a$-$a*$ sector produce $2m_a^{(1)}m_a^{(2)}(n_a^{(1)}n_a^{(2)}-1)$ 
symmetric and antisymmetric representations. We require that any 
brane stack satisfy $m_a^{(1)}m_a^{(2)}=0$ to avoid the appearance of 
exotic quantum numbers and to satisfy the RR tadpole cancellation 
conditions  (\ref{nmama}) and (\ref{nmatr}). Adding D5-branes with 
$m^{(1)}=m^{(2)}=0$ will satisfy the conditions (\ref{tcNn}) and 
(\ref{nntrg}) without modifying the fermionic matter content. 

Since $I_{12}=-I_{12*}$ under 1-cycles for which $m_a^{(1)}m_a^{(2)}=0$, 
we must set $I_{12}=\pm 3$ and choose CP factors yielding no fermions 
in the 1-$2*$ sector to get $3(3,2)$ representations of $U(3)\times U(2)$. 
This configuration makes 
it impossible to obtain just the standard model spectrum without 
$U(1)$-$U(1)$-$U(1)$ anomalies.$^{\ref{rf:IM})}$
An example is given in Table II for the case in which the orbifold group is $Z_3$.
\begin{table}
\caption{Example of D5-brane 1-cycles and CP phases 
giving rise to three generations of quarks. 
The orbifold group is $Z_3$ and the parameter $\zeta$ is $e^{2\pi i/3 }$.}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline 
\hline 
$N_a$ & $(n_a^{(1)},m_a^{(1)})$ & $(n_a^{(2)},m_a^{(2)})$ & CP phase \\
\hline
$N_1=3$ & $(n_1^{(1)},0)$ & $(n_1^{(2)},3/n_1^{(1)})$ & $\zeta^2$\\
\hline
$N_2=2$ & $(n_2^{(1)},m_2^{(1)})$ & $(1/m_2^{(1)},0)$ & $\zeta $ \\
\hline
$N_3=1$ & $(n_3^{(1)},m_3^{(1)})$ & $(-1/m_3^{(1)},0)$ & $\zeta $\\
\hline 
$N_4=1$ & $(n_4^{(1)},m_4^{(1)})$ & $(-1/m_4^{(1)},0)$ & $\zeta $ \\
\hline
$N_5=1$ & $(n_5^{(1)},0)$ & $(n_5^{(2)}, -3/n_5^{(1)})$ & $\zeta^2$\\
\hline
\end{tabular}
\end{center}
\end{table}

The spectrum under the gauge symmetry 
$SU(3)\times SU(2)\times U(1)^5$ 
is given by 
\begin{equation}
\begin{array}{c}
3(3,2)_{(1,-1,0,0,0)}+3(\overline{3},1)_{(-1,0,1,0,0)} 
+3(\overline{3},1)_{(-1,0,0,1,0)} \\ 
+3(1,2)_{(0,1,0,0,-1)} +3(1,1)_{(0,0,-1,0,1)}+3(1,1)_{(0,0,0,-1,1)}.
\end{array} \label{spd5}
\end{equation}
Although the spectrum has a $U(1)$-$U(1)$-$U(1)$ anomaly and 
$G^2$-$U(1)$ mixed anomalies that must be removed by some mechanism, 
two anomaly-free $U(1)$\ linear combinations exist. We can define 
the hypercharge as 
\begin{equation}
Y=\frac{1}{6}Q_1+\frac{1}{2}(Q_3-Q_4+Q_5).
\end{equation} and obtain identically the standard model spectrum under 
$SU(3)\times SU(2)\times U(1)_Y$ gauge symmetry 
with three generations of right-handed neutrino. 

We now consider the construction of D5-branes in the oriented theory.
In this case, any matter fermions (\ref{spd5}) for the unoriented theory are 
obtained from the $a$-$b$ sector, not from the $a$-$b*$ sector.
Adding an appropriate number of $U(1)$ branes to the stacks in Table II, 
we obtain D5-branes for the oriented theory that satisfy the 
conditions on the 1-cycles and CP phases (\ref{rrtnor1})--(\ref{rrtnor3}).   

%whose existence may be supported by solar neutrino experiments. 
% Either massless bosons or tachyons exist at the $a$-$b$ sector of D5-branes, 
% when the $a$-$th$ stack and $b$-$th$ stack have 
% the same CP phase\cite{rf:AF}.  
% The bosonic fields which are of three-dimensional representations 
% under $U(3)$ appear at the 1-$2*$, 1-$3*$, 1-$4*$ sector.  
% The rolls of these fiels are unknown. They are not tachyonic but 
% massless on conditions for 1-cycles 
% that $3n_b^{(1)}=n_1^{(1)}n_1^{(2)}m_b^{(1)}$, where $b=2,3,4$.

Configurations with intersecting D-branes cause the breakdown of 
supersymmetry. Then the string scale must be close to the weak scale 
to avoid the fine-tuning problem. The four dimensional 
Plank mass $M_p$ and string scale $M_s$ are related as 
$M_p\approx 4\pi^2M_s^4\sqrt{V_TV_{Z_N}}/\lambda _{II}$, 
where $\lambda _{II}$ is the Type II string coupling, $V_T$ is the volume of 
the tori, and $V_Z$ is the volume of the orbifold. 
While a very large value of $V_T$ leads to 
small Yukawa and gauge couplings, we can give a very large volume 
to the transverse $Z_N$ orbifold. For models of D5-branes, 
for example, we can set $M_s \approx 1 \sim 10$ TeV and 
$V_T\sim 1/M_s^2$. In order to obtain very large value of the Plank mass, 
we should choose $\sqrt{V_{Z_N}}\approx 10^9$ -- $10^{11}$ (GeV)$^{-1}$, 
i.e., $10^{-2}$ -- $10^{-4}$ cm. 

In this paper, we have not investigated all of the 
standard-like spectra that accompany intersecting D5-branes. 
In particular, unoriented models may be obtained from 
1-cycles for which $m_a^{(1)}m_a^{(2)}\neq 0$, $n_a^{(1)}n_a^{(2)}=1$. 
In such models, the $a$-$a*$ sector may produce the antisymmetric tensors of 
$N_a\wedge N_a$ under $U(N_a)$ for $N_a=2,3$ without symmetric tensors 
that are not present in the matter content of the standard model. 
% \subsection{}
% \section*{References}
% \section*{Acknowledgements}
% We would like to thank ...........

% \appendix

% \section{Second Appendix}


\begin{thebibliography}{99}
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% Some macros are available for the bibliography:
%   o for general use
%      \JL : general journals          \andvol : Vol (Year) Page
%   o for individual journal 
%      \PR  : Phys. Rev.               \PRL : Phys. Rev. Lett.
%      \NP  : Nucl. Phys.              \PL  : Phys. Lett.
%      \JMP : J. Math. Phys.           \CMP : Commun. Math. Phys.
%      \PTP : Prog. Theor. Phys.       \JPSJ: J. Phys. Soc. Jpn.
%      \JP  : J. of Phys.              \NC  : Nouvo Cim.
%      \IJMP: Int. J. Mod. Phys.       \ANN : Ann. of Phys.
% Usage:
%   \PR{D45,1990,345}            ==> Phys.~Rev.\ {\bf D45} (1990), 345
%   \JL{Phys.~Lett.,A30,1981,56} ==> Phys.~Lett.\ {\bf A30} (1981), 56
%   \andvol{B123,1995,1020}      ==> {\bf B123} (1995), 1020
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\bibitem{rf:AF} 
G. Aldazabal, S. Franco, L. E. I\'anez and R. Rabadan, A. M. Uranga, 
% {\it Intersecting Brane Worlds}, preprint 
hep-ph/0011132,
% {\it D=4 Chiral String Compactifications from Intersecting Branes}, 
hep-th/0011073.
\label{rf:AF} 
\bibitem{rf:IM} L. E. Ib\'anez, F. Marchesano, R. Rabadan,
%  {\it Getting 
% just the Standard Model at Intersecting}, preprint 
hep-th/0105155.
\label{rf:IM}  
% \bibitem{rf:arfaei}
% H. Arfaei and M.M.Sheikh Jabbari, {\it Different D-brane Intersections},
% pre-print hep-th/9608167 \label{rf:arfaei}.\\ 
\bibitem{rf:blumen}
R. Blumenhagen, L. G\"orlich, B. K\"ors and D. L\"ust,
hep-th/0007024, \\
G. Aldazabal, A. Font, L. E. Ib\'anez and G.Violero, 
hep-th/9804026,\\
S. Ishihara, H. Kataoka and H. Sato, 
Phys. Rev. D60 (1999),126005.
 % \label{rf:blumen} 
\bibitem{rf:doug} M. R. Douglas and G. Moore, hep-th/9603167.
\end{thebibliography}

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