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\begin{document}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\chapter{Superconformal field theories}
\setcounter{equation}{0}
\setcounter{footnote}{0}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{A-Maximization}
\label{Chap14:SCFT:sec:a-Maximization}





classically 
the trace of the energy--momentum tensor  for a scale-invariant theory vanishes.
\newline trace anomaly. r:
\beq
T^\mu_\mu=
\frac{1}{g^3}\tilde{\beta}(F_{\mu\nu}^b)^2-
a(\tilde R_{\mu\nu\rho\sigma})^2+\dots~,
\eeq
 {\em central charge} $a$
\newline  $\tilde{\beta}$ is the numerator of the exact NSVZ $\beta$ function
\newline $R_{\mu\nu\rho\sigma}$ is the curvature tensor

\vspace{24pt}
Cardy conjectured  that  $a$ satisfies: 
\beq
a_{IR}<a_{UV}\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{A-Maximization}

in SCFT  $a$ can determined by  `t Hooft anomalies of SC
$R$-charge
\beq
a=\frac{3}{32}\left(3 \mathrm{Tr }R^3-\mathrm{Tr }R \right)~
\label{Chap14:acharge}
\eeq
$T_{\mu\nu}$ and the $R$-current ($J_{R\mu}$) in the
same supermultiplet\newline In superspace, super-energy--momentum tensor 
$T_{\alpha {\dot \alpha}}(x,\theta,{\bar \theta})$ contains:\newline $J_{R\mu}$, $J_{\alpha \mu}$ in the $\theta$ and ${\bar \theta}$ components and $T_{\mu\nu}$
in the $\theta^2$ component

\vspace{24pt}
 superconformal $R$-charge:
\beq
R=R_0+\sum_i c_i Q_i~
\label{Chap14:superconformalR}
\eeq





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{A-Maximization}

superconformal symmetry relates  different triangle anomalies
\newline  $\langle J_R J_R J_i\rangle$  related to $\langle TTJ_i\rangle$  by
\beq
9\,\mathrm{Tr}\,R^2 Q_i=\mathrm{Tr}\,Q_i~
\eeq

\vspace{24pt} two-point function
\beq \langle J_i(x)J_k(0)\rangle \propto \tau_{ik} \,\frac{1}{x^4}~
\eeq
Unitarity $\Rightarrow$ $\tau_{ik}$ to have positive definite eigenvalues
\newline Superconformal symmetry $\Rightarrow$
\beq
  \mathrm{Tr}R Q_i Q_k  =- \frac{\tau_{ik}}{3}~
\eeq

$\Rightarrow \mathrm{Tr}\,R Q_i Q_k$ is negative definite

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{A-Maximization}

 Intriligator and Wecht: correct choice of the $R$-charge 
 \beq
R=R_0+\sum_i c_i Q_i~
\eeq

 maximizes $a$-charge:
\beq
\frac{\partial a}{\partial
c_i}&=&\frac{3}{32}\left(9\,\mathrm{Tr}\,R^2
Q_i-\mathrm{Tr}Q_I\right)=0~\\
\frac{\partial^2 a}{\partial c_i\partial c_k
}&=&\frac{27}{16}\mathrm{Tr}\,R Q_i Q_k < 0~
\eeq


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{The simplest chiral SCFT }
\label{Chap14:SCFT:sec:antisymm}







\beq 
\begin{array}{c|c|c|c|c}
  &  SU(N)  &  SU(F)   &  SU(F+N-4)  &  U(1)_R  \\
\hline  Q  & \fund &  \fund  & {\bf 1} &  R(Q)  \\
 \overline{Q}  &  \overline{\fund}   & {\bf 1} & \fund &
 R(\overline{Q})  \\
A  & \Yasymm & {\bf 1} & {\bf 1} &  R({\rm A}) \\
\end{array}~
\eeq

$F=0$, breaks SUSY
\newline  $F=1,2$,  runaway vacua
\newline $F= 3$, quantum deformed moduli space
\newline $F=4$,  s-confining
\newline $F=5$, splits into an 
 IR free and IR fixed point sectors

\vspace{24pt}
\noindent $F>5$ ?

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

parameterized by  mesons
$M=Q {\bar Q}$, $H= {\bar Q}A  {\bar Q}$
and 
baryons:
\beq
\begin{array}{cc}
N \,{\rm even} & N\,{\rm odd}\\ 
{\bar Q}^{N}&{\bar Q}^{N}\\ 
A^{{N}/{2}}&Q A^{{N-1}/{2}}\\
Q^2 A^{(N-2)/{2}}&Q^3 A^{(N-3)/{2}}\\
\vdots&\vdots\\
Q^k A^{(N-k)/{2}}& Q^k A^{(N-k)/{2}} 
\end{array}
\eeq
where $k \le min(N,F)$
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{R-charge}
anomaly cancellation for large $F,N$ $\Rightarrow$
\beq
R({\rm A})= \frac{F}{N} \left(2-R(Q)-\left(\frac{N}{F}+1\right) R(\overline{Q})\right)~
\eeq
In general
\beq
R(Q)&=&2- \frac{6}{N}   + b (N-F )+ c\\
R(\overline{Q})&=& \frac{6}{N}+b F  - c \frac{F}{F + N-4}\\
R({\rm A})&=&- \frac{12}{N} -2 b F 
\eeq
\beq
a&=& 3 (N^2 - 1 + F N (R(Q) - 1)^3 + N (F + N - 4) (R(\overline{Q}) - 1)^3 \\
&&+ N (N - 1)/2 (R(A) - 1)^3) - (N^2 - 1) - N F  (R(Q) - 1) \\
&& -
   N (F + N - 4) (R(\overline{Q}) - 1) - N (N - 1)\sfrac{1}{2} (R(A) - 1)
   \eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{R-charge}
a-maximization gives for $F,N$:
\beq
R(Q)=
R(\overline{Q})=-\frac{12-9\left(\frac{N}{F}\right)^2+\sqrt{\left(\frac{N}{F}\right)^2(-4
+\frac{N}{F}(73\frac{N}{F}-4))}}{3(-4+(\frac{N}{F}-4)\frac{N}{F})}~
\eeq
\Red{even though
 theory is chiral}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{R-charge}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{figure}[htb]
\begin{center}\includegraphics[width=0.39\hsize]{slideimages/naiveR}
\includegraphics[width=0.39\hsize]{slideimages/Dmeson}
\end{center}
\label{Chap14:fig:amax1}
\end{figure}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
(a) The $R$-charges of the fundamental fields, with $R(Q)=R(\overline{Q})$
\newline (b) The corresponding dimensions of the meson operators.

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Two Free Mesons}



reduce $F$ from the Banks--Zaks fixed point at $F\sim 2N$
\newline \Red{meson $M=Q {\bar Q}$  goes free} at 
\beq
F= F_1=\frac{9 N}{4(4+\sqrt{7})}\approx 0.3386\, N~
\eeq
meson $H= {\bar Q} A {\bar Q}$ is still interacting

Kutasov: assume only one accidental $U(1)$ for the free meson $M$ then 
\beq
a_{\rm int}&=&a-a(R(M))
\\
&=&a-\frac{3}{32} F (F+N-4)\big(3
(R(Q)+R(\overline{Q})-1)^3\\
&&-(R(Q)+R(\overline{Q})-1))~\nonumber
\eeq
\Red{meson $H= {\bar Q} A {\bar Q}$ goes free} at
$F =F_2\approx 0.2445 N$


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Two Free Mesons}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{center}
\begin{figure}[hb]
\begin{center}
\includegraphics[width=0.49\hsize]{slideimages/R}\includegraphics[width=0.49\hsize]{slideimages/D}
\end{center}
\label{Chap14:fig:amax2}
\end{figure}
\end{center}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent (a) The $R$-charges of the fundamental fields\newline (b) The corresponding dimensions of the meson operators


%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dual Description}


 for 
 $N$  odd and $F\ge 5$:
 \beq 
\begin{array}{c|cc|cc}
  & SU(F-3)  & Sp(2F-8)  & SU(F)  & SU(N+F-4)
\\
\hline
\tilde{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}   \\
a & \Yasymm & {\bf 1} & {\bf 1} & {\bf 1} \\
l & {\bf 1} & \Yfund & {\bf 1} & \overline{\Yfund}\\
B_1 & {\bf 1} & {\bf 1} & \Yfund & {\bf 1} \\
M & {\bf 1} & {\bf 1} & \Yfund & \Yfund \\
H & {\bf 1} & {\bf 1} & {\bf 1} & \Yasymm  \\
\end{array}~
\eeq
with a superpotential
  \beq W=c_1 M q l \tilde y + c_2 H
ll+ B_1 q \overline{p}+ a \tilde{y} \tilde{y} ~
\label{Chap14:oddNsup}
\eeq
\Red{ $ M=Q {\bar Q}$ and 
$H= {\bar Q} A{\bar Q}$ are mapped to elementary fields }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dual Description}
even $N$ and $F\ge 5$:
 \beq \label{finaldualNeven}
\begin{array}{c|cc|ccc}
  & SU(F-3)  & Sp(2F-8)  & SU(F)  & SU(N+F-4) & SU(2)
\\
\hline
\tilde{y} & \overline{\Yfund} & \Yfund & {\bf 1} & {\bf 1} & {\bf 1} \\
\overline{p} & \overline{\Yfund} & {\bf 1} & {\bf 1} & {\bf 1} & \Yfund \\
q &\Yfund  & {\bf 1} & \overline{\Yfund} & {\bf 1} & {\bf 1}  \\
a & \Yasymm & {\bf 1} & {\bf 1} & {\bf 1}& {\bf 1} \\
l & {\bf 1} & \Yfund & {\bf 1} & \overline{\Yfund}& {\bf 1}\\
S & {\bf 1} & {\bf 1} & \Yfund & {\bf 1}& \Yfund \\
B_0 & {\bf 1} & {\bf 1} & {\bf 1} & {\bf 1} & {\bf 1}\\
M & {\bf 1} & {\bf 1} & \Yfund & \Yfund & {\bf 1}\\
H & {\bf 1} & {\bf 1} & {\bf 1} & \Yasymm  & {\bf 1}\\
\end{array}~
\eeq 
with a superpotential 
\beq \label{dualwNeven} W=
c_1 M q l \tilde y + c_2 H ll+ S q \overline{p}+ a \tilde{y} \tilde{y} +B_0 a
\overline{p}^2 ~
\eeq 
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dual Description}

\noindent $M$ goes free at $F=F_{1}$, $c_1\to 0$ chiral operator  $q l \tilde y$ has dimension 2

\vspace{12pt}
\noindent $H$ goes free at $F=F_2$,  $c_2\to 0$  chiral operator $ll$ has dimension 2 corresponding to $R$-charge $4/3$\newline \Red{$l$ has $R$-charge $2/3$ and dimension 1? is $l$ free?}

self-consistent if  $l$ is a 
gauge-invariant operator




\Blue{recall SUSY QCD:}
\beq
W= c \,M \phi {\overline \phi}~
\eeq
when $M$ goes free, the coupling $c\to 0$ the chiral operator $\phi {\overline \phi}$ has
dimension 2\newline
$a$-maximization $\Rightarrow$ $\phi$, $ {\overline \phi}$ are free 
 if we assume
accidental axial symmetry for  dual quarks\newline accidental axial symmetry only  if 
 dual gauge group is IR-free\newline dual $\beta$ function $\Rightarrow$
 dual looses asymptotic freedom for $F<3N/2$\newline 
\Red{dual quarks are free}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dual $\beta$ function}

with $c_1$ and $c_2$ set to zero 
 \beq W=  B_1 q \overline{p}+ a \tilde{y} \tilde{y}  ~
 \label{Chap14:reducedsup}
 \eeq
$Sp(2F-8)$  has $\tilde{y}$ and $l$ with gauge interactions
\beq
\beta(g)&=&-\frac{g^3}{16\pi^2} \left[3(2F-6)-(F-3)
(1- \gamma_{\tilde{y}|_{g=0}})-(N+F-4)\right] 
\\&&+\mathcal{O}(g^5), \nonumber
\eeq
nonperturbative $SU(F-3)$ and superpotential corrections through  anomalous dimension
$\gamma_{\tilde{y}}$\newline \Red{$Sp(2F-8)$ is IR free if}
\beq
N-4F+11-(F-3)\gamma_{\tilde{y}|_{g=0}} > 0~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dual $\beta$ function}
superpotential 
has dimension 3 $\Rightarrow$
\beq
\gamma_{a}+2\gamma_{\tilde{y}}=0~~~~~~(*)
\label{Chap14:gammaeq}
\eeq
$a^{(F-3)/{2}}$ is a gauge-invariant operator $\Rightarrow$
\beq
\frac{F-3}{2}+\frac{F-3}{4}\gamma_{a}\ge 1~~~~~~(**)
\label{Chap14:gammaabound}
\eeq
Combining eqns (*) and (**)
we have that for large $F$,  $\gamma_{\tilde{y}} \le 1$
\newline large $F$ and large $N$ limit with $F< {N}/{5}$:
\beq
\beta \propto N-4F+11-(F-3)\gamma_{\tilde{y}}  > 0~,
\eeq
\Red{$Sp(2F-8)$ is IR  free}

\vspace{12pt}  assuming $l$ is free for $F<F_2$ we can check that
$Sp(2F-8)$ becomes IR free at $F=F_2$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$F<F_2$: Mixed Phase}

 theory splits into two sectors in the IR,
free magnetic sector:
\beq 
\begin{array}{c|c|cc}
 & Sp(2F-8)  & SU(F)  & SU(N+F-4)
\\
\hline
l  & \Yfund & {\bf 1} & \overline{\Yfund}\\
M & {\bf 1} & \Yfund & \Yfund \\
H & {\bf 1} & {\bf 1} & \Yasymm  \\
\end{array}~
\eeq
 interacting superconformal sector:
is
\beq 
\begin{array}{c|c|cc}
  & SU(F-3)  & Sp(2F-8)  & SU(F)  
\\
\hline
\tilde{y} & \overline{\Yfund} & \Yfund & {\bf 1}  \\
\overline{p} & \overline{\Yfund} & {\bf 1} & {\bf 1}  \\
q &\Yfund  & {\bf 1} & \overline{\Yfund}  \\
a & \Yasymm & {\bf 1} & {\bf 1} \\
B_1 & {\bf 1} & {\bf 1} & \Yfund  \\
\end{array}~
\eeq
 \beq W=  B_1 q \overline{p}+ a \tilde{y} \tilde{y}  ~
 \eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{${\mathcal N}=1$: Open Questioins}
\begin{itemize}
\item how \Red{nonperturbative effects} make $\gamma_{Q}\ne\gamma_{\bar Q}$ only 
for $F<F_1$
\item
mixed-phase first conjectured in \Red{theories with an adjoint},  still
not proven
\item
\Red{$SO$  with spinors} 
\end{itemize}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{${\mathcal N}=2$}
\label{Chap14:SCFT:sec:N2AD}


$SU(N)$ with ${\mathcal N}=2$ SUSY and $F$  hypermultiplets in $\Yfund$ 
\beq
\beta(g) =-{{g^3}\over{16\pi^2}}\frac{(3N -N(1-2\beta(g)/g) - F)}{1-N g^2/{8 \pi^2}}~
\label{Chap14:beta2NSVZ}
\eeq
adjoint  in supermultiplet with  gluon and gluino\newline  $\Rightarrow$  $Z=1/g^2$ $\Rightarrow$
$\gamma(g) = 2 \beta(g)/g$
\newline $\gamma_Q=0$,  non-renormalization of the superpotential
$\Rightarrow$ non-renormalization of the \Kahler function, both related to a prepotential

solving  for $\beta(g)$
\beq
\beta(g) =-{{g^3}\over{16\pi^2}}(2N  - F)~
\label{Chap14:beta2}
\eeq
\Red{exact at one-loop}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{${\mathcal N}=2$ SCFT}
\label{Chap14:SCFT:sec:N2AD}

\beq
\beta(g) =-{{g^3}\over{16\pi^2}}(2N  - F)~
\eeq
vanishes  for $F=2N$

$\beta$  vanishes independent  of $g$ $\Rightarrow$ line of fixed points

Seiberg--Witten analysis $\Rightarrow$ $a^i_D$ have no logarithmic corrections\newline classical relations between $a^i$ and $a^j_D$ are exact\newline  theory with $F=2N$ hypermultiplets  is  \Red{nonperturbatively conformal}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Argyres--Douglas fixed points}

massless  electrically and  magnetically charged particles at the same point in the moduli space

electric charge: $g_{\rm IR}\to 0$, magnetic charge: $g_{\rm IR}\to \infty$\newline
 \Blue{IR fixed point?  Argyres and Douglas:  yes!  }
 
 \vspace{24pt}
 ${\mathcal N}=2$  $SU(2)$ with one flavor\newline 
adjust mass and VEV so monopole and dyon points coincide 

for $m= 3 \Lambda_1/4$ and $u= 3 \Lambda^2_1/4$ 
\beq
y^2=\left(x-\frac{ \Lambda_1^2}{4} \right)^3~
\label{Chap14:SWcurveAD}
\eeq
all three roots coincide\newline Seiberg--Witten analysis shows that
$a_D$ has no logarithmic corrections, \Red{theory is  conformal}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Argyres--Douglas fixed points}

 charges in $U(1)$ theories with IR fixed points do not produce long-range fields
\newline using \beq
d \ge \frac{1}{2}[ C_2({\bf r}) + C_2(V)- C_2({\bf r^\prime}) ]~.
\label{Chap6:unitarityboundD}
\eeq
 $F_{\mu\nu}$, 
is in a $(1,0)+(0,1)$  of $SO(4)$, has a scaling dimension $d\ge 2$
\newline
at an interacting IR fixed point  generically $d>2$
\newline conformal symmetry and 
dimensional analysis $\Rightarrow$ fields  fall off as $1/x^d$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Other SCFTs}
have several different interactions and are superconformal  \newline  lines (or manifolds) of fixed points
\newline if there 
are $n$ interactions and only $p$ independent $\beta$ functions \newline then 
 $n-p$ dimensional
manifold of fixed points

\vspace{24pt}\noindent moving in manifold $\leftrightarrow$ changing coupling of
an exactly marginal operator\newline \Blue{operator in ${\mathcal L}$ has scaling dimension 4, independent of couplings}


\vspace{24pt}\Red{can also happen in  
${\mathcal N}=1$  theories }



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{${\mathcal N}=4$ SUSY
 gauge theory}
\label{Chap14:SCFT:sec:N4}

${\mathcal N}=1$ SUSY
 gauge theory with three chiral supermultiplets in the adjoint  with a particular
 superpotential\newline $\equiv$ ${\mathcal N}=2$ SUSY gauge theory with an adjoint hypermultiplet \newline In general ${\mathcal N}=4$ theories have a global $SU(4)_R \times U(1)_R$ $R$-symmetry \newline restrictted  to vector supermultiplet 
does not transform under the $U(1)_R$
 
$\lambda$, and the three adjoint fermions, $\psi$, transform as a \Blue{${\bf 4}$ of the $SU(4)_R$} \newline  real adjoint scalars $\phi$ transform as a \Blue{${\bf 6}$ of $SU(4)_R$}\newline 
 in terms of  ${\mathcal N}=1$ fields, the $SU(4)_R$ symmetry is 
 not manifest\newline
only $SU(3)\times U(1)$ subgroup is apparent
\newline for canonically normalized ${\mathcal N}=1$ superfields the superpotential is
\beq
W_{{\mathcal N}=4}=- i \sqrt{2} \,Y\,
{\mathrm Tr }\,\Phi_1 \left [\Phi_2, \Phi_3 \right] =\frac{ Y}{3 \sqrt{2}}\, \epsilon_{ijk} f^{abc}
\,\Phi^{c}_i \Phi_j^a \Phi^b_k ~
\label{Chap14:W4}
\eeq
where $a, \ldots ,e =1,\ldots,N^2-1$ are the adjoint gauge indices\newline  $i, \ldots, m=1,2,3$ are $SU(3)$ flavor indices, and $\Phi_i = T^a \Phi^a_i$
\newline for ${\mathcal N}=4$ SUSY,  \Blue{$Y=g$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{${\mathcal N}=4$ SUSY
 gauge theory}
 Lagrangian is given by
\beq
\label{eq:Chap14:N=4}
&&{\mathcal L}_{{\mathcal N}=4}=
-\frac{1}{4}F^a_{\mu\nu}F^{a\mu\nu}-i \bar{\lambda}^a \sigma^{\mu}
D_{\mu} \lambda^a -i \bar{\psi}_i^a \sigma^{\mu} D_{\mu} \psi_i^a
+D^{\mu}\phi^{\dagger  a}_i D_{\mu}\phi_i^a \nonumber \\
&& -\sqrt{2}g f^{abc}(\phi^{\dagger  c}_i \lambda^a \psi^b_i -
\bar{\psi}^c_i  \bar{\lambda}^a \phi^b_i)
-\frac{Y}{\sqrt{2}}\epsilon_{ijk} f^{abc}
(\phi^{c}_i \psi_j^a \psi^b_k +
\bar{\psi}^c_i \bar{\psi}^a_j \phi^{\dagger \,b}_k) \\
&& +\frac{g^2}{2} (f^{abc}\phi^b_i\phi^{\dagger \, c}_i)
(f^{ade}\phi^d_j\phi^{\dagger \, e}_j)-\frac{Y^2}{2}
\epsilon_{ijk}\epsilon_{ilm} (f^{abc} \phi^b_j\phi^c_k)(f^{ade}
\phi^{\dagger d}_l  \phi^{\dagger e}_m)~\nonumber
\eeq

$SU(N)$ gauge theory with ${\mathcal N}=2$ SUSY and $A$ adjoint hypermultiplets has
\beq
\beta(g) =-{{g^3}\over{16\pi^2}}(2  - 2 A)N~
\label{Chap14:beta4}
\eeq
$\Rightarrow$ ${\mathcal N}=4$ gauge theory has  \Red{$\beta=0$ $\leftrightarrow$ SCFT}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Quivers and Mooses}

\vspace{12pt}\noindent Theories with gauge groups connected by bifundamentals 
\newline
 called \Blue{``quiver"}
theories  and  \Blue{``mooses" }
\newline  matter content can be represented by a quiver/moose diagram\newline
 in certain cases a quiver/moose theory can be considered 
as a latticization (a.k.a ``deconstruction") along a discretized extra dimension


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{${\mathcal N}=4$ and orbifolds}
mod-out by a discrete subgroup $\Gamma$ of the gauge and global symmetries
\newline  $\rightarrow$ ``daughter" theory
\newline \Blue{$\rightarrow$ quiver/moose}

large $N$ limit of an $SU(N)$ 
dominated by planar diagrams\newline if   $\Gamma$ 
embedded in gauge group using  regular representation $N$ times then the planar diagrams
of  daughter $\propto$  planar diagrams of  full theory (up to a rescaling of the gauge
coupling)\newline  \Blue{ large $N$ limit, daughters of the   ${\mathcal N}=4$ gauge theory are conformal}
\newline
orbifolding in different ways  break different amounts of SUSY:
\beq
\begin{array}{ccl}
SU(4)_R \supset \Gamma, &  SU(3) \not \supset \Gamma & \Rightarrow {\mathcal N}=0 \nonumber\\
SU(3)\supset \Gamma, & SU(2) \not \supset \Gamma & \Rightarrow {\mathcal N}=1 \\
SU(2) \supset \Gamma, &   & \Rightarrow {\mathcal N}=2~. \nonumber
\end{array}
\eeq
\Red{unbroken SUSY $\leftrightarrow$ size of the $R$-symmetry  subgroup   invariant under $\Gamma$}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{${\mathcal N}=4$ and orbifolds}
simplest case: permutation group $\Gamma= Z_k$  embedded
in the gauge group\newline  regular representation of 
$Z_k$:
\beq
{\bf \gamma^a} = {\rm diag}(\omega^0, \omega^a, \omega^{2a}, \ldots, \omega^{(k-1)a})~
\eeq
where
$\omega= e^{2 \pi i/k}$ and
$a = 0,1,\ldots, k-1$,  embed $Z_k$ in $SU(kN)$        by defining
\beq
{\bf \gamma^a_N} = {\rm diag}({\bf 1_N}, {\bf 1_N}\omega^a, {\bf 1_N}\omega^{2a}, \ldots,{\bf 1_N} \omega^{(k-1)a})~
\eeq
so  adjoint transforms as
\beq
{\bf Ad} \rightarrow {\bf \gamma^a_N} {\bf Ad} ({\bf \gamma^a_N})^\dagger~
\label{Chap14:orbifoldtans}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{${\mathcal N}=4$ and orbifolds}
parts of the $kN\times kN$ matrix of gauge fields  left invariant 
are 
\beq
A_{inv}={\rm diag}({\bf A_1}, {\bf A_2}, {\bf A_3}, \ldots, {\bf A_k})~,
\eeq
where ${\bf A_i}$ t adjoint under the $i$th $SU(N)$ subgroup of
$SU(k N)$\newline \Blue{orbifolded gauge group is $\Pi_{i=1}^k\,SU(N)_i$}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Orbifold example}
example the $Z_6$ orbifold where the embedding of $Z_6$ in
the global $SU(4)$ $R$-symmetry is such that the four fermion fields  transform
as:
\beq
(\psi_1, \psi_2, \psi_3, \psi_4) \rightarrow (\omega^a\psi_1, 
\omega^{-2a}\psi_2,\omega^{3a}\psi_3 ,\omega^{4a} \psi_4)~
\eeq
under a global transformation\newline
adjoint fermion $\psi_3$ that transforms as 
\beq
\psi_3 \rightarrow \omega^{3a}  {\bf \gamma^a_N} \psi_3 ({\bf \gamma^a_N})^\dagger~
\label{Chap14:orbifoldtferm}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Orbifold example}
invariant pieces of $\psi_3$:
\beq
\begin{array}{cccccc}
0&0&0& \psi_{14} &0&0 \\
0&0&0&0&  \psi_{25}  &0 \\
0&0&0&0&0&  \psi_{36}   \\
  \psi_{41}  &0&0&0&0&0 \\
0& \psi_{52} &0&0&0 &0 \\
0&0& \psi_{63} &0&0&0   \\
\end{array}~
\eeq
 bifundamentals transforming
as ($\Yfund$, $\overline{ \Yfund}$) under 
$SU(N)_i \times SU(N)_j$\newline  similar analysis  for the
remaining fermion and scalar fields


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Orbifolds and the Hierarchy Problem}

proposed  that  orbifold theories solve the
hierarchy problem if physics was conformal above $1$~TeV\newline
\Red{exactly conformal theory has no quadratic divergences}

\vspace{24pt}
consider the effective theory below some scale $\mu$
\newline calculate the one-loop
$\beta$ functions,  set the $\beta=0$ \newline in daughter theories where  matter fields are distinct bifundamentals,  fixed points for $Y$, and  $ \lambda_i$, 
approach ${\mathcal N}=4$ SUSY: \newline \Blue{$Y=\lambda_i=g$ as $N\rightarrow \infty$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Orbifolds and the Hierarchy Problem}
at  fixed point the one-loop scalar mass is given by
\beq
m_\phi^2 = \left[ N c_i \lambda_i +3 {{N^2-1}\over{N}} g^2 - 8NY^2\right]
{{\mu^2}\over{16 \pi^2}}~
\eeq
large $N$ limit $\sum_i c_i=5$: no quadratic divergence
\newline  leading order in $1/N$:
\beq
m_\phi^2 =
{{3 g^2}\over{N}}{{\mu^2}\over{16 \pi^2}}~
\eeq
to get $m_\phi= 1$~TeV, with $\mu= M_{\rm Pl}$ we need $N=10^{28}$

\vspace{24pt}
\Red{if scalar mass term is relevant operator}
in  low-energy effective theory below  SUSY breaking scale, \Red{a large mass is generated}






\end{document}


 
 
