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\begin{document}
\begin{titlepage}
\begin{center}
{\hbox to\hsize{hep-th/9801207 \hfill UCB-PTH-98/09}}
{\hbox to\hsize{ \hfill LBNL-41341}}
{\hbox to\hsize{ \hfill BU/HEP-98-04}}
{\hbox to\hsize{ \hfill UCSD-PTH-98/05}}
\bigskip
\bigskip
{\Large \bf Gauge Theories with Tensors from \\ Branes and Orientifolds}
\bigskip
\bigskip
{\bf Csaba Cs\'aki$^{a,}$\footnote{Research Fellow, Miller Institute for
Basic Research in Science.},
Martin Schmaltz$^{b}$, Witold Skiba$^c$,
and \\
John Terning$^a$}\\
\smallskip
{\small \it $^a$ Department of Physics, University of California, Berkeley,
CA 94720}
and
{\small \it Theory Group, Lawrence Berkeley National Laboratory, Berkeley, CA
94720}
\smallskip
{\tt csaki@thwk5.lbl.gov, terning@alvin.lbl.gov}
\bigskip
{\small \it $^b$Department of Physics, Boston University,
Boston, MA 02215 }
\smallskip
{\tt schmaltz@abel.bu.edu}
\bigskip
{ \small \it $^c$Department of
Physics,
University of California at San Diego, La Jolla CA 92093 }
\smallskip
{\tt skiba@einstein.ucsd.edu}
\bigskip
\vspace*{1cm}
{\bf Abstract}\\
\end{center}
We present brane constructions in Type IIA string theory for
${\cal N}=1$ supersymmetric $SO$ and $Sp$
gauge theories with tensor representations using an orientifold 6-plane.
One limit of these set-ups corresponds to ${\cal N}=2$ theories previously
constructed by Landsteiner and Lopez,
while a different limit yields ${\cal N}=1$
$SO$ or $Sp$ theories with a massless tensor and no superpotential.
For the $Sp$-type orientifold projection comparison with the field
theory moduli space leads us to postulate two new rules governing
the stability of configurations of D-branes intersecting
the orientifold.
Lifting one of our configurations to M-theory by finding the
corresponding curves, we re-derive the ${\cal N}=1$ dualities
for $SO$ and $Sp$ groups using semi-infinite D4 branes.
\bigskip
\bigskip
\end{titlepage}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Introduction}
\setcounter{equation}{0}
A growing number of supersymmetric field theories have been realized as field
theories on the world volume of various brane configurations. While brane
dynamics provides information about field theory, one also learns about
properties of brane configurations from field theory knowledge. One considers
branes either in the context of perturbative string theory~[1-14] or embedded
in M-theory or F-theory~[15-41].
Orientifold projection is a natural
way of obtaining $SO$ and $Sp$ theories on the world volume of branes,
and it can also yield field theories with two-index tensors. Many results
have been obtained by using an orientifold four plane~\cite{Nick,N2SoSp,
CW,DSB,AOT2,Gimon,JulieKen}. In this paper we
consider brane configurations with an orientifold projection
realized by an orientifold six plane (O6-plane). These
configurations allow us to describe gauge theories which
have not been constructed previously using other brane configurations.
Our configurations can be understood as generalizations of the brane
set-up considered by Landsteiner and Lopez~\cite{LL} for ${\cal N}=2$
supersymmetric $SO$ and $Sp$ theories. Landsteiner and Lopez (LL)
considered D4 branes stretched between two parallel NS5 branes.
In between the NS5 branes there are D6 branes and an O6-plane.
The O6-plane amounts to a reflection involving the
space-time directions which are orthogonal to the orientifold.
As usual, the orientifold projection combines this spatial reflection with
a parity inversion on the string world sheet.
In this paper, we break the ${\cal N}=2$ supersymmetry of the LL
configuration by rotating the two NS5 branes. The rotation
corresponds to adding a mass term for the adjoint chiral
superfield in the field theory. A new interesting brane
configuration is obtained when the two NS5 branes are
rotated until they are parallel to the
orientifold, then the corresponding field theory has an
additional massless chiral superfield: a symmetric
(antisymmetric) tensor in the case of $SO$ ($Sp$).
We investigate the Coulomb and Higgs branches for all
our theories in detail and demonstrate agreement with
the field theory expectations.
For the $Sp$ configuration we find that two new rules
concerning the consistency of D branes and orientifolds are required.
The ``doubling rule" requires D branes which cross or
are parallel to the orientifold to come in pairs (thus
explaining why $Sp$ theories only exist for even numbers of
colors and flavors), and the ``s'-rule" which is
similar to the ``s-rule" of Hanany and Witten~\cite{HananyWitten},
and which is necessary
to reproduce the dimension of the Higgs branch of the field
theory moduli space correctly.
Finally, we also consider brane configurations corresponding to
$SO$ and $Sp$ theories with only fundamental matter fields. We
explicitly construct the corresponding M-theory curves
and use them to re-derive Seiberg's duality for $SO$ and $Sp$ gauge theories.
This derivation is similar to the derivation of Seiberg's
duality using an O4 plane in Ref.~\cite{CW}.
The paper is organized as follows. In the next section, we describe the basic
brane configuration and determine the corresponding low-energy field theory.
In Section 3, we study the Higgs branch of theories with
flavors. In Section 4, we derive duality for $SO$ and $Sp$ with fundamentals
in the context of string theory by considering brane motions. We discuss the
embedding of this set-up in M-theory and present curves describing our
brane configuration in Section 5. Finally, we conclude in the last section.
Certain elements of our paper have appeared very recently in
Refs.~\cite{kutasov,Karch}. In particular, Brunner, Hanany, Karch and
Lust~\cite{Karch}
mention the brane configurations of Sec.~\ref{sec:setup}. The counting of the
${\cal N}=2$ moduli spaces of Section~\ref{sec:higgs} together with a slightly
different statement of the s'-rule of Section~\ref{sec:higgs}
(with the same physical
consequences) appeared in the work of Elitzur, Giveon, Kutasov and
Tsabar~\cite{kutasov}.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{The Brane Set-up\label{sec:setup}}
\setcounter{equation}{0}
\begin{figure}
\PSbox{LL.eps hscale=70 vscale=70 hoffset=150 voffset=0}{13.7cm}{4.5cm}
\caption{The brane configuration in Type IIA string theory
giving rise to the pure ${\cal N}=2$ $SO$ or $Sp$ theories.
The $\otimes$ denotes the O6-plane which is perpendicular to the
NS5 branes. \label{LL}}
\end{figure}
In this section we consider brane configurations
in Type IIA string theory which give rise to $SO$ and $Sp$
gauge groups. Our starting point is the ${\cal N}=2$ configuration of
Ref.~\cite{LL} presented in Fig.~\ref{LL}. In this configuration we have
an O6-plane in the $(x_0,x_1,x_2,x_3,x_7,x_8,x_9)$ directions, which thus acts
as a mirror\footnote{We present
all of our brane configurations as embedded into the double
cover of the orientifolded space.} in $(x_4,x_5,x_6)$.
In addition, we have two NS5 branes
in the $(x_0,x_1,x_2,x_3,x_4,x_5)$ directions
which are mirror images of each other under the orientifold projection.
There are also $N$ D4 branes in the $(x_0,x_1,x_2,x_3,x_6)$
directions connecting
the two NS5 branes. This theory, as discussed in Ref.~\cite{LL},
corresponds to ${\cal N}=2$ $SO(N)$ theory or
$Sp(N)$ theory depending on the orientifold charge.
First we will discuss the ${\cal N}=1$ $SO(N)$ theory obtained
from this ${\cal N}=2$ set-up by rotation of the NS5 branes. Then we repeat the
discussion for $Sp$ groups. The analysis for the $Sp$ theories is
very similar to the case of $SO$ groups, with the exception of an
important subtlety which we discuss in detail. Throughout this
paper, we use the complex coordinates $v=x_4+ix_5$ and
$w=x_8+ix_9$. With this notation, in the ${\cal N}=2$ theory the
NS5 branes are at a point in the $w$ plane and fill out the $v$ plane,
while the O6 is at the origin of $v$ and fills out $w$.
\subsection{The $SO(N)$ Theories}
The ${\cal N}=2$ $SO(N)$ theory is given in Fig.~\ref{LL}. The moduli space
of this
theory corresponds to giving expectation values to the adjoint of the
$SO(N)$ (which is an antisymmetric tensor). In the brane language
this corresponds to sliding the D4 branes between the two parallel
NS5 branes. At a generic point of the moduli space the adjoint VEV
breaks $SO(N)$ to $U(1)^r$, where $r=N/2$ for $N$ even and $r=(N-1)/2$
for $N$ odd, which just corresponds to sliding all D4 branes apart
from each other as illustrated in Fig.~\ref{SOmoduli}. Due to the
O6 projection the D4 branes have to slide between the NS5 branes
in pairs in the opposite directions, thus for $N$ even we get
as a dimension of the moduli space $N/2$. For $N$ odd one of the
D4 branes is stuck at the O6, and the number of moduli is
given by $(N-1)/2$.
\begin{figure}
\PSbox{SOmoduli.eps hscale=70 vscale=70 hoffset=150 voffset=0}{13.7cm}{4.5cm}
\caption{The moduli space of the ${\cal N}=2$ $SO(N)$ theory. Note that the
D4 branes have to move in pairs away from the O6-plane.
\label{SOmoduli}}
\end{figure}
The ${\cal N}=2$ $SO(N)$ theory has an anomalous $U(1)_R$ symmetry, under which
the adjoint field (the antisymmetric tensor of $SO(N)$) carries charge
two. The $U(1)_R$ can be identified with rotations of the $v$ (45) plane,
$v \rightarrow e^{i\theta} v$, we call this symmetry $R_v$. Thus, the $R_v$
charge of the adjoint is two.
Let us now rotate the NS5 branes slightly out of the $v$ plane into the
$w$ plane (Fig.~\ref{rotate}).
Since the configuration must remain symmetric
under the orientifold projection, both NS5 branes have to be
rotated, but in opposite directions. As a result of the rotation, the
D4 branes are fixed at the origin, they can not slide between
the NS5 branes anymore. Thus the moduli space of the theory is lifted.
This is exactly analogous to what happens in the ${\cal N}=2$ $SU(N)$ theories
when one of the NS5 branes is rotated~\cite{HOO,Witten2}.
In the field theory a mass for the adjoint chiral superfield is
generated, breaking ${\cal N}=2$ to ${\cal N}=1$,
and lifting the Coulomb branch of the theory. The
antisymmetric tensor $A$ gets a small mass $W=\mu A^2$ when the
two NS5 branes are slightly rotated. The $\mu \to 0$ limit corresponds to the
${\cal N}=2$ theory discussed before, and for generic $\mu$ we obtain
pure ${\cal N}=1$ $SO(N)$ gauge theory with no massless chiral superfields.
This theory has no moduli space.
\begin{figure}
\PSbox{rotate.eps hscale=70 vscale=70 hoffset=150 voffset=0}{13.7cm}{4.5cm}
\caption{The brane configuration obtained by the rotation of the
NS5 branes of the ${\cal N}=2$ $SO(N)$ theory.
\label{rotate}}
\end{figure}
However, we can observe that there is another limit of the rotated
brane configuration where the $SO(N)$ theory does have a moduli space.
If the angle of rotation is $\pi /2$ the two NS5 branes become parallel
to each other and the O6-plane, and the D4 branes can
slide between the NS5 branes again (Fig.~\ref{puretensor}).
The emerging moduli space of the brane configuration must correspond
to a field whose mass goes to zero in the field theory. We conjecture
that this field transforms as a symmetric tensor under the $SO(N)$
gauge group.\footnote{In the classical
picture the trace of the symmetric tensor of $SO(N)$ appears as a
modulus as well.
This degree of freedom is presumably frozen by
an infrared divergence of the quantum theory, similar to the trace
of the adjoint of the ${\cal N}=2$ $SU(N)$ theories~\cite{Witten}.}.
We now present four different arguments in support of this
conjecture.
\begin{figure}
\PSbox{puretensor.eps hscale=70 vscale=70 hoffset=150
voffset=0}{13.7cm}{4.5cm}
\caption{The brane configuration obtained in the $\mu \to \infty$ limit,
which gives
rise to ${\cal N}=1$ $SO(N)$ theory with a massless symmetric tensor, or
$Sp(N)$ with an antisymmetric tensor.
\label{puretensor}}
\end{figure}
\begin{figure}
\PSbox{N=1SOmoduli.eps hscale=70 vscale=70 hoffset=150
voffset=0}{13.7cm}{4.5cm}
\caption{The moduli space of the ${\cal N}=1$ $SO(N)$ theory with a symmetric
tensor. If all D4 branes move apart the gauge group is completely broken.
\label{N=1SOmoduli}}
\end{figure}
First we compare the moduli space of the field theory of $SO(N)$ with a
symmetric tensor to that of the brane picture. In the brane picture
the moduli space corresponds to the sliding of the D4 branes along the
parallel NS5 branes (see Fig.~\ref{N=1SOmoduli}).
This obviously has $N$ moduli,
and since a single D4 brane in the presence of an O6 corresponds to
``$SO(1)$'', this means that on a generic point of the moduli space
the $SO(N)$ is completely broken. Let us now consider the unbroken
group if we move away only one D4 brane (Fig.~\ref{SObreaking}).
Then we still have
$N-1$ D4 branes on top of each other and thus this breaking
corresponds to $SO(N)\to SO(N-1)$. This brane picture is in complete
agreement with the field theory result. In the field theory,
giving a VEV
\begin{displaymath}
V
\left( \begin{array}{ccccc}
1 & \\
& 1 & \\
& & \ddots \\
& & & 1 \\
& & & & -(N-1) \end{array} \right)
\end{displaymath}
to the symmetric tensor breaks the gauge group to $SO(N-1)$, and there is
a massless symmetric tensor of the $SO(N-1)$ group remaining. This
can be identified with the above brane motion where one of the D4 branes is
sliding away from the remaining $N-1$. In general, the symmetric tensor
$S$ of $SO(N)$ can completely break the $SO(N)$ group, which corresponds
to moving all D4 branes apart from each other. Finally, the independent gauge
invariant operators are given by ${\rm Tr} S,{\rm Tr} S^2,
\ldots , {\rm Tr} S^N$\cite{DM}, and thus the number of moduli
agrees with the brane prediction. In summary, we find complete
agreement between the field theory and the brane moduli space.
Note, that the antisymmetric tensor
(adjoint) would not give the right dimension of moduli space, since an
${\cal N}=2$ $SO(2M)$ theory has $M$ moduli.
\begin{figure}
\PSbox{SObreaking.eps hscale=70 vscale=70 hoffset=150
voffset=0}{13.7cm}{4.5cm}
\caption{The brane configuration which corresponds to the
breaking $SO(N)\to SO(N-1)$.
\label{SObreaking}}
\end{figure}
As a second piece of evidence for establishing that the brane configuration
of Fig.~\ref{puretensor} corresponds to $SO(N)$ with a symmetric tensor
consider the brane set-up of Fig.~\ref{SU} (we will discuss this brane
configuration in more detail in Section~\ref{SUsection}).
Here we have three NS5 branes,
two of which are parallel to the O6-plane and are thus in the
$(x_0,x_1,x_2,x_3,x_8,x_9)$
directions, while the third NS5 brane is at the same point as the
O6 in the $x_6$ direction, but is perpendicular to it, since it is
in the $(x_0,x_1,x_2,x_3,x_4,x_5)$
directions. This theory corresponds to ${\cal N}=1$ $U(N)$
theory with a symmetric tensor and its conjugate. We can however
move the D4 branes away from the middle NS5 brane by sliding them
between the parallel NS5 branes, and the resulting brane
configuration is that of Fig.~\ref{puretensor}
(see Fig.~\ref{SUhiggs}). In the field theory this corresponds to the
higgsing of $U(N)$ to $SO(N)$ by a VEV for a symmetric tensor which
leaves one massless symmetric tensor transforming under the $SO(N)$
as expected from the brane picture.
\begin{figure}
\PSbox{SU.eps hscale=70 vscale=70 hoffset=150
voffset=0}{13.7cm}{4.5cm}
\caption{The brane configuration corresponding to an ${\cal N}=1$ $U(N)$
with a symmetric and a conjugate symmetric tensor. The $\otimes$ corresponds
to a third NS5 brane perpendicular to the other two NS5 branes and the
O6.
\label{SU}}
\end{figure}
\begin{figure}
\PSbox{SUhiggs.eps hscale=70 vscale=70 hoffset=150
voffset=0}{13.7cm}{4.5cm}
\caption{The brane configuration corresponding to breaking
$U(N)\to SO(N)$ by giving an expectation value to the symmetric tensor.
\label{SUhiggs}}
\end{figure}
As a third piece of evidence consider the argument in Section 2 of
the work of Landsteiner and Lopez~\cite{LL}. LL argue that $N$ D4 branes
in an orientifold background give rise to two kinds of matter
multiplets: an adjoint and a two-index tensor other than the
adjoint. The boundary conditions imposed on the states on the
D4 branes by the transverse NS5 branes project out some of
these states. Which states are projected out depends on the orientation
of the NS5 branes. If they
are perpendicular to the O6 the tensor is projected out, thus
leaving an ${\cal N}=2$ theory. If the NS5 branes are parallel to the O6 the
adjoint
and part of the tensor is projected out leaving $SO(N)$ with a
symmetric tensor.
As final piece of evidence, consider the brane configuration of
Fig.~\ref{product}. There, the theory between the middle two NS5 branes
is an ${\cal N}=2$ $SO(N)$ theory, and due to the additional $F$ D4 branes
we get $F$ hypermultiplets of this ${\cal N}=2$ theory. Because the additional
D4 branes end on two extra NS5 branes at the left and the right of the
set-up, the $U(F)$ flavor symmetry is gauged.
Thus, the field theory corresponding to Fig.~\ref{product} is given by
\[
\begin{array}{ccc}
& SO(N) & U(F) \\
\Phi & \Yasymm=Adj & 1 \\
Q& \Yfund & \Yfund \\
\bar{Q} & \Yfund & \overline{\Yfund} \end{array}.
\]
The superpotential of the theory is
\beq
W= Q\Phi \bar{Q}.
\eeq
\begin{figure}
\PSbox{product.eps hscale=70 vscale=70 hoffset=130
voffset=0}{13.7cm}{4.5cm}
\caption{The brane configuration which corresponds to an ${\cal N}=2$
$SO(N)$ theory with $F$ flavors and with
the $U(F)$ flavor symmetry gauged.
\label{product}}
\end{figure}
Now let us apply Seiberg's ${\cal N}=1$ duality to the $U(F)$ group
(assuming that $F 0$),
the extension to the other cases
($b \le 0$ or $\tilde{b}\le 0$) is straightforward and gives further
evidence for duality. For details see Ref.~\cite{MR}.}
$\tilde \Lambda_{\tilde N,F} \rightarrow 0$. To make things simpler we
rewrite equation (\ref{SOyw}) with $w_-$ expressed in terms of $w_+$ and
vice versa by use of Equation (\ref{SOwpm}). More explicitly:
\beq
\left(\mu \xi \right)^{N-1} y \prod_{i=1}^F\left( {{ {{\mu \xi}\over{w_-}} -
m_i}\over{ {{\mu \xi}\over{w_+}} - m_i}} \right) = v^2 \left( {{\mu
\xi}\over{w_-}} \right)^{2N-2}
\label{SOywsub}
\eeq
is equivalent to
\beq
\left(\mu \xi \right)^{\tilde N-1} y \prod_{i=1}^F\left( {{w_- - \mu
M_i}\over{w_+ - \mu M_i}} \right) = v^2 w_-^{2 \tilde N-2} ~,
\label{SOywdual}
\eeq
where
\beq
\tilde N = F- N +2 ~,
\eeq
corresponding to the dual gauge group $SO(\tilde N)$,
and $M_i$ is the meson VEV:
\beq
M_i = {{\xi}\over{m_i}} .
\eeq
Note that
\beq
\xi =
\left( \tilde \Lambda_{\tilde N,F}^{3(\tilde N-1) - F} \prod_i^F m_i
\right)^{{1}\over{\tilde N-1}}
\eeq
In order to hold the meson VEV fixed in the $R \rightarrow 0$ limit
we must take:
\beq
m_i^{F-N+1} \sim e^{L/R} \rightarrow 0 ~.
\eeq
Thus we see that the string theory limit of the M-theory curve with
$\Lambda_{N,F} \rightarrow \infty$ (i.e. $L < 0$) gives precisely the
string theory picture of the dual gauge theory.
\subsection{$Sp(2N)$}
Next we consider the M-theory curves for the $Sp(2N)$ theory.
The construction again includes an O6-plane (but with the opposite charge)
at a point in $v$ and
extending in the $w$ direction and $x_9$. There are two
NS5 branes, one on each side (in $x_6$) of the O6-plane
extending in the $w_\pm$ directions.
The NS5 branes have $2N$ D4 branes stretched between them, and $F$
semi-infinite
D4 branes extending to the left from the left
NS5 brane, and mirrors of these on the right (Fig.~\ref{semiinfinite}).
Again we need three equations in the complex variables $v$,
$w$, $x$ and $y$. As before we give the curves and then the evidence that they
are
correct.
The O6-plane is described by \cite{LL}:
\beq
x y = v^{-4} ~.
\label{Spxy}
\eeq
We also have
\beq
w_+ w_- = \mu \xi ~,
\label{Spwpm}\eeq
and
\beq
\left(\mu \xi \right)^{N+1} y \prod_{i=1}^F\left( {{w_+ - m_i}\over{w_- - m_i}}
\right) = v^{-2} w_+^{2N+2} ~.
\label{Spyw}
\eeq
It is straightforward to check that equations (\ref{Spxy}), (\ref{Spwpm}), and
(\ref{Spyw}) satisfy the correct symmetries and to check that decoupling a
flavor works correctly.
\subsection{The String Limit}
We now consider the string theory limit $R \rightarrow 0$ of the curve with
\beq
\xi^{N+1 } \sim e^{-L/R} ~,
\label{SpL}
\eeq
$L > 0$ and $m_i$ fixed.
We first consider fixed $w_+ \gg m_i$,
and search for solutions
with large $y \sim t = e^{-s/R}$. Equation (\ref{Spwpm}) implies that
\beq
w_- \rightarrow 0 ~,
\eeq
while equation
(\ref{Spyw}) implies that
\beq
e^{-L/R} e^{-s/R} \sim {\rm const.} \,\,w_+ ^{2N-F} ~,
\eeq
so $s = -L$ in the limit $R \rightarrow 0$.
These solutions correspond to the NS5 brane on the left. Next consider
\beq
w_+ \sim m_i + e^{-c/R}~.
\eeq
Equation
(\ref{SOyw}) implies that
\beq
e^{-L/R} e^{-s/R} e^{- c/R} \sim {\rm const.} ~,
\eeq
so $s = -L- c$. Thus we see $F$ solutions (one for each $m_i$) extending to the
left that approach $w_+ = m_i$ as $s \rightarrow - \infty$.
These are the $F$ semi-infinite D4 branes on the left. The mirror
solutions can be found on the right for large $x$.
Finally consider
\beq
w_+ \sim e^{-c/R}~,
\eeq
equation
(\ref{Spyw}) implies that for $2c < L/(N+1)$
\beq
e^{-L/R} e^{-s/R} \sim {\rm const.} \,\, e^{- (2N +2) c/R} ~,
\eeq
so $s = -L+ (2N +2) c$ and these solutions sit to the right of the NS5 brane;
they are the color D4 branes.
We can also check the curve by counting the number of solutions for large,
fixed
$y$ and comparing to the number of branes in the string picture.
Using equations (\ref{v}) and (\ref{Spwpm})
we rewrite equation (\ref{Spyw}) in terms of $y$ and $w_+$:
\beq
\left(\mu \xi \right)^{N+1} y \prod_{i=1}^F \left(w_+ - m_i \right)
w_+^{F-2N-4} \left(w_+^2 - \mu \xi \right)^2 = \prod_{i=1}^F \left( \mu \xi -
m_i w_+ \right)
\label{Sptubes}
\eeq
There are again three cases:
a) $F \ge 2N+4$, there are $2F-2N$ solutions, $F$ with $w_+ \approx m_i$ (the
``flavor-branes"), $F-2N-4$ with $w_+ \approx 0$ (this corresponds to the
NS5 brane on the right
bending to the left) and four solutions with $w_+^2 \approx \mu \xi$
(associated with the
O6-plane);
b) $2N+2 \le F < 2N+4$, there are $F+4$ solutions, $F$ with $w_+ \approx m_i$,
and four solutions with $w_+^2 \approx \mu \xi$ (associated with the O6-plane);
c) $F < 2N+2$, there are $2N+6$ solutions, $F$ with $w_+ \approx m_i$, and
$2N+2-F$
with large $w_+$ (the NS5 brane on the left bends to the left), and four
solutions with $w_+^2 \approx \mu \xi$ (associated with the O6-plane).
\subsection{Duality}
We can also see that the M-theory curve correctly describes the dual gauge
theory
by simply
rewriting Equation (\ref{Spyw}) and taking the limit $R \rightarrow 0$ with
$\Lambda \rightarrow \infty$.
The curve,
\beq
\left(\mu \xi \right)^{N+1} y \prod_{i=1}^F\left( {{ {{\mu \xi}\over{w_-}} -
m_i}\over{ {{\mu \xi}\over{w_+}} - m_i}} \right) = v^{-2} \left( {{\mu
\xi}\over{w_-}} \right)^{2N+2} ~,
\label{Spywsub}
\eeq
is equivalent to
\beq
\left(\mu \xi \right)^{\tilde N+1} y \prod_{i=1}^F\left( {{w_- - \mu
M_i}\over{w_+ - \mu M_i}} \right) = v^{-2} w_-^{2 \tilde N+2} ~,
\label{Spywdual}
\eeq
where
\beq
\tilde N = F- N - 2 ~,
\eeq
corresponding to the dual gauge group $Sp(2 \tilde N)$.
Again we see that this correctly reproduces dual gauge theory in the
string theory limit.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Conclusions}
We have investigated brane realizations of ${\cal N}=1$ $SO$ and $Sp$ gauge
theories with two-index tensor representations. Our construction
employed an orientifold six plane. By studying various limiting
cases we have established the field content and the tree-level
superpotential of our brane construction. The important limiting cases
correspond to ${\cal N}=2$ theory with tensor hypermultiplets and
${\cal N}=1$ theories with massless tensors and no superpotential.
For these cases, we have checked that the brane degrees of
freedom are in exact correspondence
with the flat directions in field theory.
Consistency requirements impose two new restrictions
on brane configurations involving an orientifold six plane. First,
orientifolds with the $Sp$-type projection can be only crossed by pairs
of D4 branes. A single D4 brane is not invariant under
the projection of $Sp$-type. Second, the s-rule needs to be generalized
to our s'-rule in the presence of an orientifold six plane
with negative charge. Our first
observation says that D4 branes crossing an O6-plane
have to group into pairs invariant
under the orientifold projection. Then, according to the s'-rule
only one of the two branes in each
such pair can end on a given D6 brane parallel to an O6-plane.
We have also studied duality for $SO$ and $Sp$ groups both in the string
theory framework and in M-theory. In string theory, we included D6 branes
which transform under flavor symmetries. The dual gauge group and meson
excitation are identified when the original brane configuration is
deformed by moving NS5 and D6 branes. In M-theory, we included semi-infinite
branes and found the equations describing the single brane configuration.
This configuration encodes the information about the dual
gauge group and the vacuum expectation values of the meson fields.
\section*{Acknowledgments}
We are grateful to Jan de Boer, Kentaro Hori, Hirosi Ooguri,
Yaron Oz, Erich Poppitz and Raman Sundrum for useful discussions.
We also thank Jan de Boer, Yaron Oz and Zheng Yin for comments on the
manuscript.
C.C. is a research fellow of the Miller Institute for Basic Research
in Science. C.C. and J.T. are supported in part by the National Science
Foundation under grant PHY-95-14797, and are also partially supported by
the Department of Energy under contract DE-AC03-76SF00098.
M.S. is supported by the U.S.
Department of Energy under grant \#DE-FG02-91ER40676.
W.S. is supported by the Department of Energy
under contract DE-FG03-97ER405046.
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\end{document}