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\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\chapter{More Seiberg duality}
\setcounter{equation}{0}
\setcounter{footnote}{0}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$SO(N)$ gauge theory}
 \label{Chap10:sec:SONmoduli}
 
with $F$ quarks in the vector representation
\beq
\begin{tabular}{c|c|cc}
& $SO(N)$ & $SU(F)$ & $U(1)_R$ \\
\hline
$Q$ & \fund & \fund &  ${{F+2-N}\over{F}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
\end{tabular}~
\eeq
discrete
for $N>3$, axial $Z_{2F}$ symmetry
\beq
Q \rightarrow e^{{2 \pi i}/2F}  Q~
\eeq
 for $N=3$ there is a discrete
axial $Z_{4F}$ symmetry

\vspace{24pt}
one-loop $\beta$ function coefficient,  for $N >4$ is
\beq
b= 3(N-2) -F~
\label{Chap10:SObeta}
\eeq

no dynamical spinors, static spinor
sources cannot be screened \newline \Red{distinction between 
area-law confining
and Higgs phases}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$SO(N)$ group theory}

adjoint of $SO(N)$ is two-index
antisymmetric tensor \newline 

odd $N$, there is one spinor 
representation

 even $N$ there are two inequivalent spinors

 \vspace{24pt}

 
for $N=4k$ the spinors are self-conjugate

  for $N=4k+2$ the
two spinors are complex conjugates

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$SO(N)$ group theory}

\beq \begin{array}{ccc} \hline 
\multicolumn{3}{c}{SO(2N+1)} \\ 
{\rm Irrep} \, {\bf r}& d({\bf r}) & 2T({\bf r}) \\ \hline
\vphantom{2^{2^2}} \Yfund & 2N+1 & 2 \\
S & 2^N & 2^{N-2} \\
\Yasymm & N(2N+1) & 4N-2 \\
\Ysymm & (N+1)(2N+1)-1& 4N+6 \\ \hline \end{array}
\label{Chap10:oddSO}
\eeq
 
\beq\begin{array}{ccc} \hline 
\multicolumn{3}{c}{SO(2N)} \\ 
{\rm Irrep} \,{\bf r}& d({\bf r}) & 2T({\bf r}) \\ \hline
\vphantom{2^{2^2}} \Yfund & 2N & 2 \\
S, {\overline S} & 2^{N-1} & 2^{N-3} \\
\Yasymm & N(2N-1) & 4N-4 \\
\Ysymm & N(2N+1)-1& 4N+4 \\ \hline \end{array}
\label{Chap10:evenSO}
\eeq
 $S$ denotes a spinor,
and ${\overline S}$ denotes the conjugate spinor


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The $SO(N)$ moduli space $F<N$}

D-flatness conditions (up to flavor transformations):
\beq
\langle \Phi\rangle =
\left( \begin{array}{ccc}
v_1 & & \\
& \ddots &  \\
& & v_F \\
0 & \ldots & 0 \\
\vdots  & & \vdots \\
0 & \ldots & 0 \\
\end{array} \right)~
\eeq

 \vspace{24pt}
 generic point in the classical moduli space  $SO(N)\rightarrow SO(N-F)$ \newline
$NF -N(N-1)+ (N-F)(N-F-1)$ massless chiral supermultiplets
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The $SO(N)$ moduli space $F \ge N$}
\beq
\langle \Phi \rangle = \left( \begin{array}{cccccc}
v_1 & & & 0 & \ldots & 0\\
& \ddots & &\vdots & & \vdots  \\
& & v_N & 0   & \ldots & 0\\
\end{array} \right)~
\eeq
 generic point in the moduli space the $SO(N)$
broken completely \newline 
$NF -N(N-1)$ massless chiral supermultiplets.

describe light degrees of freedom  by ``meson'' and 
(for $F\ge N$) ``baryon''
fields:
\beq
M_{j i} &=&  \Phi_j \Phi_i~\\
B_{[i_1,\ldots,i_N]} &=&  \Phi_{[i_1}\ldots\Phi_{i_N]}~
\eeq



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The $SO(N)$ moduli space $F \ge N$}
Up to flavor
transformations:
\beq
\langle M\rangle &=&
\left( \begin{array}{cccccc}
 v_1^2 & & & & &\\
& \ddots &  & & &   \\
& & v_N^2  &  & & \\
& & & 0 & &\\
& & & & \ddots & \\
& & & & & 0
\end{array} \right) ~\\
\langle B_{1,\ldots,N} \rangle &=&  v_1\ldots v_N~
\eeq

 rank of $M$ is at most $N$ \newline 
If the rank of $M$ is $N$, then $B = \pm \sqrt{{\rm det}^\prime M}$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The $SO(N)$ $F < N-2$}


\beq
\begin{array}{ccc}
 &U(1)_A & U(1)_R \\
W^a & 0 & 1 \\
\Lambda^b & 2F & 0 \\
{\rm det}M & 2F & 2(F+2-N) 
\end{array}~
\eeq
ADS superpotential:
\beq
W_{\rm dyn} = c_{N,F} \left( {{\Lambda^b}\over{ {\rm det}M
}}\right)^{{1}/(N-2-F)} ~
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Duality for $SO(N)$}
\label{Chap10:sec:SOduals}

$F \ge 3(N-2)$  lose asymptotic freedom\newline 
 $F$ just below $3(N-2)$ we have an IR fixed point

solution to the anomaly matching for $F >  N-2$,  is given by:
\beq\nonumber
\begin{tabular}{c|c|cc}
& $SO(F-N+4)$ & $SU(F)$ & $U(1)_R$ \\
\hline
$q$ & \fund & $\overline{\fund}$ &  ${{N-2}\over{F}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$M$ & {\bf 1} & \Ysymm &  ${{2(F+2-N)}\over{F}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
\end{tabular}~
\eeq

For $F > N-1$, $N>3$  unique superpotential
\beq
W={{M_{j i}}\over{2 \mu}} \phi^j \phi^i~
\eeq

 dual  baryon operators:
\beq
{\wtilde B}^{[i_1,\ldots,i_{\wtilde N}]} &=&  \phi^{[i_1}\ldots\phi^{i_{\wtilde N}]}~
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Hybrid ``Baryon'' Operators}

since adjoint is an antisymmetric tensor. In  $SO(N)$
we have:
\beq
h_{[i_1,\ldots,i_{N-4}]} &=& W_\alpha^2 \Phi_{[i_1}\ldots\Phi_{i_{N-4}]}~
\nonumber\\
H_{[i_1,\ldots,i_{N-2}] \alpha} &=& W_\alpha \Phi_{[i_1}\ldots\Phi_{i_{N-4}]}~
\eeq
While in the dual theory we have:
\beq
{\wtilde h}^{[i_1,\ldots,i_{{\wtilde N}-4}]} &=& 
{\wtilde W}_\alpha^2 \phi^{[i_1}\ldots\phi^{i_{{\wtilde N}-4}]}~
\nonumber\\
{\wtilde H}^{[i_1,\ldots,i_{{\wtilde N}-2}]}_\alpha &=& 
{\wtilde W}_\alpha \phi^{[i_1}\ldots\phi^{i_{{\wtilde N}-4}]}~
\eeq

 The two theories  thus have a  mapping of mesons, baryons, and hybrids:
\beq
&&M \leftrightarrow M~, \nonumber \\
&&B_{i_1, \ldots, i_N} \leftrightarrow \epsilon_{i_1,\ldots,i_F}
{\wtilde h}^{i_1,\ldots,i_{{\wtilde N}-4}}~ \nonumber \\
&&h_{i_1, \ldots, i_{N-4}} \leftrightarrow \epsilon_{i_1,\ldots,i_F}
{\wtilde B}^{i_1,\ldots,i_{{\wtilde N}}}~  \\
&&H^{[i_1,\ldots,i_{N-2}]}_\alpha \leftrightarrow \epsilon_{i_1,\ldots,i_F}
{\wtilde H}^{[i_1,\ldots,i_{{\wtilde N}-2}]}_\alpha~\nonumber
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dual one-loop $\beta$ function}

\beq
\beta(\wtilde g) \propto - {\wtilde g}^3 (3 ({\wtilde N}-2)-F)  
= -{\wtilde g}^3 (2F-3(N-2))~
\eeq
 lose asymptotic freedom when $F\le 3(N-2)/2$
 
When 
\beq
F=3 ({\wtilde N}-2) - \epsilon {\wtilde N}~
\eeq
perturbative IR fixed point in the dual theory

\vspace{24pt}
 $SO(N)$ with $F$ vectors
has an \Red{interacting IR fixed point } for
$3(N-2)/2 < F < 3(N-2)$

\vspace{24pt}
 $N-2\le F \le 3(N-2)/2$   \Red{IR free massless
composite gauge bosons, quarks, mesons, and their superpartners}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Special case:  $F \le N-5$}
\label{Chap10:sec:specialcases}

$SO(N)\rightarrow SO(N-F) \supset SO(5)$ \newline gaugino condensation, dynamical superpotential:
\beq
W_{\rm dyn} \propto \langle \lambda \lambda \rangle \propto
\left( {{16 \Lambda^{3(N-2)-F}}\over{ {\rm det}M
}}\right)^{{1}/(N-2-F)} ~
\eeq


\Red{runaway
vacua}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Special case:  $F = N-4$}

$SO(N)\rightarrow SO(4) \sim SU(2)_L \times SU(2)_R$ \newline
two gaugino condensates 
\beq
W_{\rm cond.} =2 \langle \lambda \lambda \rangle_L +2 \langle \lambda
\lambda \rangle_R ={{1}\over{2}}(\epsilon_L +\epsilon_R)
\left( {{16 \Lambda^{2N-1}}\over{ {\rm det}M
}}\right)^{{1}/{2}} ~
\eeq
\beq
\epsilon_{L,R} = \pm 1~
\eeq
two physically distinct
branches: $(\epsilon_L +\epsilon_R)= \pm 2$ and 
 $(\epsilon_L +\epsilon_R)=0$ \newline
 first branch has runaway
vacua, second has a quantum moduli space.
\newline at $M=0$, $M$ satisfies the `t Hooft 
anomaly matching

\Red{confinement without chiral
symmetry breaking, no baryons} \newline
Integrating out a flavor on  first branch gives 
runaway   ($F=N-5$)\newline  second branch
no SUSY vacua 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Special case:  $F = N-3$}
$SO(N)\rightarrow SO(4) \sim SU(2)_L \times SU(2)_R\rightarrow SU(2)_d \sim SO(3)$ 
\newline instanton effects ($\Pi_3(G/H) = \Pi_3(SU(2))=Z$) \newline and
gaugino condensation 
\beq
W_{\rm inst.+cond.}= 4(1+\epsilon) 
 {{ \Lambda^{2N-3}}\over{ {\rm det}M}} ~
\eeq
two phases of the gaugino condensate \newline 
two physically distinct
branches:  $\epsilon = 1$ and 
with $\epsilon=-1$\newline  first has runaway vacua, while
the second has a \Red{quantum moduli space}\newline   Integrating out
a flavor, we would need to find two branches again\newline so
$W \ne 0$ even on the second branch\newline
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Special case:  $F = N-3$}
must have some other fields\newline
anomaly matching given by:
\beq\nonumber
\begin{tabular}{c|cc}
& $SU(F)$ & $U(1)_R$ \\
\hline
$q$ & $\overline{\fund}$ &  ${{N-2}\over{F}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$M$ & \Ysymm &  ${{2(F+2-N)}\over{F}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
\end{tabular}
\label{Chap10:Nm3dual}
\eeq
most general superpotential
\beq
W= {{1}\over{2 \mu}} M qq\, \,f\left( {{ {\rm det} M \,
Mqq}\over{{\Lambda^{2N-2}}}}\right)~
\eeq
where $f(t)$ is an unknown function\newline 
adding a mass term   gives 
\beq
q_F=\pm i v~
\eeq
which gives  correct number of ground states
\beq
q \leftrightarrow h = Q^{N-4} W_\alpha W^\alpha~
\eeq
\Red{confinement without chiral symmetry breaking
with hybrids}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Special case:  $F = N-1$}
Starting with the $F=N$ dual which has an $SO(4)$ gauge group,
and integrating out a flavor there will be instanton effects
when we break to $SO(3)$\newline  dual superpotential is modified in the
case $F=N-1$ to be:
\beq
W={{M_{j i}}\over{2 \mu}} \phi^j \phi^i - {{1}\over{64 \Lambda^{2N-5}}} 
{\rm  det} M~
\label{Chap10:SO:FeqNm1}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Special case:  $F = N-2$}
both descriptions generically
break to $SO(2) \sim U(1)$

\Red{monopoles}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{SUSY $Sp(2N)$}
\label{Chap10:sec:Spduals}

An $Sp(2N)$ gauge theorywith $2F$ quarks ($F$ flavors)
in the fundamental representation
has a global $SU(2F) \times U(1)_R$ symmetry as follows:
\beq
\begin{tabular}{c|c|cc}
& $Sp(2N)$ & $SU(2F)$ & $U(1)_R$ \\
\hline
$Q$ & \fund & \fund &  ${{F-1-N}\over{F}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
\end{tabular}
\eeq

adjoint of $Sp(2N)$ is the  two-index
symmetric tensor
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{ $Sp(2N)$ Representations}

\beq \begin{array}{ccc} \hline
\multicolumn{3}{c}{Sp(2N)} \\ 
{\rm Irrep}\, {\bf r}& d({\bf r}) & T({\bf r}) \\ \hline
\vphantom{2^{2^2}} \Yfund & 2N & 1 \\
\Yasymm & N(2N-1) -1& 2N-2 \\
\Ysymm & N(2N+1)& 2N+2 \\
\Ythreea & \frac{N(2N-1)(2N-2)}{3}-2N & \frac{(2N-3)(2N-2)}{2}-1 \\
\Ythrees & \frac{N(2N+1)(2N+2)}{3} & \frac{(2N+2)(2N+3)}{2} \\
\vphantom{\frac{2}{2_{2_{2_2}}}} \Yadjoint & \frac{2N(2N-1)(2N+1)}{3}-2N & (2 N)^2-4 \\ \hline
\end{array}
\eeq
dimension $\Yasymm$ smaller  by  $- 1 $ than naive expectation
\newline invariant tensor of $Sp(2N)$ is $\epsilon_{ij}$\newline 
representation formed with two antisymmetric indices is reducible

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{ SUSY $Sp(2N)$}
one-loop $\beta$ function for $N >4$ is
\beq
b= 3(2N+2) -2F~
\eeq
moduli space is parameterized by a ``meson''
\beq
M_{j i} &=&  \Phi_j \Phi_i~
\eeq
antisymmetric in the flavor indices $i,j$


holomorphic intrinsic scale considered as a spurion field
\newline Pfaffian
of a $2F \times 2F$ matrix $M$ is given by
\beq
{\rm Pf}M = \epsilon^{i_1\ldots i_{2F}} M_{i_1i_2}\ldots M_{i_{2F-1}i_{2F}}~
\label{Chap10:Pfaffian}
\eeq
\beq
\begin{array}{c|cc}
 &U(1)_A & U(1)_R \\ \hline
\Lambda^{b/2} & 2F & 0 \\
{\rm Pf}M & 2F & 2(F-1-N) \\
\end{array}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{ SUSY $Sp(2N)$}
for $F<N+1$ possible to generate a dynamical superpotential
\beq
W_{\rm dyn}\propto  \left( {{\Lambda^{{b}\over{2}}}\over{ {\rm Pf}M
}}\right)^{{1}/(N+1-F)} ~
\eeq

For $F=N+1$ one finds confinement with chiral symmetry breaking
\beq
{\rm Pf}M = \Lambda^{2(N+1)}~
\label{Chap10:Spconstraint}
\eeq
For $F=N+2$ one finds s-confinement
with a superpotential:
\beq
W = {\rm Pf}M~
\label{Chap9:sconfineSp}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Duality for $Sp(2N)$}

solution to the anomaly matching for $F >  N-2$:
\beq
\begin{tabular}{c|c|cc}
& $Sp(2(F-N-2))$ & $SU(2F)$ & $U(1)_R$ \\
\hline
$q$ & \fund & $\overline{\fund}$ &  ${{N+1}\over{F}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$M$ & {\bf 1} & \asymm &  ${{2(F-1-N)}\over{F}}$ 
$\vphantom{\raisebox{3pt}{\asymm}}$\\
\end{tabular}
\label{Chap10:Spdual}
\eeq
 a unique superpotential:
\beq
W={{M_{j i}}\over{\mu}} \phi^j \phi^i~
\eeq
For $3(N+1)/2 < F<3(N+1)$ we have an \Red{ IR fixed point }\newline 
For $N+3 \le F \le 3(N+1)/2$ the dual is \Red{ IR free}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Why chiral gauge theories are interesting} 
\label{Chap10:sec:chiralgauge}


vector-like theory we can give
masses to all the matter fields \newline $\rightarrow$ pure YM, gaugino condensation
but no SUSY breaking

Witten's index argument: number of bosonic minus fermionic vacua 
does not change

\Red{If taking the mass to zero does not move some
vacua in from or out to infinity, then the massless theory has
unbroken  SUSY}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{ first example of a chiral gauge theory} 

\beq\nonumber
\begin{tabular}{c|c|cc}
& $SU(N)$ & $SU(N+4)$  \\
\hline
$\overline{Q}$ & $\overline{\fund}$ & \fund  
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$T$  & \Ysymm & {\bf 1}  
$\vphantom{\raisebox{3pt}{\asymm}}$\\
\end{tabular}~
\eeq
 dual to 
\beq\nonumber
\begin{tabular}{c|c|cc}
& $SO(8)$ & $SU(N+4)$  \\
\hline
$q$ & \fund & $\overline{\fund}$  
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$p$ & {\bf S} & {\bf 1}  
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$U \sim {\rm det}T$  & {\bf 1} & {\bf 1}  
$\vphantom{\raisebox{3pt}{\asymm}}$\\
$M\sim \overline{Q}T\overline{Q}$ & {\bf 1} & \Ysymm   
$\vphantom{\raisebox{3pt}{\asymm}}$\\
\end{tabular}~
\eeq
with a superpotential
\beq
W=Mqq +Upp~
\eeq
\Red{This dual  theory is vector-like!}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{ chiral dual of vector theory} 
The dual $\beta$ function coefficient is:
\beq
b=3(8-2)-(N+4)-1=13-N~
\eeq
So the dual is IR free for $N>13$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Csaki, Schmaltz, Skiba}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.45\hsize]{slideimages/csaki} \includegraphics[width=0.45\hsize]{slideimages/schmaltz}   \includegraphics[width=00.221\hsize]{slideimages/skiba} }
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{S-Confinement}
\label{Chap10:sec:sconfine}

$SU(N)$ with $N+1$ flavors.
\beq
W= {{1}\over{\Lambda^{2N-1}}}\left( {\rm det} M - B M \overline{B} \right)~
\eeq
meson--baryon description was valid over the whole moduli space\newline
smooth description with no phase transitions\newline 
 theory has complementarity,  static source
screened by squarks

\vspace{24pt}
 To generalize: \newline  need  fields that are fundamentals of $SU$ or $Sp$ and spinors of $SO$
 \newline only consider theories with  superpotential in the 
confined description

 \Red{Theories that satisfy these conditions are called s-confining}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{S-Confinement}
single gauge group $G$, choose $U(1)_R$ such that
\beq\nonumber
\begin{tabular}{c|c|c}
& $G$ & $U(1)_R$  \\
\hline
$\phi_i$ & ${\bf r_i}$ & q \\
$\phi_{j\ne i}$ & ${\bf r_j}$ &0 
\end{tabular}~
\eeq
 $q$ is determined by anomaly cancellation:
\beq
0 &=& (q-1)T({\bf r_i}) +T(Ad) - \sum_{j\ne i}T({\bf r_j})\\
&=&q \,T({\bf r_i}) +T(Ad) - \sum_j T({\bf r_j})~
\eeq
can do this for any field, and for each choice the superpotential
has $R$-charge 2, we have
\beq
W \propto \Lambda^3 \left[ \Pi_i \left( {{\phi_i}\over{\Lambda}}\right)^{T({\bf r_i})}
\right]^{2/(\sum_jT({\bf r_j})-T(Ad))}~
\eeq
 in general,  a sum of terms with different
contractions of gauge indices
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{S-Confinement}
Requiring superpotential be holomorphic at the origin
$\Rightarrow$ integer powers of the composites $\Rightarrow$  integer powers of the fundamental fields


\vspace{24pt} Unless
all the $T(r_i)$ have a common divisor  must have
\beq
\sum_jT(r_j)-T(Ad) &=& 
\textrm{$1$ or $2$, for $SO$ or $Sp$} \nonumber\\
2(\sum_jT(r_j)-T(Ad)) &=&\textrm{$1$ or $2$, for $SU$}~
\eeq
cases from different conventions for 
normalizing generators, for $SO$ and $Sp$  $T(\fund)=1$,
while for $SU$  $T(\fund)=1/2$


Anomaly cancellation for $SU$ and $Sp$ require that the left-hand side be
even.  

\vspace{24pt} \Red{condition is necessary for  s-confinement, but not sufficient}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{S-Confinement}
check explicitly (by exploring the moduli space) 
that for $SO$ none of the candidate theories  where the sum is 2 turn out to be s-confining

\beq
\sum_jT(r_j)-T(Ad) &=& 
\left\{ \begin{array}{l}
\textrm{$1$, for $SU$ or $SO$}  \\
\textrm{$2$, for $Sp$}
\end{array}\right.~
\eeq
gives  finite list of candidate s-confining theories

\vspace{24pt}
check  candidate theories 
 by going out in moduli space\newline  generically break
to theories with smaller gauge groups 
\newline if the sub-group theory  not s-confining the original
theory  not s-confining
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{S-Confinement}
For $SU$ one finds that the following theories are s-confinings:
\beq 
\begin{array}{l|l} 
SU(N) &\, (N+1) (\Yfund + \overline{\Yfund})\,;\,\,\, \Yasymm + N\, \overline{\Yfund} + 4\, \Yfund \,;\,\,\, 
 \Yasymm + \overline{\Yasymm} + 3 (\Yfund + \overline{\Yfund})  \\
SU(5) &\,  3 \left(\Yasymm + \overline{\Yfund}\right)  \,;\,\,\, 
  2\, \Yasymm + 2\, \Yfund + 4\, \overline{\Yfund}  \\
SU(6) &\, 2\, \Yasymm + 5\, \overline{\Yfund} + \Yfund  \,;\,\,\, 
 \Ythreea + 4 (\Yfund + \overline{\Yfund})  \\
SU(7) &\, 2 \left(\Yasymm + 3\, \overline{\Yfund}\right)
\end{array}~
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$SU(N)$ with $ \Yasymm$, $F=4$}


\beq 
\begin{array}{c|c|ccccc}
& SU(2N+1)  & SU(4) & SU(2N+1) & U(1)_1 & U(1)_2 & U(1)_R \\ \hline
A & \Yasymm & 1 & 1 & 0 & 2N+5 & 0 \\
\overline{Q} & \overline{\Yfund} & 1 & \Yfund & 4 & -2N+1 & 0 \\
Q & \Yfund & \Yfund & 1 & -2N-1 & -2N+1 & \frac{1}{2} 
\end{array}~
\eeq
confined description:
\beq
\begin{array}{c|ccccc}
 & SU(4) & SU(2N+1) & U(1)_1 & U(1)_2 & U(1)_R \\ \hline
(Q\overline{Q}) &  \Yfund & \Yfund & 3-2N & -4N+2 & \frac{1}{2} \\
(A\overline{Q}^2) & 1 &   \Yasymm & 8 & -2N+7 & 0 \\
(A^{N}Q)  & \Yfund &  1 & -2N-1 & 2N^2+3N+1 & \frac{1}{2} \\
(A^{N-1}Q^3) & \overline{\Yfund} &  1 & -6N-3 & 2N^2-3N-2 & 
\frac{3}{2} \\
(\overline{Q}^{2N+1}) &  1 & 1 & 4(2N+1) & -4N^2+1 & 0 \end{array}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$SU(N)$ with $ \Yasymm$, $F=4$}
s-confinement superpotential:
\beq
 W &=& \frac{1}{\Lambda^{2N}} \Big[ (A^NQ)(Q\overline{Q})^3 
(A\overline{Q}^2)^{N-1}+(A^{N-1}Q^3)(Q\overline{Q})(A\overline{Q}^2)^{N}
\nonumber \\ && \,\,\,\,\,\,\,\,\,\,\,\,\,+
 (\overline{Q}^{2N+1})(A^N Q)(A^{N-1} Q^3)
\Big]~
\eeq
equations of motion reproduce  classical constraints\newline 
integrating out
a flavor gives confinement with chiral symmetry breaking


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Deconfinement}
\label{Chap10:sec:deconfine}

Consider  $SU(N)$ with $ \Yasymm$
for odd $N$  with $F\ge 5$  (${\overline F}\equiv N+F-4$):
\beq
\begin{array}{c|c|ccccc}
 & SU(N)  & SU(F)  & SU({\overline F}) & U(1)_1 & U(1)_2 & U(1)_R
\\
\hline
A & \Yasymm & {\bf 1} & {\bf 1} & 0  & -2F\vbr &
{{-12}\over{N}}\vbr \\
Q & \Yfund & \Yfund & {\bf 1}  & 1  & N-F & 2-{{6}\over{N}}\vbr \\
\overline{ Q} &  \overline{\Yfund}  & {\bf 1}  & \Yfund & {{-F}\over{N+F-4}}  &
F & {{6}\over{N}}\vbr \\
\end{array}~
\eeq
take $A$ 
to be  a composite meson of a s-confining $Sp$ theory
\beq
\begin{array}{c|cc|ccccc}
 & SU(N)  & Sp(N-3)  & SU(F)  & SU(N+F-4) & U(1)_1 & U(1)_2 & U(1)_R
\\
\hline
Y & \Yfund & \Yfund & {\bf 1} & {\bf 1} & 0  & -F\vbr &
{{-6}\over{N}}\vbr \\
Z & {\bf 1} & \Yfund & {\bf 1} & {\bf 1} & 0  & FN\vbr &
8 \vbr \\
\overline{P} & \overline{\Yfund} & {\bf 1} & {\bf 1} & {\bf 1} & 0  & F-FN\vbr &
6-{{6}\over{N}}\vbr \\
Q & \Yfund & {\bf 1} & \Yfund & {\bf 1}  & 1  & N-F & 2-{{6}\over{N}}\vbr \\
\overline{ Q}  & \overline{\Yfund} & {\bf 1} & {\bf 1}  & \Yfund & {{-F}\over{N+F-4}}  &
F & {{6}\over{N}}\vbr \\
\end{array}~\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Deconfinement}
 superpotential
\beq
W=Y Z \overline{P}~
\eeq
eqm for $P$ sets the meson 
$(YZ)=0$ , also sets Pf $M=0$ in dynamical
superpotential  to zero 
\newline  $SU(N)$ gauge group of this new description has $N+F-3$ flavors, use  \Red{SUSY QCD duality 
to find another dual}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Deconfinement}
\beq
\begin{array}{c|cc|cc}
 & SU(F-3)  & Sp(N-3)  & SU(F)  & SU({\overline F}) 
\\
\hline
y & \Yfund & \Yfund & {\bf 1} & {\bf 1}  \\
\overline{p} & \overline{\Yfund} & {\bf 1} & {\bf 1} & {\bf 1}  \\
q & \Yfund & {\bf 1} & \overline{\Yfund} & {\bf 1}   \\
\overline{ q} & \overline{\Yfund} & {\bf 1}  & {\bf 1}  & \overline{\Yfund}  \\
M & {\bf 1} & {\bf 1} & \Yfund & \Yfund \\
L & {\bf 1} & \Yfund & {\bf 1} & \Yfund \\
B_1 & {\bf 1} & {\bf 1} & \Yfund & {\bf 1} \\
\end{array}~
\eeq
with a superpotential
\beq
W= M q \overline{q} + B_1 q \overline{p} +L y \overline{q}~
\eeq
 $Sp(N-3)$ with $N+2F-7$ fundamentals has an  \Red{$Sp(2F-8)$ dual }
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Deconfinement}
\beq
\begin{array}{c|cc|cc}
 & SU(F-3)  & Sp(2F-8)  & SU(F)  & SU({\overline F}) 
\\
\hline
\wtilde y & \overline{\Yfund} & \Yfund & {\bf 1} & {\bf 1}  \\
\overline{p} & \overline{\Yfund} & {\bf 1} & {\bf 1} & {\bf 1}  \\
q & \Yfund & {\bf 1} & \overline{\Yfund} & {\bf 1}   \\
\overline{ q}  & \overline{\Yfund}& {\bf 1}  & {\bf 1}  & \overline{\Yfund}  \\
M & {\bf 1} & {\bf 1} & \Yfund & \Yfund \\
l & {\bf 1} & \Yfund & {\bf 1} & \overline{\Yfund}\\
B_1 & {\bf 1} & {\bf 1} & \Yfund & {\bf 1} \\
a & \Yasymm & {\bf 1} & {\bf 1} & {\bf 1} \\
H & {\bf 1} & {\bf 1} & {\bf 1} & \Yasymm  \\
(L y) & \Yfund & {\bf 1} & {\bf 1} & \Yfund \\
\end{array}~
\eeq
with 
\beq
W= a \wtilde{y} \wtilde{y} + H ll + (L y) l \wtilde y +
M q \overline{q} + B_1 q \overline{p} + (L y) \overline{q}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Deconfinement}
after integrating out $(L y)$ and  $\overline{q}$
becomes
\beq
W= a \wtilde{y} \wtilde{y} + H ll  +
M q l \wtilde y + B_1 q \overline{p} ~
\eeq
With $F=5$ we have a gauge group $SU(2) \times SU(2)$ and
one can show (using D[scalar]$\ge 1$) that for $N>11$ 
this theory has an \Red{IR fixed point}\newline also show that
some of the fields (eg. $H$ and $l$) are \Red{IR-free}

\vspace{24pt} integrating out
one flavor completely breaks the gauge group \newline  light
degrees of freedom are just the composites of the s-confinment
for $F=4$







\end{document}



