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\begin{document}
\input{psfig.sty}

\begin{flushright}
\baselineskip=12pt
UPR-1040-T \\
\end{flushright}

\begin{center}
\vglue 1.5cm

{\Large\bf Supersymmetric $G^3$ Unification from Intersecting D6-Branes
 on Type IIA Orientifolds}

\vglue 2.0cm

{\Large
Tianjun Li$^a$\footnote{
E-mail: tli@sns.ias.edu. Phone Number: (609) 734-8024. Fax Number:  (609) 
951-4489.}
 and Tao Liu$^b$\footnote{E-mail: liutao@sas.upenn.edu. Phone Number: 
(215) 573-6105.}}
\vglue 1cm
{$^a$ School of Natural Science, Institute for Advanced Study,  \\
             Einstein Drive, Princeton, NJ 08540, USA\\
$^b$ Department of Physics and Astronomy, University of Pennsylvania, \\
Philadelphia, PA 19104, USA\\}

\end{center}

\vglue 1.5cm
\begin{abstract}
 We propose three supersymmetric $G^3$ models
with $S_3$ symmetry and minimal stacks of D6-branes
 from the intersecting D6-branes
 on Type IIA orientifolds. In those
 models, the RR tadpole can be cancelled and the 4-dimensional
$N=1$ supersymmetry is preserved elegantly. Phenomenologically,
there exists the gauge coupling unification, the Standard Model
fermions and Higgs doublets can be embedded into the bifundamental
representations, but there are no symmetric or anti-symmetric
 representations. Especially in Model I where the gauge group is
$U(4)^3$, we just have three famililes of fermions
 and three pairs of Higgs particles. We also show that
 the $G^3$ gauge symmetry can be broken down to
the Standard Model gauge symmetry by introducing the light open
 string states.
\\[1ex]
PACS: 11.25.-w; 11.25.Mj; 12.10.-g; 12.60.-i
\\[1ex]
Keywords: D6-Branes; Type IIA Orientifolds; Unification


\end{abstract}


\vspace{0.5cm}
\begin{flushleft}
\baselineskip=12pt
April 2003\\
\end{flushleft}
\newpage
\setcounter{page}{1}
\pagestyle{plain}
\baselineskip=14pt


\section{Introduction}

Since 1984, there has been a lot of work and effort devoted to the
string model building or string phenomenology, whose goal is to
obtain the Standard Model (SM) 
or Minimal Supersymmetric Standard Model (MSSM)
 as an effective theory of an string-based model.
However, there is still a big gap between the string theory and
the weak scale Standard Model. As we know, the previous string
models are mainly built in the weakly coupled heterotic string
theory with $E_8\times E_8$ gauge group~\cite{GSW, JP}, because it
naturally obtains the Grand Unified Theory (GUT) due to the elegant
$E_8$ breaking chain: $E_8 \supset E_6 \supset SO(10) \supset
SU(5)$. Even now, this is an interesting subject because of the
model buildings in M-theory on $S^1/Z_2$ [3$-$7].

In recent years, the emergence of M-theory opened up many new
avenues for the consistent string model buildings. Especially, we
can construct the open string models that are non-perturbative
from the dual heterotic string description due to the advent of
D-branes~\cite{JPEW}. The technique of conformal field theory in
describing D-branes and orientifold planes on orbifolds has played
a key role in the construction of consistent 4-dimensional
supersymmetric $N=1$ chiral models on Type II orientifolds. There
are two kinds of theories which have chiral fermions from the
D-brane constructions. In the first kind of models, the D-branes
are located at orbifold singularities, and the chiral fermions
appear on the worldvolume of D-branes [9$-$15].

The second kind of models, which have been studied extensively
during the last several years, is the Type II orientifold models
with intersecting D-branes, where the open string spectrum
contains chiral fermions which are localized at the D-brane
intersections~\cite{bdl}. In the begining, the model buildings
were developed and explored mainly for the non-supersymmetric
constructions. A lot of non-supersymmetric three-family
Standard-like models and GUT models have been obtained, satisfying
the Ramond-Ramond tadpole cancellation conditions [17$-$28].
However, the string scale of non-supersymmetric models is close to
the Planck scale since the intersecting D6-branes typically have
no common transverse in the internal space. Thus, there are
uncancelled Neveu-Schwarz-Neveu-Schwarz tadpoles and exists the
gauge hierarchy problem.

On the other hand, the supersymmetric model buildings turn out to
be very constraining. The first supersymmetric model with
intersecting D6-branes and the features of the supersymmetric
Standard-like models have been constructed in Refs.~\cite{CSU1, CSU2}.
They are based on the $Z_2\times Z_2$ orientifold with D6-branes
wrapping specific supersymmetric three-cycles of the six-torus.
The supersymmetric Standard-like models, $SU(5)$ and Pati-Salam
 models have
been discussed in detail later~\cite{CPS, CP}, as well as the
phenomenology~\cite{CLS1, CLS2, CLW}. Moreover, the supersymmetric
Pati-Salam models based on $Z_4$ and $Z_4\times Z_2$ orientifolds
with intersecting D6-branes were constructed~\cite{blumrecent,
Honecker}. In these models, the left-right symmetric gauge
structure was obtained by brane recombinations, so the final
models do not have the explicit toroidal orientifold construction,
where the conformal field theory can be applied for the
calculation of the full spectrum and couplings.

Looking back on these model buildings, we may find that people
took such philosophy: directly construct the familiar models, 
for example, the SM, Standard-like models, $SU(5)$, flipped $SU(5)$
  and Pati-Salam models, from the intersecting D-branes on type II 
orientifolds
since these models have been understood very well from traditional
phenomenological analysis. In this paper, our philosophy is
completely different. Our question is: which are the natural models
from the intersecting D6-branes on Type IIA orientifolds?
The natural models, in our opinion, have the following
five properties:\\

(1) Elegantly, the RR tadpole can be cancelled and the 4-dimensional
$N=1$ supersymmetry is preserved due to the
special symmetry;\\

(2) The Standard Model gauge group is the subgroup of
the gauge symmetry at string scale, and three families of quarks and 
leptons
and a pair of the SM Higgs doublets are included in the massless
open string spectrum; \\

(3) The gauge symmetry at string scale can be broken down to the
Standard Model gauge symmetry via Higgs mechanism or Wilson line. \\

(4) Gauge coupling unification.\\

(5) The minimal stacks of the D6-branes, which include the
D6-branes wrapping on the orientifolds.\\

By adding $S_3$ symmetry on the tadpole cancellation conditions
and supersymmetry preserving conditions, as well as on the
geometry, $T^6 = T^2 \times T^2 \times T^2$, we do obtain three
models with above five properties from $T^6 /(Z_2 \times Z_2)$
orientifolds with intersecting D6-branes.
In these models, there are three stacks of D6-branes
with the same brane numbers which are invariant under $S_3$,
and there is one stack of D6-branes which wraps on the $\Omega R$ 
orientifold
 to cancel the RR tadpole and has no intersection with other
three stacks of D6-branes.
 In Model I, the gauge
group is $U(4)\times U(4) \times U(4)$, and three tori are all
tilted. In Model II, the gauge group is also $U(4)\times U(4)
\times U(4)$, and three tori are all rectangular. In Model III,
the gauge group is $U(8)\times U(8) \times U(8)$, and three tori
are all rectangular. And we present the D6-brane configurations
and chiral open string spectrum at massless level. In all our three
models, the Standard Model fermions and Higgs particles are
embedded into the bifundamental representations, and we do not
have the symmetric and anti-symmetric representations. In
particular, we just have three families of fermions and three
pairs of Higgs particles for Model I. Moreover, we show that there
exists the gauge coupling unification in our models. We consider the
gauge symmetry breaking, too. To be concrete, we show that in
Model I, the $U(4)\times U(4) \times U(4)$ gauge symmetry can
indeed be broken down to the Standard Model gauge symmetry by
introducing the light open string states, and similar mechanism
works for the Model II and III. Furthermore, we comment on the
variations of our models, which do not have the fifth property.
The generalization and phenomenology of our models are under
investigation~\cite{TLTL}.

The paper is organized as follows. In Section 2, we review the 
constructions
of the supersymmetric models from $T^6 /(Z_2 \times Z_2)$ orientifolds 
with intersecting
D6-branes. Our model buildings are given in Section 3. In section 4,
we discuss the gauge coupling unification, gauge symmetry breaking and
variant models. Discussions and Conclusions are given in
Section 5.

\section{Supersymmetric Model Buildings from
$T^6 /(Z_2 \times Z_2)$ Orientifolds with Intersecting
D6-Branes}

The rules to construct the supersymmetric models from Type IIA 
orientifolds on
$T^6 /(Z_2 \times Z_2)$ with D6-branes at generic angles , and
to obtain the spectrum of massless open string
 states were discussed in Ref.~\cite{CSU2}.
Here, we follow the notation in Ref.~\cite{CPS}.

The starting point is Type IIA string
theory compactified on a $T^6 /(Z_2 \times Z_2)$ orientifold.
We consider $T^{6}$ to be a six-torus factorized as $T^{6} = T^{2}
\times T^{2} \times T^{2}$ whose complex coordinates are $z_i$,
$i=1,\; 2,\; 3$ for each of the 2-torus, respectively. The generators
for the orbifold group $Z_{2} \times Z_{2}$
$\theta$ and $\omega$, which are associated with their
twist vectors $(1/2,-1/2,0)$ and $(0,1/2,-1/2)$ respectively,
act on the complex coordinates of $T^6$ as
\beqa
& \theta: & (z_1,z_2,z_3) \to (-z_1,-z_2,z_3)~,~ \nonumber \\
& \omega: & (z_1,z_2,z_3) \to (z_1,-z_2,-z_3)~.~\,
\label{orbifold}
\eeqa
 The orientifold projection is implemented by gauging the
symmetry $\Omega R$, where $\Omega$ is world-sheet parity, and $R$
acts as 
\beqa
 R: (z_1,z_2,z_3) \to ({\ov z}_1,{\ov z}_2,{\ov
z}_3)~.~\, 
\label{orientifold} 
\eeqa
So, there are four kinds
of orientifold 6-planes (O6-planes) for the actions of $\Omega R$,
$\Omega R\theta$, $\Omega R \omega$, and $\Omega R\theta\omega$,
respectively. To cancel the RR charges of O6-planes, we
introduce some stacks of $N_a$ D6-branes, which wrap on the
factorized three-cycles. Meanwhile, we have two kinds of complex
structures consistent with orientifold projection for a torus --
rectangular and tilted~\cite{CSU2, CPS}. If we denote the homology
classes of the three cycles wrapped by the D6-brane stacks as
$n_a^i[a_i]+m_a^i[b_i]$ and $n_a^i[a'_i]+m_a^i[b_i]$ with
$[a_i']=[a_i]+\frac{1}{2}[b_i]$ for the rectangular and tilted
tori respectively, following the notation of Ref.~\cite{CPS}, we
can label a generic two cycle by $(n_a^i,l_a^i)$ in either case,
where in terms of the wrapping numbers $l_{a}^{i}\equiv m_{a}^{i}$
for a rectangular torus and $l_{a}^{i}\equiv
2\tilde{m}_{a}^{i}=2m_{a}^{i}+n_{a}^{i}$ for a tilted torus. Note
that for a tilted torus, $l_a^i-n_a^i$ must be even. For a stack
of $N_a$ D6-branes along the cycle $(n_a^i,l_a^i)$, we also need
to include their $\Omega R$ images $N_{a'}$ with wrapping numbers
$(n_a^i,-l_a^i)$. For D6-branes on the top of O6-planes, we count
the D6-branes and their images independently. So, the homology
three-cycles for stack $a$ of $N_a$ D6-branes and its orientifold
image $a'$ take the form 
\beq
[\Pi_a]=\prod_{i=1}^{3}\left(n_{a}^{i}[a_i]+2^{-\beta_i}l_{a}^{i}[b_i]\right),\;\;\;
\left[\Pi_{a'}\right]=\prod_{i=1}^{3}
\left(n_{a}^{i}[a_i]-2^{-\beta_i}l_{a}^{i}[b_i]\right)~,~\,
\eeq 
where $\beta_i=0$ if the $i-th$ torus is rectangular and
$\beta_i=1$ if it is tilted. And the homology 3-cycles wrapped by
the four O6-planes are \beq \Omega R: [\Pi_{\Omega R}]= 2^3
[a_1]\times[a_2]\times[a_3]~,~\, \eeq \beq \Omega R\omega:
[\Pi_{\Omega
R\omega}]=-2^{3-\beta_2-\beta_3}[a_1]\times[b_2]\times[b_3]~,~\,
\eeq \beq \Omega R\theta\omega: [\Pi_{\Omega
R\theta\omega}]=-2^{3-\beta_1-\beta_3}[b_1]\times[a_2]\times[b_3]~,~\,
\eeq \beq
 \Omega R\theta:  [\Pi_{\Omega
R}]=-2^{3-\beta_1-\beta_2}[b_1]\times[b_2]\times[a_3]~.~\,
\label{orienticycles}
\eeq
Then, the intersection numbers are
\beq
I_{ab}=[\Pi_a][\Pi_b]=2^{-k}\prod_{i=1}^3(n_a^il_b^i-n_b^il_a^i)~,~\,
\eeq
\beq
I_{ab'}=[\Pi_a]\left[\Pi_{b'}\right]=-2^{-k}\prod_{i=1}^3(n_{a}^il_b^i+n_b^il_a^i)~,~\,
\eeq
\beq
I_{aa'}=[\Pi_a]\left[\Pi_{a'}\right]=-2^{3-k}\prod_{i=1}^3(n_a^il_a^i)~,~\,
\eeq
\beq
{I_{aO6}=[\Pi_a][\Pi_{O6}]=2^{3-k}(-l_a^1l_a^2l_a^3
+l_a^1n_a^2n_a^3+n_a^1l_a^2n_a^3+n_a^1n_a^2l_a^3)}~,~\,
\label{intersections}
\eeq
where $[\Pi_{O6}]=[\Pi_{\Omega
R}]+[\Pi_{\Omega R\omega}]+[\Pi_{\Omega
R\theta\omega}]+[\Pi_{\Omega R\theta}]$ is the sum of O6-plane
homology three-cycles wrapped by the four O6-planes,
 and $k=\beta_1+\beta_2+\beta_3$ is
the total number of tilted tori.

\begin{table}[t]
\caption{General spectrum on intersecting D6-branes at
generic angles which is valid for both rectangular and tilted tori.
The representations in the table make sense to $U(N_a/2)$ due to
$Z_2\times Z_2$ orbifold projection~\cite{CSU2}. In supersymmetric 
situations,
scalars combine with the fermions to form the chiral
supermultiplets.}
\renewcommand{\arraystretch}{1.25}
\begin{center}
\begin{tabular}{|c|c|}
\hline {\bf Sector} & \phantom{more space inside this box}{\bf
Representation}
\phantom{more space inside this box} \\
\hline\hline
$aa$   & $U(N_a/2)$ vector multiplet  \\
       & 3 Adj. chiral multiplets  \\
\hline
$ab+ba$   & $I_{ab}$ $(\fund_a,\antifund_b)$ fermions   \\
\hline
$ab'+b'a$ & $I_{ab'}$ $(\fund_a,\fund_b)$ fermions \\
\hline $aa'+a'a$ &$-\frac 12 (I_{aa'} - \frac 12 I_{a,O6})\;\;
\Ysymm\;\;$ fermions \\
          & $-\frac 12 (I_{aa'} + \frac 12 I_{a,O6}) \;\;
\Yasymm\;\;$ fermions \\
\hline
\end{tabular}
\end{center}
\label{spectrum}
\end{table}

The general spectrum on intersecting D6-branes at
generic angles, which is valid for both rectangular and tilted tori, is 
given in 
Table \ref{spectrum}. And the 4-dimensional chiral supersymmetric (N=1) 
models from Type IIA Orientifolds
with intersecting D6-branes are mainly constrained in two
aspects:\\

I. Tadpole Cancellation Conditions\\

As sources of RR fields, D-branes and orientifold 6-planes are required to 
satisfy the
Gauss law in a compact space, {\it i.e.}, the total RR charges of
D-branes and O6-planes must vanish since the RR field flux lines can't 
escape.
 The RR tadpole cancellation conditions are
\begin{eqnarray}
\sum_a N_a [\Pi_a]+\sum_a N_a \left[\Pi_{a'}\right]-4[\Pi_{O6}]=0~,~\,
\end{eqnarray}
where the last contributions come from O6-planes which have $-4$ RR
charge in the D6-brane charge units by exchanging RR field while
scattering.

Tadpole cancellation directly leads to the $SU(N)^3$ cubic non-abelian
anomaly cancellation~\cite{Uranga, imr, CSU2}. And the cancellation of
U(1) mixed gauge and gravitational anomaly
or $[SU(N)]^2 U(1)$ gauge anomaly can be achieved by Green-Schwarz
mechanism mediated by untwisted RR fields~\cite{Uranga, imr, CSU2}.
By the way, the RR fields'
couplings to the U(1)s' gauge bosons will give masses to some
combinations of these U(1)s (including the anomaly-free U(1)s only if
their gauge bosons non-trivially couple to the RR fields)~\cite{Uranga, 
imr}.\\

II. Conditions for 4-Dimensional $N = 1$ Supersymmetric D6-brane
Configuration\\

The 4-dimensional $N=1$ supersymmetric models require that 1/4 
supercharges from
10-dimensional
Type I T-dual be preserved, {\it i.e}, they should survive both two
supersymmetry breaking mechanisms: orientation projection of the
intersecting D6-branes and orbifold projection on the background
manifold at the same time. Concrete analysis shows that the
N=1 supersymmetry can be preserved only if the rotation angle of any
D6-brane with respect to the $\Omega R$-plane is an element of
$SU(3)$, or in other words,
$\theta_1+\theta_2+\theta_3=0 $, where $\theta_i$ is the angle between the
$D6$-brane and the $\Omega R$-plane in the $i-th$ torus. In
Ref.~\cite{CPS}, this condition is rewritten as
\begin{eqnarray}
 -x_A l_a^1l_a^2l_a^3+x_B
l_a^1n_a^2n_a^3+x_C n_a^1l_a^2n_a^3+x_D n_a^1n_a^2l_a^3=0~,~\,
\label{susycondition1}
\end{eqnarray}
\begin{eqnarray}
-n_a^1n_a^2n_a^3/x_A+n_a^1l_a^2l_a^3/x_B+l_a^1n_a^2l_a^3/x_C+l_a^1l_a^2n_a^3/x_D<0~,~\,
\label{susycondition2}
\end{eqnarray}
where $x_A=\lambda,\; x_B=\lambda
2^{\beta_2+\beta3}/\chi_2\chi_3,\; x_C=\lambda
2^{\beta_1+\beta3}/\chi_1\chi_3,\; x_D=\lambda
2^{\beta_1+\beta2}/\chi_1\chi_2$, and
$\chi_i=R_2^i/R_1^i$ are the
complex structure moduli where
$R_1^i$ and $R_2^i$ are radii for the $i-th$ torus due to 
$T^2\equiv S^1\times S^1$. $\lambda$ is a
positive parameter without physical significance.

In fact, one can classify all possible supersymmetric brane
configurations that can be used in any model based on this
construction because unlike the tadpole cancellation conditions,
supersymmetry constrains each stack of D6-branes individually~\cite{CPS}.
It has been shown that a D6-brane wrapping a 3-cycle
which has three wrapping numbers
$n_a^i,\; l_a^i$ equal to zero is necessarily parallel to one of the
four orientifold planes. And such brane configurations can be used to
solve the tadpole cancellation
conditions. In addition, there is no supersymmetric brane
configuration along a 3-cycle with two vanishing wrapping numbers,
and hence the only other possibilities are one or no zero wrapping
numbers. Each of these gives a supersymmetric configuration
together with a constraint on the complex structure moduli
\cite{CPS}. Since there are three independent moduli parameters, we
can generically have at most three such D6-brane
configurations. Otherwise, the system is overconstrained.


\section{Supersymmetric $G^3$ Unification}
Generally speaking, the tadpole cancellation conditions and the
supersymmetry preserving conditions are too stringent to find
the realistic GUT models, and the existing GUT models always tend to
produce the extra unwanted gauge interactions (especially U(1)s) and
extra fermions beyond the SM or MSSM. Our purpose in this section
is to construct the natural GUT models with the five
properties emphasized in Introduction.

Let us look at the tadpole cancellation conditions first. If we consider
$N^{(i)}$ D6-branes wrapped along the $i-th$ orientifold plane whose
wrapping numbers are given in Table \ref{orientifold}, the tadpole
cancellation conditions are modified to \beqa -2^k N^{(1)} -
\sum_{\sigma} N_{\sigma}
 n_{\sigma}^1 n_{\sigma}^2 n_{\sigma}^3 = -16~,~\,
\label{tad4} \eeqa \beqa -2^k N^{(2)} + \sum_{\sigma} N_{\sigma}
n_{\sigma}^1 l_{\sigma}^2 l_{\sigma}^3 =-16~,~\, \label{tad1}
\eeqa \beqa -2^k N^{(3)}+\sum_{\sigma} N_{\sigma} l_{\sigma}^1
n_{\sigma}^2 l_{\sigma}^3 =-16~,~\, \label{tad2} \eeqa \beqa -2^k
N^{(4)}+\sum_{\sigma} N_{\sigma} l_{\sigma}^1 l_{\sigma}^2
n_{\sigma}^3=-16 ~.~\, \label{tad3} \eeqa We may note that there
is a $S_3$ symmetry for the upper index 1, 2 and 3 among Eqs.
(\ref{tad1}), (\ref{tad2}) and (\ref{tad3}) if
$N^{(2)}=N^{(3)}=N^{(4)}$. And the simplest case for $N^{(2)}$,
$N^{(3)}$, and $N^{(4)}$ is that all of them are zero.

\renewcommand{\arraystretch}{1.4}
\begin{table}[t]
\caption{Wrapping numbers of the four O6-planes.}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
  Orientifold Action & O6-Plane & $(n^1,l^1)\times (n^2,l^2)\times 
(n^3,l^3)$\\
\hline
    $\Omega R$& 1 & $(2^{\beta_1},0)\times (2^{\beta_2},0)\times
(2^{\beta_3},0)$ \\
\hline
    $\Omega R\omega$& 2& $(2^{\beta_1},0)\times (0,-2^{\beta_2})\times
(0,2^{\beta_3})$ \\
\hline
    $\Omega R\theta\omega$& 3 & $(0,-2^{\beta_1})\times 
(2^{\beta_2},0)\times
(0,2^{\beta_3})$ \\
\hline
    $\Omega R\theta$& 4 & $(0,-2^{\beta_1})\times (0,2^{\beta_2})\times
    (2^{\beta_3},0)$ \\
\hline
\end{tabular}
\end{center}
\label{orientifold}
\end{table}

In addition, suppose we have three stacks of D6-branes whose wrapping
numbers are invariant under $S_3$ symmetry which acts on three tori. Then,
if one stack of D6-branes preserves $N=1$ supersymmetry, all three
stacks of D6-branes will preserve the $N=1$ supersymmetry automatically.

From above discussions, we find that the models with $S_3$
symmetry and three stacks of D6-branes, {\it i.e.}, $N_a=N_b=N_c=2N$ and
$N^{(2)}=N^{(3)}=N^{(4)}=0$, may naturally satisfy the tadpole
cancellation conditions and preserve the $N=1$ supersymmetry. So we'll
consider such a setup with one auxiliary stack of D6-branes which
wraps on the first orientifold. The auxiliary stack of D6-branes
is trivial in our models since it is introduced just for tadpole 
cancellation
purpose and has no intersection with the other three stacks of
D6-branes.  Then the gauge group of our models is $G^3$ where
$G=U(N)$. Although we only need $\chi_1=\chi_2=\chi_3$ for the tadpole 
cancellation
and supersymmetry preserving conditions, we assume that
$R_1^1=R_2^1=R_1^2=R_2^2=R_1^3=R_2^3$ due to the concern of gauge
coupling unification.

For simplicity, we consider three stacks of D6-branes ($a$, $b$ and $c$)
 with one zero
wrapping number. Without loss of generality, we have two cases
with $S_3$ symmetry
\begin{eqnarray}
n_{a}^{1}=n_{b}^{2}=n_{c}^{3}=0~,~\,
\label{case1}
\end{eqnarray}
\begin{eqnarray}
l_{a}^{1}=l_{b}^{2}=l_{c}^{3}=0~.~\,
\label{case2}
\end{eqnarray}
For the first case given by Eq. (\ref{case1}), it is hard to find the
models which satisfy the tadpole cancellation conditions and
preserve the N=1 supersymmetry because the supersymmetry conditions
are very strong constraints in this case.

\renewcommand{\arraystretch}{1.4}
\begin{table}[t]
\caption{Model I. D6-brane configuration in (2p+1)-generation N=1
supersymmetric $U(4)^3$ model. This model is built on three tilted
2-tori with $Z_2\times Z_2$ orbifold symmetry and p is a non-negative
integer.}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$N_i$ & $(n_{i}^{1}, l_{i}^{1})$ & $(n_{i}^{2}, l_{i}^{2})$ & $(n_{i}^{3}, 
l_{i}^{3})$ \\
\hline
$N_a=8$ & $(2,0)$ & $(2p+1,1)$ & $(2p+1,-1)$ \\
\hline
$N_b=8$ & $(2p+1,-1)$ & $(2,0)$ & $(2p+1,1)$ \\
\hline
$N_c=8$ & $(2p+1,1)$ & $(2p+1,-1)$ & $(2,0)$ \\
\hline $N_g$ & \multicolumn{3}{c|}{$N_g n_g^1 n_g^2
n_g^3=-48(2p+1)^2+16$} \\
\hline
\end{tabular}
\end{center}
\label{sol1}
\end{table}


\renewcommand{\arraystretch}{1.4}
\begin{table}[t]
\caption{Model II. D6-brane configuration in (8p)-generation N=1
supersymmetric $U(4)^3$ model. This model is built on three
rectangular 2-tori with $Z_2\times Z_2$ orbifold symmetry and p is
a positive integer.}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$N_i$ & $(n_{i}^{1}, l_{i}^{1})$ & $(n_{i}^{2}, l_{i}^{2})$ & $(n_{i}^{3}, 
l_{i}^{3})$ \\
\hline
$N_a=8$ & $(2,0)$ & $(p,1)$ & $(p,-1)$ \\
\hline
$N_b=8$ & $(p,-1)$ & $(2,0)$ & $(p,1)$ \\
\hline
$N_c=8$ & $(p,1)$ & $(p,-1)$ & $(2,0)$ \\
\hline
$N_g$ & \multicolumn{3}{c|}{$N_g n_g^1 n_g^2 n_g^3=-48p^2+16$} \\
\hline
\end{tabular}
\end{center}
\label{sol2}
\end{table}

\renewcommand{\arraystretch}{1.4}
\begin{table}[t]
\caption{Model III. D6-brane configuration in (2p)-generation N=1
supersymmetric $U(8)^3$ model. This model is built on three rectangular 
2-tori
with $Z_2\times Z_2$ orbifold symmetry and p is a positive
integer.}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$N_i$ & $(n_{i}^{1}, l_{i}^{1})$ & $(n_{i}^{2}, l_{i}^{2})$ & $(n_{i}^{3}, 
l_{i}^{3})$ \\
\hline
$N_a=16$ & $(1,0)$ & $(p,1)$ & $(p,-1)$ \\
\hline
$N_b=16$ & $(p,-1)$ & $(1,0)$ & $(p,1)$ \\
\hline
$N_c=16$ & $(p,1)$ & $(p,-1)$ & $(1,0)$ \\
\hline
$N_g$ & \multicolumn{3}{c|}{$N_g n_g^1 n_g^2 n_g^3=-48p^2+16$} \\
\hline
\end{tabular}
\end{center}
\label{sol3}
\end{table}

Therefore, we focus on the second case. For simplicity, we only
consider the models with bifundamental representations which the
Standard Model fermions and Higgs can be embedded into. To avoid
the symmetric and anti-symmetric representations, we require that
\begin{eqnarray}
l_{a}^{2}n_{a}^{3}=-n_{a}^{2}l_{a}^{3}~;~ &
l_{b}^{3}n_{b}^{1}=-n_{b}^{3}l_{b}^{1}~;~ &
l_{c}^{1}n_{c}^{2}=-n_{c}^{1}l_{c}^{2}~,~\,
\label{vanish}
\end{eqnarray}
which are equivalent to the supersymmetry preserving conditions.

Because of the $S_3$ symmetry among the three stacks
of D6-branes or among the three 2-tori, Eq. (\ref{vanish}) implies
\begin{eqnarray}
l_{b}^{3}n_{a}^{3}=-n_{b}^{3}l_{a}^{3}~;~ &
l_{c}^{1}n_{b}^{1}=-n_{c}^{1}l_{b}^{1}~;~ &
l_{a}^{2}n_{c}^{2}=-n_{a}^{2}l_{c}^{2}~,~\,
\end{eqnarray}
and vice versa. This means that at massless level,
the representations $(N_a/2, N_b/2, 1)$,
 $(1,N_b/2,N_c/2)$, $(N_a/2, 1, N_c/2)$ (or their complex conjugations)
 will appear or disappear together with the symmetric and
anti-symmetric representations in the models with $G^3$ unification. As
for the determination of $N$ in $U(N)^3$ gauge group,
 we only have two choices:
4 or 8, which can be figured out from the tadpole cancellation
conditions in our setup:
\begin{eqnarray}
N_a n_{a}^{1}l_{a}^{2}l_{a}^{3}=-16~,~
N_b l_{b}^{1}n_{b}^{2}l_{b}^{3}=-16~,~
N_c l_{c}^{1}l_{c}^{2}n_{c}^{3}=-16~,~
\label{tadpole1}
\end{eqnarray}
\begin{eqnarray}
-(N_a n_{a}^{1}n_{a}^{2}n_{a}^{3}+N_b n_{b}^{1}n_{b}^{2}n_{b}^{3}
+N_cn_{c}^{1}n_{c}^{2}n_{c}^{3})-N_{g}n_{g}^{1}n_{g}^{2}n_{g}^{3}=-16.
\label{tadpole2}
\end{eqnarray}
where $N_a=N_b=N_c=2N$. Obviously, $N$ can't be larger than 8 since
the four O6-planes in our setup can only provide $-16$ RR charge
in the D6-brane charge units,
 while $N=2$ is ruled out due to the phenomenological
concern. We emphasize that for $U(4)^4$ model, the three
tori can be tilted, but, for $U(8)^3$ model, the three tori can
not be tilted since
$n_a^1-l_a^1$ is odd.



\renewcommand{\arraystretch}{1.4}
\begin{table}[t]
\caption{Chiral open string spectrum for the $U(N)^3$
GUT models. $N=4$ for Model I and Model II, and $N=8$ for Model III.
$N_f = 2p+1, ~8p, ~2p$ for Model I, Model II, and
Model III, respectively.}
\vspace{0.4cm}
\begin{center}
\begin{tabular}{|c||c||c|c|c|}
\hline Sector & $U(N) \times U(N) \times U(N)$ &
$Q_a$ & $Q_b$ & $Q_c$ \\
\hline
$ab + ba$ & $N_f \times (N,{\ov N},1)$ & $1$ & $-1$ & $0$ \\
\hline
$bc + cb$ & $N_f \times (1,N,{\ov N})$ & $0$ & $1$ & $-1$ \\
\hline
$ca + ac$ & $N_f \times ({\ov N},1,N)$ & $-1$ & $0$ & $1$ \\
\hline
\end{tabular}
\end{center}
\label{spectrum4}
\end{table}

There are three typical solutions corresponding to three
$G^3$  models. The D6-brane configurations for Model I,
Model II, and Model III are given in Tables \ref{sol1},
\ref{sol2}, and \ref{sol3}, respectively. 
We also present the chiral open string spectrum for those models in
Table \ref{spectrum4}. In Model I, three
tori for $T^6$ are all tilted, and in Model II and Model III,
three tori for $T^6$ are all rectangular.
In short, we have
$2p+1$, $8p$ and $2p$ generations of bifundamental representations
under $U(N)^3$ gauge symmetry which include the Standard Model
fermions and Higgs particles. In particular,
in Model I, we can only have three families of fermions and
three pairs of Higgs particles.

\section{Comments on Phenomenology and Variations of the Models}
In this section, we would like to consider the gauge coupling
unification, the gauge symmetry breaking, and the variant models.

\subsection{Gauge Coupling Unification}

The gauge couplings have been discussed in Refs.~\cite{CLS1, CLW}.
Since the gauge couplings are associated with different stacks of
D6-branes, usually they do not have a
 conventional gauge coupling  unification,
although the value of each gauge coupling at the string scale is
predicted in terms of the moduli $\chi_i$ and the ratio of the
Planck scale to string scale. Let us calculate
the 4-dimensional gauge coupling in detail, and show that in our models, 
we
do have the gauge coupling unification.

Intersecting Dp-branes provide us a world where the gauge sectors are
localized on subspaces of space-time while gravity propagates in
10-dimension (with 4-dimensional gravity eventually reproduced at low 
energy via
the standard compactification on some 6-dimensional internal space). 
Before
compactification, the gravitational and gauge interaction on Dp-brane can 
be
generally described by an effective action~\cite{CVJN}
\beqa
S_{10} \supset \int \, d^{10}x \, \frac{M_s^{\, 8}}{(2\pi)^7 g_s^2} \, 
R_{10d} \, +
\int \, d^{p+1}x \, \frac{M_s^{p-3}}{(2\pi)^{p-2} g_s} \, F_{p+1}^{\, 
2}~,~
\eeqa
where $M_s=1/\sqrt{\alpha'}$ is the string scale, $g_s$ is
the string coupling, and the powers of $g_s$ follow from the Euler
characteristic of the worldsheet which produces the interactions for
gravitons (sphere) and gauge bosons (disk).
Upon the compactification, the 4-dimensional
 action picks up volume factors and is
\beqa
S_{4} \supset \int \, d^{4}x \, \frac{M_s^{\, 8}V_{6}}{(2\pi)^7 g_s^2} \, 
R_{4d} \,
+ \int \, d^{4}x \, \frac{M_s^{\, p-3} V_{p-3}}{(2\pi)^{p-2} g_s} \,
F_{4d}^{\, 2} ~.~\,
\eeqa
This allows us to define the 4-dimensional Planck scale $M_{Pl}$ and
the 4-dimensional gauge coupling $g_{YM}^{\sigma}$
for the D6-brane stack $\sigma$ in $M^4
\times T^6/Z_2 \times Z_2$ setup
\beqa
 M_{Pl}^2 = \frac{ M_s^8 V_6
}{ (2 \pi)^7 g_s^2} ~,~\,
\label{mpms}
\eeqa
\beqa
\frac{1}{(g_{YM}^{\sigma})^2} = \frac{ M_s^3 V_3^{\sigma}}{(2
\pi)^4 g_s}~,~\,
\eeqa where
\beqa
 V_6 = \frac{ (2 \pi)^6}{4}
\prod_{i=1}^3 R_1^i R_2^i~,~\, 
\eeqa 
is the physical volume of
$T^6$ and 
\beqa
 V_3^{\sigma} = {1\over 4} (2 \pi)^3 \prod_{i=1}^3
\sqrt{\left(n_{\sigma}^{i} R_1^i\right)^2 + \left(2^{-\beta_i}
l_{\sigma}^i R_2^i\right)^2}~,~\, 
\label{v3} 
\eeqa 
is the physical
volume of three-cycle wrapped by the D6-brane stack $\sigma$.
Therefore, we can write the gauge coupling $g_{YM}^{\sigma}$ in
terms of $M_s$, $M_{Pl}$, $V_3^{\sigma}$ and $V_6$ 
\beqa
(g_{YM}^{\sigma})^2 &=& \sqrt{2\pi}{{M_s\sqrt{V_6}}\over{M_{Pl}
V_3^{\sigma}}}~.~\, 
\eeqa
In our models, 
$R_1^1=R_2^1=R_1^2=R_2^2=R_1^3=R_2^3$, so, we have
\beqa 
(g_{YM}^{\sigma})^2 =
{{{\sqrt {8\pi}} M_s}\over\displaystyle {M_{Pl}}}
{1\over\displaystyle {\prod_{i=1}^3 {\sqrt {\left(n_{\sigma}^{i}
\right)^2 + \left(2^{-\beta_i} l_{\sigma}^i \right)^2}}}}~.~\,
\eeqa
Thus, we do have the gauge coupling unification in our three
models due to the $S_3$ symmetry. To be more
interesting, we find that the unified gauge coupling is purely
determined by two fundamental physical constants: the string scale
$M_s$ and the 4-dimensional Planck scale $M_{Pl}$.

\subsection{Gauge Symmetry Breaking}

In our models, the $U(N)^3$ gauge symmetry can be broken down to
the Standard Model gauge symmetry by introducing the light open
string states. As an example, we only consider the Model I, and
similarly, one can discuss the gauge symmetry breaking in Model II
and Model III.

In Model I, we have 3 families by choosing $p=1$. The gauge
group is $U(4)\times U(4)\times U(4)$, which has subgroup
$SU(4)\times SU(2) \times SU(2)$, {\it i.e.}, the Pati-Salam
model. The left-handed fermions come from the $(4, \bar 4, 1)$
representations, the right-handed fermions come from the $(\bar 4,
1, 4)$ representations, and the pair of Higgs doublets come from
the $(1, 4, \bar 4)$ representations. Then, we will have three pairs
of Higgs doublets. However, in order to have the D-flat and F-flat
directions, we find that there are no Higgs particles at massless
state level which can break the $U(4)\times U(4)\times U(4)$ gauge
symmetry down to the $SU(4)\times SU(2) \times SU(2)$ or Standard
Model gauge symmetry. Thus, the GUT breaking Higgs fields must
arise from the light open string spectrum.

Indeed, we do have such kind of Higgs fields.
The ``$a$'' stack of D6-branes $a$ is parallel to the orientifold ($\Omega 
R$)
image $b'$ of the ``$b$'' stack of D6-branes along the
third torus, {\it i.e.},
 the ``$b$'' stack of D6-branes $b$ is parallel to the orientifold 
($\Omega R$)
image $a'$ of the ``$a$'' stack of D6-branes along the third
torus. Then, there are open strings which stretch between the
branes $a$ and $b'$ (or say $a'$ and $b$). If the minimal distance
squared $Z^2_{(ab')}$
 (in $\alpha'$ units) between these two branes on the third torus is 
small,
{\it i.e.},  the minimal length squared of the stretched string is small,
we have the light scalars with masses $Z^2_{(ab')}/(4\pi^2 \alpha')$ from 
the
NS sector, and the light fermions with the same masses from the R 
sector~\cite{Uranga, imr}.
These scalars and fermions form the 4-dimensional $N=2$ hypermultiplets.

Similarly, the ``$b$'' stack of D6-branes $b$ is parallel to the
orientifold ($\Omega R$) image $c'$ of the ``$c$'' stack of
D6-branes along the first torus, and the ``$c$'' stack of
D6-branes $c$ is parallel to the orientifold ($\Omega R$) image
$a'$ of the ``$a$'' stack of D6-branes along the second torus. Thus,
we can also have the light hypermultiplets from the open strings
which stretch in between the branes $b$ and $c'$, and between the
branes $c$ and $a'$.

The light open string spectrum is given in Table \ref{spectrum5}.
These light Higgs fields can break the $U(4)^3$ down to the
Standard Model gauge symmetry. Roughly speaking, the Higgs fields
in the $(1,4,4)$ and $(1,{\ov 4}, {\ov 4})$ representations can break
the $U(4)\times U(4)\times U(4)$ gauge symmetry down to the
$U(4)\times SU(2) \times SU(2)$ gauge symmetry, and the Higgs
fields in the $(4, 1, 4)$ and $({\ov 4}, 1, {\ov 4})$ representations
can break the $U(4)\times SU(2) \times SU(2)$ gauge symmetry down
to the Standard Model gauge symmetry. The detail symmetry breaking
pattern and phenomenology are under investigation. By the way, we
do not need the particles in the $(4,4,1)$ and $({\ov 4},{\ov 4},1)$
representations to be light because we do not need them to break
the gauge symmetry.

\renewcommand{\arraystretch}{1.4}
\begin{table}[t]
\caption{Light open string spectrum in the Model I which can break
the $U(4)^3$ gauge symmetry down to the Standard Model gauge
symmetry.} \vspace{0.4cm}
\begin{center}
\begin{tabular}{|c||c||c|c|c|c|}
\hline Sector & $U(N) \times U(N) \times U(N)$ &
$Q_a$ & $Q_b$ & $Q_c$ & Mass Square\\
\hline
$ab' + ba'$ & $4 \times (4,4,1)$ & $1$ & $1$ & $0$ & $Z^2_{(ab')}/(4\pi^2 
\alpha')$ \\
$ab' + ba'$ & $4 \times ({\ov 4},{\ov 4},1)$ & $-1$ & $-1$ & $0$
 & $Z^2_{(ab')}/(4\pi^2 \alpha')$ \\
\hline
$bc' + cb'$ & $4 \times (1,4,4)$ & $0$ & $1$ & $1$ & $Z^2_{(bc')}/(4\pi^2 
\alpha')$ \\
$bc' + cb'$ & $4 \times (1,{\ov 4}, {\ov 4})$ & $0$ & $-1$ & $-1$
&
 $Z^2_{(bc')}/(4\pi^2 \alpha')$ \\
\hline
$ca' + ac'$ & $4 \times (4, 1, 4)$ & $1$ & $0$ & $1$ & 
$Z^2_{(ca')}/(4\pi^2 \alpha')$ \\
$ca' + ac'$ & $4 \times ({\ov 4}, 1, {\ov 4})$ & $-1$ & $0$ & $-1$
& $Z^2_{(ca')}/(4\pi^2 \alpha')$ \\
\hline
\end{tabular}
\end{center}
\label{spectrum5}
\end{table}

\subsection{The Variations of $G^3$ Models}

There are some variations of our supersymmetric
$G^3$ models, which still have the first four properties,
{\it i.e}, these variant models typically have more than
four stacks of D6-branes.


\renewcommand{\arraystretch}{1.4}
\begin{table}[t]
\caption{The variant model of Model I. D6-brane configuration in
(2p+1)-generation N=1 supersymmetric $U(4)\times U(2)^4$ model.
This model is built on three tilted 2-tori with $Z_2\times Z_2$
orbifold symmetry and p is a non-negative integer.} \vspace{0.4cm}
\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline
$N_i$ & $(n_{i}^{1}, l_{i}^{1})$ & $(n_{i}^{2}, l_{i}^{2})$ & $(n_{i}^{3}, 
l_{i}^{3})$ \\
\hline
$N_a=8$ & $(2,0)$ & $(2p+1,1)$ & $(2p+1,-1)$ \\
\hline
$N_{b_1}=4$ & $(2p+1,-1)$ & $(2,0)$ & $(2p+1,1)$ \\
\hline
$N_{b_2}=4$ & $(2p+1,-1)$ & $(2,0)$ & $(2p+1,1)$ \\
\hline
$N_{c_1}=4$ & $(2p+1,1)$ & $(2p+1,-1)$ & $(2,0)$ \\
\hline
$N_{c_2}=4$ & $(2p+1,1)$ & $(2p+1,-1)$ & $(2,0)$ \\
\hline $N_g$ & \multicolumn{3}{c|}{$N_g n_g^1 n_g^2
n_g^3=-48(2p+1)^2+16$} \\
\hline
\end{tabular}
\end{center}
\label{sol8}
\end{table}

As an example, we present one variant model of Model I. We split
the ``$b$'' stack D6-branes into two equal stacks of parallel
D6-branes, ``$b_1$'' and ``$b_2$'', and split the ``$c$'' stack D6-branes
into ``$c_1$'' and ``$c_2$''. The D6-brane configuration is given
in Table \ref{sol8}. And the gauge group for this model is
$U(4)\times U(2)^4$. It is not hard to calculate the massless
chiral open string spectrum which includes the Standard Model
fermions and Higgs doublets. Moreover, similar to 
the discussions in subsection
4.2, we can break the $U(4)\times U(2)^4$ gauge symmetry down to
the Standard Model gauge symmetry by introducing the light open
string states. Therefore, we will not repeat those discussions
here.

\section{Discussions and Conclusions}
In general, we can construct the other models which have above
five properties, for instance, the models with three stacks of
D6-branes which have no zero wrapping numbers. And we might relax our
conditions $\chi_1=\chi_2=\chi_3$
 and $R_1^1=R_2^1=R_1^2=R_2^2=R_1^3=R_2^3$. Or one can
introduce more than four stacks of D6-branes. The
general model building and phenomenology are under
investigation~\cite{TLTL}.

In this paper, by adding $S_3$ symmetry on the tadpole
cancellation conditions and supersymmetry preserving conditions,
as well as on the geometry, $T^6 = T^2 \times T^2 \times T^2$, we
obtain three natural GUT models with five interesting
properties. In Model I and Model II, the gauge group is
$U(4)\times U(4) \times U(4)$, while in Model III the gauge group
is $U(8)\times U(8) \times U(8)$. The three tori of $T^6$ are all
tilted for Model I, and they are all rectangular for  Model II and
Model III. The D6-brane configurations and chiral open string spectrum
at massless level are given in Tables
\ref{sol1}$-$\ref{spectrum4}. In all our three models, the
Standard Model fermions and Higgs particles can be embedded into
the bifundamental representations, and there are no
 symmetric and anti-symmetric representations.
In particular, we only have three families of fermions and three
pairs of Higgs particles for Model I.
Moreover, we show that there exists the gauge coupling unification in
our models. We consider the gauge symmetry breaking, too. Explicitly, we 
show
that in Model I, the $U(4)\times U(4) \times U(4)$ gauge symmetry can 
indeed be
broken down to the Standard Model gauge symmetry by introducing the
light open string states, and similar mechanism works for the Model II and 
III.
Furthermore, we comment on the variations of our models, which do not
have the fifth property.



\section*{Acknowledgments}
We would like to thank Fernando Marchesano and Ioannis Papadimitriou
for helpful discussions, and thank Huiyu Albert Li for inspirational
conversations. The research of T. Li was supported by the National
 Science Foundation under Grant No.~PHY-0070928.
 And the research of T. Liu was supported in part
 by the U.S.~Department of Energy under Grant
 No.~DOE-EY-76-02-3071.

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\end{document}


