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\begin{document}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{The Affleck--Dine--Seiberg superpotential}
\setcounter{equation}{0}
\setcounter{footnote}{0}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{SUSY QCD Symmetry}
\label{Chap8:sec:SymmHolo}
 $SU(N)$  with $F$ flavors where $F <N$
\beq
\begin{tabular}{c|c|cccc}
& $SU(N)$ & $SU(F)$  & $SU(F)$ & $U(1)$ & $U(1)_R$ \\
\hline
$\Phi$, $Q$ & \fund & \fund & ${\bf 1}$ & 1 & ${{F-N}\over{F}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$\overline{\Phi}$, $\overline{Q}$ & $\overline{\fund}$   
& ${\bf 1}$  & $\overline{\fund}$
& -1 & ${{F-N}\over{F}}$ $\vphantom{\raisebox{3pt}{\asymm}}$\\
\end{tabular}
\eeq

Recall that the auxiliary $D^a$ fields:
\beq
D^a = g( \Phi^{*jn}(T^a)^m_n \Phi_{mj} - 
\overline{\Phi}^{jn}(T^a)^m_n \overline{\Phi}^*_{mj})~
\eeq
where $j =1 \ldots F$; 
$m, n=1\ldots N$,
$a=1\ldots N^2-1$, \newline $D$-term potential:
\beq
V= \frac{1}{2} D^a D^a ~
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Classical Moduli Space}

D-flat moduli space 
\beq
\langle \overline{\Phi}^{*}\rangle =
\langle \Phi \rangle = \left( \begin{array}{ccc}
v_1 & & \\
& \ddots &  \\
& & v_F \\
0 & \ldots & 0 \\
\vdots  & & \vdots \\
0 & \ldots & 0 \\
\end{array} \right)
\eeq
where $\langle \Phi \rangle$ is a $N\times F$ matrix \newline
generic point in the moduli space  $SU(N)\rightarrow SU(N-F)$
\beq
N^2-1-((N-F)^2-1)=2NF - F^2
\eeq
of $2NF$ chiral s.multiplets
only $F^2$ singlets are  massless \newline  \Blue{super
Higgs mechanism:  vector s.multiplet ``eats'' a
chiral s.multiplet}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Light ``Mesons"}

describe  $F^2$ 
light degrees of freedom in a gauge invariant
way by  $F \times F$ matrix 
\beq
M^j_i =  \overline{\Phi}^{j n}\Phi_{n i}~
\eeq
where we sum over the color index n


$M$ is  a  chiral superfield which is a product
of chiral superfields,
the only renormalization of $M$ is the product
of wavefunction renormalizations for $\Phi$ and $\overline{\Phi}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Chiral Symmetries}

 \noindent axial $U(1)_A$ symmetry  is 
explicitly broken
by instantons
\newline $U(1)_R$ symmetry remains unbroken

 mixed anomalies between
the global current and two gluons
 \newline $U(1)_R$: multiply the $R$-charge by the index\newline \indent gaugino contributes $1 \cdot N$ \newline  \indent each of
the $2 F$ quarks contributes $((F-N)/{F}-1)\cdot \sfrac{1}{2}$
\beq
A_{Rgg} = N+\left(\frac{F-N}{F}-1\right)\frac{1}{2} 2F=0  ~
\eeq
 $U(1)_A$:  gauginos do not contribute
\beq
A_{Agg} = 1 \cdot 2 F \cdot \frac{1}{2}~
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Spurious  Symmetry}
 keep track of selection rules  from the
broken $U(1)_A$ \newline define a spurious
symmetry 
\beq
Q\rightarrow e^{i\alpha}Q ~\nonumber\\
\overline{Q}\rightarrow e^{i\alpha}\overline{Q}~\nonumber\\
\theta_{\rm YM}\rightarrow\theta_{\rm YM}+2F\alpha~
\eeq
 holomorphic intrinsic scale 
transforms as
\beq
\Lambda^b \rightarrow e^{i2F \alpha}\Lambda^b~
\label{Chap8:spuriousRchargeLambda}
\eeq
 construct the effective superpotential  from: $W^a$, $\Lambda$,
and $M$
\beq
\begin{array}{c|cc}
 &U(1)_A & U(1)_R \\ \hline
W^a W^a& 0 & 2 \\
\Lambda^b & 2F & 0 \\
{\rm det}M & 2F & 2(F-N) \\
\end{array}
\eeq
${\rm det}M$ is  only $SU(F)\times SU(F)$ invariant 
made out of $M$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Effective Wilsonian Superpotential}
terms have the form
\beq
\Lambda^{bn} (W^aW^a)^m({\rm det}M)^p~
\eeq
periodicity of $\theta_{YM} \Rightarrow$ 
only
have powers of $\Lambda^{b}$ \newline (for $m=1$   perturbative
term $b \ln (\Lambda)\, W^aW^a$ because of  anomaly)

 superpotential is neutral under $U(1)_A$ and has
charge 2 under $U(1)_R$
\beq
0 = n \,2F +p\,2F ~\\
2=2 m +p\,2(F-N)~
\eeq
solution  is
\beq
n=-p= {{1-m}\over{N-F}}~
\eeq
$b=3N-F>0$, sensible $\Lambda \rightarrow 0$ limit
 if $n\ge 0$,
 implies  
$m \le 1$(because $N > F$)

  $W^aW^a$ contains derivatives,
locality requires $m \ge 0$ and integer
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Effective Wilsonian Superpotential}
only two possible
terms:
$m=0$ and $m=1$

  $m=1$ term is field
strength
term\newline 
periodicity of $\theta_{\rm YM} \Rightarrow$  coefficient
proportional to $b \ln \Lambda$.  


 $m=0$ term  is the Affleck--Dine--Seiberg (ADS) 
superpotential:
\Blue{\beq
W_{\rm ADS}(N,F) = C_{N,F} \left( {{\Lambda^{3N-F}}\over{ {\rm det}M
}}\right)^{{1}/{(N-F)}} ~
\label{Chap8:WADS}
\eeq}
where $C_{N,F}$ is renormalization scheme-dependent



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: moduli space}

Consider giving a large 
VEV, $v$, to one
flavor\newline $SU(N)\rightarrow SU(N-1)$ and one
flavor is ``eaten'' by the Higgs mechanism\newline $2N-1$ broken generators

effective theory has $F-1$ flavors
and  $2F-1$ 
gauge singlets
since 
\beq
2NF-(2N-1)-(2F-1)= 2(N-1)(F-1)~ 
\eeq


low-energy effective theory for the  $SU(N-1)$ gauge theory with 
$F-1$ flavors (gauge singlets  interact
only through irrelevant operators)
running holomorphic gauge coupling, $g_L$
\beq
{{8\pi^2}\over{g_L^2(\mu)}} = b_L \ln\left({{\mu}\over{\Lambda_L}}\right)~
\eeq
\beq
b_L= 3(N-1)-(F-1)= 3 N -F - 2~
\eeq
 $\Lambda_L$ is the holomorphic intrinsic scale of the low-energy
effective theory
\beq
\Lambda_L &\equiv& |\Lambda_L|e^{i \theta_{\rm YM}/b_L} 
= \mu e^{2 \pi i \tau_L/b_L}~
\label{Chap8:holomorphicscale_L}
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: moduli space}


low-energy coupling 
should match onto high-energy coupling
\beq
{{8\pi^2}\over{g^2(\mu)}} = b \ln\left({{\mu}\over{\Lambda}}\right)~
\eeq
at the scale  $v$ in $\overline{DR}$:
\beq
{{8\pi^2}\over{g^2(v)}} =
{{8\pi^2}\over{g_L^2(v)}}~
\eeq



\beq
\left({{\Lambda}\over{v}}\right)^b = 
\left({{\Lambda_L}\over{v}}\right)^{b_L}~
\nonumber\\
{{\Lambda^{3N-F}}\over{v^2}} = \Lambda^{3N-F-2}_{N-1,F-1}~
\label{Chap8:matchingoneflavorvev}
\eeq
subscript shows the number of colors and
flavors:
$\Lambda_{N-1,F-1} \equiv \Lambda_L$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: moduli space}


represent the light $(F-1)^2$ degrees of freedom 
as an $(F-1) \times (F-1)$ matrix $\wtilde M$
\beq
{\rm det}M = v^2{\rm det}\wtilde M +\ldots~,
\eeq
where $\ldots$ represents terms involving the decoupled gauge singlet
fields
\newline Plugging  into  $W_{\rm ADS}(N,F)$
and using 
\beq
\left({{\Lambda^{3N-F}}\over{v^2}}\right)^{{1}/(N-F)}
 = \left( \Lambda^{3N-F-2}_{N-1,F-1}\right)^{{1}/((N-1)-(F-1))}~
\eeq
 reproduce 
$W_{\rm ADS}(N-1,F-1)$ \newline provided that 
$C_{N,F}$ is only a function of $N-F$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: moduli space}

equal VEVs for all flavors 
\newline $SU(N) \rightarrow SU(N-F)$ and all  flavors are ``eaten''
\newline from matching running
couplings:
\beq
\left(\frac{\Lambda}{v}\right)^{3N-F} = 
\left(\frac{\Lambda_{N-F,0}}{v}\right)^{3(N-F)}~
\eeq
we then have 
\beq
{{\Lambda^{3N-F}}\over{v^{2F}}} = \Lambda^{3(N-F)}_{N-F,0}~
\eeq
So the effective superpotential is given by
\beq
W_{\rm eff}=C_{N,F} \Lambda^{3}_{N-F,0}~
\eeq
\Blue{reproduces holomorphy arguments for gaugino condensation}
in pure SUSY Yang-Mills





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: mass terms}
\label{Chap8:sec:masspert}
mass, $m$, for one flavor\newline   low-energy effective theory is 
 $SU(N)$  with $F-1$ flavors  
 \newline Matching 
gauge couplings
at  $m$:
\beq
\left({{\Lambda}\over{m}}\right)^b = \left({{\Lambda_L}\over{m}}\right)^{b_L}
\nonumber\\
m\Lambda^{3N-F} = \Lambda^{3N-F+1}_{N,F-1}~
\eeq
holomorphy $\Rightarrow$ superpotential must have the form
\beq
W_{\rm exact} = \left( {{\Lambda^{3N-F}}\over{ {\rm det}M
}}\right)^{{1}/(N-F)} f(t)~,
\eeq
where
\beq
t= m M^F_F\left( {{\Lambda^{3N-F}}\over{ {\rm det}M
}}\right)^{{-1}/(N-F)}~,
\eeq
 since
$m M^F_F$ is  mass term in  superpotential, it
has $U(1)_A$ charge 0, and $R$-charge 2, so $t$ has $R$-charge 0

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: mass terms}
Taking the 
limit $\Lambda\rightarrow 0$, $m \rightarrow 0$,  must recover our previous
results with the addition of a small mass term
\beq
f(t) = C_{N,F} +t~
\eeq
in  double limit $t$ is  arbitrary so this is the
exact form 
\beq
W_{\rm exact} = C_{N,F}\left( {{\Lambda^{3N-F}}\over{ {\rm det}M
}}\right)^{{1}/(N-F)} +m M^F_F~
\label{Chap8:W_exact:massterm}
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: mass terms}

equations of motion for $M^F_F$ and $M^j_F$
\beq
\frac{\partial W_{\rm exact}}{\partial M^F_F}&=&
C_{N,F}\left( {{\Lambda^{3N-F}}\over{ {\rm det}M
}}\right)^{{1}/(N-F)} \left(\frac{-1}{N-F}\right) \frac{{\rm cof}(M^F_F)}{{\rm det}M} +m =0 ~\\
\frac{\partial W_{\rm exact}}{\partial M^j_F}&=&
C_{N,F}\left( {{\Lambda^{3N-F}}\over{ {\rm det}M
}}\right)^{{1}/(N-F)} \left(\frac{-1}{N-F}\right)
\frac{{\rm cof}(M^j_F)}{{\rm det}M}=0~
\eeq
(where ${\rm cof}(M^F_i)$ is the cofactor of the matrix element $M^F_i$)
  imply that 
\beq
\frac{C_{N,F}}{N-F}\left( {{\Lambda^{3N-F}}\over{ {\rm det}M
}}\right)^{{1}/(N-F)} = m M^F_F~~~~~~~~~(*)
\label{Chap8:eq.motion_MFF}
\eeq
and  that ${\rm cof}(M^F_i)=0$.
Thus, $M$ has 
the block diagonal form
\beq
 M =
 \left( \begin{array}{cc}
{\wtilde M} &0 \\
0 &  M^F_F 
\end{array} \right)~
\eeq



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: mass terms}
 Plugging  (*) into 
the exact superpotential we find
\beq
W_{\rm exact}(N,F-1) = \left(N-F+1\right)
\left({{C_{N,F}}\over{N-F}}\right)^{(N-F)/(N-F+1)} \nonumber \\ 
\times
\left( {{\Lambda^{3N-F+1}_{N,F-1}}\over{ {\rm det}{\wtilde M}
}}\right)^{{1}/(N-F+1)}~
\eeq
$\propto W_{\rm ADS}(N,F-1)$.
For consistency, we have a recursion relation:
\beq
C_{N,F-1}=
\left(N-F+1\right)
\left({{C_{N,F}}\over{N-F}}\right)^{(N-F)/(N-F+1)}~
\eeq
 instanton calculation  reliable 
for $F=N-1$ (gauge group is completely broken), 
determines $C_{N,N-1}=1$ in the
$\overline{\rm DR}$ scheme
\Blue{\beq
C_{N,F}=N-F~
\eeq}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: mass terms}

masses for all  flavors
\newline Holomorphy $\Rightarrow$

\beq
W_{\rm exact} = C_{N,F}\left( {{\Lambda^{3N-F}}\over{ {\rm det}M
}}\right)^{{1}/(N-F)} +m^i_j M^j_i~
\eeq
where $m^i_j$ is the quark mass matrix.
Equation of motion for $M$ 
\beq
M^j_i &=& (m^{-1})^j_i \left( {{\Lambda^{3N-F}}\over{ {\rm det}\, M
}}\right)^{{1}/(N-F)} ~~~~~~~(**)
\label{Chap8:MVEV}
\eeq
taking the determinant and plugging the result back in to 
(**) gives
\beq
 {\bar \Phi}^j \Phi_i   =
M^j_i& =& (m^{-1})^j_i \left({\rm det}m\, \Lambda^{3N-F}\right)^{{1}/{N}}~
\label{Chap8:vacsol}
\eeq
 result involves  $N$th root $\Rightarrow$  \Blue{$N$ distinct
vacua},
differ by the phase of $M$
 
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Consistency of $W_{\rm ADS}$: mass terms}

Matching the holomorphic gauge coupling at mass thresholds 
\beq
 \Lambda^{3N-F}{\rm det}\,m = \Lambda_{N,0}^{3N}~
\eeq
So
\beq
W_{eff}= N \Lambda_{N,0}^3~
\eeq
reproduces holomorphy result for gaugino condensation
and determines coefficient (up to  phase)
\beq
\langle \lambda^a \lambda^a \rangle=  -32 \pi^2 e^{2 \pi i k/N}
\Lambda_{N,0}^3 ~
\label{Chap8:gauginocondensate2}
\eeq
where $k=1...N$.
Starting with $F=N-1$ flavors
 can  derive the correct
ADS
effective superpotential for $0\le F < N-1$, and gaugino 
condensation for $F=0$

\Blue{ justifies the assumption
that there was a mass gap in SUSY YM}





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Generating $W_{\rm ADS}$ from instantons}
Recall  ADS superpotential
\beq
W_{\rm ADS} \propto \Lambda^{{b}/(N-F)}
\eeq
 instanton effects are suppressed by
\beq
e^{-S_{\rm inst}} \propto \Lambda^b~
\eeq
So for $F=N-1$ it is possible that instantons can generate
$W_{\rm ADS}$ 

$SU(N)$ can be completely broken\newline 
 allows for reliable instanton calculation
 
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Generating $W_{\rm ADS}$ from instantons}

With equal VEVs 
$W_{\rm ADS}$ predicts quark masses of order
\beq
{{\partial^2 W_{\rm ADS}}\over{\partial \Phi_i \partial{\overline \Phi}^j}} \sim
{{\Lambda^{2N+1}}\over{v^{2N}}}~
\eeq
and a vacuum energy density of order
\beq
\left| {{\partial W_{\rm ADS}}\over{\partial \Phi_i}} \right|^2 \sim
\left|{{\Lambda^{2N+1}}\over{v^{2N-1}}}  \right|^2~
\eeq



 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Generating $W_{\rm ADS}$ from instantons}
single instanton vertex  has $2N$ gaugino legs 
 and $2F=2N-2$ quark legs
 \vspace{-24pt}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.4\hsize]{slideimages/instanton}
\end{center}
\end{figure}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
 quark legs connected to gaugino legs by 
a scalar VEV, two gaugino legs  converted to 
quark legs by the insertion of  VEVs 

\Blue{ fermion mass
is generated}

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Generating $W_{\rm ADS}$ from instantons}

 instanton calculation $\rightarrow$ quark mass 
\beq
m &\sim& e^{- 8 \pi^2/g^2(1/\rho)} v^{2 N}\rho^{2N-1}
\sim \left(\Lambda \rho\right)^{b}
v^{2 N} \rho^{2N-1}
\sim \Lambda^{2N+1}
 v^{2 N}\rho^{4N} ~
\eeq
dimensional analysis works because integration
over $\rho$  dominated by 
\beq
\rho^2 ={{b}\over{16\pi^2 v^2}}~
\eeq
quark legs  ending at the same spacetime point gives 
${\mathcal F}$ component of $M$, and  vacuum energy of the right size


can derive the ADS
superpotential
 for smaller values of $F$ from  $F=N-1$, so we
can derive gaugino condensation for zero flavors  from the instanton
calculation with $N-1$ flavors



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Generating $W_{\rm ADS}$ from $\langle \lambda \lambda \rangle$}
\label{Chap8:sec:WADSgaugino}
For $F<N-1$ can't use instantons since at  generic
point in  moduli space
 $SU(N)\rightarrow SU(N-F) \supset SU(2)$

 IR effective theory splits into and $SU(N-F)$ gauge
theory and $F^2$ gauge singlets described by $M$\newline 
two sectors coupled by irrelevant operators
\newline 
 $SU(N-F)$ gauginos have an anomalous $R$-symmetry
 \newline
$R$-symmetry    spontaneously broken by squark
VEVs
 not  anomalous

\vspace{24pt}
QCD Analogy:

 $SU(2)_L \times SU(2)_R$  spontaneously broken
 \newline
axial anomaly of the quarks is reproduced in the low-energy
theory by an irrelevant operator (the \Blue{Wess--Zumino term}) 
which gives $\pi^0 \rightarrow \gamma \gamma$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Generating $W_{\rm ADS}$ from $\langle \lambda \lambda \rangle$}

In SUSY QCD the  correct term is
present since 
\beq
\tau = 
\frac{3(N-F)}{2 \pi i}\ln\left( {{\Lambda_{N-F,0}}\over{\mu}}\right) ~
\label{Chap8:tau_effective}
\eeq
depends on $\ln {\rm det}M$ through
 matching condition
\beq
 \Lambda^{3N-F} = \Lambda_{N-F,0}^{3(N-F)}{\rm det}M~
\label{Chap8:matching}
\eeq
relevant term in effective theory $\propto$
\beq
&&\int d^2 \theta \ln {\rm det}(M)\, W^a W^a +h.c.\nonumber \\
&&= \left[ Tr({\mathcal F}_M M^{-1}) \lambda^a \lambda^a + 
{\rm Arg}({\rm det}M)
F^{a\mu\nu} {\wtilde F}^a_{\mu\nu} + \ldots\right] +h.c.
\label{Chap9:WZtermSUSYQCD}
\eeq
where ${\mathcal F}_M$ is the auxiliary field for $M$


 second term can be seen to arise through triangle diagrams involving
the fermions in the massive gauge supermultiplets

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Generating $W_{\rm ADS}$ from $\langle \lambda \lambda \rangle$}

 ${\rm Arg}({\rm det}M)$ transforms under  chiral rotation
as
Nambu--Goldstone boson of the spontaneously broken $R$-symmetry:
\beq
{\rm Arg}({\rm det}M)\rightarrow {\rm Arg}({\rm det}M)+2 F \alpha~
\eeq
equation of motion for ${\mathcal F}_M$ gives
\beq
{\mathcal F}_M &=& \frac{\partial W}{\partial M} 
=M^{-1} \langle \lambda^a \lambda^a \rangle
\propto M^{-1} \Lambda^3_{N-F,0}
\propto  M^{-1} 
\left({{\Lambda^{3N-F}}\over{ {\rm det}M}}\right)^{{1}/(N-F)}~
\eeq
 gives  vacuum energy density that agrees with  the  
ADS
calculation


potential energy implies that a nontrivial superpotential 
generated,  only superpotential consistent with
holomorphy and symmetry is $W_{\rm ADS}$

\Blue{for 
$F<N-1$ flavors, gaugino condensation generates $W_{\rm ADS}$}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Vacuum structure}
\label{Chap8:secVacuumstructure}

 
\beq
V_{\rm ADS}&=& \sum_i  |\frac{\partial W_{\rm ADS}}{\partial Q_i}|^2+ |\frac{\partial W_{\rm ADS}}{\partial {\overline Q_i}}|^2 \nonumber\\
&=&  \sum_i   |{\mathcal F}_i|^2 + |{\overline {\mathcal F}_i}|^2~,
\eeq
is
minimized as ${\rm det}M \rightarrow \infty$, so there is a ``run-away
vacuum"   

 potential loop-hole: wavefunction renormalization
effects not included, could produce  local minima,
could not produce new vacua unless renormalization factors were
singular


  cannot happen unless 
 particles are massless at  point in the moduli space, also produces  singularity in the superpotential


At ${\rm det} M = 0$, 
 massive gauge supermultiplets become massless

\Blue{ recent progress on 
theory without VEVs}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Intriligator---Seiberg---Shih}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.35\hsize]{slideimages/Intriligator} \includegraphics[width=0.3275\hsize]{slideimages/SeibergFace}  \includegraphics[width=0.407\hsize]{slideimages/Shih}   }
\end{figure}

\vspace{12pt}
\Blue{  hep-th/0602239}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Intriligator---Seiberg---Shih}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.6\hsize]{slideimages/ISSPotential}}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

tunnelling $\propto e^{-S}$

\vspace{24pt}
$S \gg 1$ if $F<3N/2$




\end{document}