%%%%%%% Doctor thesis of Seiji Terashima
%%%%%%% Title 'Seiberg-Witten Geometry via Confining Phase superpotential'
%%%%%%% version 98/10/27, 11/15, 12/7, 12/11, 12/22, 12/24 final ver. in het02

\documentstyle[12pt,epsf]{report}
\textheight=21.5cm
\textwidth=16cm
\topmargin=0cm
\oddsidemargin=0cm
\evensidemargin=0cm
\newcommand{\beq}{\begin{equation}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqa}{\begin{eqnarray}}
\newcommand{\eeqa}{\end{eqnarray}}
\newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}}
\newcommand{\CR}{\nonumber \\}
\newcommand{\pa}{\partial}
\newcommand{\A}{\alpha}
\newcommand{\B}{\beta}
\newcommand{\D}{\delta}          
\newcommand{\G}{\gamma}
\newcommand{\E}{\epsilon}
\newcommand{\La}{\Lambda}
\newcommand{\p}{\Phi}
\newcommand{\bra}{\langle}
\newcommand{\ket}{\rangle}
\newcommand{\lm}{\lambda}
\newcommand{\s}{\sigma}
\newcommand{\tQ}{\tilde{Q}}
\newcommand{\rep}{{\cal R}}
\newcommand{\la}{{\langle}}
\newcommand{\ra}{{\rangle}}
\newcommand{\ul}{\underline}
\newcommand{\Tr}{{\rm Tr}}
\newcommand{\de}{{\rm det}}
\renewcommand{\thefootnote}{\fnsymbol{footnote}}

\newcommand{\Lm}{\Lambda}

\begin{document}


\title{
{\center{\huge {
Seiberg-Witten Geometry \\ \vspace{.3cm}
via Confining Phase Superpotential
}}} \vspace{2cm}}

\author{ 
{\Large Seiji TERASHIMA}
\vspace{1cm}
\\
A dissertation submitted to the Doctoral Program 
\\
in Physics, the University of Tsukuba 
\\
in partial fulfillment of the requirements for the
\\
degree of Doctor of Philosophy (Science)
\\  
\\
January, 1999 }

\date{}
\baselineskip=0.7cm
\maketitle



\newpage

\begin{abstract}

We study Seiberg-Witten Geometry 
to describe the non-perturbative low-energy behavior of 
$N=2$ supersymmetric gauge theories in four dimensions.
The method of $N=1$ confining phase superpotential is employed for this purpose.
It is shown that the 
ALE space of type ADE fibered over ${\bf CP}^1$ is natural geometry
for the $N=2$ supersymmetric gauge theories with ADE gauge groups.
Furthermore, we obtain in this approach previously unknown Seiberg-Witten geometry 
for four-dimensional $N=2$ gauge theory
with gauge group $E_6$ with massive fundamental hypermultiplets.
By considering the gauge symmetry breaking in this $E_6$ gauge theory,
we also obtain Seiberg-Witten geometries
for $N=2$ gauge theory 
with $SO(2N_c)$ $(N_c \leq 5)$ with massive spinor and vector 
hypermultiplets.
In a similar way the Seiberg-Witten geometry is determined for $N=2$ $SU(N_c)$
$(N_c \leq 6)$ gauge theory with massive antisymmetric and fundamental 
hypermultiplets. 

\end{abstract}


\newpage

\tableofcontents

\newpage


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Introduction}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For almost 25 years
four-dimensional supersymmetric gauge field theories 
have been investigated very intensively.
One reason for this is that 
supersymmetric theories have a remarkable property of
canceling out the divergence in the self-energies which is desirable
to construct a more natural phenomenological model at high energies
beyond the standard model.
Furthermore a proposal of 
the unification of the gauge groups of the standard model
seems to be more attractive by requiring 
the theory to have softly broken supersymmetry. 

Supersymmetric gauge theories have also been considered
as theoretical models to understand the strong coupling effects.
These effects such as color confinement and chiral symmetry breaking
are difficult to study analytically
in the theories without the supersymmetry.
On the other hand 
the action of the supersymmetric field theory is highly constrained
by its supersymmetry and
in some cases even the exact descriptions of the low-energy theories
of these have been obtained on the basis of the idea of 
duality and holomorphy 
\cite{Se1}-\cite{SeWi2}.
Consequently, the non-perturbative effects
in the supersymmetric theory can be evaluated quantitatively.

Another reason for the importance of the study of 
supersymmetric gauge field theories 
is their close relation to the superstring theory.
Superstring theories receive a lot of current research interest
since they are the only known unified models including 
quantum gravity in a consistent manner and 
have enough gauge symmetries to contain the standard model.
Moreover the superstring theory predicts the spacetime supersymmetry.
Therefore supersymmetric gauge field theories naturally
appear in the study of the superstring theory.

In the superstring theory, the supersymmetric gauge field theories
appear in two ways.
A conventional way is to have a supersymmetric field theory on the 
lower dimensional spacetime after the ten-dimensional superstring is compactified.
The other novel way is 
that supersymmetric gauge theories in various dimensions are
realized on the world volume of D-branes 
which are higher dimensional objects
on which the open strings can end.
In the framework of the superstring theory,
the gauge field theories with extended supersymmetry
\footnote{Here the extended supersymmetric theory has 
more supercharges than minimal supersymmetric theory ($N=1$ supersymmetry).
For example, the $N=2$ supersymmetry is two times as
large as the $N=1$ supersymmetry}
are important because 
ten-dimensional superstring theories have 
more supercharges than 
lower-dimensional $N=1$ (i.e. minimal) supersymmetric theory and
some or all of these supercharges are unbroken
if we compactify the superstring theory on
a suitably chosen manifold.

In the case of $N=2$ supersymmetry, a substantial progress
was made by Seiberg and Witten \cite{SeWi1,SeWi2}.
They have shown that 
the low-energy effective theory of 
the Coulomb phase of four-dimensional $N=2$ supersymmetric 
$SU(2)$ gauge theory can be described by an
auxiliary complex curve, called the Seiberg-Witten curve,
whose shape depends on the vacuum moduli $u={\rm Tr \, \p^2}$.
In this beautiful mathematical description,
massless solitons are recognized as vanishing cycles 
associated with the degeneracy of the curves and their masses are
obtained as the integral of certain one-form, 
which is called the Seiberg-Witten form,
over these cycles.
Soon after these works, generalizations to the other $N=2$ supersymmetric 
gauge theory with the classical gauge groups have been carried out
by several groups \cite{ArFa}-\cite{Ha}.
However all these generalizations are based on the assumption 
that auxiliary complex curves are of hyperelliptic type.
Without this assumption, 
simple extensions of the original work \cite{SeWi1,SeWi2} are not promising to
determine the curves.
Thus 
it is desired to invent other methods for deriving the curve 
without the assumption on the types of curves.

To this end, we notice the fact that 
the singularity of quantum moduli space of the vacua of 
the theory corresponds to 
the appearance of massless solitons.
Near the singularity, therefore, we observe interesting non-perturbative 
properties of the theories.
Moreover 
the Seiberg-Witten curves are determined almost completely
from the information of the locations of singularities on the moduli space.
In order to explore physics near $N=2$ singularities 
the microscopic superpotential explicitly breaking $N=2$ to $N=1$ 
supersymmetry is often considered 
\cite{SeWi1,SeWi2,InSe,ArDo}.
Examining the resulting superpotential 
for a low-energy effective Abelian theory it is found that 
the generic $N=2$ vacuum is lifted and only the singular loci
of moduli space remain as the $N=1$ vacua where monopoles or dyons 
can condense.
The resulting $N=1$ theory is shown to be in the confining phase
in accordance with the old idea of the confinement via 
the condensation of monopole.

This observation suggests that 
we may start with a microscopic $N=1$ theory which 
we introduce by perturbing an $N=2$ theory by adding a tree-level
superpotential built out of the Casimirs of the adjoint field
in the vector multiplet \cite{InSe,ElFoGiRa,ElFoGiRa2}
toward the construction of the $N=2$ curves.
Let us concentrate on a phase with a single confined
photon in our $N=1$ theory which 
corresponds to the classical $SU(2) \times U(1)^{r-1}$
vacua with $r$ being the rank of the gauge group.
Then the low-energy effective theory
containing non-perturbative effects provides us with the
data of the vacua with massless solitons \cite{In,InSe}.
From this we can identify the singular points in the Coulomb phase 
of $N=2$ theories and construct the $N=2$ Seiberg-Witten curves.
This idea, called ''confining phase superpotential technique'', 
has been successfully applied to 
$N=2$ supersymmetric $SU(N_c)$ pure Yang-Mills theory \cite{ElFoGiRa}.
We extend their result to the case of 
$N=2$ supersymmetric pure Yang-Mills theory with
arbitrary classical gauge group \cite{TeYa1} as well as
$N=2$ supersymmetric QCD \cite{KiTeYa} (see also \cite{InSe}-\cite{Ki}). 
The resulting curves are hyperelliptic type and agree with those of 
\cite{DaSu}-\cite{Ha}.

On the other hand, 
for exceptional gauge groups there were proposals based on
the relation between Seiberg-Witten theory and the integrable systems that
Seiberg-Witten curves are not realized 
by hyperelliptic curves \cite{MaWa,LW,WY}.
In \cite{MaWa} it is claimed that
the Seiberg-Witten curves for
the $N=2$ supersymmetric pure Yang-Mills theory 
with arbitrary simple gauge group 
are given by the spectral curves for the affine Toda lattice which has a form of
a foliation over ${\bf CP}^1$.
For $G_2$ gauge group, 
this has been confirmed by the confining phase superpotential \cite{LaPiGi} and 
the one instanton calculation \cite{Ito}.

The application of the confining phase superpotential technique 
to the $E_6,E_7,E_8$ gauge groups 
seems to be difficult at first sight 
because of the complicated structure of the $E_n$ groups.
Nonetheless we have shown that this technique can be applied in a
unified way 
in determining the singularity structure of moduli space of the Coulomb phase in
supersymmetric pure Yang-Mills theories with ADE gauge groups \cite{TeYa2}. 
Not only the classical
case of $A_r, D_r$ groups but the exceptional case of $E_6, E_7, E_8$ 
groups can be treated on an equal footing since our discussion is based on
the fundamental properties of the root system of the simply-laced Lie
algebras.
The resulting Riemann surface 
is described as a foliation over ${\bf CP}^1$ and satisfies the
singularity conditions we have obtained 
from the $N=1$ confining phase superpotential.
This Riemann surface is not of hyperelliptic type for exceptional gauge groups.

In the consideration within the scope of four-dimensional field theory,
it was unclear 
if the Riemann surface in the exact description is an auxiliary object
for mathematical setup or a real physical object.
It turns out that four-dimensional $N=2$ gauge theory on ${\bf R}^4$ 
is realized in the type IIA superstring theory
by an Neveu-Schwarz fivebrane on ${\bf R}^4 \times \Sigma$ 
where $\Sigma$ is the Seiberg-Witten curve \cite{KlLeMaVaWa}.
(This fivebrane description of the gauge theory is more transparent in view 
of 11 dimensional M theory \cite{Wi}.)
The T-dual of this curved fivebrane configuration 
is obtained as
type IIB superstring theory compactified on a Calabi-Yau three-fold
which is a compact complex K\"ahler manifold of 
complex dimension three with vanishing first Chern class.
Here we should take 
this Calabi-Yau three-fold to be a form of $K3$ fibration
over ${\bf CP}^1$ with a certain limit 
which implies the decoupling of gravity.
The singularities of $K3$, where some two-cycles get shrinked,
are classified by the ADE singularity types
and the gauge group of four-dimensional theory corresponds to
these ADE singularities of $K3$.
From the point of view of four-dimensional theory,
this limiting Calabi-Yau three-fold is considered as 
a higher dimensional generalization of the auxiliary Seiberg-Witten curve 
and called Seiberg-Witten geometry.
For ADE type gauge groups, this Seiberg-Witten geometry may be a
more natural object than the curve 
since the curve depends on the representation of the gauge group,
furthermore,
there are the Seiberg-Witten geometries which are difficult to be reduced 
to the curve.
Surprisingly it has been shown that 
this Seiberg-Witten geometry of the form of ADE singularity fibration
over ${\bf CP}^1$ naturally appears
in the framework of 
the confining phase superpotential \cite{TeYa2,TeYa3,TeYa4}
despite that this method has no relation to $K3$ or Calabi-Yau manifold
at first sight.

Some extension to include matter hypermultiplets in
representations other than the fundamentals can be also considered as
the compactification on the Calabi-Yau three-fold \cite{Br,AgGr}.
In his approach, however, only massless matters 
have been treated and
the representation of matters are very restricted.
On the other hand, the technique of 
confining phase superpotential can be also applied to 
supersymmetric theories with matter hypermultiplets and 
be used to investigate wider class of 
the theory.
Indeed
we have succeeded in deriving previously unknown Seiberg-Witten geometries for
the $N=2$ theory with $E_6$ gauge group with
the massive fundamental hypermultiplets \cite{TeYa3}.
Moreover breaking the $E_6$ symmetry down to $SO(2 N_c)$ $(N_c \leq 5)$,
we derive the Seiberg-Witten geometry for $N=2$ $SO(2 N_c)$ theory with massive 
spinor and vector hypermultiplets \cite{TeYa4}.
In the massless limit, our $SO(10)$ result is in complete agreement with the
one obtained in \cite{AgGr}.
Breaking of $E_6$ to $SU(N_c)$ $(N_c \leq 6)$ is also considered in \cite{TeYa4},
and the Seiberg-Witten geometry for the $N=2$ $SU(N_c)$ theory
with antisymmetric matters have been obtained.
The singularity structure exhibited by the 
complex curve obtained by M-theory fivebrane \cite{LaLo,LaLoLo2} 
is realized in our result.
This is regarded as non trivial evidence for the validity of our results.

As we have described so far the 
four-dimensional $N=2$ supersymmetric gauge field theories 
have very rich physical content and 
their relation to 
the superstring theory renders them further interesting 
subjects to study.
In particular the Seiberg-Witten geometry plays a very important role 
to control the dynamics of $N=2$ theories.
Our aim in this thesis is to understand 
the Seiberg-Witten geometry for various $N=2$ supersymmetric theories
in the systematic way.
In particular,
we study the Seiberg-Witten curve and Seiberg-Witten geometry of the $N=2$ supersymmetric theory
using the confining phase superpotential.

The organization of this thesis is as follows.
In chapter two, 
we review the exact description of the 
low-energy effective theory of 
the Coulomb phase of four-dimensional $N=2$ supersymmetric gauge theory
in terms of the Seiberg-Witten curve or Seiberg-Witten geometry.
In chapter three, 
we derive the Seiberg-Witten curves 
of $N=2$ supersymmetric gauge theories
by means of the $N=1$ confining phase superpotential.
In chapter four, 
we apply the confining phase superpotential method
to the $N=1$ supersymmetric pure Yang-Mills theory
with an adjoint matter with classical or ADE gauge groups. 
The results can be used to derive the Seiberg-Witten curves
for $N=2$ supersymmetric pure Yang-Mills theory 
with classical or ADE gauge groups
in the form of a foliation over ${\bf CP}^1$.
Transferring the critical points in the $N=2$ Coulomb phase 
to the $N=1$ theories we find non-trivial $N=1$ SCFT with 
the adjoint matter field governed by a superpotential.
In chapter five, 
using the confining phase superpotential we determine
the curves describing 
the Coulomb phase of $N=2$ supersymmetric gauge theories
with matter multiplets.
For $N=2$ supersymmetric QCD with classical gauge groups,
our results recover the known curves.
We also obtain previously unknown Seiberg-Witten geometry 
for four-dimensional $N=2$ gauge theory
with gauge group $E_6$ with massive fundamental hypermultiplets.
By considering the gauge symmetry breaking in this $E_6$ gauge theory,
we also obtain Seiberg-Witten geometries
for $N=2$ gauge theory 
with $SO(2N_c)$ $(N_c \leq 5)$ with massive spinor and vector 
hypermultiplets.
In a similar way the Seiberg-Witten geometry is determined for $N=2$ $SU(N_c)$
$(N_c \leq 6)$ gauge theory with massive antisymmetric and fundamental 
hypermultiplets. 
Finally, chapter six is devoted to our conclusions. 


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Seiberg-Witten Geometry} \label{SWgeo}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this chapter we review the exact description of the 
low-energy effective theory of 
the Coulomb phase of four-dimensional $N=2$ supersymmetric gauge theory
in terms of the Seiberg-Witten curve or the Seiberg-Witten geometry.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Seiberg-Witten curve}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us consider $N=2$ supersymmetric pure Yang-Mills theory
with the gauge group $G$.
This theory contains only an $N=2$ vectormultiplet 
in the adjoint representation of $G$ 
which consists of
an $N=1$ vector multiplet $W_{\alpha}$ and an $N=1$ chiral multiplet $\p$.
The scalar field $\phi$ belonging to $\p$ has the potential
\beq
V(\phi)\ =\ {\rm Tr}[\phi,\phi^\dagger]^2.
\eeq
This is minimized by taking $\phi=\sum \phi_i H^i$,
where $H^i$ belongs to the Cartan subalgebra,
and thus the classical vacua of 
this theory are degenerate and parametrized by the 
Casimirs built out from $\phi_i$ after being 
divided by the gauge transformation.
The set of Casimirs is a gauge invariant coordinate of the 
space of inequivalent 
vacua which is called the moduli space.

The generic classical vacua of 
the theory have unbroken $U(1)^{r}$ 
gauge groups and are called the Coulomb phase where $r={{\rm rank} \, G}$.
At the singularity of the classical moduli space of vacua,
there appears a non-Abelian unbroken gauge group which implies that
massless gauge bosons exist there.
For the Abelian gauge group case,
it is known from supersymmetry that the general low energy effective
Lagrangian up to two derivatives
is completely determined by a holomorphic prepotential ${\cal F}$ and must be
of the form
\beq
{\cal L}= {1\over4\pi}{\rm Im}\,\left[\, \int \!d^4\theta\,K(\p,\bar \p)
+ \int \!d^2\theta\, \left(\frac{1}{2}\sum
\tau(\p)W^{\alpha }W_{\alpha }\right)\right],
\label{effL}
\eeq
in the $N=1$ superfield language.
Here, $\Phi= \sum_{i=1}^r \p_i \,H^i$, and
\beq
K(\p,\bar \p)\ =\ {\partial {\cal F}(\p)\over\partial \p_i}\bar \p_i
\eeq
is the K\"ahler potential which prescribes a supersymmetric non-linear
$\sigma$-model for the field $\p$, and
\beq
\tau(\p)_{ij}\ =\ 
{\partial^2 {\cal F}(\p)\over\partial\p_i \partial\p_j}\ .
\eeq
This Lagrangian (\ref{effL}) contains
the terms ${\rm Im}(\tau_{ij})  F_i\cdot F_j
+ {\rm Re}(\tau_{ij})\, F_i \cdot \tilde{ F}_j$, from which we see that
\beq
\tau(\phi)\ \equiv\ {\theta(\phi)\over 2 \pi}+{4 \pi i\over {g}^2(\phi)}
\label{taudef}
\eeq
represents the complexified effective gauge coupling.
Classically, ${\cal F}(\p)=\frac{1}{2} \tau_0 {\rm Tr} \, \p^2$, 
where $\tau_0$ is the bare coupling
constant.

How is this classical moduli space of vacua modified by
the quantum effects?
Seiberg and Witten have proposed 
for the $SU(2)$ pure Yang-Mills theory
on the basis of
holomorphy and duality that 
the quantum moduli space of vacua is still parametrized by
the Casimirs, but all the vacua have only $U(1)$ \cite{SeWi1}.
Although there are still singularities in the moduli space,
the singularities in the quantum moduli space 
correspond to the 
appearance of the massless monopoles or dyons,
not to the massless gauge bosons.
Moreover it has been shown that 
the prepotential ${\cal F}$, in particular the coupling constant $\tau$, 
and also the mass of the BPS saturated state
are computed from the geometric data of 
the auxiliary complex curve, called the Seiberg-Witten curve, and a certain 
meromorphic one-form over it,
called the Seiberg-Witten form $\lambda_{SW}$.
Here the Seiberg-Witten curve is 
determined as the function over the moduli space of vacua.
The Seiberg-Witten type solutions for other $N=2$ theories with
larger gauge groups and matters have been obtained in \cite{SeWi2}-\cite{Ha}.

As an illustration of the basic idea of Seiberg-Witten,
we briefly review the case of the 
$N=2$ supersymmetric $SU(2)$ pure Yang-Mills theory.
In this case, there are two singularities corresponding to 
the appearance of 
the massless monopole or dyon at $u=\pm 2 \La^2$,
where $u=\frac{1}{2} {\rm Tr} \phi^2$ and $\La$ is the scale of the theory. 
Note that the classical singularity at the origin $u=0$ disappears.
The Seiberg-Witten curve is a torus and given by
\beq
y^2=\left( x^2-u \right)^2-4 \La^4=
\left(x^2-u +2 \La^2\right) \left(x^2-u-2 \La^2\right),
\label{su2SW}
\eeq
which is degenerate as
$y^2=x^2 (x^2\mp 4 \La^2)$ 
at the singular point $u=\pm 2 \La^2$.
The Seiberg-Witten one-form takes the form
\beq
\lm_{SW}\ =\ {1\over\sqrt2\pi}x^2{dx\over y(x,u)}.
\label{lamdef}
\eeq
The mass of the BPS state 
which has electric charge $p$ and magnetic charge $q$ 
is given in terms of the integral of $\lm_{SW}$ over
the canonical basis homology cycles of the
torus  $\alpha, \beta$ as  
\beq
m =| p a  +q a_D|,
\eeq
where the period integrals
\beqa
a(u)&=&\oint_{\alpha} \lm_{SW},  \\
a_D(u)&=& \oint_{\beta} \lm_{SW}
\eeqa
are associated with the chiral superfields belonging to the 
electric $U(1)$ multiplet and its dual magnetic $U(1)$ multiplet respectively.
The coupling constant of the low-energy theory is 
identified with the period matrix of this torus which is written as
\beq
\tau={\pa a_D(u) \over \pa a(u)}
\eeq
which has the required properties ${\rm Im}(\tau) >0$.

The Seiberg-Witten curves for the other classical gauge groups
are also proposed and verified by the one instanton calculation.
One for the $N=2$ $SU(N_c)$ gauge theory is 
\beq
y^2= P(x)^2 -4 A(x),
\label{sucurve}
\eeq
where $P(x)=\left \langle {\rm det} \left ( x- \p \right ) \right \rangle$
is the characteristic equation of $\p$ which is chosen as 
the $N_c \times N_c$ matrix of the fundamental representation.
For the $N=2$ $SU(N_c)$ pure Yang-Mills theory, $A(x)\equiv \Lm^{2 N_c}$ 
\cite{ArFa,KlLeYaTh} and
for the $N=2$ $SU(N_c)$ theory 
with $N_f$ fundamental flavors (QCD) \cite{HaOz,ArPlSh}, 
\beq
A(x) \equiv \Lm^{2 N_c-N_f} \,
{\rm det}_{N_f} \left (  x+m \right ),
\eeq
where $m$ is the $N_f \times N_f$ mass matrix of the fundamental flavors.

The Seiberg-Witten curves for 
$N=2$ $SO(2 N_c)$ gauge theory read \cite{BrLa} 
\beq
y^2= P(x)^2 -4 x^2 A(x),
\label{soecurve}
\eeq
where $P(x)=\left \langle {\rm det} \left ( x- \p \right ) \right \rangle=P(-x)$
is the characteristic equation of $\p$ which is chosen as 
the $2 N_c \times 2 N_c$ matrix of the fundamental representation.
Here for the pure Yang-Mills case $A(x)\equiv \Lm^{4( N_c-1)}$ and
for the QCD case
\beq
A(x) \equiv \Lm^{4( N_c-1)-2 N_f} \,
{\rm det}_{2 N_f} \left ( x+m \right ) =A(-x).
\eeq

For $SO(2 N_c+1 )$ gauge groups, the curves are 
\beq
y^2=\left( \frac{1}{x} P(x) \right )^2-4 x^2 A(x) ,
\label{soocurve}
\eeq
with $A(x)\equiv \Lm^{2( 2 N_c-1)}$ 
for the pure Yang-Mills theory \cite{DaSu} and
\beq
A(x) \equiv \Lm^{2( 2 N_c-1- N_f)} \,
{\rm det}_{2 N_f} \left ( x+m \right ) =A(-x),
\eeq
for QCD \cite{ArSh,Ha}.

The curves for $Sp(2 N_c)$ theory are slightly 
different from the ones for the other gauge groups.
They are given by
\beq
x^2 y^2 = \left( x^2 P(x) +2 B(x) \right)^2-4 A(x),
\label{spcurve1}
\eeq
with $B=\Lm^{2 N_c+2}$ and $A(x)\equiv \Lm^{2( 2 N_c+2)}$ 
for the pure Yang-Mills theory \cite{ArSh}, whereas
\beq
B(x)=\Lm^{2 N_c+2- N_f} {\rm Pf} m
\eeq
and 
\beq
A(x) \equiv \Lm^{ 2(2 N_c+2- N_f)} \,
{\rm det}_{2 N_f} \left ( x+m \right ) =A(-x)
\eeq
for QCD \cite{ArSh}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{Affine Toda curve}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

There is an interesting connection between 
the four-dimensional $N=2$ pure Yang-Mills theory 
and the integrable systems.
The connection is that the Seiberg-Witten curve 
for the $N=2$ pure Yang-Mills theory with the gauge group $G$ 
is identified with the spectral curve for the periodic Toda theory 
for the group $G$ \cite{MaWa}.
Moreover the Seiberg-Witten form and relevant one-cycles 
can be also read from the spectral curve.
What we want to emphasize here is that
this correspondence is true for the arbitrary simple groups,
especially for the exceptional groups.
However, as we will see just below, 
for the exceptional gauge group case 
the Seiberg-Witten curve is not of hyperelliptic type.
Introducing the characteristic polynomial 
in $x$ of order ${\rm dim}\, {\cal R}$ 
\beq
P_{\cal R}(x,u_k)={\rm det} (x-\Phi_{\cal R}),
\label{charapol1}
\eeq
where ${\cal R}$ is an arbitrary representation of $G$,
the spectral curve is given by
\beq
\tilde P_{\cal R}(x,z,u_k) \equiv
P_{\cal R} \left( x,u_k+\delta_{k,r} \left( z+\frac{\mu}{z}\right) \right) =0,
\label{spectcurve}
\eeq
which has a form of a foliation over ${\bf CP}^1$.
Here $\Phi_{\cal R}$ is a representation 
matrix of ${\cal R}$ and $u_k$ are Casimirs built out of $\Phi_{\cal R}$.
If we choose ${\cal R}$ as a large representation of $G$,
however, the genus of the curve is larger than the rank of $G$.
This means that we should suitably choose 
$2 r$ cycles 
to define $a$ and $a_D$ since the unbroken gauge group is $U(1)^r$.
In particular, for the exceptional gauge groups,
the ${\rm dim}\, {\cal R}$ is always much larger than $r$.
Although this problem is solved just in terms of the integrable system,
it seems somewhat unnatural and we expect that 
there exists a more transparent formulation.
Indeed the generalization of the Seiberg-Witten curve to
the complex dimension three manifold,
which is called Seiberg-Witten geometry, is motivated by the string theory
and is recognized to provide us with a desired formulation.
This description is equivalent to the one using the curve for 
the theory considered in this section and more interestingly 
available to the $N=2$ exceptional gauge theory with the matter flavors and
$N=2$ classical gauge theory with matter flavors in
the non fundamental representation.
They have not been described in term of the curve so far.
We will discuss this generalization in the following sections.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Seiberg-Witten geometry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

To see how the Seiberg-Witten geometry arises from the 
string theory,
we first consider the $E_8 \times E_8$ heterotic string theory on 
$K3 \times T^2$.
In the low energy region,
this theory becomes effectively four dimensional $N=2$ supersymmetric theory
with possibly non-Abelian gauge bosons and gravitons.
To obtain the four dimensional non-Abelian gauge field theory without gravity,
we should take the limit $\A' \rightarrow 0$ 
and simultaneously the weak string coupling limit as
\beq
\frac{1}{g^2_{het}}=-{b } \, {\log} \left( \sqrt{\A'} \La \right)
\rightarrow \infty,
\label{weaks}
\eeq
where $g_{het}$ is a coupling constant of the heterotic string theory,
which is considered as the four dimensional gauge coupling at
the plank scale ${\A'}^{-{1 \over 2}}$ and
$b$ is the coefficient of the one loop beta function of the gauge field.
The condition (\ref{weaks}) is required to make
the dynamical scale of the non-Abelian gauge theory
$\La$ fixed at a finite value.
Although in this setting we can obtain the 
four dimensional $N=2$ supersymmetric non-Abelian gauge field theory,
it is still difficult to compute the prepotential ${\cal F}$
of the Coulomb phase of the theory
if the coupling constant $g_{het}$ (more precisely 
$ \La$) is not small.

Fortunately there is a duality between
the heterotic string theory on 
$K3 \times T^2$ and 
the type IIA string theory on a Calabi-Yau three-fold $X_3$
\cite{KaVa,FeHaStVa}.
What is important is that
the type IIA dilaton, whose expectation value is the
type IIA string coupling,
is in hypermultiplet.
Therefore in the type IIA side 
the exact moduli space of the Coulomb phase can be determined from 
classical computation.
Here we have used the fact that 
the $N=2$ supersymmetry prevents couplings 
between neutral vector and hypermultiplets 
in the low energy effective action \cite{ArPlSe}.
Note that the heterotic string coupling constant is converted to 
the geometrical data, K\"ahler structure moduli.
Since the K\"ahler structure moduli is corrected by 
the string world sheet instantons,
the type IIA description is not sufficiently simple to deal with.
Remember here that the mirror symmetry
maps the type IIA superstring on $X_3$ to a type IIB superstring
on the mirror Calabi-Yau three-fold $\tilde{X}_3$ with
interchanging the K\"ahler structure moduli and
the complex structure moduli.
Thus in this type IIB description 
classical string sigma model answer 
for the original vector moduli space 
is already the full exact result.
This implies that 
the Seiberg-Witten geometry for the gauge field theory 
is identified with the compactification manifold $\tilde{X}_3$.
The Calabi-Yau three-fold has the canonical holomorphic three-form $\Omega$
and $a,a_D$ are obtained as the integration of $\Omega$ over
the three-cycles ${\Gamma_{\A_I}},\ ,{\Gamma_{\B^J}}$, $I,J=1,\dots,
h_{11}(\tilde X_3)+1$, which span a integral symplectic basis of $H_3$,
with the $\A$-type of cycles being dual to the $\B$-type of cycles,
\beq
a_i\ = \int_{\Gamma_{\A_i}}\! \Omega , \qquad
{a_D}_j\ = \int_{\Gamma_{\B^j}} \! \Omega.
\label{CYperiods}
\eeq
Here $i,j$ runs from one to
the rank of the gauge group of the heterotic string theory.
The other cycles is not relevant in the field theory limit 
since its integration diverges in the limit ${\A'} \rightarrow 0$.

To obtain the Seiberg-Witten geometry, we should take the limit $\A' \rightarrow 0$ of
$\tilde{X}_3$.
To this end, we introduce the asymptotically local Euclidean space
(ALE space) $ W_{ADE}(x_i)=0$ with ADE singularity at the origin.
Here the polynomial $W_{ADE}(x_i)$ is given as follows
\beqa
W_{A_r}(x_1,x_2,x_3; v)
&=&x_1^{r+1}+x_2 x_3 +v_2{x_1}^{r-1}+v_3 x_1^{r-2}+\cdots+v_r x_1+v_{r+1}, \\
W_{D_r}(x_1,x_2,x_3; v)
&=&{x_1}^{r-1}+x_1{x_2}^2-{x_3}^2 \CR
&&\hspace{.4cm}+v_2{x_1}^{r-2}+v_4x_1^{r-3}+
\cdots+v_{2(r-2)} x_1+v_{2(r-1)}+v_r x_2, \\
W_{E_6}(x_1,x_2,x_3;w)
&=&x_1^4+x_2^3+x_3^2\CR
&&\hspace{.4cm}+w_2\, x_1^2 x_2 +w_5\, x_1x_2
+w_6\, x_1^2+w_8\, x_2+w_9\, x_1+w_{12}, \\
W_{E_7}(x_1,x_2,x_3;w) 
&=& {x_1}^3+x_1{x_2}^3+x_3^2 - w_2 x_1^2 x_2-w_6 x_1^2\CR
&&\hspace{.4cm}-w_8
x_1 x_2  -w_{10} x_2^2 -w_{12} x_1-w_{14} x_2-w_{18}, \\
W_{E_8}(x_1,x_2,x_3;w) 
&=& {x_1}^3+{x_2}^5+x_3^2 - w_2 x_1 x_2^3-w_8 x_1 x_2^2\CR
&&\hspace{.4cm}-w_{12} x_2^3 -w_{14} x_1 x_2 -w_{18} x_2^2-w_{24}
x_2-w_{30}, 
\label{ALEdef1}
\eeqa
where $v_k$ and $w_k$ correspond to the degree $k$ Casimirs 
which resolve the singularity at the origin.
Then the mirror Calabi-Yau three-fold $\tilde{X}_3$ 
for the $N=2$ pure Yang-Mills theory is written as
\beq
W_{\tilde X_3}(x_j,z;w_k) =
\epsilon\Big(z+{\Lambda^{2h}\over z}+W_{ADE}(x_j,w_k)
\Big)+{\it o}(\epsilon^2) = 0 ,
\label{ALEfibr1}
\eeq
where $\epsilon={\A'}^{\frac{h}{2}}$ and 
the gauge group is represented by the ADE singularity.
Here $h$ is the dual Coxeter number for the ADE Lie algebra.
Therefore the Seiberg-Witten geometry for the $N=2$ pure Yang-Mills theory
is obtained as
\beq
z+{\Lambda^{2h}\over z}+W_{ADE}(x_j,w_k)=0.
\label{ALEfibr2}
\eeq

It is relatively easier for
the $SU(N_c)$ gauge group case 
to see the equivalence of the description using this Seiberg-Witten geometry 
and the Seiberg-Witten curve \cite{KlLeMaVaWa}. From the fact that
the variables $x_2,x_3$ are both quadratic in $W_{A_{N_c-1}}$,
it was shown that 
these variables 
can be ''integrated out'' from $W_{A_{N_c-1}}$ \cite{KlLeMaVaWa}.
Then changing the coordinate $y=-2 z+P$,
we see that the curve (\ref{sucurve}) is equivalent to the 
corresponding Seiberg-Witten geometry.
For $SO(2 N_c)$ case almost the same procedure can be applied,
while for $E_n$ case the Seiberg-Witten geometry (\ref{ALEfibr2}) does not
resemble to the curve (\ref{spectcurve}).
This problem is solved by 
finding a certain 
transformation of (\ref{ALEfibr2}) to get(\ref{spectcurve})
\cite{LW,WY}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Confining Phase Superpotential} \label{CPS}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


In this chapter, we will apply 
the confining phase superpotential technique to the 
$N=2$ supersymmetric pure Yang-Mills theories.
A simplest example of the application of this is
the $N=2$ $SU(2)$ gauge theory \cite{InSe}.
We will see that only for this case 
the confining phase superpotential technique is exact and for other cases
this technique is applicable under a mild assumption.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Simplest example: $SU(2)$ gauge theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

For the $SU(2)$ gauge group, we take a tree-level superpotential
$ W= m u$, where $u={1 \over 2} \Tr \, \p^2$ and
$m$ is a mass parameter of the adjoint chiral superfield $\p$.
If $m$ is very small, 
we can consider this theory as the 
$N=2$ supersymmetric $SU(2)$ pure Yang-Mills theory perturbed 
by the $N=1$ small mass term $W$.
The exact low-energy theory of the $N=2$ theory 
near the massless monopole singularity
has a $U(1)$ vector multiplet and a monopole hypermultiplet
with a superpotential determined by the requirement of $N=2$ supersymmetry
\beq
W_{N=2}^{eff}=A_D \tilde{M} M,
\eeq
where $A_D$ is the dual $U(1)$ vector multiplet and
the $\tilde{M}, M$ are monopole hypermultiplet \cite{SeWi1}.
Note that the bosonic part of $A_D$ is $a_D$ and
its VEV determines the mass of the monopole.
Thus the equation of motion, 
which should be
satisfied for a supersymmetric ground state, 
of the theory perturbed by $W$ becomes
\beqa
0 &=& \frac{\pa W^{eff}}{\pa M} =A_D \tilde{M}, \label{eqm1} \\
0 &=& \frac{\pa W^{eff}}{\pa \tilde{M}} = A_D M, \label{eqm2} \\
0 &=& \frac{\pa W^{eff}}{\pa u} =
 {\pa A_D \over \pa u} \tilde{M} M +m, \label{eqm3}
\eeqa
where $W^{eff}=W_{N=2}^{eff} +W$.
The equations (\ref{eqm1}), (\ref{eqm2}) 
may be reduced to $0 =\langle A \rangle=\langle A_{D} \rangle$, which means
that only the $N=2$ vacuum where the monopole becomes massless remains
as $N=1$ vacuum. From the equation (\ref{eqm2}), 
we see that there is a non-zero monopole 
condensation $\la \tilde{M} M \ra = -m / {\pa A_D \over \pa u}$.
The non zero monopole condensation is regarded as the source of 
confinement.

On the other hand, if mass $m$ is very large, then we can integrate out 
the adjoint chiral superfield $\p$ and low-energy effective theory
becomes the $N=1$ supersymmetric $SU(2)$ pure Yang-Mills theory
which is believed to be in the confining phase.
The relation between the high-energy scale $\La$ and the low-energy 
scale $\La_L$ is determined by matching the scale at the 
adjoint mass $m$ as
\beq
\La^{2 \cdot 2}=
{\La_L}^{3 \cdot 2} 
(m)^{-2}.
\eeq
Since the gaugino condensation dynamically generates the 
superpotential in the $N=1$ $SU(2)$ theory
the low-energy effective superpotential takes the form 
\beq
W_L= \pm 2 m \La^{2}.
\label{wlsu2}
\eeq
Although this effective superpotential is evaluated in the region of
large $m$, it is shown that (\ref{wlsu2}) is exact for all values of $m$
\cite{InSe} by virtue of holomorphy, symmetry and asymptotic
dependence on the parameter of the theory \cite{Se1}.
Thus the relation $\la u \ra=\pa W_L/ \pa m=
\pm 2 \La^{2}$ holds exactly.
Finally taking the $N=2$ limit $m \rightarrow 0$,
we obtain the correct singularities of the moduli space of 
the $N=2$ supersymmetric $SU(2)$ Yang-Mills theory at $u=\pm 2 \La^{2}$.

The $N=2$ supersymmetric $SU(2)$ QCD, 
which has fundamental hypermultiplets,
has been studied in an analogous way 
and shown to yields the known singularity 
structure of the curve \cite{ElFoGiRa,ElFoGiRa2}. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Outline of 
confining phase superpotential}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this section, we generalize the above method to the case of 
other gauge groups.
Let us consider the low energy theory for a generic vacuum in the Coulomb
phase of $N=2$ supersymmetric gauge theory with the gauge group $G$.
Generically the low energy behavior of this theory is described by
$N=2$ supersymmetric $U(1)^{{\rm rank} G}$ pure Yang-Mills theory. 
As in the previous section, 
we add a tree level superpotential 
$W=\sum_{k} g_k u_k$, where $u_k$ are 
the Casimirs built out of the adjoint chiral superfield $\p$,
to this $N=2$ theory.
According to 
the technique called the confining phase superpotential \cite{ElFoGiInRa},
we concentrate on investigating
the vicinity of a singular point of the $N=2$ moduli space of vacua 
where a single monopole or dyon becomes massless.
The low energy $N=1$ theory has a superpotential which is approximately 
given by 
\beq
W^{eff}=A_D(u_k) \tilde{M} M +\sum_{k} g_k u_k,
\eeq
as in the $SU(2)$ case.
The equation of motion
of this perturbed theory becomes
\beqa
A_D(u_k) &=& 0, \\
{\pa A_D(u_k) \over \pa u_k} \tilde{M} M &=&  - g_k.
\label{vaccond3}
\eeqa
It is important in this equation that only the $N=2$ vacua with,
at least, a single massless monopole or dyon remain as the $N=1$ vacua.
In these $N=1$ vacua, the monopole or dyon can condense so as to 
confine a single $U(1)$ photon.

Conversely, 
we can start with a 
microscopic $N=1$ 
gauge theory which is obtained from an $N=2$ gauge
theory perturbed by $W$.
If we can calculate the low energy effective superpotential
as the function of the scale $\La$ of the original theory and $g_i$,
then by taking the $N=2$ limit $g_i \rightarrow 0$ 
we can find the location of the 
singularity in the moduli space of the $N=2$ theory.

Let us consider
$N=2$ $SU(3)$ Yang-Mills theory as an illustration of the method.  
Perturbing
by $W=m u+g v$, where $u={1 \over 2} \Tr \Phi^2$ and $v={1\over 3} \Tr
\Phi^3$, leads to classical vacua with $\Phi =0$, in which $SU(3)$ is
unbroken, and 
\beq
\Phi =
{\rm diag}\left({m \over g},{m \over g}, -2 {m \over g} \right),
\label{pclsu3}
\eeq
in which there is a
classically unbroken $SU(2)\times U(1)$.  
We focus on the vacuum with
unbroken $SU(2)\times U(1)$ gauge group.
In the semiclassical approximation,
the low-energy theory for this vacuum consists of  
the $N=1$ $SU(2)$ Yang-Mills theory with a superpotential
$\tilde{W}$ 
and a decoupled $N=2$ $U(1)$ Yang-Mills theory.
(This $U(1)$ theory is free and 
we can ignore it in the following consideration.)
The scale $\tilde{\Lambda}$ of this $SU(2)$ theory is related to 
the high-energy $SU(3)$ scale $\Lambda$ by
\beq
\tilde{\La}^{2 \cdot 2}=
\left( \frac{3 m}{g} \right)^{-2} \La^{2 \cdot 3},
\eeq
which is obtained by matching the $SU(3)$ scale to the $SU(2)$ scale at the
scale $(m /g)-( -2 m / g)=(3 m / g)$ 
of the $W$ bosons which become massive
by the Higgs effect.
The superpotential $\tilde{W}$ may be evaluated as
\beq
\tilde{W}=\frac{1}{2} W''(m/g) \,\, \Tr \p_{SU(2)}^2+
\frac{1}{3 \cdot 2} W'''(m/g) \,\, \Tr \p_{SU(2)}^3
= \frac{3 m}{2} \Tr \p_{SU(2)}^2+\frac{g}{3} \Tr \p_{SU(2)}^3,
\eeq
where $W(x)=\frac{m}{2} x^2+\frac{g}{3} x^3$ 
and $\p_{SU(2)}$ is an unbroken $SU(2)$ part of $\p$.
Note that in $\tilde{W}$ we suppress 
the terms which are
not relevant to the $SU(2)$ theory.
Therefore the adjoint chiral superfield $\p_{SU(2)}$
has a mass $3 m$ and can be integrated out.
We are then left with an $N=1$ $SU(2)$ pure Yang-Mills theory
with a scale $\Lambda _L$ which is related to the scale $\Lambda$ by
\beq
\Lambda_L^{3 \cdot 2}=(3m)^2 \tilde{\La}^{2 \cdot 2}=g^2 \Lambda ^6.
\eeq
Since the gaugino condensation dynamically generates the 
superpotential in the $N=1$ $SU(2)$ pure Yang-Mills theory
the low-energy effective superpotential finally takes the form 
\beq
W_L={m^3\over g^2} \pm 2 \Lambda_L^3={m^3\over g^2} \pm 2 g \Lambda ^3,
\label{wli}
\eeq
where the first term is the tree level term $W$ evaluated for
$\Phi={\rm diag} (m/g,m/g,-2m/g)$.
We note that to obtain (\ref{wli}) we should integrate out
all the fields in the original theory then no dynamical fields are 
remained.

The superpotential
(\ref{wli}) is certainly correct in the limit $m \gg \Lambda$ and $m/g\gg
\Lambda$, where the original theory is broken to our low energy theory
at a very high scale.  
In the case of (\ref{wli}), however, we can not directly
rule out additive corrections of the form $W_{\Delta}=\sum
_{n=1}^\infty a_n(m^3/g^2)(g\Lambda/m)^{6n}$.  
We will simply assume that (\ref{wli}) is exact for all
values of the parameters \cite{ElFoGiInRa}.  
This assumption is referred to as the
assumption of vanishing $W_{\Delta}$ \cite{In}.  
We will see in the following that
this assumption is correct at least for the theory we have investigated.
However there is a subtle point concerned with the 
choice of the basis of the Casimirs of the gauge group.
This point is discussed later.
It will be seen also that the statement that
$W_{\Delta}=0$ seems to reflect the absence of mixing of various 
classical vacua like $\theta$ vacua in QCD.

Once assuming (\ref{wli}) is exact,
we obtain
\beqa
\langle u \rangle&=&\frac{\partial W_L }{\partial m}=
3\left({m\over g}\right), \\
\langle v \rangle&=&\frac{\partial  W_L}{\partial g}=
-2\left({m\over g}\right) ^3\pm 2\Lambda ^3.
\label{uvvev}
\eeqa
In the $N=2$ limit $m,g \rightarrow 0$,
these two vacua of the perturbed theory must lie on the 
singularities of the moduli space of 
the Coulomb phase of the $N=2$ theory
since in this limit the vacuum condition (\ref{vaccond3}) is valid.
Therefore the vacua (\ref{uvvev}) must 
parameterize 
the singularities of the Seiberg-Witten curve 
$y^2=(x^3-xu-v)^2-4\Lambda ^6$ for the $N=2$ $SU(3)$ pure Yang-Mills 
theory.
The singularities of the curve are indicated by the discriminant locus
\beq
\Delta_{SU(3)}=4 u^3-27 v^2 -108 \La^6 \mp 108 v \La^3=0.
\eeq
Indeed, if we eliminate $m/g$ from (\ref{uvvev}) then 
we obtain $ \Delta_{SU(3)}=0$.
We have thus confirmed that the proposed Seiberg-Witten curve
for $SU(3)$ pure Yang-Mills theory is correct using the
confining phase superpotential.
Note that the parameter of the singularities of the $N=2$ moduli space
corresponds to the ratio $m/g$. 

In the following chapters, we will apply this confining phase superpotential technique
to various $N=2$ supersymmetric gauge theories 
in order to verify the proposed Seiberg-Witten geometries or
derive the new Seiberg-Witten geometries if they are unknown.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{$N=2$ Pure Yang-Mills Theory} \label{YM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Classical gauge groups}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now we apply the confining phase superpotential method to 
$N=2$ supersymmetric pure Yang-Mills theories with
classical gauge groups.

First we begin with 
the $SU(N_c)$ gauge theory \cite{ElFoGiInRa}.
The gauge symmetry breaks down to $U(1)^{N_c-1}$ in the 
Coulomb phase of $N=2$ $SU(N_c)$ Yang-Mills theories.
Near the singularity of a single massless dyon 
we have a photon coupled to the light dyon hypermultiplet
while the photons for the rest $U(1)^{N_c-2}$ factors remain free.
We now perturb the theory by adding a tree-level superpotential
\beq
W=\sum_{n=1}^{N_c} g_n u_n, 
\hskip10mm  u_n=\frac{1}{n} {\rm Tr}\,  \p^n ,
\label{r1}
\eeq
where $\p$ is the adjoint $N=1$ superfield in the $N=2$ vectormultiplet and
$g_1$ is an auxiliary field implementing ${\rm Tr}\, \p =0$.
In view of the macroscopic theory, we see that under the perturbation 
by (\ref{r1}) only the $N=2$ singular loci survive as the 
$N=1$ vacua where a single photon is confined
and the $U(1)^{N_c-2}$ factors decouple.

The result should be directly recovered when we start with the 
microscopic $N=1$ $SU(N_c)$ 
gauge theory which is obtained from $N=2$ $SU(N_c)$ Yang-Mills
theory perturbed by (\ref{r1}).
For this we study the vacuum with unbroken $SU(2) \times U(1)^{N_c-2}$.
The classical vacua of the theory are determined by the equation of motion
$W'(\p)=\sum_{i=1}^{N_c} g_i \p^{i-1}=0$. Then the roots $a_i$ of 
\beq
W'(x)=\sum_{i=1}^{N_c} g_i x^{i-1}=
g_{N_c} \prod_{i=1}^{N_c-1} (x-a_i)
\label{suneigen}
\eeq
give the eigenvalues of $\p$. In particular the unbroken 
$SU(2) \times U(1)^{N_c-2}$ vacuum is described by 
\beq
\Phi ={\rm diag} (a_1, a_1, a_2, a_3, \cdots , a_{N_c-1}).
\eeq
In the low-energy limit the adjoint superfield for $SU(2)$ becomes massive and
will be decoupled. We are then left with an $N=1$ $SU(2)$ Yang-Mills theory 
which is in the confining phase and the
photon multiplets for $U(1)^{N_c-2}$ are decoupled.

The relation between the high-energy $SU(N_c)$ scale $\La$ and the low-energy 
$SU(2)$ scale $\La_L$ is determined by first matching at the
scale of $SU(N_c)/SU(2)$ $W$ bosons and then by matching at the $SU(2)$
adjoint mass $M_{\rm ad}$. One finds \cite{KuScSe}, \cite{ElFoGiInRa}
\beq
\La^{2 N_c}=
{\La_L}^{3 \cdot 2} 
\left ( \prod_{i=2}^{N_c-1} (a_1-a_i) \right )^2
(M_{\rm ad})^{-2}.
\label{matching}
\eeq
To compute $M_{\rm ad}$ we decompose 
\beq
\Phi=  \Phi_{cl} + \delta \Phi + \delta \tilde{\Phi},
\label{decompose}
\eeq
where 
$ \delta \Phi $ denotes the fluctuation along the unbroken $SU(2)$ 
direction and $ \delta \tilde{\Phi}$ along the other directions.
Substituting this into $W$ we have
\beqa
W & = & 
W_{cl} + \sum_{i=2}^{N_c} g_i \frac{i-1}{2}
\; {\rm Tr}\, ( \delta \Phi^2  \Phi_{cl}^{i-2} )+\cdots \CR
& =& W_{cl} + \frac{1}{2} W''(a_1) \; {\rm Tr}\, \D \Phi^2+\cdots \CR
& =& W_{cl} + \frac{1}{2} \; g_{N_c} \prod^{N_c}_{i=2} (a_1-a_i)\;
{\rm Tr}\, \D \Phi^2 +\cdots ,
\eeqa
where $[ \D \Phi,  \Phi_{cl} ]=0$ has been used and $W_{cl}$ is the 
tree-level superpotential evaluated in the classical vacuum.
Hence, $M_{\rm ad}=g_{N_c} \prod^{N_c-1}_{i=2} (a_1-a_i)$ and the relation
(\ref{matching}) reduces to
\beq
{\La_L}^6 = g_{N_c}^2 \La^{2 N_c}.
\eeq
Since the gaugino condensation dynamically generates the 
superpotential in the $N=1$ $SU(2)$ theory
the low-energy effective superpotential finally takes the form 
\cite{ElFoGiInRa}
\beq
W_L= W_{cl} \pm  2 {\La_L}^3 = W_{cl} \pm 2 g_{N_c} \La^{N_c}.
\label{wl}
\eeq

We simply assume here that the superpotential (\ref{wl}) is exact for any 
values of the parameters. (This is equivalent to assume 
$W_{\Delta}=0$ \cite{In}, \cite{ElFoGiInRa}.) From (\ref{wl}) we obtain
\beq
\langle u_n \rangle = {\partial W_L \over \partial g_n}
=u_n^{cl}(g) \pm 2 \La^{N_c} \delta_{n, N_c} 
\label{qvacua}
\eeq
with $u_n^{cl}$ being a classical value of $u_n$. As we argued above these 
vacua should correspond to the singular loci of $N=2$ massless dyons. This
can be easily confirmed by plugging (\ref{qvacua}) in the $N=2$ $SU(N_c)$
curve \cite{KlLeYaTh}, \cite{ArFa}
\beq
y^2 = {\left \langle {\rm det} (x-\Phi) \right \rangle}^2- 4 \La^{2 N_c} 
= \left( x^{N_c}- \sum_{i=2}^{N_c} \bra s_i \ket x^{N_c-i} \right)^2
- 4 \La^{2 N_c},
\eeq
where
\beq
ks_k+\sum_{i=1}^k i s_{k-i} u_i=0, \hskip10mm k=1,2,\cdots
\eeq
with $s_0=-1$ and $s_1=u_1=0$. We have
\beqa
y^2 &=& \left ( x^{N_c}- s_2^{cl} x^{N_c-2}- \cdots -s_{N_c}^{cl} \right ) 
\left ( x^{N_c}- s_2^{cl} x^{N_c-2}- \cdots 
-s_{N_c}^{cl} \pm 4 \La^{N_c} \right ) \CR
 &=& (x-a_1)^2 (x-a_2) \cdots (x-a_{N_c-1}) \, 
\Big( (x-a_1)^2 \cdots (x-a_{N_c-1}) \pm 4 \La^{N_c} \Big).
\label{su}
\eeqa
Since the curve exhibits the quadratic degeneracy we are exactly at the
singular point of a massless dyon in the $N=2$ $SU(N_c)$ Yang-Mills vacuum.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{$SO(2 N_c)$ case}

Let us now apply our procedure to the $N=2$ $SO(2 N_c)$ Yang-Mills theory.
We take a tree-level superpotential to break $N=2$ to $N=1$ as 
\beq
W=\sum_{n=1}^{N_c-1} g_{2 n} u_{2 n} + \lm v,
\label{soeventree}
\eeq
where 
\beqa
&& u_{2 n} =\frac{1}{2 n} {\rm Tr}\, \Phi^{2 n}, \CR
&& v ={\rm Pf}\, \Phi=\frac{1}{2^{N_c} N_c !} \E_{i_1 i_2 j_1 j_2 \cdots}
\Phi^{i_1 i_2} \Phi^{j_1 j_2} \cdots 
\eeqa
and the adjoint superfield $\Phi$ is an antisymmetric 
$2 N_c \times 2 N_c$ tensor.
This theory has classical vacua which satisfy the condition
\beq
W'(\Phi)=\sum_{i=1}^{N_c-1} g_{2 i} (\Phi^{2 i-1})_{ij}-
\frac{\lm}{2^{N_c} (N_c-1) !} \E_{\: i\: j \: k_1 k_2 l_1 l_2 \cdots}
\Phi^{k_1 k_2} \Phi^{l_1 l_2} \cdots=0.
\label{wd}
\eeq
For the skew-diagonal form of $\p$
\beq \p={\rm diag}
(\s_{2}e_0,\; \s_{2}e_1,\; \s_{2}e_2, \cdots, \s_{2}e_{N_c-1}), \hskip10mm
\s_{2}=i \left( \begin{array}{cc} 0 &  -1 \\ 1 & 0 \end{array} \right )
\eeq
the vacuum condition (\ref{wd}) becomes
\beq
\sum_{i=1}^{N_c-1} g_{2 i} (-1)^{i-1} {e_n}^{2 i-1}+ (-i)^{N_c}
\frac{\lm}{2 e_n} \prod_{i=0}^{N_c-1} e_i=0 \;, \hskip10mm 0 \leq n \leq N_c-1.
\label{ai}
\eeq
Thus we see that $e_n \, (\not= 0)$ are the roots of $f(x)$ defined by 
\beq
f(x)=\sum_{i=1}^{N_c-1} g_{2 i} x^{2 i}+d,
\label{fpoly}
\eeq
where we put $d=(-i)^{N_c} \frac{1}{2} \lm \prod_{i=0}^{N_c-1} e_i $.

Since our main concern is the vacuum with a single confined photon 
we focus on the unbroken $SU(2) \times U(1)^{N_c-1}$ vacuum. Thus writing  
(\ref{fpoly}) as
\beq
f(x)=g_{2(N_c-1)} \prod_{i=1}^{N_c-1} (x^2-a_i^2),
\eeq
we take
\beq
\Phi  =  
{\rm diag} (\s_{2}a_1,\; \s_{2}a_1,\;  \s_{2}a_2, \cdots, \s_{2}a_{N_c-1})
\eeq 
with $d = (-i)^{N_c} \frac{1}{2} \lm a_1^2 \prod_{i=2}^{N_c-1} a_i$.
We then make the scale matching between the 
high-energy $SO(2 N_c)$ scale $\La$ and the low-energy $SU(2)$ scale 
$\La_L$. Following the steps as in the $SU(N_c)$ case yields
\beq
\La^{2 \cdot 2( N_c-1)}={\La_L}^{3 \cdot 2}
\left ( \prod_{i=2}^{N_c-1} (a_1^2-a_i^2) \right )^2
(M_{\rm ad})^{-2}, 
\label{so2nmatching}
\eeq
where the factor arising through the Higgs mechanism is easily calculated 
in an explicit basis of $SO(2 N_c)$. In order to evaluate the $SU(2)$ adjoint 
mass $M_{\rm ad}$ we first substitute the decomposition (\ref{decompose})
in $W$ and proceed as follows:
\beqa
W
& = & W_{cl} + \sum_{i=1}^{N_c-1} g_i \frac{2 i-1}{2}
\; {\rm Tr}\, ( \delta \Phi^2  \Phi_{cl}^{2 i-2} )
+ \lm \left ( {\rm Pf}_4 \D \Phi \right ) 
\left({\rm Pf}_{2(N_c-2)} \Phi_{cl} \right ) +\cdots \CR
& = & W_{cl} + \sum_{i=1}^{N_c-1} g_i \frac{2 i-1}{2}
\; {\rm Tr}\, ( \delta \Phi^2  \Phi_{cl}^{2 i-2} )
+ \lm \left ( \frac{1}{4} {\rm Tr}\, \D \Phi^2 \right ) 
\left ( \prod_{k=2}^{N_c-1} (-i a_k) \right ) +\cdots \CR
& =& W_{cl} + \left. \frac{1}{2} \frac{{\rm d}}{{\rm d} x} 
\left( \frac{f(x)}{x} \right ) \right |_{x=a_1} 
{\rm Tr}\, \D \Phi^2 +\cdots \CR
& =& W_{cl} +  g_{2(N_c-1)} \prod^{N_c-1}_{i=2} (a_1^2-a_i^2)
\, {\rm Tr}\, \D \Phi^2  +\cdots, 
\eeqa
where ${\rm Pf}_4$ is the Pfaffian of a upper-left $4 \times 4$ sub-matrix and
${\rm Pf}_{2(N_c-2)} $ is the Pfaffian of a lower-right 
$ 2(N_c-2)\times 2(N_c-2)$ sub-matrix. Thus we observe that $M_{\rm ad}$
cancels out the Higgs factor in (\ref{so2nmatching}), which leads to
${\La_L}^6 =  g_{2(N_c-1)}^2 \La^{4(N_c-1)}$. The low-energy 
superpotential is now given by 
\beq
W_L= W_{cl} \pm 2 {\La_L}^3 = W_{cl} \pm 2 g_{2(N_c-1)} \La^{2(N_c-1)},
\eeq
where the second term is due to the gaugino condensation in the low-energy
$SU(2)$ theory. 

The vacuum expectation values of gauge invariants are obtained from $W_L$ as
\beqa
&& \langle u_{2 n} \rangle = {\pa W_L \over \pa g_{2n}}=
u_{2 n}^{cl}(g,\lambda) \pm 2 \La^{2(N_c-1)} \delta_{n, N_c-1}, \CR
&& \langle v \rangle ={\pa W_L \over \pa \lm }=v_{cl}(g,\lambda).
\label{so2nsingular}
\eeqa
The curve for  $N=2\;SO(2 N_c)$ is known to be \cite{BrLa}
\beqa
y^2 &=& {\left \langle {\rm det} (x-\Phi) \right \rangle}^2
-4 \La^{4( N_c-1)} x^4  \CR
&=& \Big( x^{2N_c}- \sum_{i=1}^{N_c-1} \bra s_{2i} \ket x^{2(N_c-i)}
+\bra v \ket^2 \Big)^2-4 \La^{4( N_c-1)} x^4,
\eeqa
where
\beq
ks_k+\sum_{i=1}^k i s_{k-i} u_{2i}=0, \hskip10mm k=1,2,\cdots
\label{defs}
\eeq
with $s_0=-1$. At the values (\ref{so2nsingular}) of the moduli coordinates
we see the quadratic degeneracy 
\beqa
y^2 &=& \left ( x^{2 N_c}-   s_2^{cl}   x^{2 (N_c-1)}- \cdots 
-  s^{cl}_{2 (N_c-1)} x^2 +v_{cl}^2   \right ) \CR 
 & & \;\; \times 
\left ( x^{2 N_c}-   s_2^{cl}   x^{2 (N_c-1)}- \cdots 
-  s_{2 (N_c-1)}^{cl} x^2  +v_{cl}^2  \pm 4 \La^{2 (N_c-1)} x^2 \right ) \CR
 &=& (x^2-a_1^2)^2 (x^2-a_2^2) \cdots (x^2-a_{N_c-1}^2) \CR
& & \;\; \times
\Big((x^2-a_1^2)^2 (x^2-a_2^2) \cdots (x^2-a_{N_c-1}^2) 
\pm 4 \La^{2 (N_c-1)} x^2 \Big).
\eeqa
This is our desired result. Notice that the apparent singularity at 
$\langle v \rangle=0$ is not realized in our $N=1$ theory. Thus the point
$\langle v \rangle=0$ does not correspond to massless solitons in agreement
with the result of \cite{BrLa}.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{$SO(2 N_c+1)$ case}

Our next task is to study the $SO(2 N_c+1)$ gauge theory.
A tree-level superpotential breaking $N=2$ to $N=1$ is assumed to be 
\beq
W=\sum_{n=1}^{N_c} g_{2 n} u_{2 n}, \hskip10mm 
u_{2 n} = \frac{1}{2 n} {\rm Tr}\, \Phi^{2 n}.
\label{sooddtree}
\eeq
The classical vacua obey
$W'(\Phi)=\sum_{i=1}^{N_c} g_{2 i} \Phi^{2 i-1}=0$.
The eigenvalues of $\p$ are given by the roots $a_i$ of
\beq
W'(x)=\sum_{i=1}^{N_c} g_{2 i} x^{2 i-1}=
g_{2 N_c} x \prod_{i=1}^{N_c-1} (x^2-a_i^2).
\label{sooddmotion}
\eeq
As in the previous consideration we take 
the $SU(2) \times U(1)^{N_c-1}$ vacuum.
Notice that there are two ways of  breaking  $SO(2 N_c+1)$ to 
$SU(2) \times U(1)^{N_c-1}$. One is to take all the eigenvalues distinct
(corresponding to $SO(3) \times U(1)^{N_c-1}$). The other is to choose
two eigenvalues coinciding and the rest distinct
(corresponding to $SU(2) \times U(1)^{N_c-1}$ with $a_i \not= 0$).
Here we examine the latter case
\beq
\Phi=
{\rm diag}(\s_{2}a_1,\; \s_{2}a_1,\; \s_{2}a_2, \cdots, \s_{2}a_{N_c-1},\; 0),
\hskip10mm 
\s_{2}=i \left( \begin{array}{cc} 0 &  -1 \\ 1 & 0 \end{array} \right ).
\eeq
In this vacuum the high-energy $SO(2 N_c+1)$ scale $\La$ and the low-energy
 $SU(2)$ scale $\La_L$ are related by
\beq
\La^{2 \cdot (2 N_c-1)}={\La_L}^{3 \cdot 2}
a_1^2 \left ( \prod_{i=2}^{N_c-1} (a_1^2-a_i^2) \right )^2
(M_{\rm ad})^{-2},
\eeq
where the $SU(2)$ adjoint mass $M_{\rm ad}$ is read off from
\beqa
W
& = & W_{cl} + \sum_{i=1}^{N_c} g_{2 i} \frac{2 i-1}{2}
\; {\rm Tr}\, ( \delta \Phi^2  \Phi_{cl}^{2 i-2} )+\cdots \CR
& =& W_{cl} + \frac{1}{2} W''(a_1) \; {\rm Tr}\, \D \Phi^2+\cdots \CR
& =& W_{cl}+g_{2 N_c} a_1^2 \prod^{N_c-1}_{i=2} (a_1^2-a_i^2)\,
{\rm Tr}\, \D \Phi^2  +\cdots.
\eeqa
So, we obtain ${\La_L}^6 =  g_{2 N_c}^2 a_1^2 \La^{2(2 N_c-1)}$. The low-energy
effective superpotential becomes
\beq
W_L= W_{cl} \pm 2 {\La_L}^3 = 
W_{cl} \pm 2 g_{2 N_c} a_1 \La^{2 N_c-1}.
\label{wll}
\eeq
If we assume $W_{\Delta}=0$ the expectation values $\bra u_{2i}\ket$
are calculated from $W_L$ by expressing $a_1$ as a function of $g_{2 i}$.

For the sake of illustration let us discuss the $SO(5)$ theory 
explicitly. From (\ref{wll}) we get
\beqa
\langle u_{2} \rangle & = & 2 a_1^2 \pm \frac{1}{a_1} \La^3, \CR
\langle u_{4} \rangle & = &  a_1^4  \mp a_1 \La^3 
\label{r3}
\eeqa
and $a_1^2=-g_2/g_4$. We eliminate $a_1 $ from (\ref{r3}) to obtain 
\beq
27 \La^{12} - \La^6 u_2^3 + 36 \La^6 u_2 u_4 -u_2^4 u_4 
+8 u_2^2 u_4^2 -16 u_4^3=0.
\label{r4}
\eeq
This should be compared with the $N=2$ $SO(5)$ discriminant \cite{DaSu}
\beq
s_2^2 (27 \La^{12} - \La^6 s_1^3-36 \La^6 s_1 s_2 +s_1^4 s_2 
+8 s_1^2 s_2^2 +16 s_2^3)^2=0,
\label{r6}
\eeq
where $s_1=u_2$ and $s_2=u_4-u_2^2/2$ according to (\ref{defs}). Thus we
see the discrepancy between (\ref{r4}) and (\ref{r6}) which implies
that our simple assumption of $W_{\Delta} = 0$ does not work.
Inspecting (\ref{r4}) and (\ref{r6}), however, we notice how to remedy the
difficulty. Instead of (\ref{sooddtree}) we take a tree-level superpotential
\beq
W=g_{2} s_1+g_4 s_2=
g_2 u_2+g_4 \left(u_4-\frac{1}{2} u_2^2 \right).
\label{ws2}
\eeq
The classical vacuum condition is
\beq
 W'(\Phi)=(g_2 -g_4 u_2) \Phi +g_4 \Phi^3 =0.
\eeq
To proceed, therefore, we can make use of the results obtained in the
foregoing analysis just by making the replacement
\beqa
g_4 & \rightarrow & \tilde{g_4}=g_4, \CR
g_2 & \rightarrow & \tilde{g_2}=g_2-u_2 g_4.
\label{u2}
\eeqa
(especially evaluation of $M_{\rm ad}$ is not invalidated because
${\rm Tr}\, \D \Phi=0$.)
The eigenvalues of $\Phi$ are now determined in a self-consistent manner by
\beq
W'(x)=\tilde{g_2} x+\tilde{g_4} x^3=
\tilde{g_4} x \left( x^2+\frac{\tilde{g_2}}{\tilde{g_4}} \right)
=\tilde{g_4} x (x^2-a_1^2)=0.
\eeq
Then we have $u_2^{cl}=2 a_1^2=-2 \tilde{g_2}/\tilde{g_4}$ and
$\tilde{g_2} =-g_2$ from (\ref{u2}), which leads to
\beq
a_1^2=\frac{g_2}{g_4}.
\eeq
Substituting this in (\ref{wll}) we calculate $\bra s_i \ket$ and 
find the relation of $s_i$ which is precisely the discriminant (\ref{r6})
except for the classical singularity at $\bra s_2\ket =0$.

The above $SO(5)$ result indicates that an appropriate mixing term with
respect to $u_{2i}$ variables in a microscopic superpotential will be
required for $SO(2 N_c+1)$ theories. We are led to assume
\beq
W=\sum_{i=1}^{N_c-1} g_{2 i} u_{2 i} + g_{2 N_c} s_{N_c}
\label{ws}
\eeq
for the gauge group $SO(2 N_c+1)$ with $N_c \geq 3$. Then the following 
analysis is analogous to the $SO(5)$ theory. First of all notice that
\beq
s_{N_c}=u_{2 N_c} -u_{2(N_c-1)} u_2 +
(\hbox{polynomials of $u_{2 k}, \;1 \leq k < N_c-1$}).
\eeq
Therefore the eigenvalues of $\Phi$ are given by the roots of 
(\ref{sooddmotion}) with the replacement
\beqa
g_{2 N_c} & \rightarrow & \tilde g_{2 N_c}=g_{2 N_c}, \CR
g_{2 (N_c-1)} & \rightarrow & \tilde g_{2 (N_c-1)}=
g_{2 (N_c-1)} - u_2 g_{2 N_c}.
\eeqa
Then we have $u_2=a_1^2+\sum_{k=1}^{N_c-1} a_k^2=a_1^2-
 \tilde g_{2 (N_c-1)}/\tilde g_{2 N_c}$ and find
\beq
a_1^2=\frac{g_{2 (N_c-1)}}{g_{2 N_c}}.
\eeq
It follows that the effective superpotential is given by
\beq
W_L=W_L^{cl} \pm 2 \sqrt{g_{2 N_c} g_{2(N_c-1)}} \La^{2 N_c-1}.
\eeq

The vacuum expectation values of gauge invariants are obtained from $W_L$ as 
\beqa
\langle s_n \rangle & = &  s_n^{cl}(g), \hskip10mm  1 \leq n \leq N_c-2 \CR 
\langle s_{N_c-1} \rangle & = &  s_{N_c-1}^{cl}(g) 
\pm \frac{1}{a_1} \La^{2 N_c-1}, \CR
\langle s_{N_c} \rangle & = &  s_{N_c}^{cl}(g)
\pm a_1 \La^{2 N_c-1} .
\eeqa
For these $\bra s_i\ket$ the curve describing the $N=2\;SO(2 N_c+1)$ 
theory \cite{DaSu} is shown to be degenerate as follows:
\beqa
y^2 & = & {\left \langle {\rm det} (x-\Phi) \right \rangle}^2
- 4 x^2 \La^{2( 2 N_c-1)}  \CR
& = & (x^{2 N_c} - \langle s_1 \rangle x^{2(N_c-1)} -\cdots 
 - \langle s_{N_c-1} \rangle x^2- \langle s_{N_c} \rangle
+ 2 x \La^{2 N_c-1} ) \CR
&  & \; \times (x^{2 N_c} - \langle s_1 \rangle x^{2(N_c-1)} -
\cdots 
 - \langle s_{N_c-1} \rangle x^2- \langle s_{N_c} \rangle
- 2 x \La^{2 N_c-1} ) \CR
& = & \left\{ (x^2-a_1^2)^2 (x^2-a_2^2) \cdots (x^2-a_{N_c-1}^2) \pm
\La^{2 N_c-1} \left( -\frac{x^2}{a_1} -a_1 +2 x \right) \right\} 
\CR
&  & \; \times
\left\{ (x^2-a_1^2)^2 (x^2-a_2^2) \cdots (x^2-a_{N_c-1}^2) \pm
\La^{2 N_c-1} \left( -\frac{x^2}{a_1} -a_1 -2 x \right) \right\} 
\CR
& = & (x^2-a_1^2)^2 
\left( (x+a_1)^2 (x^2-a_2^2) \cdots (x^2-a_{N_c-1}^2) \mp 
\frac{\La^{2 N_c-1}}{a_1} \right) \CR
&  & \; \times
\left( (x-a_1)^2 (x^2-a_2^2) \cdots (x^2-a_{N_c-1}^2) \mp 
\frac{\La^{2 N_c-1}}{a_1} \right).
\eeqa
Thus we see the theory with the superpotential (\ref{ws}) recover the
$N=2$ curve correctly with the assumption $W_{\Delta}=0$. 
As in the $SO(2 N_c)$ case, 
the singularity at $\langle s_{ N_c} \rangle=0$, which corresponds to the
classical $SO(3) \times U(1)^{N_c-1}$ vacuum,
does not arise in our theory.

We remark that the particular form of superpotential (\ref{ws}) is not 
unique to derive the singularity manifold.
In fact we may start with a superpotential
\beq
W=\sum_{i=1}^{N_c-1} g_{2 i} \left(u_{2 i} +h_i(s) \right)+ 
g_{2 N_c} \left(s_{N_c}+h_{N_c}(s) \right), 
\label{r7}
\eeq
where $h_i(s)$ are arbitrary polynomials of $s_j$ with $j \geq N_c-2$,
to verify the $N=2$ curve. However, we are not allowed to take 
a superpotential such as $W=\sum_{i=1}^{N_c} g_{2 i} s_i$, 
because there are no $SU(2) \times U(1)^{N_c-1}$ vacua
(there exist no solutions for $\tilde g_{2( N_c-1)}$).
Note also that there are no $SO(3) \times U(1)^{N_c-1}$ vacua
in the theory with superpotential (\ref{r7}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{$Sp(2 N_c)$ case}

Finally we discuss the $Sp(2 N_c)$ gauge theory.
The adjoint superfield $\Phi$ is a $2 N_c \times 2 N_c$ tensor which is
subject to
\beq
{}^t \Phi = J \Phi J \quad \Longleftrightarrow \quad J \Phi 
\;\; \hbox{is symmetric},
\eeq
where $J={\rm diag}(i\sigma_2, \cdots, i\sigma_2)$.
Let us assume a tree-level superpotential 
\beq
W=\sum_{n=1}^{N_c} g_{2 n} u_{2 n}, \hskip10mm
u_{2 n} = \frac{1}{2 n} {\rm Tr}\, \Phi^{2 n}.
\label{sptree}
\eeq
Then our analysis will become quite similar to that for $SO(2 N_c+1)$. The
classical vacuum with unbroken $SU(2) \times U(1)^{N_c-1}$ gauge group
corresponds to
\beq
J \Phi=
{\rm diag}(\s_{1}a_1,\; \s_{1}a_1,\; \s_{1}a_2,\cdots,\s_{1}a_{N_c-1}) ,
\hskip10mm \s_{1}
=\left( \begin{array}{cc} 0 &  1 \\ 1 & 0 \end{array} \right ).
\eeq
The scale matching relation becomes
\beq
\La^{2 \cdot (N_c+1)}={\La_L}^{3 \cdot 2 \cdot \frac{1}{2}} 
a_1^4 
\left ( \prod_{i=2}^{N_c-1} (a_1^2-a_i^2) \right )^2 
 (M_{\rm ad})^{-1}.
\eeq
Since the $SU(2)$ adjoint mass is given by 
$M_{\rm ad}=g_{2 N_c} a_1^2 \prod^{N_c-1}_{i=2} (a_1^2-a_i^2)$ we get
$ {\La_L}^3 = g_{2 N_c} \La^{2(N_c+1)}/a_1^2$. The low-energy
effective superpotential thus turns out to be
\beq 
W_L=W_{cl}+2  \frac{g_{2 N_c}}{a_1^2} \La^{2(N_c+1)}.
\eeq

Checking the result with $Sp(4)$ we encounter the same problem as in the
$SO(5)$ theory. Instead of (\ref{sptree}), thus, we take a superpotential
in the form (\ref{ws2}), reproducing the $N=2$ $Sp(4)$ curve \cite{ArSh}.
Similarly, for $Sp(2 N_c)$ we study a superpotential 
(\ref{ws}). It turns out that $\bra s_i\ket$ are calculated as
\beqa
\langle s_n \rangle & = &  s_n^{cl}(g), \hskip12mm 1 \leq n \leq N_c-2, \CR 
\langle s_{N_c-1} \rangle & = &  s_{N_c-1}^{cl}(g)
- \frac{2}{a_1^4} \La^{2(N_c+1)}, \CR
\langle s_{N_c} \rangle & = &  s_{N_c}^{cl}(g)
+  \frac{4}{a_1^2} \La^{2(N_c+1)}.
\eeqa
These satisfy the $N=2\; Sp(2 N_c)$ singularity condition \cite{ArSh} 
since the curve exhibits the quadratic degeneracy
\beqa
x^2 y^2 & = & \left( x^2 \left \langle {\rm det} (x-\Phi) 
\right \rangle +\La^{2(N_c+1)} \right)^2
- \La^{4(N_c+1)}  \CR
& = & (x^{2 (N_c+1)} - \langle s_1 \rangle x^{2 N_c} -\cdots 
 - \langle s_{N_c-1} \rangle x^4- \langle s_{N_c} \rangle x^2
+ 2  \La^{2 (N_c+1)} ) \CR
&  & \; \times (x^{2 (N_c+1)} - \langle s_1 \rangle x^{2 N_c} -
\cdots 
 - \langle s_{N_c-1} \rangle x^4- \langle s_{N_c} \rangle x^2) \CR
& = & \left\{ x^2 (x^2-a_1^2)^2 (x^2-a_2^2) \cdots (x^2-a_{N_c-1}
^2)+
2 \La^{2 (N_c+1)} 
\left( \left( \frac{x}{a_1} \right)^4 -
2 \left( \frac{x}{a_1} \right)^2 +1 \right) \right\} \CR
&  & \; \times
\left( x^2 {\rm det} (x-\Phi_{cl}) \right)\CR
& = & (x^2-a_1^2)^2 
\left( x^2 (x^2-a_2^2) \cdots (x^2-a_{N_c-1}^2) +
 \frac{\La^{2( N_c+1)}}{a_1^4}
\right) \! \times\!
\left( x^2 {\rm det} (x-\Phi_{cl}) \right).
\eeqa
It should be mentioned that our remarks on $SO(2 N_c+1)$ theories also
apply here.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ADE gauge groups}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Our purpose in this section is to show that, under 
an appropriate ansatz, the low-energy effective superpotential for 
the Coulomb phase is obtained in a unified way for all ADE gauge groups just 
by using the fundamental properties of the root system $\Delta$. 
Let us consider the case of 
the gauge group $G$ is simple and simply-laced, namely, 
$G$ is of ADE type. 
Our notation
for the root system is as follows. The simple roots of $G$ are denoted as
$\alpha_i$ where $1 \leq i \leq r$ with $r$ being the rank of $G$. Any root
is decomposed as $\alpha =\sum_{i=1}^{r} a^i \alpha_i$. The component indices
are lowered by $a_i=\sum_{j=1}^{r} A_{ij} a^j$ where $A_{ij}$ is the ADE
Cartan matrix. The inner product of two roots $\alpha$, $\beta$ are then
defined by
\beq
\A \cdot \B = \sum_{i=1}^{r} a^i b_i = \sum_{i,j=1}^{r} a^i A_{ij} b^j,
\eeq
where $\B=\sum_{i=1}^{r} b^i \A_i$. For ADE all roots have the equal norm and
we normalize $\alpha^2=2$.

In our $N=1$ theory we take a tree-level superpotential 
\beq
W=\sum_{k=1}^{r} g_{k} u_{k}(\Phi),
\label{tree}
\eeq
where $u_k$ is the $k$-th Casimir of $G$ constructed from $\Phi$
and $g_k$ are coupling constants. The mass dimension of $u_k$ is $e_k+1$
with $e_k$ being the $k$-th exponent of $G$.
When $g_k=0$ $\Phi$ is considered as the
chiral field in the $N=2$ vector multiplet and we have $N=2$ ADE supersymmetric
gauge theory. 

We first make a classical analysis of the theory with the superpotential 
(\ref{tree}).
The classical vacua are determined by the equation of motion
$\frac{\pa W}{\pa \Phi}=0$ and the $D$-term equation.
Due to the $D$-term equation,
we can restrict $\Phi$ to take the values in the Cartan subalgebra
by the gauge rotation.
We denote the vector in the Cartan subalgebra corresponding to the
classical value of $\Phi$ as $a=\sum_{i=1}^{r} a^i \A_i$.
Then the superpotential becomes
\beq    
W(a)=\sum_{k=1}^{r} g_{k} u_{k}(a),
\label{tree2}
\eeq
and the equation of motion reads
\beq
\frac{\pa W(a)}{\pa a^i}=
\sum_{k=1}^{r} g_{k} \frac{\pa u_{k}(a)}{\pa a^i}=0.
\label{eq1}
\eeq
For $g_k \not\equiv 0$ we must have
\beq
J(a) \equiv {\rm det} \left(  \frac{\pa u_{j}(a)}{\pa a^i} \right) =0.
\label{Jzero}
\eeq
According to \cite{Hu} it follows that
\beq
J(a) =c_1 \prod_{\A \in \Delta^+} a \cdot \A ,
\label{J}
\eeq
where $\Delta^+$ is a set of positive roots and $c_1$ is a certain constant. 

The condition $J(a)=0$ means that the vector $a$ hits a wall of the
Weyl chamber and there occurs enhanced gauge symmetry. Suppose that the vector
$a$ is orthogonal to a root, say, $\A_1$
\beq
a \cdot \A_1=0,
\label{su2sol}
\eeq
where $\A_1$ may be taken to be a simple root.
In this case we have the unbroken gauge group
$SU(2) \times U(1)^{r-1}$ where the $SU(2)$ factor is spanned by 
$\{ \A_1 \cdot H, E_{\A_1}, E_{-\A_1} \}$ in the Cartan-Weyl basis.
If some other factors of $J$ vanish besides $a \cdot \A_1$
the gauge group is further enhanced from $SU(2)$. 
Since $SU(2) \times U(1)^{r-1}$
is the most generic unbroken gauge group we shall restrict ourselves to
this case in what follows.

We remark here that there is the case in which the $SU(2) \times U(1)^{r-1}$
vacuum is not generic. As a simple, but instructive example consider
$SU(4)$ theory. Casimirs are taken to be
\beqa
u_1 &=& {1\over 2} {\rm Tr}\, \Phi^2, \CR
u_2 &=& {1\over 3} {\rm Tr}\, \Phi^3, \CR
u_3 &=& {1\over 4} {\rm Tr}\, \Phi^4 
-\alpha \left( {1\over 2}{\rm Tr}\, \Phi^2 \right)^2, 
\eeqa
where $\alpha$ is an arbitrary constant. If we set $\alpha =1/2$ it is observed
that the $SU(2) \times U(1)^2$ vacuum exists only for the special values
of coupling constants, $(g_2/g_3)^2=g_1/g_3$. Thus, for $\alpha =1/2$, the
$SU(2) \times U(1)^2$ vacuum is not generic though it does so for 
$\alpha \not= 1/2$. This points out that we have to choose the appropriate
basis for Casimirs when writing down (\ref{tree}) to have the 
$SU(2) \times U(1)^{r-1}$ vacuum generically \cite{TeYa1}.

Now we assume that
there is no mixing between the $SU(2) \times U(1)^{r-1}$ vacuum and
other vacua with different unbroken gauge groups.
According to the arguments of \cite{InSe2},
we should not consider the broken gauge group instantons.
We thus expect that there is only perturbative effect 
in the energy scale above the scale $\La_{YM}$ of the 
low-energy effective $N=1$ supersymmetric 
$SU(2)$ Yang-Mills theory. 


%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Higgs mass}
%%%%%%%%%%%%%%%%%%%%%%%%%

Our next task is to evaluate the Higgs scale
associated with the spontaneous breaking of the 
gauge group $G$ to $SU(2) \times U(1)^{r-1}$.
For this purpose we decompose the adjoint representation of $G$ to 
irreducible representations of $SU(2)$. We fix the $SU(2)$ direction by taking
a simple root $\alpha_1$. It is clear that the spin $j$ of every representation
obtained in this decomposition satisfies $j \leq 1$ since all roots have the
same norm and the $SU(2)$ raising (or lowering) operator shifts a root $\alpha$
to $\alpha +\alpha_1$ (or $\alpha -\alpha_1$). The fact that there is no
degeneration of roots indicates that the $j=1$ multiplet has the
roots $(\alpha_1,0,-\alpha_1)$ corresponding to the unbroken $SU(2)$
generators. The roots orthogonal to $\alpha_1$ represent the $j=0$ multiplets.
The $j=1/2$ multiplets have the roots $\alpha$ obeying 
$\A \cdot \A_1=\pm 1$. Let us define a set of these roots by
$\Delta_d=\{ \alpha | \alpha \in \Delta,\, \alpha \cdot \alpha_1 =\pm 1 \}$.
For each root $\alpha \in \Delta_d$ there appears a massive gauge boson.
These massive bosons pair up in $SU(2)$ doublets with weights $(\alpha,
\alpha \pm \alpha_1)$ which indeed have the same mass $|a\cdot \alpha |=
|a\cdot (\alpha \pm \alpha_1)|$ since $a\cdot \alpha_1=0$.

We now integrate out the fields that become massive by the Higgs mechanism.
The massless $U(1)^{r-1}$ degrees of freedom are decoupled. 
The resulting theory characterized by the scale $\La_H$ is $N=1$ 
$SU(2)$ theory with an adjoint chiral multiplet.
The Higgs scale $\La_H$ is related to the high-energy scale $\La$ through
the scale matching relation
\beq
\La^{2 h}= \La_H^{2\cdot 2} \left( \prod_{\B \in \Delta_d,\, \B >0} 
a \cdot \B \right)^\ell,
\label{summaa} 
\eeq
where $2h=4+\ell n_d/2$, $n_d$ is the number of elements in $\Delta_d$ 
and $h$ stands for the 
dual Coxeter number of $G$; $h=r+1, 2r-2, 12, 18,30$ for $G=A_r, D_r, 
E_6, E_7, E_8$ respectively. The reason for $\beta >0$
in (\ref{summaa}) is that weights $(\beta ,\beta \pm \alpha_1)$ of an
$SU(2)$ doublet are either both positive or both negative 
since $\alpha_1$ is the simple
root, and gauge bosons associated with $\beta <0$ and $\beta >0$ have the
same contribution to the relation (\ref{summaa}).

To fix $\ell$ we calculate $n_d$ by evaluating the quadratic Casimir $C_2$
of the adjoint representation in the following way. Taking hermitian generators
we express $C_2$ in terms of the structure constants $f_{abc}$ through
$\sum_{a,b}f_{abc}f_{abc'}=-C_2 \, \delta_{cc'}$. From the commutation 
relation 
$[\alpha_1\cdot H, E_\alpha ]=(\alpha_1 \cdot \alpha) E_\alpha$ one can check
\beq
C_2={1\over 2}\sum_{\A \in \Delta} ( \A_1 \cdot \A )^2
={1\over 2} \left( \sum_{\A \in \Delta_d} ( \A_1 \cdot \A )^2 + 
2 (\alpha_1 \cdot \alpha_1)^2 \right)= {1\over 2} \left ( n_d +8 \right).
\eeq
On the other hand, the dual Coxeter number $h$ is given by
$h=C_2/\theta^2$ with $\theta$ being the highest root. We thus find
\beq
n_d = 4 (h-2)
\label{numbd}
\eeq
and (\ref{summaa}) becomes
\beq
\La^{2 h}= \La_H^{2 \cdot 2} \prod_{\B \in \Delta_d,\, \B >0} a \cdot \B .
\label{summa}
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Adjoint mass}
%%%%%%%%%%%%%%%%%%%%%%%%%%%

After integrating out the massive fields due to the Higgs mechanism we are
left with $N=1$ $SU(2)$ theory with the massive adjoint.
In order to evaluate the mass of the adjoint chiral multiplet $\Phi$
we need to clarify some properties of Casimirs.
Let $\s_\B$ be an element of the Weyl group of $G$ 
specified by a root $\B= \sum_{i=1}^r b^i \A_i$.
The Weyl transformation of a root $\alpha$ is given by
\beq
\s_\B (\A) = \A - (\A \cdot \B) \B .
\label{Weyl}
\eeq
When $\s_\B$ acts on the Higgs v.e.v. vector $a=\sum_{i=1}^r a^i \A_i$ we have
\beq
{a'}^i = \sum_{j=1}^r {S_\B}^i_{\; j} \; a^j, \hskip5mm
{S_\B}^i_{\; j} \equiv \D^i_{\; j}-b^i b_j,
\eeq
where $\s_\B (a)=\sum_{i=1}^r a'^i \A_i$. 
Since the Casimirs $u_k(a)$ are Weyl invariants it is obvious to see
\beq
\frac{\pa}{\pa a^i} u_k(a) = \frac{\pa}{\pa a^i} u_k(a')=
\sum_{j=1}^r {S_\B}^j_{\; i} 
\left. \left( \frac{\pa}{\pa {a}^j} u_k(a) \right) \right|_{a \rightarrow a'}.
\eeq
Let $\bar a$ be a particular v.e.v. which is fixed under the action 
of $\s_\B$, then we find the identity
\beq
\left. \sum_{j=1}^r \left( \D^j_{\; i} - {S_\B}^j_{\; i} \right)
\frac{\pa}{\pa {a}^j} u_k(a) \right|_{a=\bar a} =0
\eeq
for all $i$, and thus
\beq
\left. \sum_{j=1}^r b_j \frac{\pa}{\pa {a}^j} u_k(a) \right|_{a=\bar a} =0.
\eeq
This implies that for any v.e.v. vector $a$ 
and root $\beta$ we can write down 
\beq
\sum_{j=1}^r b^j \frac{\pa}{\pa {a}^j} u_k(a)= (a \cdot \B) \; u_k^\B (a),
\label{diff}
\eeq
where $u_k^\B (a)$ is some polynomial of $a^i$.
If we set $\beta =\alpha_i$, a simple root, we obtain a useful formula
\beq
\frac{\pa}{\pa {a}^i} u_k(a)= a_i  \; u_k^{\A_i} (a).
\label{diff2}
\eeq

As an immediate application of the above results, for instance, we point out
that (\ref{J}) is derived from (\ref{diff}) and the fact that 
the mass dimension of $J(a)$ is given by
\beq
\sum_{k=1}^r e_k={1\over 2}\, ({\rm dim}\,  G-r),
\eeq
where $e_k$ is the $k$-th exponent of $G$.

Let us further discuss the properties of $u_k^{\A_j} (a)$. Define $D_{mn}$ as
\beq
D_{mn}  \equiv (-1)^{n+m} {\rm det} 
\left( \frac{\pa u_{\tilde{j}}(a)}{\pa a^{\tilde{i}}} \right),
\hskip5mm 1 \leq m,n \leq r ,
\eeq
where $1 \leq \tilde{i},\tilde{j} \leq r$ with $\tilde{i} \not= m,
\tilde{j} \not= n$,
then $D_{1n}$ is a homogeneous polynomial of $a^i$ with the mass dimension
$\sum_{k=1}^{r} e_k -e_n$.
We also denote $\Delta_e$ as a set of positive roots where $\A_1$ and 
$SU(2)$ doublet roots $\A$ with $\A+\A_1 \not\in \Delta^+$ are excluded.
If we set $a_1=0$ and $a \cdot \B=0$ where $\B$ is 
any root in $\Delta_e$
we see $D_{1n}=0$ from the identity (\ref{diff}).
Consequently we can expand 
\beq
D_{1n}=h_n(a) \prod_{\B \in \Delta_e} (a \cdot \B)+a_1 f_n(a),
\eeq
where $h_n(a)$, $f_n(a)$ are polynomials of $a_i$. In particular
\beq
D_{1r}=c_2 \prod_{\B \in \Delta_e} (a \cdot \B)+a_1 f_r(a),
\label{Dexpr}
\eeq
where $c_2$ is a constant. Notice that the first term on the rhs has the 
correct mass dimension since the number of roots in $\Delta_e$ reads
\beq
{1\over 2}\, ({\rm dim}\,  G-r) -1-{n_d\over 4}=\sum_{k=1}^r e_k -(h-1),
\eeq
where we have used (\ref{numbd}) and $e_r=h-1$.

We are now ready to evaluate the 
mass of $\Phi$ in intermediate $SU(2)$ theory.
The fluctuation of $W(a)$ around the classical vacuum yields the adjoint mass.
To find the mass relevant for the scale matching we should only 
consider the components of $\Phi$ which are coupled to the 
unbroken $SU(2)$. The mass $M_\Phi$ of these components is then given by
\beq
2 M_\Phi=\left. \frac{\pa^2}{(\pa a^1)^2} W(a) \right |
=\left. \frac{\pa}{\pa a^1} (a_1 W_1) \right | =
\left. \left( a_1 \frac{\pa}{\pa a^1} W_1+2 W_1 \right) \right| 
=\left. 2 W_1 \right| ,
\label{massofadj}
\eeq
where $W_1=(\sum_{k=1}^r g_k u_k^{\A_1}) (a) $
and $a^i$ are understood as solutions of the equation of motion (\ref{eq1}).

To proceed further it is convenient to rewrite the equation motion (\ref{eq1})
and the vacuum condition (\ref{su2sol}) with the simple root 
$\alpha_1$ as follows:
\beqa
g_1 : g_2 : \cdots : g_r &=& D_{11} : D_{12} : \cdots : D_{1r},
\CR
a_1& =& 0.
\label{eqmoratio}
\eeqa
The solutions of these equations are expressed as functions 
of the ratio $g_i/g_r$. Then we notice that $J(a)$ defined in 
(\ref{Jzero}) turns out to be
\beq
J=\sum_{k=1}^r \frac{\pa u_k}{\pa a^1} D_{1k} =
\frac{D_{1r}}{g_r} \sum_{k=1}^r g_k\, \frac{\pa u_k}{\pa a^1} =
D_{1r}\, a_1 \frac{W_1}{g_r}.
\eeq
Combining (\ref{J}) and (\ref{Dexpr}) we obtain
\beq
M_\Phi^2 = \left( W_1| \right)^2=
\left( \frac{c_1}{c_2} \right)^2 g_r^2 \prod_{\B \in \Delta_d,\, \B>0} 
a \cdot \B .
\label{MPhi}
\eeq

Upon integrating out the massive adjoint we
relate the scale $\La_H$ with the scale $\La_{YM}$ of the
low-energy $N=1$ $SU(2)$ Yang-Mills theory by
\beq
\La_H^{2 \cdot 2 } = \La_{YM}^{ 3 \cdot 2}/M_\Phi^2 .
\eeq
We finally find from this and (\ref{summaa}), (\ref{MPhi}) that 
the scale matching relation becomes
\beq
\La_{YM}^{ 3 \cdot 2} =g_r^2 \La^{2 h},
\label{reg}
\eeq
where the top Casimir $u_r$ has been rescaled so that we can set $c_1/c_2=1$.

Following the previous discussions and the perturbative 
nonrenormalization theorem for the superpotential, 
we derive the low-energy effective superpotential
\beq
W_L= W_{cl}(g) \pm 2 {\La_{YM}}^3 = W_{cl}(g) \pm 2 g_r \La^{h},
\label{exactW}
\eeq
where the term $\pm 2 {\La_{YM}}^3$ appears as a result of the gaugino 
condensation in low-energy $SU(2)$ theory
and $W_{cl}(g)$ is the tree-level superpotential evaluated at the
classical values $a^i(g)$. We will assume that (\ref{exactW}) is the exact 
effective superpotential valid for all values of parameters.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Determination of singularities and $N=2$ curves}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

The vacuum expectation values of gauge invariants are obtained from $W_L$ 
\beq
\langle u_k \rangle = {\pa W_L \over \pa g_k}=u_k^{cl}(g) 
\pm 2 \La^h \delta_{k, r}.
\label{vevofun}
\eeq
We now wish to show that the expectation values (\ref{vevofun}) parametrize
the singularities of algebraic curves. For this let us introduce
\beq
P_{\cal R}(x,u_k^{cl})={\rm det} (x-\Phi_{\cal R})
\label{charapol}
\eeq
which is the characteristic polynomial in $x$ of order ${\rm dim}\, {\cal R}$ 
where ${\cal R}$ is an irreducible representation of $G$. 
Here $\Phi_{\cal R}$ is a representation 
matrix of ${\cal R}$ and $u_k^{cl}$ are Casimirs built out of $\Phi_{\cal R}$.
The eigenvalues of $\Phi_{\cal R}$ are given in terms of the weights 
$\lambda_i$ of the representation ${\cal R}$. Diagonalizing $\Phi_{\cal R}$
we may express (\ref{charapol}) as
\beq
P_{\cal R}(x,a)=\prod_{i=1}^{{\rm dim}\, {\cal R}} (x-a \cdot \lambda_i),
\eeq
where $a$ is a Higgs v.e.v. vector, the discriminant of which takes the form
\beq
\Delta_{\cal R}
=\left( \prod_{i \neq j} a \cdot (\lambda_i-\lambda_j) \right)^2.
\eeq
It is seen that, for $a$ which is a solution to (\ref{eq1}), 
we have $\Delta_{\cal R}=0$, that is
\beq
P_{\cal R}(x, u_k^{cl}(a))=\partial_x P_{\cal R}(x, u_k^{cl}(a))=0
\label{clasin}
\eeq
for any representation. The solutions of the classical equation of
motion thus give rise to the singularities of the level manifold
$P_{\cal R}(x, u_k^{cl})=0$.

In order to include the quantum effect what we should do is to modify the
top Casimir $u_r$ term so that the gluino condensation in (\ref{vevofun}) is
properly taken into account. We are then led to take a curve
\beq
\tilde P_{\cal R}(x,z,u_k) \equiv
P_{\cal R} \left( x,u_k+\delta_{k,r} \left( z+\frac{\mu}{z}\right) \right) =0,
\label{curve}
\eeq
where $\mu=\La^{2 h}$ and an additional complex variable $z$ has been 
introduced. Let us check the degeneracy of the curve at the expectation
values (\ref{vevofun}), which means to check if
the following three equations hold
\beqa
\tilde P_{\cal R}(x,z,\bra u_k\ket) & =& 0, \\
\partial_x \tilde P_{\cal R}(x,z,\bra u_k\ket) & =& 0, \\
\partial_z \tilde P_{\cal R}(x,z,\bra u_k\ket) & =& 
\left(1-\frac{\mu}{z^2} \right) \pa_{u_r}
\tilde P_{\cal R}(x,z,\bra u_k\ket)=0.
\label{difxi}
\eeqa
The last equation (\ref{difxi}) has an obvious solution $z=\mp \sqrt{\mu}$. 
Substituting this into the first two equations we see that the singularity
conditions reduce to the classical ones (\ref{clasin})
\beqa
\tilde P_{\cal R}(x,\mp \sqrt{\mu},\bra u_k\ket) 
& =& P_{\cal R} \left( x,\bra u_k\ket \mp \delta_{k,r} 2 \sqrt{\mu} \right)
=P_{\cal R} ( x,u_k^{cl} )=0,    \\
\pa_x \tilde P(x,\mp \sqrt{\mu},\bra u_k\ket)
 & =& \pa_x P_{\cal R} \left( x,\bra u_k\ket \mp \delta_{k,r} 
2 \sqrt{\mu} \right)=\pa_x P_{\cal R} ( x,u_k^{cl} )=0.
\eeqa
Thus we have shown that (\ref{vevofun}) parametrize the singularities of
the Riemann surface described by (\ref{curve}) irrespective of the
representation ${\cal R}$.

Let us take the $N=2$ limit by letting all $g_i \rightarrow 0$ 
with the ratio $g_i/g_r$ fixed, then 
(\ref{curve}) is the curve describing the Coulomb phase of 
$N=2$ supersymmetric Yang-Mills theory with ADE gauge groups.
Indeed the curve (\ref{curve}) in this particular form of foliation agrees 
with the one obtained systematically in \cite{MaWa} in view of integrable 
systems \cite{Gor},\cite{NT},\cite{IM}. For $E_6$ and $E_7$ see 
\cite{LW},\cite{WY}. 

Finally we remark that there is a possibility of
(\ref{difxi}) having another solutions besides
$z=\mp \sqrt{\mu}$. If we take the fundamental representation such
solutions are absent for $G=A_r$, and for $G=D_r$ there is a solution
with vanishing degree $r$ Casimir (i.e. Pfaffian),
but it is known that this is an apparent singularity \cite{BrLa}. For
$E_r$ gauge groups there could exist additional solutions.  We expect that 
these singularities are apparent and do not represent physical massless
solitons.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$N=1$ superconformal field theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We will discuss non-trivial fixed points in our $N=1$ theory characterized
by the microscopic superpotential (\ref{tree}). To find critical points we 
rely on the construction of $N=2$ superconformal field theories
realized at particular points in the moduli space of the Coulomb
phase \cite{ArDo},\cite{APSW},\cite{EHIY},\cite{EH}. 
At these $N=2$ critical points mutually non-local massless dyons
coexist. Thus the critical points lie on the singularities in the moduli
space which are parametrized by the $N=1$ expectation values (\ref{vevofun})
as was shown in the previous section. This enables us to adjust the
microscopic parameters in $N=1$ theory to the values of $N=2$ non-trivial
fixed points. Doing so in $N=2$ $SU(3)$ Yang-Mills theory Argyres and Douglas
found non-trivial $N=1$ fixed points \cite{ArDo}. 
We now show that this class of $N=1$
fixed points exists in all ADE $N=1$ theories in general. 

Let us start with rederiving $N=2$ critical behavior based on the curve 
(\ref{curve}). An advantage of using the curve (\ref{curve}) is that one can
identify higher critical points and determine the critical exponents 
independently of the details of the curve. 

If we set $z= \mp \sqrt{\mu}$ the condition for higher critical points is
\beq
P_{\cal R}(x,u_k^{cl})=\pa_x^n P_{\cal R}(x,u_k^{cl})=0
\eeq
with $n>2$. Hence there exist higher critical points at 
$u_k=u_k^{sing} \pm 2 \La^h \D_{k,r}$ where $u_k^{sing}$ are the classical 
values of $u_k$ for which the gauge group $H$ with rank larger than one is
left unbroken. The highest critical point corresponding to the unbroken $G$
is located at $u_k=\pm 2 \La^h \D_{k,r}$.

Near the highest critical point the curve (\ref{curve}) behaves as
\beq
u_r+z+{\mu \over z}=c\, x^h+\delta u_k \, x^j,
\label{critcurve}
\eeq
where the second term on the rhs with $j=h-(e_k+1)$ represents a small
perturbation around the criticality at $\delta u_k=0$. 
A constant $c$ is irrelevant
and will be set to $c=1$. Let $u_r=\pm 2\La^h, x=\delta u_k^{1/(h-j)}s$ and
$z \pm \La^h=\rho$, then (\ref{critcurve}) becomes
\beq
\rho \simeq \delta u_k^{h \over 2(h-j)} 
(\mp \La^h)^{1\over 2} (s^h+s^j)^{1\over 2}.
\eeq
We now apply the technique of \cite{EHIY} to verify the scaling behavior of
the period integral of the Seiberg-Witten differential $\lambda_{SW}$.
For the curve (\ref{curve}) it is known that $\lambda_{SW}=xdz/z$. Near the
critical value $z=\mp \sqrt{\mu}$ we evaluate 
\beqa
\oint \lambda_{SW}&=&\oint x{dz \over z} \simeq \oint x d\rho  \CR
&\simeq & \delta u_k^{h+2 \over 2(h-j)} \oint ds 
{hs^h+js^j \over (s^h+s^j)^{1/2}}.
\label{period}
\eeqa
Since the period has the mass dimension one we read off critical exponents
\beq
{2\, (e_k+1) \over h+2}, \hskip10mm k=1,2,\cdots ,r
\eeq
in agreement with the results obtained earlier for $N=2$ ADE Yang-Mills 
theories \cite{EHIY},\cite{EH}.

When our $N=1$ theory is viewed as $N=2$ theory perturbed by the tree-level
superpotential (\ref{tree}) we understand that the mass gap in $N=1$ theory
arises from the dyon condensation \cite{SeWi1}. Let us show that the dyon
condensate vanishes as we approach the $N=2$ highest critical point under
$N=1$ perturbation. The $SU(2)\times U(1)^{r-1}$ vacuum in $N=1$ theory 
corresponds to the $N=2$ vacuum where a single monopole or dyon becomes
massless. The low-energy effective superpotential takes the form
\beq
W_m=\sqrt{2} AM\tilde M+\sum_{k=1}^rg_kU_k,
\eeq
where $A$ is the $N=1$ chiral superfield in the $N=2$ $U(1)$ vector multiplet,
$M, \tilde M$ are the $N=1$ chiral superfields of an $N=2$ dyon hypermultiplet
and $U_k$ represent the superfields corresponding to Casimirs $u_k(\Phi)$.
We will use lower-case letters to denote the lowest components of the
corresponding upper-case superfields. Note that $\bra a\ket =0$ in the vacuum
with a massless soliton.

The equation of motion $dW_m=0$ is given by
\beq
-{g_k\over \sqrt{2}}={\pa A\over \pa U_k}M\tilde M, \hskip10mm 1\leq k\leq r
\label{eqmo}
\eeq
and $AM=A\tilde M=0$, from which we have
\beq
{g_k \over g_r}={\pa a / \pa u_k \over
\pa  a / \pa  u_r  }, \hskip10mm 1\leq k\leq r-1,
\eeq
when $\bra a\ket =0$. The vicinity of $N=2$ highest criticality may be
parametrized by
\beq
\bra u_k\ket =\pm 2\La^h \delta_{k,r}+c_k \, \epsilon^{e_k+1}, \hskip10mm 
c_k={\rm constant},
\eeq
where $\epsilon$ is an overall mass scale. From (\ref{period}) one obtains
\beq
{\pa  a \over \pa  u_k } \simeq \epsilon^{{h\over 2}-e_k},
\hskip10mm 1\leq k\leq r,
\eeq
so that
\beq
{g_k \over g_r} \simeq \epsilon^{h-e_k-1} \longrightarrow 0, 
\hskip10mm 1\leq k\leq r-1
\eeq
as $\epsilon \rightarrow 0$. The scaling behavior of dyon condensate
is easily derived from (\ref{eqmo})
\beq
\bra m\ket =\Big( -{g_r \over \sqrt{2} \pa  a / \pa  u_r  }
\Big)^{1/2} \simeq \sqrt{g_r}\, \epsilon^{(h-2)/4} \longrightarrow 0.
\eeq
Therefore the gap in the $N=1$ confining phase vanishes. We thus find that
$N=1$ ADE gauge theory with an adjoint matter with a tree-level superpotential
\beq
W_{\rm crit}= g_{r} u_r(\Phi) 
\label{Wcrit}
\eeq
exhibits non-trivial fixed points. The higher-order polynomial $u_r(\Phi)$
is a dangerously irrelevant operator which is irrelevant at the UV gaussian
fixed point, but affects the long-distance behavior significantly
\cite{KuScSe}.





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{$N=2$ Gauge Theory with  Matter Multiplets} \label{IMM}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this chapter, we extend our analysis to describe the Coulomb phase of 
$N=1$ supersymmetric gauge theories 
with $N_f$ flavors of 
chiral matter multiplets $Q^i,\tQ_j$ ($1 \leq i,j \leq N_f$) in addition to
the adjoint matter $\Phi$.
Here $Q$ belongs to an irreducible representation 
${\cal R}$ of the gauge group $G$ with the dimension $d_R$ 
and $\tQ$ belongs to the conjugate representation of ${\cal R}$.
A tree-level superpotential consists of 
the Yukawa-like term $\tilde{Q} \Phi^{l} Q$ in addition to the Casimir
terms built out of $\Phi$, and we shall consider arbitrary classical gauge
groups and ADE gauge groups. 
In the appropriate limit the theory is reduced to $N=2$ supersymmetric 
QCD. 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Classical gauge groups and fundamental matters}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


We start with discussing $N=1$ $SU(N_c)$ supersymmetric gauge theory
with an adjoint matter field $\p$, $N_f$ flavors of fundamentals $Q$ and
anti-fundamentals $\tilde{Q}$.
We take a tree-level superpotential 
\beq
W=\sum_{n=1}^{N_c} g_n u_n
+\sum_{l=0}^{r} {\rm Tr}_{N_f} \, \lm_l \, \tilde{Q} \p^l Q, 
\hskip10mm  u_n=\frac{1}{n} {\rm Tr}\,  \p^n ,
\label{r1a}
\eeq
where ${\rm Tr}_{N_f} \, \lm_l \, \tilde{Q} \p^l Q = 
\sum_{i,j=1}^{N_f} (\lm_l)^i_j  \tilde{Q}_i \p^l Q^j$ and $r \leq N_c-1$.
If we set $(\lm_0)^i_j=m^i_j$ with $[m, m^{\dagger}]=0$, 
$(\lm_1)^i_j=\D^i_j , \, 
(\lm_l)^i_j=0 $ for $l>1$ and all $g_i=0$, eq.(\ref{r1a}) recovers the
superpotential in $N=2$ $SU(N_c)$ supersymmetric QCD with quark mass $m$.
The second term in (\ref{r1a}) was considered in a recent work \cite{Ka}.

Let us focus on the classical vacua with $ Q=\tilde{Q}=0$ and an
unbroken $SU(2) \times U(1)^{N_c-2}$ symmetry
which means $\p  ={\rm diag} (a_1, a_1, a_2, a_3, \cdots , a_{N_c-1})$ 
up to gauge transformations. 
(Note that the superpotential (\ref{r1a}) has no classical vacua with unbroken 
$U(1)^{N_c-1}$.) In this vacuum, we will evaluate semiclassicaly 
the low-energy effective superpotential.
Our procedure is slightly different from that adopted in \cite{ElFoGiInRa}
upon treating $Q$ and $\tilde{Q}$.
We investigate the tree-level parameter region where
the Higgs mechanism occurs at very high energies and 
the adjoint matter field $\p$ is quite heavy.
Then the massive particles are integrated out and 
the scale matching relation becomes
\beq
{\Lm_L}^{6-N_f} = g_{N_c}^2 \Lm^{2 N_c-N_f},
\label{sumatch}
\eeq
where $\Lm$ is the
dynamical scale of high-energy $SU(N_c)$ theory with $N_f$ flavors
and $\Lm_L$ is the scale of low-energy $SU(2)$ theory with $N_f$ flavors.
Eq.(\ref{sumatch}) is derived by following the $SU(N_c)$ 
Yang-Mills case \cite{ElFoGiInRa} while taking into account the existence of 
fundamental flavors at low energies \cite{KuScSe}.

The semiclassical superpotential in low-energy $SU(2)$ theory with $N_f$ 
flavors reads
\beq
W=\sum_{n=1}^{N_c} g_n u_n^{cl} +
\sum_{l=0}^{r} a_1^l \, {\rm Tr}_{N_f} \, \lm_l \, \tilde{Q} Q
\label{w1}
\eeq
which is obtained by substituting the classical values of $\p$
and integrating out all the fields except for those coupled to 
the $SU(2)$ gauge boson.
Here, the constraint ${\rm Tr} \p^{cl}=a_1+\sum_{i=1}^{N_c-1} a_i=0$ and
the classical equation of motion $\sum_{i=1}^{N_c-1} a_i
=-g_{N_c-1}/g_{N_c}$ yield \cite{Ki}
\beq
a_1= \frac{g_{N_c-1}}{g_{N_c}}.
\eeq
Below the flavor masses which can be read off from the superpotential 
(\ref{w1}), the low-energy theory becomes $N=1$ $SU(2)$ Yang-Mills theory 
with the superpotential
\beq
W=\sum_{n=1}^{N_c} g_n u_n^{cl}.
\label{w2}
\eeq
This low-energy theory 
has the dynamical scale $\Lm_{YM}$ which is related to $\Lm$ through
\beq
{\Lm_{YM}}^{6} = {\rm det} \left ( \sum_{l=0}^{r} \lm_l a_1^l \right ) \,
g_{N_c}^2 \Lm^{2 N_c-N_f}.
\label{sc1}
\eeq

As in the previous literatures \cite{ElFoGiInRa},\cite{TeYa1}
we simply assume here that 
the superpotential (\ref{w2}) and the scale matching relation (\ref{sc1}) 
are exact for any values of the tree-level parameters.
Now we add to (\ref{w2}) a dynamically generated piece which arises from 
gaugino condensation in $SU(2)$ Yang-Mills theory.
The resulting effective superpotential $W_L$ where all the matter fields have
been integrated out is thus given by
\beqa
W_L & = & \sum_{n=1}^{N_c} g_n u_n^{cl} \pm 2 \Lm_{YM}^3 \CR
  & = & \sum_{n=1}^{N_c} g_n u_n^{cl} \pm 2 g_{N_c} \sqrt{A(a_1)}
\label{w3}
\eeqa
with $A$ being defined as $A(x) \equiv \Lm^{2 N_c-N_f} \,
{\rm det} \left ( \sum_{l=0}^{r} \lm_l x^l \right ) $. From 
$\langle u_n \rangle = \partial W_L/\partial g_n$ we find
\beq
\langle u_n \rangle = u_n^{cl} (g) \pm \D_{n,N_c-1} 
\frac{A'(a_1)}{\sqrt{A(a_1)}}
\pm \D_{n,N_c}  \frac{1}{\sqrt{A(a_1)}} \left ( 2 A(a_1) -a_1 A'(a_1) \right).
\label{v1}
\eeq

If we define a hyperelliptic curve
\beq
y^2= P(x)^2 -4 A(x),
\label{c1}
\eeq
where $P(x)=\left \langle {\rm det} \left ( x- \p \right ) \right \rangle$
is the characteristic equation of $\p$, the curve is quadratically 
degenerate at the vacuum expectation values (\ref{v1}).
This can be seen by plugging (\ref{v1}) in $P(x)$
\beq
P(x)=P_{cl} (x) \mp x  \frac{A'(a_1)}{\sqrt{A(a_1)}}
\mp \frac{1}{\sqrt{A(a_1)}} \left ( 2 A(a_1) -a_1 A'(a_1) \right),
\eeq
where $P_{cl}(x)={\rm det} \left ( x- \p_{cl} \right )$, and hence
\beq
P(a_1)= \mp 2 {\sqrt{A(a_1)}} \, , \hskip10mm
P'(a_1)= \mp \frac{A'(a_1)}{\sqrt{A(a_1)}}.
\eeq
Then the degeneracy of the curve is confirmed by checking
$ y^2|_{x=a_1}=0$ and 
$ \frac{\pa}{\pa x} y^2 |_{x=a_1} = 0$.

The transition points from the confining to the Coulomb phase are reached by
taking the limit $g_{i} \rightarrow 0$ while keeping the ratio $g_i/g_{j}$ 
fixed \cite{ElFoGiInRa}. 
These points correspond to the singularities in the moduli space.
Therefore the curve (\ref{c1}) is regarded as 
the curve relevant to describe the Coulomb phase of the theory with the
tree-level superpotential 
$W=\sum_{l=0}^{r} {\rm Tr}_{N_f} \, \lm_l \, \tilde{Q} \p^l Q$.
Indeed, the curve (\ref{c1}) agrees with the one obtained in \cite{Ka}.
Especially in the parameter region that has $N=2$ supersymmetry,
we find agreement with the curves for $N=2$ $SU(N_c)$ QCD with 
$N_f<2 N_c-1$ \cite{HaOz},\cite{ArPlSh},\cite{ArSh}.\footnote[2]{For 
$N_f=2 N_c-1$ an instanton may generate a mass term and shift
the bare quark mass in $A(x)$. If we include this effect the curve (\ref{c1})
completely agrees with the result in \cite{ArSh}.}

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{$SO(2 N_c)$ case}

The procedure discussed above can be also applied to the other classical
gauge groups.
Let us consider $N=1$ $SO(2 N_c)$ supersymmetric gauge theory
with an adjoint matter field $\p$ 
which is an antisymmetric $2 N_c \times 2 N_c$ tensor, and $2 N_f$ flavors 
of fundamentals $Q$. We assume a tree-level superpotential 
\beq
W=\sum_{n=1}^{N_c-2} g_{2 n} u_{2 n} + g_{2 (N_c-1)} s_{N_c-1}+\lm v
+{1\over 2}\sum_{l=0}^{r} {\rm Tr}_{2 N_f} \, \lm_l \, Q \p^l Q, 
\label{w4}
\eeq
where $r \leq 2 N_c-1$,
\beq
u_{2 n} =\frac{1}{2 n} {\rm Tr}\, \Phi^{2 n}, \hskip10mm
v ={\rm Pf}\, \Phi=\frac{1}{2^{N_c} N_c !} \E_{i_1 i_2 j_1 j_2 \cdots}
\Phi^{i_1 i_2} \Phi^{j_1 j_2} \cdots  
\eeq
and
\beq
ks_k+\sum_{i=1}^k i s_{k-i} u_{2i}=0, \hskip10mm s_0=-1, 
\hskip10mm k=1,2,\cdots .
\eeq
Here, ${}^t \lm_l=(-1)^l \lm_l$ and the $N=2$ supersymmetry is present
when we set $(\lm_0)^i_j=m^i_j$ where $[m, m^{\dagger}]=0$, 
$(\lm_1)^i_j={\rm diag} (i \s_2,i \s_2, \cdots )$ with 
$\s_{2} = \pmatrix{0 & -i \cr i & 0}, \, 
(\lm_l)^i_j=0 $ for $l>1$ and all $g_i=0$. 

As in the case of $SU(N_c)$, 
we concentrate on the unbroken $SU(2) \times U(1)^{N_c-1}$ vacua with
$\p  ={\rm diag} 
(a_1 \s_2, a_1 \s_2 , a_2 \s_2, a_3 \s_2, \cdots , a_{N_c-1} \s_2)$
and $Q=0$.
By virtue of using $s_{N_c}$ instead of $u_{2 N_c}$ in (\ref{w4})
the degenerate eigenvalue of $\p_{cl}$ is expressed by $g_i$
\beq
a_1^2=\frac{g_{2(N_c-2)}}{g_{2(N_c-1)}}
\eeq
as found for the $SO(2 N_c+1)$ case \cite{TeYa1}.
Note that the superpotential (\ref{w4}) has
no classical vacua with unbroken $SO(4) \times U(1)^{N_c-1}$ 
when $g_{2 (N_c-2)} \neq 0$.
We also note that 
the fundamental representation of $SO(2 N_c)$ is decomposed into 
two fundamental representations of $SU(2)$ under the above embedding.
It is then observed that 
the scale matching relation between the high-energy $SO(2 N_c)$ scale $\Lm$ and
the scale $\Lm_L$ of low-energy $SU(2)$ theory with $2 N_f$ fundamental 
flavors is given by
\beq
{\Lm_L}^{6-2 N_f} = g_{2(N_c-1)}^2 \Lm^{4( N_c-1) -2 N_f}.
\eeq

The superpotential for low-energy $N=1$ $SU(2)$ QCD with $2 N_f$ flavors
can be obtained in a similar way to the $SU(N_c)$ case, but the duplication
of the fundamental flavors are taken into consideration. 
After some manipulations it turns out that the superpotential for 
low-energy $N=1$ $SU(2)$ QCD with $2 N_f$ flavors is written as
\beq
W=\sum_{n=1}^{N_c-2} g_{2 n} u_{2 n}^{cl} 
+ g_{2 (N_c-1)} s_{N_c-1}^{cl}+\lm v^{cl}
+\sum_{l=0}^{r} a_1^l {\rm Tr}_{2 N_f} \, \lm_l \, \widetilde{{\bf Q}} {\bf Q},
\label{w5}
\eeq
where
\beq
{\bf Q}^j = {1\over \sqrt{2}} \pmatrix{Q^j_1-iQ^j_2 \cr
                     Q^j_3-iQ^j_4},     \hskip10mm
\widetilde{\bf Q}_j= {1\over \sqrt{2}}\pmatrix{Q^j_1+iQ^j_2 \cr
                          Q^j_3+iQ^j_4}.
\eeq
Upon integrating out the $SU(2)$ flavors we have
the scale matching  between 
$\Lm$ and $\Lm_{YM}$ for $N=1$ $SU(2)$ Yang-Mills theory 
\beq
{\Lm_{YM}}^{6} = {\rm det} \left ( \sum_{l=0}^{r} \lm_l a_1^l \right ) \,
g_{2(N_c-1)}^2 \Lm^{4( N_c-1)-2 N_f},
\label{sc2}
\eeq
and we get the effective superpotential 
\beqa
W_L & = & \sum_{n=1}^{N_c-2} g_n u_n^{cl} 
+ g_{2 (N_c-1)} s_{N_c-1}^{cl}+\lm v^{cl}
\pm 2 \Lm_{YM}^3 \CR
  & = & \sum_{n=1}^{N_c-2} g_n u_n^{cl} 
+ g_{2 (N_c-1)} s_{N_c-1}^{cl}+\lm v^{cl} \pm 2 g_{2(N_c-1)} \sqrt{A(a_1)},
\label{wso}
\eeqa
where $A$ is defined by $A(x) \equiv \Lm^{4( N_c-1)-2 N_f} \,
{\rm det} \left ( \sum_{l=0}^{r} \lm_l x^l \right ) =A(-x)$.

The vacuum expectation values of gauge invariants are obtained from $W_L$ as
\beqa
\langle s_{ n} \rangle & =&  s_{ n}^{cl} (g) 
\pm \D_{n,N_c-2} \frac{A'(a_1)}{\sqrt{A(a_1)}}
\pm \D_{n,N_c-1}  \frac{1}{\sqrt{A(a_1)}} 
\left ( 2 A(a_1) -a_1^2 A'(a_1) \right),    \CR
\langle v \rangle & =&  v^{cl} (g),
\label{vso}
\eeqa
where $A'(x)=\frac{\pa}{\pa x^2} A(x)$.
It is now easy to see that a curve 
\beq
y^2=P(x)^2-4 x^4 A(x) 
\eeq
with $P(x)=\left \langle {\rm det} \left ( x-\p \right ) \right \rangle$
is degenerate at these values of $\langle s_n \rangle, \, \langle v \rangle$, 
and reproduces the known $N=2$ curve \cite{Ha}, \cite{ArSh}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{$SO(2 N_c+1)$ case}

The only difference between $SO(2N_c)$ and $SO(2 N_c+1)$ is that 
the gauge invariant ${\rm Pf}\, \p$ vanishes for $SO(2 N_c+1)$.
Thus, taking a tree-level superpotential 
\beq
W=\sum_{n=1}^{N_c-1} g_{2 n} u_{2 n} + g_{2 N_c} s_{N_c}
+{1\over 2} \sum_{l=0}^{r} {\rm Tr}_{2 N_f} \, \lm_l \, Q \p^l Q , 
\hskip10mm  r \leq 2N_c ,
\label{wsoo}
\eeq
we focus on the unbroken $SU(2) \times U(1)^{N_c-1}$ vacuum which has 
the classical expectation values 
$\p  ={\rm diag} (a_1 \s_2, a_1 \s_2 , a_2 \s_2, \cdots , a_{N_c-1} \s_2,0)$ 
and $Q=0$ \cite{TeYa1}.
As in the $SO(2 N_c)$ case we make use of
the scale matching relation between the high-energy scale $\Lm$ and
the low-energy $N=1$ $SU(2)$ Yang-Mills scale $\Lm_{YM}$ 
\beq
{\Lm_{YM}}^{6} = {\rm det} \left ( \sum_{l=0}^{r} \lm_l a_1^l \right ) \,
g_{2 N_c} g_{2(N_c-1)} \Lm^{2(2 N_c-1- N_f)}.
\label{sc3}
\eeq
As a result we find the effective superpotential 
\beqa
W_L & = & \sum_{n=1}^{N_c-1} g_{2n} u_n^{cl} + g_{2 N_c} s_{N_c}^{cl}
\pm 2 \Lm_{YM}^3 \CR
  & = & \sum_{n=1}^{N_c-1} g_{2n} u_n^{cl} + g_{2 N_c} s_{N_c}^{cl}
\pm 2 \sqrt{g_{2 N_c} g_{2(N_c-1)} A(a_1)},
\label{wsoo2}
\eeqa
where $A$ is defined as $A(x) \equiv \Lm^{2( 2 N_c-1- N_f)}\,
{\rm det} \left ( \sum_{l=0}^{r} \lm_l x^l \right )$.

Noting the relation $a_1^2=g_{2(N_c-1)}/g_{2 N_c}$ \cite{TeYa1}
we calculate the vacuum expectation values of gauge invariants 
\beqa
\langle s_{ n} \rangle =  s_{ n}^{cl} (g) 
& \pm & \D_{n,N_c-1} \frac{1}{\sqrt{A(a_1)}} 
\left (  \frac{A(a_1)}{a_1}+a_1 A'(a_1) \right) \CR
& \pm & \D_{n,N_c}  \frac{1}{\sqrt{A(a_1)}} 
\left ( a_1 A(a_1) -a_1^3 A'(a_1) \right).
\label{vsoo}
\eeqa
For these $\langle s_n \rangle$ we observe 
the quadratic degeneracy of the curve
\beq
y^2=\left( \frac{1}{x} P(x) \right )^2-4 x^2 A(x) ,
\eeq
where $P(x)=\left \langle {\rm det} \left ( x-\p \right ) \right \rangle$.
In the $N=2$ limit we see agreement with the curve constructed in 
\cite{Ha},\cite{ArSh}. The confining phase superpotential for the $SO(5)$ gauge
group was obtained also in \cite{LaPiGi}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\section{$Sp(2 N_c)$ case}

Let us now turn to $Sp(2 N_c)$ gauge theory.
We take for matter content an adjoint field $\p$
and $2 N_f$ fundamental fields $Q$.
The $2N_c \times 2N_c$ tensor $\p$ is subject to ${}^t \Phi = J \Phi J $ with
$J={\rm diag}(i\sigma_2, \cdots, i\sigma_2)$.
Our tree-level superpotential reads
\beq
W=\sum_{n=1}^{N_c-1} g_{2 n} u_{2 n} + g_{2 N_c} s_{N_c}
+{1\over 2} \sum_{l=0}^{r} {\rm Tr}_{2 N_f} \, \lm_l \, Q J \p^l Q,
\label{wsp}
\eeq
where ${}^t \lm_l=(-1)^{l+1} \lm_l$ and $r \leq 2N_c-1$.
The classical vacuum with the unbroken $SU(2) \times U(1)^{N_c-1}$ gauge group
corresponds to
\beq
J \Phi=
{\rm diag}(\s_{1}a_1,\; \s_{1}a_1,\; \s_{1}a_2,\cdots,\s_{1}a_{N_c-1}) ,
\hskip10mm \s_{1}
=\left( \begin{array}{cc} 0 &  1 \\ 1 & 0 \end{array} \right ),
\eeq
where $a_1^2=g_{2(N_c-1)}/g_{2N_c}$.
The scale $\Lm_L$ for low-energy $SU(2)$ theory with $2 N_f$ flavors
is expressed as \cite{TeYa1}
\beq
{\Lm_L}^{6-2 N_f} = \left( \frac{g_{2 N_c}^2}{g_{2 (N_c-1)}} \right)^2 
\Lm^{2(2N_c+2- N_f)}.
\label{embed}
\eeq

There exists a subtle point in the analysis of $Sp(2 N_c)$ theory. When
$Sp(2 N_c)$ is broken to $SU(2) \times U(1)^{N_c-1}$ the instantons in
the broken part of the gauge group play a role since the index of the embedding
of the unbroken $SU(2)$ in $Sp(2 N_c)$ is larger than one 
(see eq.(\ref{embed})) \cite{InSe2},\cite{Aff}. 
The possible instanton contribution to $W_L$ will be of the 
same order in $\Lm$ as low-energy $SU(2)$ gaugino condensation.
Therefore even in the lowest quantum corrections
the instanton term must be added to $W_L$.

For clarity we begin with discussing $Sp(4)$ Yang-Mills theory.
In this theory by the symmetry and holomorphy
the effective superpotential is determined to take the form 
$W_L=f \left( \frac{g_4}{g_2} \Lm^2 \right) \frac{g_4^2}{g_2} \Lm^6$
with $f$ being certain holomorphic function.
If we assume that there is  only one-instanton effect,
the precise form of $W_L$ including the low-energy gaugino 
condensation effect may be given by
\beq
W_L= 2 \frac{g_4^2}{g_2} \Lm^6 \pm 2 \frac{g_4^2}{g_2} \Lm^6,
\eeq 
as in the case of $SO(4) \simeq SU(2) \times SU(2)$ breaking to 
the diagonal $SU(2)$. This is due to the fact $Sp(4) \simeq SO(5)$ and the
natural embedding of $SO(4)$ in $SO(5)$. Our low-energy $SU(2)$ gauge
group is identified with the one diagonally embedded in 
$SO(4) \simeq SU(2) \times SU(2)$ \cite{InSe2},\cite{ILS}. Accordingly, 
in $Sp(2 N_c)$ Yang-Mills theory, we first make the matching at the scale of 
$Sp(2 N_c)/Sp(4)$ $W$ bosons by taking all the $a_1-a_i$ large. 
Then the low-energy superpotential is found to be
\beq
W_L=W_{cl}+2  \frac{g_{2 N_c}}{a_1^2} \Lm^{2(N_c+1)}
\pm 2 \frac{g_{2 N_c}}{a_1^2} \Lm^{2(N_c+1)}.
\eeq 

Let us turn on the coupling to fundamental flavors $Q$ and evaluate
the instanton contribution. When flavor masses vanish there is a global
$O(2N_f) \simeq SO(2N_f)\times {\bf Z}_2$ symmetry. The couplings $\lambda_l$
and instantons break a ``parity'' symmetry ${\bf Z}_2$. We treat this 
${\bf Z}_2$ as unbroken by assigning odd parity to the instanton factor
$\Lm^{2 N_c+2- N_f}$ and $O(2N_f)$ charges to $\lambda_l$. Symmetry
consideration then leads to the one-instanton factor proportional to
$B(a_1)$ where
\beq
B(x)=\Lm^{2 N_c+2- N_f} {\rm Pf}
\left(\sum_{l\, {\rm even}} \lambda_{l}x^{l}\right).
\eeq
Note that $B(x)$ is parity invariant since Pfaffian has odd parity.
Thus the superpotential for low-energy $N=1$ $SU(2)$ QCD with $2 N_f$ flavors
including the instanton effect turns out to be
\beq
W=\sum_{n=1}^{N_c-1} g_{2 n} u_{2 n}^{cl} + g_{2 N_c} s_{N_c}^{cl}
+\sum_{l=0}^{r} a_1^l {\rm Tr}_{2 N_f} \, \lm_l \, \widetilde{{\bf Q}} {\bf Q}
+2  \frac{g_{2 N_c}^2}{g_{2 (N_c-1)}} B(a_1),
\label{wsp2}
\eeq
where
\beq
{\bf Q}^j = \pmatrix{Q^j_1 \cr
                     Q^j_3},     \hskip10mm
\widetilde{\bf Q}_j= \pmatrix{Q^j_2 \cr
                          Q^j_4}.
\eeq
When integrating out the $SU(2)$ flavors,
the scale matching relation between $\Lm$ and the scale $\Lm_{YM}$ 
of $N=1$ $SU(2)$ Yang-Mills theory becomes
\beq
{\Lm_{YM}}^{6} = {\rm det} \left ( \sum_{l=0}^{r} \lm_l a_1^l \right ) \,
\left(  \frac{g_{2 N_c}^2}{g_{2 (N_c-1)}} \right)^2 
\Lm^{2(2N_c+2- N_f)},
\label{sc4}
\eeq
and we finally obtain the effective superpotential 
\beqa
W_L & = & \sum_{n=1}^{N_c-1} g_n u_n^{cl} + g_{2 N_c} s_{N_c}^{cl}
\pm 2 \Lm_{YM}^3 +2  \frac{g_{2 N_c}^2}{g_{2 (N_c-1)}} B(a_1) \CR
  & = & \sum_{n=1}^{N_c-1} g_n u_n^{cl} + g_{2 N_c} s_{N_c}^{cl}
+2 \frac{g_{2 N_c}^2}{g_{2 (N_c-1)}}  
\left(B(a_1) \pm \sqrt{A(a_1)} \right),
\label{wsp3}
\eeqa
where $A(x) \equiv \Lm^{2( 2 N_c+2- N_f) }\,
{\rm det} \left ( \sum_{l=0}^{r} \lm_l x^l \right )$.

The gauge invariant expectation values $\langle s_n \rangle$ are 
\beqa 
\langle s_{ n} \rangle =  s_{ n}^{cl} (g) 
& +& \D_{n,N_c-1} \frac{1}{a_1^4} \left(
-2 B(a_1)+2a_1^2B'(a_1)  \pm \frac{1}{\sqrt{A(a_1)}} 
\left( -2 A(a_1) + a_1^2 A'(a_1) \right) \right) \CR
& + & \D_{n,N_c}  \frac{1}{a_1^2} \left( 
4 B(a_1)-2a_1^2B'(a_1) \pm  \frac{1}{\sqrt{A(a_1)}} 
\left( 4 A(a_1)  - a_1^2 A'(a_1) \right) \right).
\label{vsp}
\eeqa
Substituting these into a curve 
\beq
x^2 y^2 = \left( x^2 P(x) +2 B(x) \right)^2-4 A(x),
\label{spcurve}
\eeq
we see that the curve is degenerate at (\ref{vsp}). In this case too 
our result (\ref{spcurve}) agrees with the $N=2$ curve obtained in \cite{ArSh}.

Before concluding this section, we should note that 
the effective superpotentials $W_L$ obtained in this section 
are also confirmed in the approach based on the brane dynamics
\cite{DeOz,Te}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{ADE gauge groups and various matters} \label {secade}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let us consider $N=1$ gauge theory with the ADE gauge group
and $N_f$ flavors of chiral matter multiplets $Q^i,\tQ_j$
in addition to
the adjoint matter $\Phi$.
We take a tree-level superpotential
\beq
W = \sum_{k=1}^{r} g_{k} u_{k}(\Phi) + 
\sum_{l=0}^{q} {\rm Tr}_{N_f} \, \gamma_l \, \tilde{Q} \Phi_R^l Q,
\label{treem}
\eeq
where $\p_R$ is a $d_R \times d_R$ matrix representation of $\Phi$ in 
${\cal R}$ and $(\gamma_l)_{ij}$, $1 \leq i,j \leq N_f$, are
the coupling constants
%${\rm Tr}_{N_f} \, \gamma_l \, \tilde{Q} \p_R^l Q = 
%\sum_{i,j=1}^{N_f} (gamma_l)^i_j  \tilde{Q}_i \p_R^l Q^j$
and $q$ should be restricted so that $\tilde{Q} \p_R^l Q$ is irreducible
in the sense that it cannot be factored into gauge invariants.
If we set $(\gamma_0)^i_j=m^i_j$ with $[m, m^{\dagger}]=0$, 
$(\gamma_1)^i_j=\sqrt{2} \D^i_j , \, 
(\gamma_l)^i_j=0 $ for $l>1$ and all $g_i=0$, (\ref{treem}) reduces to the
superpotential in $N=2$ supersymmetric Yang-Mills theory with massive
$N_f$ hypermultiplets. 

Let us focus on the classical vacua of the Coulomb phase with $Q=\tilde{Q}=0$ 
and an unbroken $SU(2) \times U(1)^{r-1}$ gauge group symmetry.
The vacuum condition for $\Phi$ is given by (\ref{eqmoratio})
and the classical vacuum takes the form as in the Yang-Mills case
\beq
\p_R = {\rm diag} (a \cdot \lm_1,a \cdot \lm_2, \cdots, a \cdot \lm_{d_R}),
\eeq
where $\lm_i$ are the weights of the representation ${\cal R}$.
In this vacuum, we will evaluate semiclassically 
the low-energy effective superpotential
in the tree-level parameter region where
the Higgs mechanism occurs at very high energies and 
the adjoint matter field $\Phi$ is quite heavy.
Then the massive particles are integrated out and 
we get low-energy $SU(2)$ theory with flavors. 

This integrating-out process
results in the scale matching relation which is essentially the same as the
the Yang-Mills case (\ref{reg}) except that we here have to take into 
account flavor loops. The one instanton factor in high-energy theory 
is given by  $\La^{2 h- l({\cal R}) N_f}$.
Here the index $l({\cal R})$ of the representation ${\cal R}$ is defined by 
$l({\cal R}) \D_{ab}={\rm Tr}( T_a T_b)$ where $T_a$ is
the representation matrix of ${\cal R}$ with root vectors normalized as
$\alpha^2=2$. The index is always an integer \cite{Sla}.
The scale matching relation becomes
\beq
\La_L^{ 3 \cdot 2-l({\cal R}) N_f} =g_r^2 \La^{2 h - l({\cal R}) N_f},
\label{sumatcha}
\eeq
where $\La_L$ is the scale of low-energy $SU(2)$ theory with massive flavors.

To consider the superpotential for low-energy $SU(2)$ theory with $N_f$
flavors we decompose the matter representation ${\cal R}$ of $G$ in terms
of the $SU(2)$ subgroup. We have
\beq
{\cal R} =  \bigoplus_{s=1}^{n_{\cal R}} {\cal R}_{SU(2)}^s
\oplus {\rm singlets},
\eeq
where ${\cal R}_{SU(2)}^s$ stands for a non-singlet $SU(2)$ representation.
Accordingly $Q^i$ is decomposed into $SU(2)$ singlets and ${\bf Q}^i_s$
($1 \leq i \leq N_f$, $1 \leq s \leq n_{\cal R}$) 
in an $SU(2)$ representation ${\cal R}_{SU(2)}^s$. $\tilde{Q}_i$ is
decomposed in a similar manner. The singlet components are decoupled in
low-energy $SU(2)$ theory.

The semiclassical superpotential for $SU(2)$ theory with $N_f$ 
flavors is now given by
\beq
W=\sum_{k=1}^{r} g_k u_k^{cl} +
\sum_{s=1}^{n_{\cal R}} 
\sum_{l=0}^{q} (a \cdot \lm_{{\cal R}_s})^l \, 
{\rm Tr}_{N_f} \, \gamma_l \, \tilde{{\bf Q}}_s {\bf Q}_s,
\label{w1a}
\eeq
where $\lm_{{\cal R}_s}$ is a weight of ${\cal R}$ which branches to 
the weights in ${\cal R}_{SU(2)}^s$. Here we assume that ${\cal R}$ is
a representation which does not break up into integer spin representations
of $SU(2)$; otherwise we would be in trouble when $\gamma_0=0$.
The fundamental representations of ADE groups
except for $E_8$ are in accord with this assumption.

We now integrate out massive flavors to obtain low-energy $N=1$ $SU(2)$ 
Yang-Mills theory with the dynamical scale $\La_{YM}$. Reading off the flavor
masses from (\ref{w1a}) we get the scale matching
\beqa
\La_{YM}^{3\cdot 2} & =&  g_r^2 A(a), \CR
A(a) & \equiv & 
\La^{2 h - l({\cal R}) N_f}
\prod_{s=1}^{n_{\cal R}} \left\{ {\rm det} \left( 
\sum_{l=0}^{q} \gamma_l (a \cdot \lm_{{\cal R}_s})^l
\right)^{l({{\cal R}_{SU(2)}^s})} \right\},
\label{smr}
\eeqa
where $l({{\cal R}_{SU(2)}^s})$ is the index of ${\cal R}_{SU(2)}^s$ which is
related to $l({\cal R})$ through
\beq
l({\cal R}) = \sum_{s=1}^{n_{\cal R}} l({{\cal R}_{SU(2)}^s}).
\eeq
The index of the spin $m/2$ representation of $SU(2)$ is given by
 $m(m+1)(m+2)/6$. 

Including the effect of $SU(2)$ gaugino condensation we finally arrive at 
the effective superpotential for low-energy $SU(2)$ theory
\beq
W_L  = W_{cl}(g) \pm 2 \La_{YM}^3
     = W_{cl}(g) \pm 2g_r \sqrt{A(a)},
\label{Wflavor}
\eeq
The expectation values $\langle u_k \rangle =\pa W_L/ \pa g_k$ are found to be
\beqa
\langle u_j \rangle & = &  u_j^{cl} 
\pm 2 \frac{ \pa \sqrt{A}}{\pa g_j'} , \hskip10mm  1 \leq j \leq r-1, \CR
\langle u_r \rangle & = & u_r^{cl} \pm 2 \left(
\sqrt{A} +g_r \sum_{k=1}^{r-1} \frac{\pa g_k'}{\pa g_r} 
\frac{ \pa \sqrt{A}}{\pa g_k'} \right) \CR
 & =& u_r^{cl} \pm 2 \left( \sqrt{A} -
\sum_{k=1}^{r-1} g_k' \frac{ \pa \sqrt{A}}{\pa g_k'} \right),
\label{vevmat}
\eeqa
where we have set $g_k' = g_k/ g_r$ and used the fact that
$u_k^{cl}$ and $A$ are functions of $g_k'$ since $a_i$ in (\ref{Wflavor}) 
are solutions of (\ref{eq1}) (see also (\ref{eqmoratio})). 

Let us show that the vacuum expectation values (\ref{vevmat}) obey 
the singularity condition for the family of $(r-1)$-dimensional complex 
manifolds defined by ${\cal W}=0$ with coordinates $z, x_1,\cdots ,x_{r-1}$ 
where
\beq
{\cal W} = z+ \frac{A(x_n)}{z}-
\sum_{i=1}^{r} x_i \left( u_i-u_i^{cl}(x_n) \right).
\label{mfd}
\eeq
Here we have introduced the variables $x_i$ ($1\leq i \leq r-1$) instead of
$g_i'$ to express $A(g_n')$ and $u_i^{cl} (g_n')$, $x_r=1$ and $u_i$ are
moduli parameters.
The manifold ${\cal W}=0$ is singular when
\beq
{\pa {\cal W}\over \pa z}=0, \hskip10mm {\pa {\cal W}\over \pa x_i}=0.
\label{Wsing}
\eeq
Then, if we set $ z =\pm \sqrt{A(x_k)}$, $x_k=g_k'$ and $u_j=\bra u_j\ket$
it is easy to show that the singularity conditions are satisfied
\beqa
\left. {\cal W} \right |  & = & \pm 2 \sqrt{A(g_k')} - 
\sum_{i=1}^{r} g_i' \left( \langle u_i \rangle -u_i^{cl}(g_k') \right)=0, \CR
\left. {\pa {\cal W}\over \pa z} \right |  &=& 0,    \CR
\left. \frac{\pa {\cal W}}{\pa x_j} \right | & =&
\pm \frac{1}{\sqrt{A(g_k')}} \frac{\pa A(g_k')}{\pa g_j'}-
\langle u_j \rangle +
\frac{\pa}{\pa g_j'}  \left( \sum_{i=1}^{r} g_i' u_i^{cl} (g_k') \right) \CR
& =& -u_j^{cl} (g_k') + g_r \frac{\pa}{\pa g_j} 
\left( \frac{W_{cl}(g)}{g_r} \right) =0, \hskip10mm 1\leq j\leq r-1.
\eeqa
Thus the singularities of the manifold defined by
${\cal W}=0$ are parametrized by the expectation values $\bra u_k \ket$.

Let us explain how the known curves for $SU(N_c)$ and $SO(2N_c)$
supersymmetric QCD are reproduced from (\ref{mfd}). First we consider 
$SU(N_c)$ theory with $N_f$ fundamental flavors.
Here we denote the degree $i$ Casimir by $u_i$ and correspondingly change
the notations for $x_j$ and $g_j'$. It is shown in \cite{Ki},\cite{KiTeYa}
that 
\beq
A= \La^{2 N_c-N_f}
{\rm det}_{N_f} \left( \sum_{l=0}^q (a^1)^l \gamma_l \right), \hskip10mm
a^1=g_{N_c-1}',
\eeq
and hence (\ref{mfd}) becomes
\beq
{\cal W} = z+ \frac{A(x_{N_c-1})}{z} - 
\sum_{i=2}^{N_c} x_i (u_i-u_i^{cl}(x_n)).
\label{wsu}
\eeq
Since $A$ depends only on $x_{N_c-1}$ one can eliminate other 
variables $x_1,\cdots,x_{N_c-2}$ by imposing $\pa {\cal W} / \pa x_j=0$
to get the relation
\beq
u_j^{cl} (x_n) = u_j
\label{impose}
\eeq
for $2 \leq j \leq N_c-2$, and then
\beq
{\cal W} = z+ \frac{A(x_{N_c-1})}{z} -
(u_{N_c} -u_{N_c}^{cl} (x_n)) -x_{N_c-1} (u_{N_c-1}-u_{N_c-1}^{cl}(x_n)).
\label{wsu2}
\eeq

Remember that
\beq
0={\rm det} \left( a^1 - \Phi_{cl} \right) 
=(a^1)^{N_c} -s_2^{cl} (a^1)^{N_c-1}-\cdots -s_{N_c}^{cl},
\label{vacdet}
\eeq
where 
\beq
ks_k+\sum_{i=1}^kis_{k-1}u_i=0, \hskip10mm  u_n={1\over n}{\rm Tr}\, \Phi^n,
\hskip10mm k=1,2,\cdots
\eeq
with $s_0=-1$ and $s_1=u_1=0$. We see with the aid of (\ref{vacdet}) that
\beqa
u_{N_c}^{cl}+ x_{N_c-1} u_{N_c-1}^{cl}  &=&
(u_{N_c}^{cl}-s_{N_c}^{cl} ) +x_{N_c-1} ( u_{N_c-1}^{cl} -s_{N_c-1}^{cl})+
(s_{N_c}^{cl}+ x_{N_c-1} s_{N_c-1}^{cl}) \CR
&=& (u_{N_c}-s_{N_c} ) +x_{N_c-1} ( u_{N_c-1}-s_{N_c-1}) \CR
 && + \left( (x_{N_c-1})^{N_c} -s_2 (x_{N_c-1})^{N_c-1}-
\cdots -s_{N_c-2} \right),
\eeqa
where (\ref{impose}) and the fact that 
$s_{N_c}= u_{N_c} + (\hbox{polynomial of $u_{k}, \;2 \leq k \leq N_c-2$})$
have been utilized. We now rewrite (\ref{wsu2}) as
\beqa
{\cal W} &= & z+ \frac{A(x)}{z} - 
(u_{N_c}+ x u_{N_c-1} ) +(u_{N_c}^{cl}+ x u_{N_c-1}^{cl} ) \CR
 & =& z+ \frac{A(x)}{z} +
x^{N_c} -s_2 x^{N_c-1}-\cdots -s_{N_c},
\label{wsu2f}
\eeqa
where $x_{N_c-1}$ was replaced by $x$ for notational simplicity.
This reproduces the hyperelliptic curve derived in \cite{Ka},\cite{KiTeYa} 
after making a change of variable $y=z-A(x)/z$ and 
agrees with the $N=2$ curve obtained in \cite{HaOz},\cite{ArPlSh},\cite{ArSh} 
in the $N=2$ limit .

Next we consider $SO(2 N_c)$ theory with $2 N_f$ fundamental flavors $Q$.
Following \cite{KiTeYa} we take a tree-level superpotential 
\beq
W=\sum_{n=k}^{N_c-2} g_{2 k} u_{2 k} + g_{2 (N_c-1)} s_{N_c-1}+\lm v
+{1\over 2}\sum_{l=0}^{q} {\rm Tr}_{2 N_f} \, \gamma_l \, Q \p^l Q, 
\label{w4a}
\eeq
where
\beqa
u_{2 k} &=& \frac{1}{2 k} {\rm Tr}\, \Phi^{2 k}, 
\hskip10mm 1 \leq k \leq N_c-1,  \CR
v &=& {\rm Pf}\, \Phi=\frac{1}{2^{N_c} N_c !} \E_{i_1 i_2 j_1 j_2 \cdots}
\Phi^{i_1 i_2} \Phi^{j_1 j_2} \cdots  
\eeqa
and
\beq
ks_k+\sum_{i=1}^k i s_{k-i} u_{2i}=0, \hskip10mm s_0=-1, 
\hskip10mm k=1,2,\cdots .
\eeq
According to \cite{TeYa1} we have
\beq
(a^1)^2=g_{2(N_c-2)}', \hskip10mm \lm'=2 \prod_{j=2}^{N_c-1} (-i a^j),
\hskip10mm v^{cl}=-g_{2(N_c-2)}' \lm' /2
\label{vcla}
\eeq
and \cite{KiTeYa}
\beq
A= \La^{4(N_c-1)-2 N_f} 
{\rm det}_{2 N_f} \left( \sum_{l=0}^q (a^1)^l \gamma_l \right),
\label{Aa1}
\eeq
and thus
\beq
{\cal W} = z+ \frac{A(x_{N_c-2})}{z} - 
\sum_{i=1}^{N_c-1} x_i (u_{2 i}-u_{2 i}^{cl}(x_n))-x (v-v^{cl}(x_n)),
\label{wsoa}
\eeq
where $\lm'=\lm/g_{2(N_c-1)}$ was replaced by $x$ and $g_{2i}/g_{2(N_c-1)}$
by $x_i$.

In view of (\ref{Aa1}) we again notice that there are redundant variables 
which can be eliminated
by imposing the condition $\pa {\cal W} / \pa x_j=0$ so as to obtain
\beq
u_{2 j}^{cl} (x_n) = u_{2 j}
\eeq
for $1 \leq j \leq N_c-3$. We then find
\beqa
{\cal W} = z+ \frac{A(x_{N_c-2})}{z} &- &
(u_{2 (N_c-1)} -u_{2(N_c-1)}^{cl} (x_n)) 
-x_{N_c-2} (u_{2(N_c-2)}-u_{2(N_c-2)}^{cl}(x_n))  \CR
&-& x (v-v^{cl}(x_n)).
\label{wso2}
\eeqa
Using $\det (a^1-\Phi_{cl})=0$ we proceed further as in the $SU(N_c)$ case.
The final result reads
\beqa
{\cal W} &= & z+ \frac{A(y)}{z} +
\frac{1}{y} \left( 
y^{N_c} -s_1 y^{N_c-1}-\cdots -s_{N_c-1}y+{ v^{cl}(x_n)}^2 \right) \CR
&& -x (v-v^{cl}(x_n)) \CR
&=& z+ \frac{A(y)}{z} -
\frac{1}{4} x^2 y+y^{N_c-1} -s_1 y^{N_c-2}-\cdots -s_{N_c-1}-v x,
\label{dnfinal}
\eeqa
where we have set $y=x_{N_c-2}$ and used (\ref{vcla}). 
It is now easy to check
that imposing $\pa {\cal W}/\pa x=0$ to eliminate $x$ yields the known curve
in \cite{KiTeYa} which has the correct $N=2$ limit \cite{Ha},\cite{ArSh}.

It should be noted here that adding gaussian variables in (\ref{wsu2f})
and (\ref{dnfinal}) we have
\beqa
{\cal W}_{A_{n-1}} &= & z+ \frac{A(y_1)}{z}+y_1^n-s_2y_1^{n-1}-\cdots -s_n
+y_2^2+y_3^2,   \CR
{\cal W}_{D_{n}} &=& z+ \frac{A(y_1)}{z} -
\frac{1}{4} y_2^2 y_1+y_1^{n-1} -s_1 y_1^{n-2}-\cdots -s_{n-1}-vy_2+y_3^2.
\eeqa
These are equations describing ALE spaces of AD type fibered over ${\bf CP}^1$.
Inclusion of matter hypermultiplets makes fibrations more complicated than
those for pure Yang-Mills theory.
For $A_n$ the result is rather obvious, but for $D_n$ it may be interesting
to follow how two variables $y_1,\, y_2$ come out naturally from (\ref{mfd}).
These variables are traced back to coupling constants $g_{2(n-2)}/g_{2(n-1)},\,
\lambda /g_{2(n-1)}$, respectively, and their degrees indeed agree
$[y_1]=[g_{2(n-2)}/g_{2(n-1)}]=2,\, [y_2]=[\lambda /g_{2(n-1)}]=n-2$.

This observation suggests a possibility that even in the $E_n$ case we may
eliminate redundant variables and derive the desired ALE form of Seiberg-Witten geometry
directly from (\ref{mfd}).
This issue is considered in the next subsection.


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{$E_6$ theory with fundamental matters}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this subsection we will show that 
an extension of \cite{TeYa2} enables us to obtain 
exceptional Seiberg-Witten geometry with fundamental hypermultiplets.
The resulting manifold  takes the form of a fibration of the ALE space of 
type $E_6$.

Let us consider $N=1$ $E_6$ gauge theory with $N_f$ fundamental matters
$Q^i,\tQ_j$ ($1 \leq i,j \leq N_f$) and an adjoint matter $\Phi$. $Q^i,\tQ_j$
are in ${\bf 27}$ and $\overline {\bf 27}$, and $\Phi$ in ${\bf 78}$ of $E_6$.
The coefficient of the one-loop beta function is given by $b=24-6N_f$, and 
hence the theory is asymptotically free for $N_f=0, 1, 2, 3$ and finite
for $N_f=4$. We take a tree-level superpotential
\beq
W = \sum_{k \in {\cal S}} g_{k} s_{k}(\Phi) +
{\rm Tr}_{N_f} \, \gamma_0 \, \tilde{Q} Q +
{\rm Tr}_{N_f} \, \gamma_1 \, \tilde{Q} \Phi Q,
\label{treeme6}
\eeq
where ${\cal S}=\{2,5,6,8,9,12\}$ denotes the set of degrees of 
$E_6$ Casimirs $s_k(\Phi)$ and
$g_k$, $(\gamma_a)_i^j$ $(1 \leq i,j \leq N_f)$ are coupling 
constants. A basis for the $E_6$ Casimirs will be specified momentarily.
When we put $(\gamma_0)^i_j=\sqrt{2} m^i_j$ with $[m, m^{\dagger}]=0$,
$(\gamma_1)^j_i=\sqrt{2} \D^j_i$ and all $g_k=0$, (\ref{treeme6}) is reduced to 
the superpotential in $N=2$ supersymmetric Yang-Mills theory with massive
$N_f$ hypermultiplets.

We now look at the Coulomb phase with $Q=\tilde{Q}=0$.
Since $\Phi$ is restricted to take the values in the Cartan subalgebra we
express the classical value of $\Phi$ in terms of a vector \footnote{Our
notation is slightly different from \cite{TeYa2}. Here we use $a_i$ with
lower index instead of $a^i$ in \cite{TeYa2}.}
\beq
a=\sum_{i=1}^6 a_i\alpha_i
\eeq
with $\alpha_i$ being the simple roots of $E_6$. 
Then the classical vacuum is parametrized by
\beq
\Phi^{cl}={\rm diag}\, (a\cdot \lm_1, a\cdot \lm_2,\cdots , a\cdot \lm_{27}),
\eeq
where $\lm_i$ are the weights for ${\bf 27}$ of $E_6$. 
For the notation of roots and weights we follow \cite{Sla}. 
We define a basis for the $E_6$ Casimirs $u_k(\Phi)$ by
\beqa
&& u_2=-{1\over 12}\, \chi_2, \hskip10mm u_5=-{1\over 60}\, \chi_5,
\hskip10mm u_6=-{1\over 6}\, \chi_6+{1\over 6\cdot 12^2}\, \chi_2^3,   \CR
&& u_8=-{1\over 40}\, \chi_8+{1\over 180}\, \chi_2\chi_6
-{1\over 2\cdot 12^4}\, \chi_2^4, \hskip10mm
u_9=-{1\over 7\cdot 6^2}\, \chi_9+{1\over 20\cdot 6^3}\, \chi_2^2\chi_5, \CR
&& u_{12}=-{1\over 60}\, \chi_{12}+{1\over 5\cdot 6^3}\, \chi_6^2
+{13\over 5\cdot 12^3}\, \chi_2\chi_5^2 \CR
&& \hskip11mm +{5\over 2\cdot 12^3}\, \chi_2^2\chi_8
-{1\over 3\cdot 6^4}\, \chi_2^3\chi_6+{29\over 10\cdot 12^6}\, \chi_2^6,
\eeqa
where $\chi_n={\rm Tr}\, \Phi^n$. 
The standard basis $w_k(\Phi)$ are written in terms of $u_k$ as follows
\beqa
&& w_2={1\over 2}u_2, \hskip10mm w_5=-{1\over 4}u_5, 
\hskip10mm w_6={1\over 96}\left( u_6-u_2^3\right), \CR
&& w_8={1\over 96}\left( u_8+{1\over 4}u_2u_6-{1\over 8}u_2^4 \right),
\hskip10mm w_9=-{1\over 48} \left( u_9-{1\over 4} u_2^2u_5\right), \CR
&& w_{12}={1\over 3456} \left( u_{12}+{3\over 32}u_6^2-{3\over 4}u_2^2u_8
-{3\over 16}u_2^3 u_6+{1\over 16}u_2^6 \right).
\label{casimiro}
\eeqa
The basis $\{u_k\}$ and (\ref{casimiro}) were first introduced 
in \cite{LW}.\footnote{The Casimirs $u_1,u_2,u_3,u_4,u_5,u_6$
in \cite{LW} are denoted
here as $u_2,u_5,u_6,u_8,u_9,u_{12}$, respectively.}
In our superpotential (\ref{treeme6}) we then set
\beq
s_2=w_2, \;\; s_5=w_5, \;\; s_6=w_6, \;\; s_8=w_8, \;\; s_9=w_9, \;\; 
s_{12}=w_{12}-\frac{1}{4} w_6^2.
\label{casimir}
\eeq
We will discuss later why this particular form is assumed.

The equations of motion are given by
\beq
\frac{\pa W(a)}{\pa a_i}=
\sum_{k \in {\cal S}} g_{k} \frac{\pa s_{k}(a)}{\pa a_i}=0.
\label{eq1a}
\eeq
Let us focus on the classical vacua with
an unbroken $SU(2) \times U(1)^{5}$ gauge symmetry. Fix the $SU(2)$ 
direction by choosing the simple root $\alpha_1$, then we have the vacuum 
condition
\beq
a\cdot \alpha_1=2a_1-a_2=0.
\label{vaccon}
\eeq
It follows from (\ref{eq1a}), (\ref{vaccon}) that
\beqa
&& {g_9\over g_{12}}={D_{1,9}\over D_{1,12}}   \CR
&&  \hskip7mm = -{1\over 8} 
\Big( 2 a_1 a_5 a_4-a_4 a_3^2+a_5^2 a_4+a_4^2 a_3-a_3 a_6^2+a_3^2 a_6  \CR
&& \hskip20mm -2 a_1 a_5^2+2 a_1 a_6^2-2 a_4^2 a_1-a_5 a_4^2
-2 a_1 a_3 a_6+2 a_4 a_3 a_1  \Big), \CR
&& {g_8\over g_{12}}={D_{1,8}\over D_{1,12}}  \CR
&&  \hskip7mm = -{1\over 48}        
\Big( 12 a_1 a_5^2 a_4-6 a_1^2 a_5^2-6 a_1^2 a_6^2-4 a_1^3 a_3+4 a_3^3 a_1
+2 a_3^3 a_4+2 a_3^3 a_6-a_4^4  \CR
&& \hskip20mm -3 a_3^2 a_6^2-3 a_3^2 a_4^2-a_3^4-a_6^4
-12 a_1 a_5 a_4^2-2 a_5 a_4 a_6^2+8 a_1 a_3 a_6^2+3 a_1^4 \CR
&& \hskip20mm  +6 a_1^2 a_4 a_3-8 a_4 a_3^2 a_1-2 a_5 a_4 a_3^2-2 a_4^2 a_3 a_6
+6 a_1^2 a_5 a_4-2 a_4 a_3 a_6^2  \CR
&& \hskip20mm +2 a_5 a_4^2 a_3-2 a_5^2 a_3 a_6
+2 a_4 a_3^2 a_6-a_5^4-2 a_5^2 a_4 a_3-2 a_1^2 a_3^2-6 a_1^2 a_4^2  \CR
&& \hskip20mm +2 a_5^2 a_6^2+2 a_4^2 a_6^2+2 a_5^2 a_3^2
+6 a_1^2 a_3 a_6-8 a_1 a_3^2 a_6+8 a_4^2 a_3 a_1-4 a_5^2 a_1 a_3  \CR
&& \hskip20mm +2 a_5 a_4 a_3 a_6
+4 a_5 a_4 a_1 a_3+2 a_3 a_4^3-3 a_4^2 a_5^2
+2 a_4 a_5^3+2 a_4^3 a_5+2 a_6^3 a_3 \Big) ,   \CR
&& {g_6\over g_{12}}={D_{1,6}\over D_{1,12}}   \CR
&&  \hskip7mm = {1\over 192} \left( 4 a_3^3 a_1 a_5^2-18 a_4^3 a_1^2 a_5+
13 a_3^4 a_1^2-a_3^4 a_5^2-7 a_3^3 a_6^3 +9 a_1^2 a_6^4
+\cdots \right) ,
\label{solgg}
\eeqa
where $D_{1,k}$ is the cofactor for a $(1,k)$ element of the $6\times 6$
matrix $[\pa s_i(a)/\pa a_j]$, $i\in {\cal S}$ and
$j=1, \ldots, 6$ \cite{TeYa2}. In (\ref{solgg}) the explicit expression for
$g_6/g_{12}$ is too long to be presented here, and hence suppressed.
Denoting $y_1=g_9/g_{12},\, y_2=g_8/g_{12},\, y_3=g_6/g_{12} $,
we find that the others are expressed in terms of $y_1,\, y_2$
\beq
 {g_2\over g_{12}}={D_{2,2}\over D_{2,12}}=y_1^2 y_2, \hskip10mm
{g_5\over g_{12}}={D_{2,5}\over D_{2,12}}=y_1 y_2.
\label{solgg2}
\eeq
This means that our superpotential specified with Casimirs (\ref{casimir})
realizes the $SU(2)\times U(1)^5$ vacua only when the coupling constants
are subject to the relation (\ref{solgg2}).

Notice that reading off degrees of $y_1,\, y_2, \, y_3$ from (\ref{solgg}) 
gives $[y_1]=3,\, [y_2]=4,\, [y_3]=6$. 
Thus, if we regard $y_1,\, y_2,\, y_3$ as variables to
describe the $E_6$ singularity, (\ref{solgg}) and (\ref{solgg2}) may be
identified as relevant monomials in versal deformations of the $E_6$
singularity. In fact we now point out an intimate relationship between 
classical solutions corresponding to the symmetry breaking 
$E_6 \supset SU(2)\times U(1)^5$ and the $E_6$ singularity.
For this we examine the superpotential (\ref{treem})
at classical solutions 
\beqa
W_{cl}&=&g_{12}\sum_{k\in {\cal S}} 
\left( {g_k\over g_{12}} \right) s_k^{cl}(a) \CR
&=& g_{12}\, \left( s_2^{cl}y_1^2 y_2+s_5^{cl}y_1y_2+s_6^{cl}y_3
+s_8^{cl} y_2+s_9^{cl} y_1+s_{12}^{cl} \right).
\label{wcl1}
\eeqa
Evaluating the RHS with the use of (\ref{vaccon})-(\ref{solgg2}) leads to
\beq
W_{cl}= -g_{12}\, \left(2 y_1^2 y_3+y_2^3-y_3^2 \right).
\label{wcl2}
\eeq
It is also checked explicitly that
\beqa
 -4 y_1 y_3 &=& 2s_2^{cl}y_1 y_2+s_5^{cl}y_2+s_9^{cl} , \CR
 -3 y_2^2 &=& s_2^{cl}y_1^2+s_5^{cl}y_1+s_8^{cl}, \CR
 -2 y_1^2+2 y_3 &=& s_6^{cl}.
\label{wcl3}
\eeqa

To illustrate the meaning of (\ref{wcl1})-(\ref{wcl3}) let us recall
the standard form of versal deformations of the $E_6$ singularity 
\beq
W_{E_6}(x_1,x_2,x_3;w)=x_1^4+x_2^3+x_3^2+w_2\, x_1^2 x_2+w_5\, x_1x_2
+w_6\, x_1^2+w_8\, x_2+w_9\, x_1+w_{12},
\eeq
where the deformation parameters $w_k$ are related to the $E_6$ Casimirs via
(\ref{casimiro}) \cite{LW}. Then what we have observed in
(\ref{wcl1})-(\ref{wcl3}) is that when we express $w_k$ in terms of $a_i$
as $w_k=w_k^{cl}(a)$ the equations
\beq
W_{E_6}={\pa W_{E_6}\over \pa x_1}={\pa W_{E_6}\over \pa x_2}
={\pa W_{E_6}\over \pa x_3}=0
\eeq
can be solved by
\footnote{We have observed a similar relation between 
the symmetry breaking solutions $SU(r+1)$ (or $SO(2r)$) 
$\supset SU(2)\times U(1)^{r-1}$ and the $A_r$ (or $D_r$) singularity.}
\beq
x_1=y_1(a), \hskip10mm x_2=y_2(a), \hskip10mm
x_3=i \left( y_3(a)-y_1(a)^2-{s_6^{cl}(a) \over 2}\right)
\eeq
under the condition (\ref{vaccon}).
This observation plays a crucial role in our analysis.

When applying the technique of confining phase superpotentials we usually take
all coupling constants $g_k$ as independent moduli parameters. To deal with
$N=1$ $E_6$ theory with fundamental matters, however, we find it 
appropriate to proceed as follows. First of all, motivated by the above
observations for classical solutions, we keep three coupling constants 
$g_6'=g_6/g_{12}$, $g_8'=g_8/g_{12}$ and $g_9'=g_9/g_{12}$ adjustable 
while the rest is fixed as
$g_2'=g_8'g_9'^2,\, g_5'=g_8'g_9'$ with $g_k'=g_k/g_{12}$.
Taking this parametrization it is seen that the equations of motion are
satisfied  by virtue of (\ref{solgg2}) in the $SU(2)\times U(1)^5$ vacua 
(\ref{vaccon}). Note here that originally there exist six classical
moduli $a_i$ among which one is fixed by (\ref{vaccon}) and three are
converted to $g_9'=y_1(a)$, $g_8'=y_2(a)$ and $g_6'=y_3(a)$, 
and hence we are left with two
classical moduli which will be denoted as $\xi_i$. Without loss of generality
one may choose $\xi_2=s_2^{cl}(a)$ and 
$\xi_5=s_5^{cl}(a)$.

We now evaluate the low-energy effective superpotential in the 
$SU(2)\times U(1)^5$ vacua. $U(1)$ photons decouple in the integrating-out 
process. The standard procedure yields the effective superpotential 
for low-energy $SU(2)$ theory \cite{ElFoGiInRa},\cite{TeYa2}
\beq
W_L  = -g_{12} \left(2 y_1^2 y_3+y_2^3-y_3^2 \right) \pm 2 \La_{YM}^3,
\label{Wflavora}
\eeq
where the second term takes account of $SU(2)$ gaugino condensation
with $\La_{YM}$ being the dynamical scale for low-energy $SU(2)$ Yang-Mills
theory.
The low-energy scale $\La_{YM}$ is related to the high-energy scale $\La$ 
through the scale matching \cite{TeYa2}
\beqa
\La_{YM}^{6} & =&  g_{12}^2 A(a), \CR
A(a) & \equiv &
\La^{24 - 6 N_f}
\prod_{s=1}^{6}  {\rm det}_{N_f} \left(
\gamma_0+ \gamma_1 (a \cdot \lm_{s}) \right) ,
\label{smra}
\eeqa
where $\lm_{s}$ are weights of ${\bf 27}$ which branch to six $SU(2)$ doublets
respectively under $E_6 \supset SU(2)\times U(1)^5$.
Explicitly they are given in the Dynkin basis as
\beqa
&& \lm_1=(1,\, 0,\, 0,\, 0,\, 0,\, 0),   
\hskip17mm \lm_2=(1,\, -1,\, 0,\, 0,\, 1,\, 0),  \CR
&& \lm_3=(1,\, -1,\, 0,\, 1,\, -1,\, 0), 
\hskip10mm \lm_4=(1,\, -1,\, 1,\, -1,\, 0,\, 0), \CR
&& \lm_5=(1,\, 0,\, -1,\, 0,\, 0,\, 1),  
\hskip14mm \lm_6=(1,\, 0,\, 0,\, 0,\, 0,\, -1).
\eeqa
Notice that $\sum_{s=1}^6\lm_s=3\alpha_1$.

Let us first discuss the $N_f=0$ case, i.e. $E_6$ pure Yang-Mills theory,
for which $A(a)$ in (\ref{smra}) simply equals $\La^{24}$. The vacuum
expectation values are calculated from (\ref{Wflavora})
\beqa
{\pa W_L\over \pa g_{12}}&=&
\langle \widetilde W(y_1,y_2,y_3;s) \rangle 
=-\left(2 y_1^2 y_3+y_2^3-y_3^2\right)
\pm 2 \La^{12}, \CR
{1\over g_{12}}{\pa W_L\over \pa y_1}
&=& \langle {\pa \widetilde W(y_1,y_2,y_3;s)\over \pa y_1} \rangle 
=-4 y_1 y_3, \CR
{1\over g_{12}}{\pa W_L\over \pa y_2}
&=& \langle {\pa \widetilde W(y_1,y_2,y_3;s)\over \pa y_2} \rangle 
=-3 y_2^2, \CR
{1\over g_{12}}{\pa W_L\over \pa y_3}
&=& \langle {\pa \widetilde W(y_1,y_2,y_3;s)\over \pa y_3} \rangle 
=-2 y_1^2+2 y_3,
\label{vevym}
\eeqa
where $y_1, y_2, y_3$ and $g_{12}$ have been treated as independent
parameters as discussed before and
\beq
\widetilde W (y_1,y_2,y_3;s)=s_2\, y_1^2 y_2+s_5\, y_1 y_2
        +s_6\, y_3+s_8\, y_2+s_9\, y_1+s_{12}.
\eeq
Define a manifold by ${\cal W}_0=0$ with four coordinate variables
$z,\, y_1,\, y_2, \, y_3 \in {\bf C}$ and
\beq
{\cal W}_0 \equiv z+ {\La^{24}\over z}
-\left( 2 y_1^2 y_3+y_2^3-y_3^2
+ \widetilde W(y_1,y_2,y_3;s ) \right)=0.
\label{mfda}
\eeq
It is easy to show that the expectation values (\ref{vevym}) parametrize the 
singularities of the manifold where
\beq
{\pa {\cal W}_0\over \pa z}={\pa {\cal W}_0\over \pa y_1}
={\pa {\cal W}_0 \over \pa y_2}={\pa {\cal W}_0\over \pa y_3}=0.
\eeq
Making a change of variables $y_1=x_1,\; y_2=x_2,\; y_3=-ix_3+x_1^2+s_6/2$ 
in (\ref{mfda}) we have
\beq
z+{\La^{24}\over z}- W_{E_6}(x_1,x_2,x_3;  w )=0.
\label{e6ym}
\eeq
Thus the ALE space description of $N=2$
$E_6$ Yang-Mills theory \cite{KlLeMaVaWa},\cite{LW} is obtained from 
the $N=1$ confining phase superpotential.

We next turn to considering the fundamental matters. In the $N=2$ limit we
have $A(a)=\La^{24-6N_f} \cdot 8^{N_f} \prod_{i=1}^{N_f} f(a, m_i)$ 
with $f(a,m)=\prod_{s=1}^6 (m+a\cdot \lm_s)$. After some algebra we find 
\beq
f(a,m)=m^6+2\xi_2 m^4-8m^3y_1+\left( \xi_2^2-12 y_2 \right) m^2
+4\xi_5 m-4 y_2 \xi_2-8y_3 ,
\eeq
where we have used (\ref{casimir})-(\ref{solgg}).
Let us recall that, in viewing (\ref{Wflavora}),
we think of $(y_1, y_2, y_3, \xi_2, \xi_5, g_{12})$ as six independent
parameters. Then the quantum expectation values are given by
\beqa
{\pa W_L\over \pa g_{12}}&=&
\langle \widetilde W(y_1,y_2,y_3;s) \rangle 
=-\left( 2 y_1^2 y_3+y_2^3-y_3^2\right)\pm 2 \sqrt{A(y_1,y_2,y_3; \xi ,m)}, \CR
{1\over g_{12}}{\pa W_L\over \pa y_1}
&=& \langle {\pa \widetilde W(y_1,y_2,y_3;s)\over \pa y_1} \rangle
=-4 y_1 y_3 \pm 2 \frac{ \pa }{\pa y_1}\sqrt{A(y_1,y_2,y_3; \xi ,m)}, \CR
{1\over g_{12}}{\pa W_L\over \pa y_2}
&=& \langle {\pa \widetilde W(y_1,y_2,y_3;s)\over \pa y_2} \rangle
=-3 y_2^2 \pm 2 \frac{ \pa }{\pa y_2}\sqrt{A(y_1,y_2,y_3; \xi ,m)}, \CR
{1\over g_{12}}{\pa W_L\over \pa y_3}
&=& \langle {\pa \widetilde W(y_1,y_2,y_3;s)\over \pa y_3} \rangle
=-2 y_1^2+2 y_3 \pm 2 \frac{ \pa }{\pa y_3}\sqrt{A(y_1,y_2,y_3; \xi ,m)}.
\label{vevmata}
\eeqa
Similarly to the $N_f=0$ case one can check that these expectation values 
satisfy the singularity condition for a manifold defined by
\beq
z+ {1 \over z} A(y_1,y_2,y_3;\xi,m)-\left( 2 y_1^2 y_3+y_2^3-y_3^2
+ \widetilde W(y_1,y_2,y_3; s) \right)=0.
\label{ye6ale}
\eeq

Note that $s_k$ in $\widetilde W$ 
are quantum moduli parameters. What about $\xi_2,\, \xi_5$ in 
the one-instanton factor $A$? Classically we have $\xi_i=s_i^{cl}$
as was seen before. The issue is thus whether the classical relations
$\xi_i=s_i^{cl}$ receive any quantum corrections at the singularities.
If there appear no quantum corrections, $\xi_i$ in $A$ can be replaced by
quantum moduli parameters $s_i$. Let us simply assume here that
$\xi_i= s_i^{cl}=\bra s_i\ket$ for $i=2,5$ in the $N=1$ $SU(2)\times U(1)^5$ 
vacua. This assumption seems quite plausible as long as we have inspected 
possible forms of quantum corrections due to gaugino condensates.

Now we find that Seiberg-Witten geometry of $N=2$ supersymmetric QCD 
with gauge group $E_6$ is described by
\beq
z+{1\over z}A(x_1,x_2,x_3; w,m)
- W_{E_6}(x_1,x_2,x_3; w)=0,
\label{e6ale}
\eeq
where a change of variables from $y_i$ to $x_i$ as in (\ref{e6ym}) 
has been made in (\ref{ye6ale}) and 
\beqa
&& A(x_1,x_2,x_3; w,m)  \CR
&=& \La^{24-6N_f} \cdot 8^{N_f} 
\prod_{i=1}^{N_f} \left( {m_i}^6+2 w_2 {m_i}^4-8{m_i}^3x_1 
+\left( w_2^2-12 x_2 \right) {m_i}^2 \right. \CR
&& \left. \hspace{3cm} 
+4 w_5 {m_i}-4 w_2 x_2 -8(x_1^2-ix_3+w_6/2) \right).
\eeqa
The manifold takes the form of ALE space of type $E_6$ fibered over 
the base ${\bf CP}^1$. Note an intricate dependence of the fibering data over
${\bf CP}^1$ on the hypermultiplet masses. This is in contrast with the
ALE space description of $N=2$ $SU(N_c)$ and $SO(2N_c)$ gauge theories with
fundamental matters. 
In (\ref{e6ale}), letting $m_i \rightarrow \infty$ while keeping 
$\La^{24-6N_f} \prod_{i=1}^{N_f} m_i^6 \equiv \La_0^{24}$ finite we
recover the pure Yang-Mills result (\ref{e6ym}).

As a non-trivial check of our proposal (\ref{e6ale}) let us examine
the semi-classical singularities. In the semi-classical limit $\Lambda
\rightarrow 0$ the discriminant $\Delta$ for (\ref{e6ale}) is expected
to take the form $\Delta \propto \Delta_G \Delta_M$ where $\Delta_G$  is a
piece arising from the classical singularities associated with the gauge
symmetry enhancement and $\Delta_M$ represents the semi-classical singularities
at which squarks become massless. When the $N_f$ matter hypermultiplets belong
to the representation $\rep$ of the gauge group $G$ we have
\beq
\Delta_M=\prod_{i=1}^{N_f}{\rm det}_{d \times d}(m_i{\bf 1}-\Phi^{cl})
=\prod_{i=1}^{N_f}P_G^\rep(m_i; u),
\label{delM}
\eeq
where $d={\rm dim}\,\rep$, $m_i$ are the masses, $\Phi^{cl}$ denotes the 
classical Higgs
expectation values and $P_G^\rep(x; u)$ is the characteristic polynomial
for $\rep$ with $u_i$ being Casimirs constructed from $\Phi^{cl}$.

For simplicity, let us consider the case in which all the quarks have equal 
bare masses. Then we can change a variable $x_3$ to $\tilde{x}_3$ so that
$A=A(\tilde{x}_3; w,m)$ is independent of $x_1$ and $x_2$.
Eliminating $x_1$ and $x_2$ from (\ref{e6ale}) by the use of
\beq
{\pa W_{E_6}\over \pa x_1}={\pa W_{E_6}\over \pa x_2}=0,
\label{aaa}
\eeq
we obtain a curve which is singular at the discriminant locus of (\ref{e6ale}).
The curve is implicitly defined through
\beq
\overline{W}_{E_6} \left(\tilde{x}_3; w_i-\delta_{i, 12} 
\left( z+\frac{A \left(\tilde{x}_3; w, m \right)}{z} \right) \right)\
=0,
\eeq
where $\overline{W}_{E_6}(\tilde{x}_3; w_i)
=W_{E_6}(x_1(\tilde{x}_3, w_i),x_2(\tilde{x}_3, w_i),\tilde{x}_3; w_i)$ and
$x_1(\tilde{x}_3, w_i)$, $x_2(\tilde{x}_3, w_i)$ are solutions
of (\ref{aaa}). Now the values of $\tilde{x}_3$ and $z$ at singularities 
of this
curve can be expanded in powers of $\La^{\frac{24-6 N_f}{2}}$. Then it is more
or less clear that the classical singularities corresponding to massless
gauge bosons are produced. Furthermore, if we denote as $R(\overline{W},A)$ 
the resultant of $\overline{W}_{E_6}(\tilde{x}_3; w_i)$
and $A \left(\tilde{x}_3;w, m \right)$, then $R(\overline{W},A)=0$ yields
another singularity condition of the curve in the limit 
$\Lambda \rightarrow 0$.
We expect that $R(\overline{W},A)=0$ corresponds to the semi-classical
massless squark singularities as is observed in the case of $N=2$ $SU(N_c)$
QCD \cite{HaOz},\cite{GPR}.
Indeed, we have checked this by explicitly computing 
$R(\overline{W},A)$ at sufficiently many points in the
moduli space. For instance, taking $w_2=2,w_5=5,w_6=7,w_8=9,w_9=11$ and 
$w_{12}=13$ in the $N_f=1$ case,  we get
\beqa
&& R(\overline{W},A) \CR
& = & m^2 \left( 3\,m^{10}+12\,m^{8} +\cdots \right)
\left( 26973 m^{27}+258552\,m^{25}+\cdots \right)^3 \CR
& & \left( m^{27} + 24\,m^{25} + 240\,m^{23} + 240\,m^{22} 
+ 2016\,m^{21} + 3360\,m^{20} + 16416\,m^{19} \right.  \CR
 & & + 34944\,m^{18}+ 88080\,m^{17} + 216576\,m^{16} + 448864\,m^{15} +
607488\,m^{14} \CR
 & &  + 2198272\,m^{13}- 296000\,m^{12}+ 4177792\,m^{11}
- 3407104\,m^{10} + 7796224\,m^{9} \CR
 & &  + 10664448\,m^{8}- 31708160\,m^{7}+ 41183232\,m^{6}
- 21889792\,m^{5} + 15575040\,m^{4} \CR
 & &  \left. - 17125120\,m^{3}- 38456320\,m^{2}- 3461120\,m + 9798656 \right) ,
\label{bbb}
\eeqa
while the $E_6$ characteristic polynomial for ${\bf 27}$ is given by
\beqa
&& P_{E_6}^{\bf 27}(x; u)  \CR
&=& x^{27}+12 w_2 x^{25}+60 w_2^2 x^{23}+48 w_5 x^{22}
+\left( 96w_6+168w_2^3 \right) x^{21}+336w_2 w_5 x^{20} \CR
&& +\left( 528w_2w_6+294w_2^4+480w_8 \right) x^{19}
+\left( 1344w_9+1008w_2^2w_5 \right) x^{18}+\cdots .
\eeqa
We now find a remarkable result that the last factor of (\ref{bbb})
precisely coincides with $P_{E_6}^{\bf 27}(m; u)$! Hence the manifold described
by (\ref{e6ale}) correctly produces all the semi-classical singularities
in the moduli space of $N=2$ supersymmetric $E_6$ QCD.

If we choose another form of the superpotential (\ref{treeme6}), say, the 
superpotential with $s_i=w_i$ for $i \in {\cal S}$ instead of (\ref{casimir})
we are unable to obtain $\Delta_M$ in (\ref{delM}). As long as we have
checked the choice made in (\ref{casimir}) is judicious in order to pass the
semi-classical test. At present, we have no definite recipe to fix the
tree-level superpotential which produces the correct semi-classical 
singularities, though it is possible to proceed by trial and error. In fact
we can find Seiberg-Witten geometry for $N=2$ $SO(2 N_c)$ gauge theory with
spinor matters and $N=2$ $SU(N_c)$ gauge theory with antisymmetric
matters \cite{TeYa3}.

In our result (\ref{e6ale}) it may be worth mentioning that 
the gaussian variable $x_3$
of the $E_6$ singularity appears in the fibering term.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Gauge symmetry breaking in Seiberg-Witten geometry}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


Staring with the $N=2$ Seiberg-Witten geometry with $E_6$ gauge group with 
massive fundamental matters, 
we construct the Seiberg-Witten geometry with $SU(N_c)$ 
and $SO(2 N_c)$ gauge groups
with various matter contents, in the rest of this section. 
All these geometries we will obtain take
the form of a fibration of the ALE spaces over a sphere.

To this end, we first discuss how to implement the gauge symmetry breaking 
in the general Seiberg-Witten geometry 
by giving appropriate VEV to the adjoint scalar field in
the $N=2$ vector multiplet.

Classically the VEV of the adjoint Higgs $\Phi$ is chosen to take the 
values in the Cartan subalgebra. The classical moduli space is then 
parametrized by a Higgs VEV vector $a=\sum_{i=1}^r a^i \A_i$.
At the generic points in the classical moduli space, the gauge group $G$ is 
completely broken to $U(1)^r$. However there are singular points 
where $G$ is broken only partially to $\prod_{i} G'_i \times U(1)^l$ 
with $G'_i$ being a simple subgroup of $G$.
If we fix the gauge symmetry breaking scale to be large,
the theory becomes $N=2$ supersymmetric gauge theory with 
the gauge group $\prod_{i} G'_i \times U(1)^l$ and
the initial Seiberg-Witten geometry reduces to the 
one describing the gauge group $G'_i$
after taking an appropriate scaling limit.

We begin with the case of $N=2$ supersymmetric
$SU(r+1)$ gauge theory with fundamental flavors.
The Seiberg-Witten curve for this theory is given by \cite{ArFa,HaOz}
\beq
y^2={\rm det}_{r+1} \left( x-\Phi_{\cal R} \right)^2 - 
\La^{2(r+1)-N_f } \prod_{i=1}^{N_f} (m_i-x).
\eeq
Choosing the classical value $\langle \Phi_{\cal R}  \rangle_{cl}$ as
\beqa
\langle \Phi_{\cal R} \rangle_{cl} 
& =& {\rm diag} \left(\langle a^1 \rangle , \langle a^2 \ra-\la a^1 \rangle,
\langle a^3 \ra -\la a^2 \rangle, \cdots ,
\langle a^{r} \ra -\la a^{r-1} \rangle, -\langle a^{r} \rangle \right) \CR
     & = & {\rm diag} (M, M, M, \cdots , M, -r M),
\eeqa
where $M$ is a constant,
we break the gauge group $SU(r+1)$ down to $SU(r) \times U(1)$.
Note that this parametrization 
is equivalent to $\la a^j \ra = j M$ which means
$\la a_j \ra = \D_{j,r} (r+1) M$.
Setting $a_i = \D_{j,r} (r+1) M+\D a_i$ and $m_i=M+m'_i$,
we take the scaling limit $M \rightarrow \infty$ with
$\La'^{2 r-N_f}=\frac{ \La^{2(r+1)-N_f } }{(r+1) M^2}$ held fixed.
Then we are left with the Seiberg-Witten curve corresponding to the gauge group $SU(r)$
\beq
(y')^2=\left\{ \left( x'-\D a^1 \right) 
\left( x'-(\D a^2-\D a^1) \right) \cdots
\left( x'-(-\D a^{r-1}) \right) \right\}^2-
\La'^{2 r-N_f } \prod_{i=1}^{N_f} (m'_i-x'),
% +O \left( \frac{1}{M^2} \right)
\label{sur}
\eeq
where $y'=\frac{y}{\sqrt{r+1} M}$ and $x'=x-M$.
Notice that we must shift the masses $m_i$ to obtain the finite masses 
of hypermultiplets in the $SU(r)$ theory with $N_f$ flavors.

Now we consider the case of $N=2$ theory with a simple gauge group $G$.
When we assume the nonzero VEV of the adjoint scalar, the largest non-Abelian
gauge symmetry which is left unbroken has rank $r-1$.
As we will see shortly, this largest unbroken gauge symmetry is realized 
by choosing
\beq
\la a_i \ra =M \, \D_{i,i_0}, \hskip10mm  1 \leq i \leq r,
\label{cond}
\eeq
where $M$ is an arbitrary constant and $i_0$ is some fixed value.
Under this symmetry breaking (\ref{cond}),
a gauge boson which corresponds to a generator $E_b$, where 
the subscript $b=\sum_{i} b^i \A_i$ indicates a corresponding root, 
has a mass proportional to $\la a \ra \cdot b=M \, b^{i_0}$.
This is seen from $[ \la a \ra \cdot H , E_b ] =(\la a \ra \cdot b) \, E_b$
where $H_i$ are the generators of the Cartan subalgebra.
Thus the massless gauge bosons correspond to 
the roots which satisfy $b^{i_0}=0$
and the unbroken gauge group becomes $G'_i \times U(1)$ 
where the Dynkin diagram of $G'$ is obtained by removing 
a node corresponding to the $i_0$-th simple root in the Dynkin diagram of $G$.
The Cartan subalgebra of $G$ is decomposed into 
the Cartan subalgebra of $G'$ and the additional $U(1)$ factor.
The former is generated by $E_{\A_k} \in G$ obeying
$[ E_{\A_k} , E_{\A_{-k}} ] \simeq \A_k \cdot H$ with $k \neq i_0$,
while the latter is generated by $\A_{i_0} \cdot H$. Therefore, we set
\beq
a^i = \left(A^{-1} \right)^{i \, i_0} M  + \D a^{i},
\eeq
where scalars corresponding to $G'$ have been denoted as $\D a$ 
with $\D a^{i_0} =0$. 
Note that the $U(1)$ sector decouples completely from the $G'$ sector
and the Weyl group of $G'$ naturally acts on $\D a$
out of which the Casimirs of $G'$ are constructed.

When the gauge symmetry is broken as above,
we have to decompose the matter representation ${\cal R}$ of $G$ in terms
of the subgroup $G'$ as well. We have
\beq
{\cal R} =  \bigoplus_{s=1}^{n_{\cal R}} {\cal R}_{s},
\eeq
where ${\cal R}_{s}$ stands for an irreducible representation of $G'$.
Accordingly $Q^i$ is decomposed into ${\bf Q}^i_s$
($1 \leq i \leq N_f$, $1 \leq s \leq n_{\cal R}$) 
in a $G'$ representation ${\cal R}_{s}$. $\tilde{Q}_i$ is
decomposed in a similar manner. 
After the massive components in $\Phi$ are integrated out,
the low-energy theory becomes $N=2$ $G' \times U(1)$ gauge theory.
The $U(1)$ sector decouples from the $G'$ sector and 
we consider the $G'$ sector only.
The semiclassical superpotential for this theory can be read off from 
(\ref{treem}). We have
\beq
W=\sum_{i=1}^{N_f} \left( \sqrt{2} \, \sum_{s=1}^{n_{\cal R}} 
\, ( \la a \ra \cdot \lm_{{\cal R}_{s}} +  m_i ) \, 
\tilde{{\bf Q}}_{i s}  {\bf Q}_s^i+\sqrt{2} \sum_{s=1}^{n_{\cal R}} 
\, \tilde{{\bf Q}}_{i s} \Phi_{{\cal R}_{s}} \, {\bf Q}_s^i \right) ,
\label{w1b}
\eeq
where $\lm_{{\cal R}_{s}}$ is a weight of ${\cal R}$ which branches to 
the weights in ${\cal R}_{s}$.
This implies that we should shift the mass $m_i$ as 
\beq
m_i=- \la a \ra \cdot \lm_{{\cal R}_{s_i}}+m'_i=
-M {\left(\lm_{{\cal R}_{s_i}} \right) }^{i_0} +m'_i
\label{ms}
\eeq
to obtain the $G'$ theory with appropriate matter hypermultiplets.
Note that we can choose ${{\cal R}_{s_i}}$ for each hypermultiplet 
separately. This enables us to obtain the $N_f$ matters in different 
representations of $G'$ from the $N_f$ matters in a single representation 
of $G$. In the limit $M \rightarrow \infty$, 
some hypermultiplets have infinite masses and decouple from the theory.
Then the superpotential (\ref{w1b}) becomes 
\beq
W=\sqrt{2} \, \sum_{i=1}^{N_f} 
\,  m'_i  \, 
\tilde{{\bf Q}}_{i \, s_i}  {\bf Q}^i_{s_i}+\sqrt{2} \sum_{i=1}^{N_f} 
\, \tilde{{\bf Q}}_{i \, s_i} \Phi_{{\cal R}_{s_i}} \, {\bf Q}^i_{s_i},
\eeq
%${\bf Q}^i={\bf Q}^i_{s_i}$ and 
%$\tilde{{\bf Q}}_{i}=\tilde{{\bf Q}}_{i \, s_i}$
and the resulting theory becomes $N=2$ theory with gauge group $G'$
with hypermultiplets belonging to the representation ${\cal R}_{s_i}$.
Note that $\la a \ra \cdot \lm_{{\cal R}_{s}}$ is proportional to 
its additional $U(1)$ charge.

In the known cases, 
the low-energy effective theory in the Coulomb phase is described
by the Seiberg-Witten geometry
which is described by a three-dimensional complex manifold 
in the form of the ALE space of ADE type fibered over ${\bf CP^1}$
\beq
z+\frac{1}{z} \La^{2 h - l({\cal R}) N_f} \prod_{i=1}^{N_f} 
{X_{G}^{{\cal R}} (x_1,x_2,x_3;a,m_i)} 
- W_G(x_1,x_2,x_3;a)=0,
\label{swg}
\eeq
where $z$ parametrizes ${\bf CP^1}$,
$h$ is the dual Coxeter number of $G$ and $l({\cal R})$
is the index of the representation  ${\cal R}$ of the matter.
Here $W_G(x_1,x_2,x_3;a)=0$ is a simple singularity of type $G$ and 
$X_{G}^{{\cal R}}(x_1,x_2,x_3;a,m_i)$ is some polynomial of 
the indicated variables. Note that
the simple singularity $W_G$ depends only on the gauge group $G$, but 
the $X_{G}^{{\cal R}}(x_1,x_2,x_3;a,m_i)$ 
depends on the matter content of the theory.

Starting with (\ref{swg}) let us consider the symmetry breaking in the 
Seiberg-Witten geometry. In the limit $M \rightarrow \infty$, the gauge symmetry $G$
is reduced to the smaller one $G'$. The Seiberg-Witten geometry is also reduced 
to the one with gauge symmetry $G'$ in this limit.
We can see this by substituting  $a= \la a \ra +\D a$ into (\ref{swg}) 
and keeping the leading order in $M$.
To leave the $j$-th flavor of hypermultiplets in the $G'$ theory,
its mass $m_j$ is also shifted as in (\ref{ms}).
After taking the appropriate coordinate $(x'_1,x'_2,x'_3)$ we should have 
\beqa
W_G(x_1,x_2,x_3;a) & =& 
M^{h-h'} W_{G'} (x'_1,x'_2,x'_3;\D a) + {\it o}(M^{h-h'}), \CR
X_{G}^{{\cal R}}(x_1,x_2,x_3;a, m_j) & =& 
M^{l({\cal R}) -l({\cal R}_{s_j})} X_{G'}^{{\cal R}_{s_j}} 
(x'_1,x'_2,x'_3;\D a,m'_j) 
+ {\it o}(M^{l({\cal R}) -l({\cal R}_{s_j})}), \CR
\eeqa
where $W_{G'}$ is a simple singularity of type $G'$,
$X_{G'}^{{\cal R}_{s_j}}$ is some polynomial of the indicated variables,
$h'$ is the dual Coxeter number of $G'$ and $l({\cal R}_{s_j})$ is 
the index of the representation ${\cal R}_{s_j}$ of $G'$.
The dependence on $M$ can be understood from 
the scale matching relation between theories with gauge group $G$ and $G'$
\beq
\La'^{\,\, 2 h'-\sum_{j=1}^{N_f} l({\cal R}_{s_j}) } =
\frac{ \La^{2 h- l({\cal R}) N_f } }{M^{2(h-h') - 
( l({\cal R}) N_f - \sum_{j} l({\cal R}_{s_j}) ) } },
\label{scale}
\eeq
where $\La'$ is the scale of the $G'$ theory.
Thus, in the limit $M \rightarrow \infty$, the Seiberg-Witten geometry becomes
\beq
z'+\frac{1}{z'} \La'^{\,\, 2 h'-\sum_{j=1}^{N_f} l({\cal R}_{s_j}) }
\prod_{j=1}^{N_f} X_{G'}^{{\cal R}_{s_j}} (x'_1,x'_2,x'_3;\D a,m'_j) 
- W_{G'} (x'_1,x'_2,x'_3;\D a)=0,
\label{swg2}
\eeq
where $z'=z / M^{h-h'}$.

Next, we will apply this reduction procedure explicitly
to the $N=2$ gauge theory with gauge group $E_6$ with
$N_f$ fundamental hypermultiplets. 



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Breaking $E_6$ gauge group to $SO(10)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

There are two ways of removing a node from the Dynkin diagram of $E_6$ 
to obtain a simple group $G'$ (see fig.\ref{fig:E6D}).
When a node corresponding to $\A_5$ (or $\A_6$) is removed,
we have $G'=SO(10)$ (or $SU(6)$).
\begin{figure}
\epsfxsize=100mm
%\epsfysize=70mm
\hspace{3cm} \epsfbox{e6Dynkin2.eps}
%\vspace{-3cm}
\caption{$E_6$ Dynkin diagram}
\label{fig:E6D}
\end{figure}
The former corresponds to the case of $G'=SO(10)$ and the latter to $G'=SU(6)$.
First we consider the breaking of $E_6$ gauge group 
down to $SO(10)$ by tuning VEV of $\Phi$ as
$\la a_i \ra = M \D_{i,5}$.
Using the inverse of the Cartan matrix we get
$\la a^i \ra = 
(\frac{2}{3} M ,\frac{4}{3} M ,\frac{6}{3} M,\frac{5}{3} M,\frac{4}{3} M,M)$.

The Seiberg-Witten geometry for $N=2$ gauge theory with 
gauge group $E_6$ with $N_f$ fundamental matters is proposed in \cite{TeYa3}
\beq
z+{1\over z} \La^{24-6N_f} 
\prod_{i=1}^{N_f} X_{E_6}^{\bf 27} (x_1,x_2,x_3;w,m_i)
- W_{E_6}(x_1,x_2,x_3;w)=0,
\label{e6aleA}
\eeq
where
\beq
W_{E_6}(x_1,x_2,x_3;w)=x_1^4+x_2^3+x_3^2+w_2\, x_1^2 x_2+w_5\, x_1x_2
+w_6\, x_1^2+w_8\, x_2+w_9\, x_1+w_{12},
\eeq
and
\beqa
&& X_{E_6}^{\bf 27} (x_1,x_2,x_3;w,m_i)  \CR
&=& 8 \left( {m_i}^6+2 w_2 {m_i}^4-8{m_i}^3x_1 
+\left( w_2^2-12 x_2 \right) {m_i}^2 \right. \CR
&& \left. \hspace{3cm} 
+4 w_5 {m_i}-4 w_2 x_2 -8(x_1^2-ix_3+w_6/2) \right).
\eeqa
Here $w_k=w_k(a)$ is the degree $k$ Casimir of $E_6$ made out of $a_j$
and the degrees of $x_1,x_2$ and $x_3$ are $3,\, 4$ and $6$ respectively.
Now, substituting $a_i= M \D_{i,5}+\D a_i$ into $w_k(a)$ 
and setting $\D a^5=0$,
we expand $W_{E_6}$ and $X_{E_6}^{\bf 27}$ in $M$.
As discussed in the previous section,
there should be coordinates $( x'_1,x'_2,x'_3 )$ which can eliminate the 
terms depending upon $M^l$ $(5 \leq l \leq 12)$ in $W_{E_6}$.
Indeed, we can find such coordinates as,
\beqa
x_1 & =& -\frac{2}{27} M^3-\frac{1}{4} M x'_1 -\frac{1}{6} M w_2, \CR
x_2 & =& \frac{1}{54} M^4+\frac{1}{12} M^2 x'_1 +\frac{1}{9} M^2 w_2
+\frac{1}{8} x'_2 +\frac{1}{6} w_2^2, \CR
x_3 & =& - i \frac{1}{16} M^2 x'_3.
\label{coorda}
\eeqa
Then the $E_6$ singularity $W_{E_6}$ is written as
\beq
W_{E_6}(x_1,x_2,x_3;w)=\left( \frac{1}{4} M \right)^4 W_{D_5}(x'_1,x'_2,x'_3;v)
+{\it O}(M^3),
\label{a1}
\eeq
where 
\beq
W_{D_5}(x_1,x_2,x_3; v)
={x_1}^4+x_1{x_2}^2-{x_3}^2+v_2{x_1}^3+v_4x_1^2+v_6x_1+v_8+v_5x_2, 
\label{WD5}
\eeq
and $v_k=v_k(\D a)$ is 
the degree $k$ Casimir of $SO(10)$ constructed from $\D a_i$.
If we represent $\Phi$ as a $10 \times 10$ matrix of the
fundamental representation of $SO(10)$, we have
 $v_{2 l}=\frac{1}{2l} {\rm Tr} \Phi^{2 l}$ and $v_5=2 i {\rm Pf} \Phi$.
Thus we see in the $M \rightarrow \infty$ limit that
the Seiberg-Witten geometry for $N=2$ pure Yang-Mills theory with 
gauge group $E_6$ becomes that with gauge group $SO(10)$.

Next we consider the effect of symmetry breaking in the matter sector.
The fundamental representation ${\bf 27}$ of $E_6$
is decomposed into the representations of $SO(10) \times U(1)$ as
\beq
{\bf 27} ={\bf 16}_{-\frac{1}{3}} \oplus {\bf 10}_{\frac{2}{3}} 
\oplus {\bf 1}_{-\frac{4}{3}},
\eeq
where the subscript denotes the $U(1)$ charge 
$ \A_5 \cdot \lm_i \, (1 \leq i \leq 27)$.
The indices of the spinor representation ${\bf 16}$
and the vector representation ${\bf 10}$ are 
four and two, respectively. Let us first take the scaling limit in such a 
way that the spinor matters of $SO(10)$ survive. Then 
the terms with $M^l$ $(l \geq 3)$ in $X_{E_6}^{27}$ must be
absent after a change of variables (\ref{coorda}) 
and the mass shift $m_i=\frac{1}{3} M + {m_{s}}_i$ (see (\ref{ms})).
In fact we find that
\beq
X_{E_6}^{\bf 27}(x_1,x_2,x_3;w,m_i) 
=M^2 X_{D_5}^{\bf 16}(x'_1,x'_2,x'_3;v,{m_s}_i)
+{\it O}(M),
\label{a2}
\eeq
where 
\beq
X_{D_5}^{\bf 16}(x_1,x_2,x_3;v,m)=m^4+\left( x_1+\frac{1}{2} v_2 \right) m^2
-m x_2+\frac{1}{2} x_3-\frac{1}{4} \left( v_4-\frac{1}{4} v_2^2 \right) 
-\frac{1}{4} v_2 x_1-\frac{1}{2} x_1^2.
\label{AD5}
\eeq
In order to make the vector matter of $SO(10)$ survive,
we shift masses as $m_i=-\frac{2}{3} M + {m_{v}}_i$. The result reads 
\beq
X_{E_6}^{\bf 27}(x_1,x_2,x_3;v,m_i) 
=M^4 X_{D_5}^{\bf 10}(x'_1,x'_2,x'_3;v,{m_v}_i)
+{\it O}(M^3),
\label{a3}
\eeq
where 
\beq
X_{D_5}^{\bf 10}(x_1,x_2,x_3;v,m)=m^2-x_1.
\eeq

Assembling (\ref{a1}), (\ref{a2}), (\ref{a3}) and
taking the limit $M \rightarrow \infty$ with 
\beq
\La^{16-4 N_s-2 N_v}_{SO(10) N_s N_v}=
2^{16+3 N_s+3 N_v} M^{-(8-2 N_s-4 N_v)} \La^{24-6 N_f}
\eeq
kept fixed, we now obtain the Seiberg-Witten geometry for $N=2$ $SO(10)$ gauge theory with 
$N_s$ spinor and $N_v$ vector hypermultiplets
\beqa
&&z  + {1\over z} \La^{16-4 N_s-2 N_v}_{SO(10) N_s N_v} 
\prod_{i=1}^{N_s} X_{D_5}^{\bf 16}(x_1,x_2,x_3;v,{m_s}_i)
\prod_{j=1}^{N_v} X_{D_5}^{\bf 10}(x_1,x_2,x_3;v,{m_v}_j) \CR
&& \hspace{6cm} - W_{D_5}(x_1,x_2,x_3;v)=0,
\label{so10ale}
\eeqa
where $N_f=N_s+N_v$.
In the massless case $m_{s_i}=m_{v_j}=0$, our result agrees with that 
obtained from the analysis of the compactification of Type IIB string theory
on the suitably chosen Calabi-Yau threefold \cite{AgGr}.
This is non-trivial evidence in support of (\ref{e6aleA}).
Moreover the Seiberg-Witten geometry derived in \cite{AgGr} is  only for 
the massless matters with $N_s-N_v=-2$. Here our expression is valid for 
massive matters of arbitrary number of flavors.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Breaking $SO(10)$ to $SO(8)$ and $SO(6)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Next we examine the gauge symmetry breaking in the 
$N=2$ $SO(10)$ gauge theory with spinor matters.
When $\Phi$ acquires the VEV $\la a_i \ra = M \D_{i,1}$,
namely $\la a^i \ra = \Big(M,M,M,{M \over 2},{M \over 2}\Big)$, the gauge group
$SO(10)$ breaks to $SO(8)$. (we rename $\D a_i$ to $a_i$ henceforth.)
Note that the spinor representation of $SO(10)$ reduces to the 
spinor ${\bf 8s}$ and its conjugate ${\bf 8c}$ of $SO(8)$.
Upon taking the limit $M \rightarrow \infty$
with $a_i = \la a_i \ra +\D a_i$,
we make a change of variables in (\ref{WD5}) 
\beqa
x_1 & =& x'_1, \CR
x_2 & =& i M x'_2, \CR
x_3 & =& M x'_3.
\label{cood1}
\eeqa
In terms of these variables, the $D_5$ singularity is shown to be
\beq
W_{D_5}(x_1,x_2,x_3;v)=\left( - M^2 \right) W_{D_4}(x'_1,x'_2,x'_3;u)
+{\it O}(M),
\eeq
where 
\beq
W_{D_4}(x_1,x_2,x_3; u)
={x_1}^3+x_1{x_2}^2+{x_3}^2+u_2 {x_1}^2+v_4 x_1+u_6+2 i \tilde{v_4} x_2, 
\label{WD4}
\eeq
$u_k$ is 
the degree $k$ Casimir of $SO(8)$ constructed from $\D a_i$
and $\tilde{v_4}={\rm Pfaffian}$.
The contribution (\ref{AD5}) coming from the matters becomes
\beq
X_{D_5}^{\bf 16}(x_1,x_2,x_3;v,{m_s}_i) 
=M^2 X_{D_4}^{\bf 8s}(x'_1,x'_2,x'_3;u,{m'_s}_i)+{\it O}(M^3),
\eeq
where 
\beq
X_{D_4}^{\bf 8s}(x_1,x_2,x_3;u,m)
=m^2+\frac{1}{2} x_1-i \frac{1}{2} x_2+\frac{1}{4} u_2.
\eeq
In the above limit, we have taken ${m_s}_i=\frac{1}{2} M +{m'_s}_i$ which
corresponds to the spinor representation of $SO(8)$.
If we instead take ${m_s}_i=-\frac{1}{2} M +{m'_s}_i$, which
corresponds to the conjugate spinor representation, then 
$x_2$ is replaced with $-x_2$ in $X_{D_4}^{\bf 8s}$.

If we consider the vector matters of $SO(10)$,
we see that a change of variables (\ref{cood1}) without the shift of mass 
does not affect ${m_v}_i-x_1$.
Therefore,
in taking the limit $M \rightarrow \infty$ with 
\beq
\La^{12-2 N_s-2 N_v}_{SO(8) N_s N_v}=
M^{-(4-2 N_s)} \La^{16-4 N_s-2 N_v}_{SO(10) N_s N_v}
\eeq
being fixed, we conclude that the Seiberg-Witten geometry for $N=2$ $SO(8)$ gauge theory
with $N_s$ spinor and $N_v$ vector flavors is
\beqa
&&z  + {1\over z} \La^{12-2 N_s-2 N_v}_{SO(8) N_s N_v}
\prod_{i=1}^{N_s} X_{D_4}^{\bf 8s}(x_1,x_2,x_3;u,{m'_s}_i)
\prod_{j=1}^{N_v} X_{D_4}^{\bf 8v}(x_1,x_2,x_3;u,{m_v}_j) \CR
&& \hspace{6cm} - W_{D_4}(x_1,x_2,x_3;u)=0,
\label{so8ale}
\eeqa
where $X_{D_4}^{\bf 8v}(x_1,x_2,x_3;u,m)=m^2-x_1$.

There is a ${\bf Z}_2$ action in the triality of $SO(8)$ which 
exchanges the vector representation and the spinor representation. Accordingly
the $SO(8)$ Casimirs are exchanged as
\beqa
v_2 & \leftrightarrow &  v_2, \CR
v_4 & \leftrightarrow &  -\frac{1}{2} v_4+3 {\rm Pf}+\frac{3}{8} v_2^2, \CR
{\rm Pf} & \leftrightarrow &   
\frac{1}{2} {\rm Pf} +\frac{1}{4} v_4-\frac{1}{16} v_2^2, \CR
v_6 & \leftrightarrow &  
v_6+\frac{1}{16} v_2^3-\frac{1}{4} v_4 v_2+\frac{1}{2} {\rm Pf} \,\, v_2.
\label{ex}
\eeqa
Thus the ${\bf Z}_2$ action is expected to
exchange $X_{D_4}^{\bf 8s}$ and $X_{D_4}^{\bf 8v}$ in (\ref{so8ale})
after an appropriate change of coordinates $x_i$.
Actually, using the new coordinates $(x'_1,x'_2)$ introduced by
\beqa
x_1 &=& -\frac{1}{2} x'_1+i \frac{1}{2}x'_2-\frac{1}{4} v_2, \CR
x_2 &=& -i \frac{3}{2} x'_1+ \frac{1}{2} x'_2-i \frac{1}{4} v_2,
\eeqa
we see that the $D_4$ singularity (\ref{WD4}) remains intact except
for (\ref{ex}) and
$X_{D_4}^{\bf 8s} \leftrightarrow X_{D_4}^{\bf 8v}$.

%Note that a ${\bf Z}_2$ action in the triality of $SO(8)$ which 
%exchange the spinor representation and the conjugate spinor representation
%is trivial.
%Thus the Seiberg-Witten geometry (\ref{so8ale}) is triality invariant.

One may further break the gauge group $SO(8)$ to $SO(6)$ following the breaking
pattern $SO(10)$ to $SO(8)$. Suitable coordinates are found 
to be $x_1=x'_1, x_2= i M x'_2$ and $x_3=M x'_3$.
The resulting Seiberg-Witten geometry for $N=2$ $SO(6)$ gauge theory with 
$N_s$ spinor flavors and $N_v$ vector flavors is
\beqa
&&z  + {1\over z} \La^{8-N_s-2 N_v}_{SO(6) N_s N_v}
\prod_{i=1}^{N_s} (\frac{1}{2} x_2 \pm {m_s}_i)
\prod_{j=1}^{N_v} ( {m_v}_j^2-x_1 ) \CR
&& \hspace{6cm} - W_{D_3}(x_1,x_2,x_3;u)=0,
\label{so6ale}
\eeqa
where $W_{D_3}(x_1,x_2,x_3; u)
={x_1}^2+x_1{x_2}^2+{x_3}^2+u_2 {x_1}+u_4+2 i {\rm Pf} \Phi x_2$. The sign 
ambiguity in (\ref{so6ale}) arises from the two possible choices of the 
shift of masses in $SO(8)$ theory.

When $N_s=0$, it is seen that the present $SO(2N_c)$ results yield the
well-known curves for $SO(2N_c)$ theory with vector matters \cite{ArSh,Ha}.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection{Breaking $E_6$ gauge group to $SU(6)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Now we wish to break the $E_6$ gauge group down to $SU(6)$ 
by giving the VEV
$\la a_i \ra = M \D_{i,6}$ to $\Phi$, that is, 
$\la a^i \ra = (M,2 M,3 M,2 M,M,2 M)$.
As in the previous section,
we first substitute $a_i= M \D_{i,6}+\D a_i$ into $w_k(a)$ in (\ref{e6aleA}) 
and set $\D a^6=0$.
Then we expand $W_{E_6}$ and $X_{E_6}^{\bf 27}$ in $M$, and
look for the coordinates $( x'_1,x'_2,x'_3 )$ which eliminate the 
terms depending on $M^l$ $(7 \leq l \leq 12)$ in (\ref{e6aleA}).
We can find such coordinates as 
\beqa
&& x_1  = - \frac {5}{8} \,M^{2}\,{ x'_1} - 
{\displaystyle \frac {3}{4}} \,{ x'_1}\,{ w_2}, \CR
&& x_2  = {\displaystyle \frac {1}{16}} \,M^{4} 
+ ({\displaystyle \frac {1}{4}} \,{ x'_2} 
+ {\displaystyle \frac {1}{4}} \,{ x'_1}^{2} + 
{\displaystyle \frac {1}{12}} \,{ w_2})\,M^{2},
\CR
&& x_3  =\lefteqn{{\displaystyle \frac {1}{160}} \,M^{6} + ( - 
{\displaystyle \frac {1}{8}} \,{ x'_2} + {\displaystyle \frac {3
}{160}} \,{ w_2})\,M^{4}} \CR
 & & \hskip2mm + {\displaystyle \frac {1}{8}} \, ( { x'_3} - 
 \,{ x'_2}^{2} - 3 { x'_2}\,{ x'_1}^{2} - 
{ x'_2}\,{ w_2} + {\displaystyle \frac {2}{15}} \,
{ w_2}^{2} - 3 { x'_1}^{4})\,M^{2
} + {\displaystyle \frac {1}{2}} { w_5}{ x'_1} \!-\!
{\displaystyle \frac {1}{10}}{ w_6}, 
\label{ec}
\eeqa
in terms of which the $E_6$ singularity $W_{E_6}$ is represented as
\beq
W_{E_6}(x_1,x_2,x_3;w)=\left( \frac{1}{2} M \right)^6 W_{A_5}(x'_1,x'_2,x'_3;v)
+{\it O}(M^5),
\eeq
where 
\beq
W_{A_r}(x_1,x_2,x_3; v)
=x_1^r+x_2 x_3 +v_2{x_1}^{r-1}+v_3x_1^{r-2}+\cdots+v_r x_1+v_{r+1}, 
\eeq
and $v_k=v_k(\D a)$ is 
the degree $k$ Casimir of $SU(6)$ build out of $\D a_i$.
Hence it is seen in the $M \rightarrow \infty$ limit that 
the Seiberg-Witten geometry for $N=2$ pure Yang-Mills theory with 
gauge group $E_6$ becomes that with gauge group $SU(6)$.

The fundamental representation ${\bf 27}$ of $E_6$
is decomposed into the representations of $SU(6) \times U(1)$ as
\beq
{\bf 27} ={\bf 15}_{0} \oplus {\bf 6}_{1} \oplus {\bf \bar{6}}_{-1},
\eeq
where the subscript denotes the $U(1)$ charge 
$ \A_6 \cdot \lm_i \, (1 \leq i \leq 27)$.
The indices of the antisymmetric representation ${\bf 15}$
and the fundamental representation ${\bf 6}$ are 
four and one, respectively.
Thus the terms with $M^l$ $(l \geq 3)$ in $X_{E_6}^{\bf 27}$ must be
absent after taking the coordinates $( x'_1,x'_2,x'_3 )$ defined in (\ref{ec}).
Note that there is no need to shift the mass
to make the antisymmetric matter survive. 
We indeed obtain a desired expression
\beq
X_{E_6}^{\bf 27}(x_1,x_2,x_3;w,m_i) 
=-M^2 X_{A_5}^{\bf 15}(x'_1,x'_2,x'_3;v,m_i)+{\it O}(M),
\eeq
where 
\beqa
&&X_{A_5}^{\bf 15}(x_1,x_2,x_3;v,m)  = 
m^4-2 m^3 x_1+ 3 \left( \frac{1}{3} v_2 +x_1^2+x_2 \right) m^2
\CR && \hspace{2cm}
+m v_3-x_3+x_1^4+2 v_2 x_1^2+3 x_2 x_1^2+v_3 x_1+x_2^2+v_2 x_2+v_4.
\eeqa
If we shift the mass as $m_i=M + {m_{f}}_i$
in order to make the vector matter survive,
we find that
\beq
X_{E_6}^{\bf 27}(x_1,x_2,x_3;v,m_i) 
=2 M^5 X_{A_5}^{\bf 6}(x'_1,x'_2,x'_3;v,{m_f}_i)+{\it O}(M^4),
\label{aa}
\eeq
where $X_{A_5}^{\bf 6}(x_1,x_2,x_3;v,m)=m+x_1$.
The shift of masses $m_i=-M + {m_{f}}_i$ also
corresponds to making the vector matter survive, but
the factor $(-1)$ is needed in the RHS of (\ref{aa}).

{}From these observations we can obtain the Seiberg-Witten geometry for $N=2$ $SU(6)$ 
gauge theory with $N_a$ antisymmetric and $N'_f$ fundamental matters
by taking the limit $M \rightarrow \infty$ while
\beq
\La^{12-4 N_a-N'_f}_{SU(6) N_a N'_f}=
(-1)^{N_a} 2^{12+2 N'_f} M^{-(12-2 N_a-5 N'_f)} \La^{24-6 N_f}
\eeq
held fixed. Our result reads
\beqa
&&z  + {1\over z} \La^{12-4 N_a-N'_f}_{SU(6) N_a N'_f} 
\prod_{i=1}^{N_a} X_{A_5}^{\bf 15}(x_1,x_2,x_3;v,{m_a}_i)
\prod_{j=1}^{N'_f} X_{A_5}^{\bf 6}(x_1,x_2,x_3;v,{m_f}_j) \CR
&& \hspace{6cm} - W_{A_5}(x_1,x_2,x_3;v)=0,
\label{su6ale}
\eeqa
where $N_f=N_a+N'_f$.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\subsection{Breaking $SU(6)$ to $SU(5)$, $SU(4)$ and $SU(3)$}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

We are now able to break $SU(r+1)$ gauge group to $SU(r)$ successively
by putting $\la a_i \ra = M \D_{i,r}$.
In sect.2 we have seen that 
the proper coordinates are chosen to be $x_1=x'_1+M/(r+1), x_2=x'_2$ and 
$x_3=M x'_3$ in terms of which  
$W_{A_{r}}(x_1,x_2,x_3; v)=M W_{A_{r-1}}(x'_1,x'_2,x'_3; v')+{\it O}(M^0)$.
Note that the degrees of $x_1, x_2$ and $x_3$ are $1,2$ and $r-1$, 
respectively. The antisymmetric representation of $SU(r+1)$
is decomposed into the antisymmetric and fundamental representations of 
$SU(r) \times U(1) $ as follows
\beq
{\bf \frac{r (r+1)}{2} } =
{\bf \frac{(r-1) r}{2}}_{\frac{2}{r+1}} \oplus {\bf r}_{-\frac{r-1}{r+1}},
\eeq
where the subscript denotes the $U(1)$ charge.
After some computations we can see that
the Seiberg-Witten geometry for $N=2$ $SU(r+1)$ $(r \leq 5)$ gauge theory 
with $N_a$ antisymmetric and $N'_f$ fundamental hypermultiplets
turns out to be
\beqa
&&z  + {1\over z} \La^{2(r+1)-(r-1) N_a-N'_f}_{SU(r+1) N_a N'_f} 
\prod_{i=1}^{N_a} X_{A_r}^{\bf \frac{r(r+1)}{2}}(x_1,x_2,x_3;v,{m_a}_i)
\prod_{j=1}^{N'_f} (x_1-{m_f}_j) \CR
&& \hspace{6cm} - W_{A_r}(x_1,x_2,x_3;v)=0,
\label{sur+1ale}
\eeqa
where $X_{A_r}^{\bf \frac{r(r+1)}{2}}$ is defined as
\beq
X_{A_r}^{\bf \frac{r(r+1)}{2}} \left( x_j;v,{m_a}_i=\frac{2 M}{r+1}
+{m'_a}_i \right)
= M X_{A_{r-1}}^{\bf \frac{(r-1)r}{2}}(x'_j;v',{m'_a}_i) +{\it O}(M^0),
\eeq
and $\La^{2(r+1)-(r-1) N_a-N'_f}_{SU(r+1) N_a N'_f}
=M^{2-N_a} \La^{2 r-(r-2) N_a-N'_f}_{SU(r) N_a N'_f}$.
Explicit calculations yield
\beqa
X_{A_4}^{\bf 10}(x_j;v,{m_a}_i) \!\!\!& =&\!\!\! m^3-m^2 x_1
+(2 x_2+2 x_1^2+v_2) m
+2 x_1^2-x_3+x_2 x_1  +v_2 x_1+v_3, \CR
X_{A_3}^{\bf 6}(x_j;v,{m_a}_i) \!\!\!& =&\!\!\! m^2+x_2-x_3+2 x_1^2+v_2, \CR
X_{A_2}^{\bf 3}(x_j;v,{m_a}_i) \!\!\!& =&\!\!\! m+x_1-x_3. 
\eeqa
We also see that 
\beqa
&& X_{A_5}^{\bf 15}\Big(x_j;v,{m_a}_i=-\frac{2}{3} M+{m'_f}_i \Big)
=M^3 (x'_1-{m'_f}_i)
+{\it O}(M^2), \CR
&& X_{A_4}^{\bf 10}\Big(x_j;v,{m_a}_i=-\frac{3}{5} M+{m'_f}_i \Big)
=-M^2 (x'_1-{m'_f}_i)
+{\it O}(M^1), \CR
&& X_{A_3}^{\bf 6}\Big(x_j;v,{m_a}_i=-\frac{1}{2} M+{m'_f}_i \Big)
=M (x'_1-{m'_f}_i-x'_3)+{\it O}(M^0)
\eeqa
by shifting masses in such a way that the fundamental matters remain.

We now check our $SU(N_c)$ results. First of all, 
for $SU(3)$ gauge group, the antisymmetric representation is identical to
the fundamental representation. 
Thus (\ref{sur+1ale}) should be equivalent to the well-known $SU(3)$ curve.
In fact, if we integrate out variables $x_2$ and $x_3$, 
the Seiberg-Witten geometry (\ref{sur+1ale}) yields the $SU(3)$ curve with
$N_a+N_f'$ fundamental flavors.

Let us next turn to the case of $SU(4)$ gauge group.
Since the Lie algebra of $SU(4)$ is the same as that of $SO(6)$,
the antisymmetric and fundamental representations of $SU(4)$ correspond to 
the vector and spinor representations of $SO(6)$ respectively.
This relation is realized in (\ref{sur+1ale}) and (\ref{so6ale}) as follows.
If we set $x_1=\frac{1}{2} x'_2$, $x_2=i x'_3- \frac{1}{2}x'_1-
\frac{1}{4} {x'_2}^2-\frac{1}{2} v_2$ and $x_3=i x'_3+\frac{1}{2} x'_1+
\frac{1}{4} {x'_2}^2+\frac{1}{2} v_2$,
we find 
\beq
W_{A_3}(x_i;v)=-\frac{1}{4} W_{D_3} (x'_i;u),
\eeq
where $u$ is related to $v$ through
$u_2=2 v_2, u_4=-4 v_4+ v_2^2$ and ${\rm Pf} =i v_3$.
Moreover we obtain $X_{A_3}^{\bf 6}(x_j;v,{m_a}_i) = {m_a}_i^2-x'_1$ and
$x_1-{m_f}_j=\frac{1}{2} x'_2-{m_f}_j$. Thus our $SU(4)$ result is in
accordance with what we have anticipated. This observation provides a
consistency check of our procedure since both $SO(6)$ and $SU(4)$ results
are deduced from the $E_6$ theory via two independent routes associated
with different symmetry breaking patterns.

Checking the $SU(5)$ gauge theory result is most intricate. 
Complex curves describing $N=2$ $SU(N_c)$ gauge theory with
matters in one antisymmetric representation and fundamental representations
are obtained in \cite{LaLo,LaLoLo2} using brane configurations.
Let us concentrate on $SU(5)$ theory with one massless antisymmetric matter
and no fundamental matters in order to compare with our result (\ref{so6ale}).
The relevant curve is given by \cite{LaLo}
\beqa
& & y^3+x y^2 (x^5+v_2 x^3-v_3 x^2+v_4 x-v_5) \CR
& & \hspace{0.5cm} 
-y \La^7 (3 x^5+3 v_2 x^3-v_3 x^2+3 v_4 x-v_5) +2 \La^{14} (x^4+v_2 x^2+v_4)=0.
\label{curL}
\eeqa
The discriminant of (\ref{curL}) has the form
\beq
\Delta_{Brane}
=F_0(v) \La^{105} (27 \La^7v_4^2+v_5^3) ( H_{50}(v,L) )^2 ( H_{35}(v,L) )^6,
\eeq
where $F_0$ is some polynomial in $v$, $H_{n}$ is a degree $n$ 
polynomial in $v$ and $L=-\La^7/4$.
If we set $v_2=v_3=0$ for simplicity, then
\beqa
H_{50}(v,L) &=& 65536\,{v_4}^{10}\,{v_5}^{2} + 1048576\,{v_4}^{
9}\,{L}^{2} - 33587200\,{v_4}^{7}\,{v_5}^{3}\,{L} 
+ 1600000\,
{v_4}^{5}\,{v_5}^{6} \CR
& & - 539492352\,{v_4}^{6}\,{v_5}\,{L}^{3} + 
3261440000\,{v_4}^{4}\,{v_5}^{4}\,{L}^{2} + 390000000\,
{v_4}^{2}\,{v_5}^{7}\,{L} \CR 
& & + 9765625\,{v_5}^{10} + 143947517952\,{v_4}^{3}\,{v_5}^{2}\,{L}^{4} + 
5378240000\,{v_4}\,{v_5}^{5}\,{L}^{3} \CR
& & + 1457236279296\, {v_4}^{2}\,{L}^{6} + 53971714048\,{v_5}^{3}\,{L}^{5}, \CR
H_{35}(v,L)& =& 32 v_5^7+432 L v_4^2 v_5^2+17496 L^3 v_4 v_5^2+177147 L^5.
\eeqa
We have also calculated the discriminant $\Delta_{ALE}$ of our expression
(\ref{sur+1ale}) with $r=4$ and found it in the factorized form.
Evaluating $\Delta_{Brane}$ and $\Delta_{ALE}$ at sufficiently many points 
in the moduli space, we observe that $\Delta_{ALE}$ also contains a factor
$H_{50}(v,L)$ with $\La_{SU(4) 1,0}^7=L$. This fact may be regarded as a
non-trivial check for the compatibility of the M-theory/brane dynamics 
result and our ALE space description. It is thus inferred that only the zeroes
of a common factor $H_{50}(v,L)$ in the discriminants represent the
physical singularities in the moduli space.\footnote{A similar phenomenon
is observed in $SU(4)$ gauge theory. We have checked that the discriminant 
of the curve for $SU(4)$ theory with one massive antisymmetric hypermultiplet
proposed in \cite{LaLo} and that of our ALE formula (\ref{sur+1ale}) 
with $r=3$ carry a common factor. See also \cite{OdToSaSa}}.

Moreover it is shown that the Seiberg-Witten geometries 
obtained in this section by breaking the $E_6$ Seiberg-Witten geometry
can be rederived using the method of $N=1$ confining phase 
superpotentials \cite{TeYa4}.



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{Conclusions}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

In this thesis, we have studied the 
$N=2$ Seiberg-Witten Geometry via the Confining Phase superpotential technique.
In particular, we have shown that the 
ALE spaces of type ADE fibered over ${\bf CP}^1$ is natural geometry
for the $N=2$ supersymmetric gauge theories with ADE gauge groups.

In chapter two, 
we have reviewed the exact description of the 
low-energy effective theory of 
the Coulomb phase of four-dimensional $N=2$ supersymmetric gauge theory
in terms of the Seiberg-Witten curve or Seiberg-Witten geometry.
The Seiberg-Witten geometry has been derived from the superstring theory 
compactified on the suitably chosen Calabi-Yau three-fold.

In chapter three, 
we have shown how to derive the Seiberg-Witten curves for the Coulomb phase 
of $N=2$ supersymmetric gauge theories
by means of the $N=1$ confining phase superpotential.
To put it concretely, 
we have obtained a low-energy effective superpotential for a phase with a 
single confined photon in $N=1$ gauge theory.
The expectation values of gauge invariants built out of
the adjoint field parametrize the singularities of moduli space of the $N=2$
Coulomb phase. 
According to this derivation 
it is clearly observed
that the quantum effect in the Seiberg-Witten curve has its origin in the $SU(2)$ gluino
condensation in view of $N=1$ gauge theory dynamics.

In chapter four, 
we have applied the confining phase superpotential 
to the $N=1$ supersymmetric pure Yang-Mills theory
with an adjoint matter with classical or ADE gauge groups. 
The results can be used to derive the Seiberg-Witten curves
for $N=2$ supersymmetric pure Yang-Mills theory 
with classical or ADE gauge groups
in the form of a foliation over ${\bf CP}^1$ which is identical 
to the spectral curves for the 
periodic Toda lattice.
Transferring the critical points in the $N=2$ Coulomb phase 
to the $N=1$ theories we have found non-trivial $N=1$ SCFT with 
the adjoint matter field governed by a superpotential.

In chapter five,
using the technique of confining phase superpotential we have determined
the curves describing the Coulomb phase of $N=1$ supersymmetric gauge theories
with adjoint and fundamental matters with classical gauge groups.
In the $N=2$ limit our results recover the curves for the Coulomb phase in
$N=2$ QCD. 
For the gauge group $Sp(2N_c)$, in particular, we have observed
that taking into account the instanton effect in addition to $SU(2)$ gaugino
condensation is crucial to obtain the effective superpotential for the phase
with a confined photon. This explains in terms of $N=1$ theory a peculiar
feature of the $N=2$ $Sp(2N_c)$ curve when compared to the $SU(N_c)$ and
$SO(N_c)$ cases.

Next we have proposed Seiberg-Witten geometry 
for $N=2$ supersymmetric gauge theory with gauge group $E_6$ 
with massive $N_f$ fundamental hypermultiplets
employing the confining phase superpotentials method.
The resulting manifold takes the form of a fibration of the ALE space of 
type $E_6$.

Starting with the Seiberg-Witten geometry for $N=2$ supersymmetric gauge theory with
gauge group $E_6$ with massive fundamental hypermultiplets, we have obtained 
the Seiberg-Witten geometry for $SO(2 N_c)$ $(N_c \leq 5)$ theory with massive spinor
and vector hypermultiplets by implementing the gauge symmetry breaking in
the $E_6$ theory. The other symmetry breaking pattern has been used to
derive the Seiberg-Witten geometry for $N=2$ $SU(N_c)$ $(N_c \leq 6)$ theory with
massive antisymmetric and fundamental hypermultiplets. All the Seiberg-Witten geometries
we have obtained are of the form of ALE fibrations over a sphere.
Whenever possible our results have been compared with those obtained in
the approaches based on the geometric engineering and the brane 
dynamics. 
It is impressive to find an agreement in spite of the fact that
the methods are fairly different. 

Thus our study of the confining
phase superpotentials supports 
that Seiberg-Witten geometry of the form of 
ALE fibrations over ${\bf CP}^1$ is a canonical description
for wide classes of the four-dimensional 
$N=2$ supersymmetric gauge field theories.
It is highly desirable 
to develop such a scheme explicitly for non-simply-laced gauge groups.

Although we have not discussed in this thesis, 
in order to analyze the mass of the BPS states and 
other interesting properties of 
the theory, one has to know the Seiberg-Witten three-form and 
appropriate cycles in the ALE fibration space.
For $N=2$ $SO(10)$ theory with massless spinor and vector hypermultiplets,
these objects may be obtained in principle from the 
Calabi-Yau threefold on which the string theory is compactified \cite{AgGr}.
It is important to find the Seiberg-Witten three-form and appropriate cycles for the
Seiberg-Witten geometry when the massive hypermultiplets exist. 






\newpage

\chapter*{Acknowledgements}

I am deeply indebted to Professor S.-K. Yang for continuous guidance, advice
and discussions.
His great helps have made it possible for me to finish this thesis.
I also would like to thank all staff and students in the elementary particle theory group
of the Institute for Theoretical Physics of the University of Tsukuba.

Finally, I would like to thank the Research Fellowships of the Japan Society 
for the Promotion of Science for financial support.

\newpage

%%%%%%%  References

\begin{thebibliography}{99}

\bibitem{Se1} N. Seiberg, Phys. Rev. {\bf D49} (1994) 6857, hep-th/9402044 

\bibitem{Se2} N. Seiberg,
Nucl. Phys. {\bf B435} (1995) 129, hep-th/9411149 

\bibitem{SeWi1} N. Seiberg and E. Witten, 
Nucl. Phys. {\bf B426} (1994) 19, hep-th/9407087 

\bibitem{SeWi2} N. Seiberg and E. Witten, 
Nucl. Phys. {\bf B431} (1994) 484, hep-th/9408099

\bibitem{ArFa} P. Argyres and A. Faraggi, 
Phys. Rev. Lett. {\bf 74} (1995) 3931, hep-th/9411057

\bibitem{KlLeYaTh} A. Klemm, W. Lerche, S. Yankielowicz and S. Theisen,
Phys. Lett. {\bf B344} (1995) 169, hep-th/9411048 

\bibitem{DaSu} U. Danielsson and B. Sundborg,
Phys. Lett. {\bf B358} (1995) 273, hep-th/9504102

\bibitem{BrLa} A. Brandhuber and K. Landsteiner,
Phys. Lett. {\bf B358} (1995) 73, hep-th/9507008

\bibitem{HaOz} A. Hanany and Y. Oz, Nucl. Phys. {\bf B452} (1995) 283,
hep-th/9505075

\bibitem{ArPlSh} P.C. Argyres, M.R. Plesser and A.D. Shapere,
Phys. Rev. Lett. {\bf 75} (1995) 1699, hep-th/9505100

\bibitem{ArSh} P.C. Argyres and A.D. Shapere, 
Nucl. Phys. {\bf B461} (1996) 437, hep-th/9509175

\bibitem{Ha} A. Hanany, Nucl. Phys. {\bf B466} (1996) 85, hep-th/9509176


\bibitem{InSe} K. Intriligator and N. Seiberg, 
Nucl. Phys. {\bf B431} (1994) 551, hep-th/9408155 

\bibitem{ArDo} P.C. Argyres and M.R. Douglas, 
Nucl. Phys. {\bf B448} (1995) 93, hep-th/9505062 

\bibitem{ElFoGiRa} S. Elitzur, A. Forge, A. Giveon and E. Rabinovici,
Phys. Lett. {\bf B353} (1995) 79, hep-th/9504080 

\bibitem{ElFoGiRa2} S. Elitzur, A. Forge, A. Giveon and E. Rabinovici,
Nucl. Phys. {\bf B459} (1996) 160, hep-th/9509130 

\bibitem{In} K. Intriligator, 
Phys. Lett. {\bf B336} (1994) 409, hep-th/9407106

\bibitem{ElFoGiInRa} S. Elitzur, A. Forge, A. Giveon, K. Intriligator and 
E. Rabinovici, Phys. Lett. {\bf B379} (1996) 121, hep-th/9603051

\bibitem{TeYa1} S. Terashima and S.-K. Yang, 
Phys. Lett. {\bf B391} (1997) 107, hep-th/9607151 

\bibitem{Ki}  T. Kitao, 
Phys. Lett. {\bf B402} (1997) 290, hep-th/9611097

\bibitem{KiTeYa} T. Kitao, S. Terashima and S.-K. Yang,
Phys. Lett. {\bf B399} (1997) 75, hep-th/9701009 

\bibitem{GPR} A. Giveon, O. Pelc and E. Rabinovici, 
Nucl. Phys. {\bf B499} (1997) 100, hep-th/9701045


\bibitem{MaWa} E.J. Martinec and N.P. Warner, 
Nucl. Phys. {\bf B459} (1996) 97, hep-th/9509161 

\bibitem{LW} W. Lerche and N.P. Warner, 
Phys. Lett. {\bf B423} (1998) 79, hep-th/9608183

\bibitem{WY} N.P. Warner and S.-K. Yang, private communication


\bibitem{LaPiGi} K. Landsteiner, J.M. Pierre and S.B. Giddings, 
Phys. Rev. {\bf D55} (1997) 2367, hep-th/9609059 

\bibitem{Ito} K. Ito, 
Phys. Lett. {\bf B406} (1997) 54, hep-th/9703180,

\bibitem{TeYa2} S. Terashima and S.-K. Yang, Nucl. Phys. {\bf B519} (1998) 453,
hep-th/9706076



\bibitem{KlLeMaVaWa} A. Klemm, W. Lerche, P. Mayr, C. Vafa and N. Warner,
Nucl. Phys. {\bf B477} (1996) 746, hep-th/9604034 

\bibitem{Wi} E. Witten,
Nucl. Phys. {\bf B500} (1997) 3, hep-th/9703166


\bibitem{TeYa3} S. Terashima and S.-K. Yang, Phys. Lett. {\bf B430} (1998) 102,
hep-th/9803014

\bibitem{TeYa4} S. Terashima and S.-K. Yang, 
{\it Seiberg-Witten Geometry with Various Matter Contents},
hep-th/9808022, to appear in Nucl. Phys. B.


\bibitem{Br} J.H. Brodie, Nucl. Phys. {\bf B506} (1997) 183, hep-th/9705068

\bibitem{AgGr} M. Aganagic and M. Gremm, 
Nucl. Phys. {\bf B524} (1998) 207-223, hep-th/9712011


\bibitem{LaLo} K. Landsteiner and E. Lopez,
Nucl. Phys. {\bf B516} (1998) 273, hep-th/9708118

\bibitem{LaLoLo2} K. Landsteiner, E. Lopez and D.A. Lowe, 
JHEP. {\bf 7} (1998) 11, hep-th/9805158


\bibitem{KaVa} S. Kachru and C. Vafa,
Nucl. Phys. {\bf B450} (1995) 69, hep-th/9505105

\bibitem{FeHaStVa} S. Ferrara, J. A. Harvey, A. Strominger and C. Vafa,
Phys. Lett. {\bf B361} (1995) 59, hep-th/9505162

\bibitem{ArPlSe} P.~C.~Argyres, M.~R.~Plesser and N.~Seiberg,
Nucl. Phys. {\bf B471} (1996) 159,  hep-th/9603042  


\bibitem{KuScSe} D. Kutasov, A. Schwimmer and N. Seiberg,
Nucl. Phys. {\bf B459} (1996) 455, hep-th/9510222


\bibitem{Hu} J.E. Humphreys, {\it Reflection Groups and Coxeter Groups},
Cambridge University Press (1990)

\bibitem{Gor} A. Gorskii, I. Krichever, A. Marshakov, A. Mironov and 
A. Morozov, Phys. Lett. {\bf B355} (1995) 466, hep-th/9505035

\bibitem{NT} T. Nakatsu and K. Takasaki, Mod. Phys. Lett. {\bf A11} (1996) 157

\bibitem{IM} H. Itoyama and A. Morozov, 
Nucl. Phys. {\bf B477} (1996) 855, hep-th/9511126;
Nucl. Phys. {\bf B491} (1997) 529, hep-th/9512161


\bibitem{APSW} P.C. Argyres,  M.R. Plesser, N. Seiberg and E. Witten,
Nucl. Phys. {\bf B461} (1996) 71, hep-th/9511154

\bibitem{EHIY} T. Eguchi, K. Hori, K. Ito and S.-K. Yang,
Nucl. Phys. {\bf B471} (1996) 430, hep-th/9603002

\bibitem{EH} T. Eguchi and K. Hori, {\it N=2 Superconformal Field Theories
in 4 Dimensions and A-D-E Classification}, hep-th/9607125


\bibitem{Ka} A. Kapustin, Phys. Lett. {\bf B398} (1997) 104, hep-th/9611049


\bibitem{InSe2} K. Intriligator and N. Seiberg, 
Nucl. Phys. {\bf B444} (1995) 125, hep-th/9503179

\bibitem{Aff}  I. Affleck, M. Dine and N. Seiberg, 
Nucl. Phys. {\bf B256} (1985) 557

\bibitem{ILS} K. Intriligator, R.G. Leigh and N. Seiberg,
Phys. Rev. {\bf D50} (1994) 1092, hep-th/9403198


\bibitem{Sla} R. Slansky, Phys. Reports {\bf 79} (1981) 1

\bibitem{DeOz} J. de Boer and Y. Oz,
Nucl. Phys. {\bf B511} (1998) 155, hep-th/9708044

\bibitem{Te} S. Terashima, 
Nucl. Phys. {\bf B526} (1998) 163, hep-th/9712172

\bibitem{OdToSaSa} H. Oda, S. Tomizawa, N. Sakai and T. Sakai,
Phys. Rev. {\bf D58} (1998) 66002,
hep-th/9802004








%\bibitem{review} 
%W. Lerche, {\it Introduction to Seiberg-Witten Geometry
%and its Stringy Origin}, hep-th/9611190;
%C. Vafa, {\it Lectures on Strings and Dualities}, hep-th/9702201;
%A. Klemm, {\it On the Geometry behind N=2 Supersymmetric Effective Actions
%in Four Dimensions}, hep-th/9705131




%\bibitem{GiKu} A. Giveon and D. Kutasov, 
%{\it Brane Dynamics and Gauge Theory},
%hep-th/9802067



\end{thebibliography}

\end{document}

















