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

\chapter{Introduction to AdS/CFT}
\setcounter{equation}{0}
\setcounter{footnote}{0}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{D-branes}
\label{Chap17:AdSCFT:sec:dbranes}

\vspace{24pt} 

Type IIA string theory: Dp-branes $p$ even (0,2,4,6,8)


\vspace{24pt} Type IIB string theory: Dp-branes 
$p$ odd (1,3,5,7,9)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{ 10D Type IIB}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.45\hsize]{slideimages/DbraneN4}}
\label{Chap17:fig:DbraneN4}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


\noindent two parallel D3-branes
\newline low-energy effective
description: Higgsed ${\mathcal N}=4$ SUSY gauge theory

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{ Two parallel D3-branes}

 \noindent lowest energy string stretched between D3-branes: $m\propto L T$   \newline $L\to 0$
massless particle $\subset$ 4D effective theory
\newline Dirichlet BC's $\rightarrow$ gauge boson and  superpartners\newline D3-branes are BPS invariant under
half of the SUSY charges\newline  $\Rightarrow$ low-energy effective theory is  \Red{${\mathcal N}=4$ SUSY gauge
theory}


\vspace{24pt}
six extra dimensions, move branes apart in six different ways\newline moduli space 
$\leftrightarrow$ $\langle \phi\rangle$   six scalars in the ${\mathcal N}=4$ SUSY gauge multiplet

\vspace{24pt} \Red{moduli space  is encoded geometrically}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$N$ parallel D3-branes}

low-energy effective theory is an \Red{${\mathcal N}=4$, $U(N)$
gauge  theory}
\newline $N^2$ ways to connect  oriented strings 

\vspace{24pt}\noindent Moving one of the branes
$\rightarrow$ mass for  $2N-1$ of the gauge bosons 
\newline $\leftrightarrow$ $\langle \phi\rangle$   breaks $U(N)\to$ $U(N-1)$


\vspace{24pt}\noindent gauge coupling 
related to  string coupling $g_s$ 
\beq
g^2 = 4 \pi g_s~
\label{Chap17:gaugestringcoupling}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Type IIA D4-branes}


 5D gauge theory, compactify 1 dimension
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.9\hsize]{slideimages/DbraneN2}}
\end{figure}
 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


D4-brane shares three spatial directions with the 5-brane
 

\beq
g_4^2=\frac{g_5^2}{L}~
\label{Chap17:54gaugecoupling}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Type IIA D4-branes}

3D end of the D4-brane has two coordinates on the 5-brane \newline $\leftrightarrow$ two real  scalars 

\vspace{24pt}\noindent 
two sets of parallel BPS states:  D4-branes and  5-branes\newline each set  invariant under one half of the SUSYs\newline
 low-energy effective theory has \Red{${\mathcal N}=2$ SUSY}

\vspace{24pt}\noindent   two real scalars $\leftrightarrow$
scalar component of  ${\mathcal N}=2$ vector supermultiplet 


\vspace{24pt}\noindent 
\Blue{moduli space is reproduced by the geometry}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{D-brane constructions}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.9\hsize]{slideimages/DbraneN2}}
\label{Chap17:fig:DbraneN2}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\noindent (a)  ${\mathcal N}=2$ SUSY (b) non-parallel NS5-branes $\leftrightarrow$ ${\mathcal N}=1$ SUSY 

\vspace{24pt}\noindent rotate one of the NS5-branes 
$\rightarrow$  D4-branes can't move $\leftrightarrow$ massive scalar\newline
breaks
\Red{ ${\mathcal N}=2$ $\to$ ${\mathcal N}=1$ SUSY}
 
 
\Blue{ the non-parallel NS5-branes preserve different SUSYs}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Adding Flavors}
$F$ D6-branes 
$||$ one of NS5-branes  along  2D  of the NS5 $\perp$ D4-branes
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.9\hsize]{slideimages/DbraneN1}}
\label{Chap17:fig:DbraneN1}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

(a)  $SU(N)$ ${\mathcal N}=1$,  $F$ flavors.  (b) Higgsing the gauge group

\vspace{24pt}\noindent
strings between
 D4 and  D6 have $SU(N)$ color index and $SU(F)$ flavor index, 
two orientations $\rightarrow$ \Red{chiral supermultiplet and conjugate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Adding Flavors}

Moving  D6 in $\perp$ direction, string between D6 and  D4
has finite length $\leftrightarrow$ adding a \Red{mass term  for  flavor}

\npar
break the D4-branes at  D6-brane and move  section of the D4
between $||$ NS5 and D6-brane 
 $\leftrightarrow$ squark VEV $\langle \phi \rangle \ne 0$,   $\langle {\overline  \phi} \rangle \ne 0$ $\leftrightarrow$ \Red{Higgsing }
 
 \npar counting \# of ways of moving segments
\newline $\rightarrow$  dimension of the the moduli space $=2NF-N^2$ \newline correct result for
classical $U(N)$ gauge theory 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Seiberg Duality}
(a) move NS5'  through the D6  (b) move NS5'  around the NS5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\hsize]{slideimages/DbraneDual}}
\label{Chap17:fig:DbraneDual}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


$N$ D4s between NS5s join up, leaving $(F-N)$ D4s, \Blue{$\#R-\#L$ fixed}

$\leftrightarrow$  $SU(F-N)$ ${\mathcal N}=1$ SUSY gauge theory with $F$ flavors 
\newline D4s between $||$ NS5 and  D6-branes  move  without Higgsing  $SU(F-N)$
\newline $\#$ ways of 
moving  $=F^2$ complex dof  \Red{$\leftrightarrow$} meson in  classical limit \newline dual quarks 
\Red{$\leftrightarrow$} strings from
 $(F-N)$ D4s to $F$ D4s \newline stretched to  finite length \Red{$\leftrightarrow$} meson VEV $\rightarrow$  
dual quark mass 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Lift to M-theory}

\noindent to get quantum corrections\newline Type IIA string theory  $\leftrightarrow$ compactification of M-theory on a circle 
\beq
g_s= (R_{10}M_{\rm Pl})^{3/2}~
\label{Chap15:IIAstringcouping}
\eeq
 finite string coupling $g_s$ $\leftrightarrow$ to a finite radius $R_{10}$ 
\newline eg. ${\mathcal N}=2$
 $SU(2)$ gauge theory $\leftrightarrow$  two D4-branes between $||$ NS5s
\newline  NS5 is low-energy description of M5-brane
\newline D4 is  low-energy description of M5-brane wrapped on circle
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Lift to M-theory}
D4s ending on NS5s  $\to$
 single M5
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.7\hsize]{slideimages/M5brane}}
\label{Chap17:fig:M5brane}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

M-theory curve describes  a 6D space, 4D spacetime
\newline remaining 2D given by the \Red{elliptic curve of Seiberg-Witten}
\newline \Blue{
larger gauge groups,  more D4-branes, surface has more handles}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{M-theory brane bending}

M5s not  $||$, bend toward or away from each other depending on the $\#$ branes
 ``pulling" on either side\newline move one D4 $\leftrightarrow$ Higgsing by a  $v=\langle\phi\rangle $ \newline  probe $g(v)$ 
 \beq
g_4^2=\frac{g_5^2}{L}~
\eeq
\Red{ bending of M5-brane $\leftrightarrow$ to running coupling}
 \newline at large $v$ bending reproduces 
$\beta$ 

M-theory  not completely developed\newline  \Red{not understood:}\newline get  quantum moduli space for ${\mathcal N}=1$  \Blue{$SU(N)$ rather than $U(N)$}
\newline dimension  of  dual quantum moduli space 
reduced \newline \Blue{from $F^2$ to $F^2-((F-N)^2-1)$ }

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$N$ D3 branes of Type IIB}
\label{Chap17:AdSCFT:sec:sugra}

 $E \ll 1/\sqrt{\alpha^\prime}$, effective theory:
\beq
S_{\rm eff} = S_{\rm brane}+S_{\rm bulk}  +S_{\rm int}~
\eeq
 $S_{\rm brane}=$ gauge theory\newline $S_{\rm bulk}=$ closed string loops $=$ Type IIB  sugra $+$ higher dimension ops\newline 10D graviton fluctuations $h$: \beq g_{MN}=\eta_{MN}+\kappa_{\rm IIB} \,h_{MN}\eeq
where $\kappa_{\rm IIB}\sim g_s \alpha^{\prime 2}$,  10D Newton's constant, has mass dimension -4
\beq
S_{\rm bulk} = \frac{1}{2 \kappa_{\rm IIB}^2} \int \sqrt{g} R\sim  \int (\partial h)^2 + \kappa_{\rm IIB}  (\partial h)^2 h +\ldots~
\eeq
$E\to 0$ $\equiv$ drop terms with positive powers of $\kappa_{\rm IIB}$, leaves  kinetic term \newline all terms in 
$S_{\rm int}$ can be neglected $\rightarrow$ free graviton

Equivalently, hold $E$, $g_s$, $N$ fixed  take $\alpha^\prime\rightarrow 0$ ($\kappa_{\rm IIB}\rightarrow 0$) \newline \Red{$\rightarrow$ free IIB sugra and  4D $SU(N)$, ${\mathcal N}=4$ SUSY gauge theory}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Supergravity Approximation}

low-energy effective theory: Type IIB supergravity with $N$ D3-branes, source for
gravity, warps the 10D space
\newline  solution for the metric:
\beq
ds^2 &=&f^{-{1}/{2}}\left(-dt^2 + dx_1^2 +dx_2^2 +dx_3^2\right) + f^{{1}/{2}}\left(dr^2
+r^2 d\Omega_5^2\right) \label{Chap17:maldacenametric} \\
f&=& 1 +\left({{R}\over{r}}\right)^4~,~~
R^4 = 4 \pi g_s \alpha^{\prime 2} N~
\label{Chap17:AdScurvature}
\eeq
where $r$ is  radial distance from  branes, and $R$ is  curvature radius \newline 
observer at  $r$ measures red-shifted $E_r$,  observer at $r=\infty$ measures
\beq
E= \sqrt{g_{tt}} \, E_r = f^{-1/4} E_r~
\eeq
$E\to 0$ $\leftrightarrow$ keep states  with $r\rightarrow 0$ or bulk states with  $\lambda \rightarrow \infty$\newline \Red{two sectors decouple since long wavelengths cannot probe short-distances} \newline agreement with previous analysis\newline  \Red{states  with $r\rightarrow 0$  $\leftrightarrow$ gauge theory,  bulk states $\leftrightarrow$ free Type IIB sugra}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Near-Horizon Limit}

study the states near  D-branes, $r\rightarrow 0$,  by change of coordinate
\beq
u = {{r}\over{\alpha^\prime}}
\eeq
hold finite as $\alpha^\prime \rightarrow 0$\newline low-energy (near-horizon) limit:
\beq
{{ds^2}\over{\alpha^\prime}} = {{u^2}\over{\sqrt{4 \pi g_s N}}} \left(dt^2 + dx_i^2\right)
+\sqrt{4 \pi g_s N}\left({{du^2}\over{u^2}} + d\Omega_5^2\right)~
\label{eqn:Chap17:nearhorizon}
\eeq
\Red{metric of AdS$_5\times S^5$}  \newline identify the gauge theory 
with  supergravity  near horizon  limit \newline Maldacena's conjecture: 
Type IIB string theory on AdS$_5 \times S^5$ $\equiv$
4D $SU(N)$
gauge  theory with ${\mathcal N}=4$ SUSY, a CFT

\npar
so much circumstantial evidence, called \Red{AdS/CFT correspondence}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Supergravity Approximation}

Sugra on AdS$_5\times S^5$ is
good approximation string theory
 when  \newline $g_s$ is weak and $R/\alpha^{\prime 1/2}$  is large:
\beq
g_s \ll 1~,\,\,\,
g_s N \gg 1~
\eeq
Perturbation theory is a good description of a gauge theory when
\beq
g^2 \ll 1~,\,\,\,
g^2 N \ll 1~
\eeq
 AdS/CFT correspondence: \newline \Red{ weakly coupled gravity} $\leftrightarrow$ \Red{large $N$, strongly coupled gauge theory}
 
 \npar hard to prove but also potentially quite useful



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{AdS$_5\times S^5$}

$S^5$  can be embedded in a flat 6D space with  constraint:
\beq
R^2 = \sum_{i=1}^6 Y_i^2~,
\eeq
$S^5$  space with
constant positive curvature,
 \newline \Red{$SO(6)$ isometry $\leftrightarrow$  $SU(4)_R$} symmetry of
 ${\mathcal N}=4$ gauge theory
 
 \npar
AdS$_5$ can be embedded in 6D:
\beq
ds^2 = -dX_0^2 -dX_5^2 + \sum_{i=1}^4 d X_i^2~
\eeq
with the constraint:
\beq
R^2=X_0^2 + X_5^2 -\left(\sum_{i=1}^4 X_i^2 \right)~
\eeq
AdS$_5$  space with a constant negative curvature and $\Lambda <0$
\newline isometry   is \Red{$SO(4,2)$ $\leftrightarrow$  conformal symmetry}
 in 3+1 D 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{AdS Space}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.65\hsize]{slideimages/hyperboloid}}
\label{Chap17:fig:hyperboloid}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
hyperboloid embedded in a higher dimensional space

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{AdS$_5$}

change to ``global" coordinates:
 \beq
 X_0 &=& R \cosh \rho \,\cos \tau~\,\,\,
 X_5 = R \cosh \rho \, \sin \tau~ \\
X_i &=& R \sinh \rho \, \Omega_i, \,\,i=1,\ldots,4~, \,\,\,
\sum_i \Omega_i^2=1~
\eeq
\beq
ds^2 = R^2(-\cosh^2 \rho \,d\tau^2 + d\rho^2 + \sinh^2 \rho \, d\Omega^2)~
\eeq
periodic coordinate $\tau$ going around the ``waist"  at $\rho=0$\newline while $\rho\ge 0$ is the $\perp$ coordinate in the horizontal direction 

\npar to get causal (rather than periodic) structure \newline cut hyperboloid at $\tau=0$, paste together an infinite 
number of copies so that $\tau$ runs from $-\infty$ to $+\infty$\newline \Blue{causal universal covering spacetime}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{AdS$_5$: ``\Poincare coordinates"}
  \beq
 X_0 &=& {{1}\over{2 u}}\left(1 + u^2(R^2+\vec{x}^2 - t^2)\right),\,\,\,
 X_5 = R \,u\, t ~\\
X_i &=& R \,u \, x_i, \,\,i=1,\ldots,3 ~;\,\,\,
 X_4 = {{1}\over{2 u}}\left(1 - u^2(R^2-\vec{x}^2 + t^2)\right)
\eeq
\beq
ds^2 = R^2\left({{du^2}\over{u^2}} +u^2(-dt^2+ d\vec{x}^2)\right)
\eeq
cover half of the space covered by the global coordinates
\newline Wick rotate to  Euclidean
\beq
\tau\rightarrow \tau_E = -i \tau~, \,\,\,
{\rm or} \,\,\,
t \rightarrow t_E = - i t~
\eeq
\beq
ds_E^2 &=& R^2\left(\cosh^2 \rho d\tau_E^2 + d\rho^2 + \sinh^2 \rho d\Omega^2\right)\\
&=& R^2\left({{du^2}\over{u^2}}  +u^2(dt_E^2+ d\vec{x}^2)\right)
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{AdS$_5$: ``\Poincare coordinates"}
 another coordinate choice (also referred to as \Poincare coordinates)  
\beq
u={{1}\over{z}}~,\,\,\,
x_4 = t_E~
\eeq
metric is  conformally flat:
\beq
ds_E^2 
= {{R^2}\over{z^2}}\left(dz^2 + \sum_{i=1}^4 dx_i^2\right)~
\label{Chap17:eq:conformallyflat}
\eeq
 boundary of this space is $R^4$ at $z=0$, 
 Wick rotation of 4D Minkowski, and a point $z=\infty$

 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{AdS/CFT correspondence}
partition functions of  CFT and the string theory are related 
\beq
\langle \exp \int d^4 x \phi_0(x) {\mathcal O}(x) \rangle_{\rm CFT}
=Z_{\rm string}\left[\phi(x,z)|_{z=0} = \phi_0(x)\right]
\label{Chap17:partitionfunccorresp}
\eeq
 \Red{${\mathcal O}$ $\subset$ CFT  $\leftrightarrow$ $\phi$  AdS$_5$ field}, $\phi_0(x)$ is boundary value

\npar
For large $N$ and $g^2 N$,  use the supergravity approximation
\beq
Z_{\rm string} \approx e^{-S_{\rm sugra}\left[\phi(x,z)|_{z=0} = \phi_0(x)\right]}~
\label{Chap17:partitionapprox}
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{CFT Operators }
\label{Chap17:AdSCFT:sec:operatorsKK}

 \Blue{${\mathcal O}$ $\subset$ CFT  $\leftrightarrow$ $\phi$  AdS$_5$ field}\newline
scaling
dimensions of \Blue{chiral operators} can be calculated from 
 $R$-charge
 
 \npar 
\Blue{primary operators}  annihilated by  lowering
operators
$S_\alpha$ and $K_\mu$
\newline \Blue{descendant operators} obtained by 
raising operators $Q_\alpha$ and $P_\mu$\newline  interested in the mapping of \Red{chiral primary operators}

\npar
 ${\mathcal N}=4$ multiplet  $SU(4)_R$ representations:\newline
($A_\mu$, ${\bf 1}$), ($\lambda_\alpha$, $\fund$), ($\phi$, $\asymm$)
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Chiral Primary Operators}


 \begin{tabular}{c|cc} 
Operator& $SU(4)_R$ & Dimension\\ \hline 
$T^{\mu \nu}$ &  ${\bf 1}$ & 4\\
$J^\mu_R$ & $\Yoneoone$  &3 \\
${\rm Tr}(\Phi^{I_1}...\Phi^{I_k})$, $k\ge2$ &  $(0,k,0)$
$\Ysquare$, $\Ytwotwotwo$, $\Ytwotwotwotwo$, $\ldots$ \vphantom{\Yfoura} &$k$\\
${\rm Tr}(W^{\alpha}W_{\alpha}\Phi^{I_1}...\Phi^{I_k})$ & $(2,k,0)$ 
\symm, $\Ytwooneone$, $\Ytwotwooneone$, $\ldots$ \vphantom{\Yfoura} & $k+3$\\
${\rm Tr}\, \phi^k F^{\mu\nu}F_{\mu\nu}+...$ & $(0,k,0)$ 
${\bf 1}$, \asymm, \Ysquare, $\ldots$ & \vphantom{\Yfoura} $k+4$
\end{tabular}




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Corresponding Type IIB KK modes}



\noindent harmonics  
on $S^5$, 
 \Blue{masses determined
by $SU(4)_R$ irrep }

\vspace{24pt}
\hspace{-30pt} \begin{tabular}{c|ccc} 
Spin& $SU(4)_R\sim SO(6)$ & $m^2R^2$&Operator\\ \hline 
2 &  
${\bf 1}$, \asymm, \Ysquare, $\ldots$ & $k(k+4)~,\,\,\, k \geq 0$ &$$k=0$,~T^{\mu \nu}$\\
1 &  $\Yoneoone$, $\Ythreetwoone$, $\Ythreetwotwoone$, $\ldots$  \vphantom{\Yfoura} &$(k-1)(k+1)~,\,\,\, k \geq 1$ &$k=1,~J^\mu_R$ \\
0 &  
$\Ysquare$, $\Ytwotwotwo$, $\Ytwotwotwotwo$, $\ldots$ \vphantom{\Yfoura} &$k(k-4)~,\,\,\, k \geq 2$&${\rm Tr}(\Phi^{I_1}...\Phi^{I_k})$ \\
 0 & 
\symm, $\Ytwooneone$, $\Ytwotwooneone$, $\ldots$ \vphantom{\Yfoura} &$ (k-1)(k+3)~,\,\,\, k \geq 0$&${\rm Tr}(W^{\alpha}W_{\alpha}\Phi^{I_1}...\Phi^{I_k})$\\
 0 & 
${\bf 1}$, \asymm, \Ysquare, $\ldots$ & \vphantom{\Yfoura} $k(k+4)~,\,\,\, k \geq 0$&${\rm Tr}\, \phi^k F^{\mu\nu}F_{\mu\nu}+...$
\end{tabular}







\npar
 lowest states  form  graviton supermultiplet of $D=5$, gauged sugra





%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Waves on AdS$_5$}
\label{Chap17:AdSCFT:sec:waves}



massive scalar field  in AdS$_5$:
\beq
S= {{1}\over{2}}\int d^4 x \, dz \sqrt{g} (g^{\mu\nu} \partial_\mu \phi
\partial_\nu \phi +m^2 \phi^2)~
\label{eq:Chap17:scalaraction}
\eeq
Using the conformally flat Euclidean metric
 \beq
ds_E^2 
= {{R^2}\over{z^2}}\left(dz^2 + \sum_{i=1}^4 dx_i^2\right)~
\eeq
and assuming a factorized solution:
\beq
\phi(x,z) = e^{ip .  x} f(p \,z)~
\eeq
eqm reduces to
\beq
z^5 \partial_z \left( {{1}\over{z^3}} \partial_z f \right) -z^2p^2 f - m^2 R^2 f=0~
\label{Chap17:scalareqm}
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Waves on AdS$_5$}
Writing $y=pz$ the solutions are modified Bessel functions:
\beq
f(y) =\left\{ \begin{array}{ccc}
y^2 I_{\Delta -2}(y) & 
\sim  & y^\Delta, \,\, {\rm as} \,\,y\rightarrow 0\\
y^2 K_{\Delta -2}(y) &\sim  & y^{4-\Delta} , \,\, {\rm as} \,\,y\rightarrow 0
\label{Chap17:scalarBesselsolutions}
\end{array} \right.~,
\eeq
$\Delta$ is determined by the mass
\beq
\Delta = 2 + \sqrt{4 + m^2 R^2}~
\label{Chap17:eq:scalingdim}
\eeq

 $y^2 I_{\Delta -2}(y)$ blows up as $y\rightarrow \infty$: not normalizable 
 \beq
x \rightarrow {{x}\over{\rho}}~, \,\,\,
p \rightarrow \rho p~
\eeq
then  the scalar field transforms as
\beq
\phi(x,z) \rightarrow \rho^{4-\Delta} e^{ip . x} f(p z)~
\eeq
conformal weight $4-\Delta$,
$\leftrightarrow$ CFT ${\mathcal O}$
must have dimension $\Delta$

\npar
\Red{bulk mass, $m$ $\leftrightarrow$ scaling dimension, $\Delta$}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Propagators on AdS$_5$}


propagate  boundary $\phi_0$ into the interior:
\beq
\phi(x,z) = c\int d^4 x^\prime {{z^\Delta}
\over{\left(z^2 +|x-x^\prime|^2\right)^\Delta}} \phi_0(x^\prime)~
\eeq
for small $z$ the bulk field scales as $z^{4-\Delta} \phi_0(x)$ 
\beq
\partial_z \phi(x,z) = c \Delta \int d^4 x^\prime {{z^{\Delta-1}}
\over{|x-x^\prime|^{2 \Delta}}} \phi_0(x^\prime) + {\mathcal O}(z^{\Delta+1})~~~~~~~(*)
\label{eqChap17:scalarderiv}
\eeq
integrating  action by parts  $+$ eqm yields:
\beq
S&=&{{1}\over{2}}\int d^4 x dz \, \partial_5\left( {{R^3}\over{z^3}} \phi  \partial_5
\phi\right)= {{1}\over{2}}\int d^4 x  \, \left( {{R^3}\over{z^3}} \phi  \partial_5
\phi\right)|_{z=0} ~
\eeq
Using the boundary condition $\phi(x,0)=\phi_0(x)$ and (*) 
\beq
\Blue{S=
{{c  R^3 \Delta}\over{2}} \int d^4 x d^4 x^\prime {{ \phi_0(x) \phi_0(x^\prime) }
\over{|x-x^\prime|^{2 \Delta}}} }~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Two-Point Function of  CFT}
for  corresponding operator ${\mathcal O}$ derived from \beq
\langle \exp \int d^4 x \phi_0(x) {\mathcal O}(x) \rangle_{\rm CFT}
\approx e^{-S_{\rm sugra}\left[\phi(x,z)|_{z=0} = \phi_0(x)\right]}~
\eeq
\beq
\langle {\mathcal O}(x) {\mathcal O}(x^\prime) \rangle 
&=&  {{\delta^2 S}\over{\delta \phi_0(x) \, \delta \phi_0(x^\prime) }}
=  {{c R^3 \Delta }\over{|x-x^\prime|^{2 \Delta}}}~
\eeq

\npar
\Red{correct scaling for dimension $\Delta$ in  4D CFT}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dimension $\leftrightarrow$ Mass}

In AdS$_{d+1}$:
\beq
\begin{array}{rl}
{\rm scalars:} & \Delta_\pm = \frac{1}{2} (d \pm \sqrt{d^2 + 4 m^2 R^2}) \\ \\
{\rm spinors:} & \Delta =  \frac{1}{2}  (d +2 |m | R) \\ \\
{\rm vectors:} & \Delta_\pm = {{1}\over{2}} (d \pm \sqrt{(d-2)^2 + 4 m^2 R^2}) \\ \\
p\textrm{-forms:} & \Delta_\pm = {{1}\over{2}} (d \pm \sqrt{(d-2p)^2 + 4 m^2 R^2}) \\ \\
{\rm massless} \,\,{\rm spin}\,\,2: & \Delta = d
\end{array}~.
\eeq
for scalar requiring  $\Delta_\pm$ is real $\Rightarrow$ Breitenlohner--Freedman bound \beq
-\frac{d^2}{4} < m^2 R^2~
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Dimension $\leftrightarrow$ Mass}

 relation  is expected to hold for
stringy states:
\beq
m \sim {{1}\over{l_s}} \leftrightarrow \Delta \sim (g^2 N)^{{1}/{4}}~\\
m \sim {{1}\over{l_{\rm Pl}}} \leftrightarrow \Delta \sim N^{{1}/{4}}~
\eeq

\npar
\Blue{large $N$ and large $g^2 N$ $\leftrightarrow$  very large dimension 
${\mathcal M}$}  \newline neglected in the supergravity approximation


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$(N+1)$ D3-branes}
%\subsection{Static Quark Potential}
\label{Chap17:AdSCFT:sec:quarkpotential}


$SU(N+1)$, ${\mathcal N}=4$
SUSY gauge theory\newline pull one of the branes  distance $u$ away $SU(N+1)\to SU(N)$ \newline stretched string states $\leftrightarrow$
massive gauge bosons 
\beq
m_W = {{u}\over{\alpha^\prime}}~
\eeq
$\fund + \overline{\fund}$ of $SU(N)$ 

\npar
$u \rightarrow \infty$  $\leftrightarrow$ static quark 

\npar
consider static quark--antiquark pair at distance
$r$ on $\partial$AdS$_5$\newline
\Red{minimum  action: string stretching from the quark to 
the antiquark}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Wilson Loops}

in AdS$_5$
\beq
 \left<W(C)\right> =  e^{-\alpha(D)}~
 \label{Chap17:WilsonLoopexp}
\eeq
where $D$ is surface of minimal area  $\partial D=C$, 
\npar
 surface
$D$ $\leftrightarrow$ to the worldsheet of the string 

\npar $\alpha(D)$ is a regularized area\newline
subtract a term $\propto$ the circumference of $C$
 $\leftrightarrow$ action of the widely separated  static quarks
 
 If $C$ is a  square in Euclidean, width $r$ and height $T$ (along the Euclidean time direction)
\beq
 \left<W(C)\right> =  e^{-T V(r)}~
\eeq
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Nonperturbative Coulomb potential}


Using the conformally flat Euclidean metric \beq
ds_E^2 
= {{R^2}\over{z^2}}\left(dz^2 + \sum_{i=1}^4 dx_i^2\right)~
\eeq
scale  size of $C$
by 
\beq
x_i \rightarrow \rho \, x_i~
\eeq
 keep $\alpha(D)$ fixed by 
scaling  $D$: 
\beq
x_i \rightarrow & \rho \, x_i~\,\,\,
z \rightarrow  & \rho \, z ~
\eeq
 $\alpha(D)$ is independent of  $\rho$, $\alpha(D)\not \propto  C \sim \rho^2$


\npar\beq
V(r) \sim - {{\sqrt{g^2 N}}\over{r}}~
\eeq
$1/r$ behavior required by conformal symmetry
\newline $\sqrt{g^2 N}$ behavior is different from perturbative result


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Breaking SUSY: finite temperature}
\label{Chap17:AdSCFT:sec:finiteT}


 take Euclidean time ($t_E=-i t$)
to be periodic:
\beq
t_E \sim t_E + \beta~\,\,\,
e^{itE} \rightarrow e^{-\beta E}~
\eeq
$\leftrightarrow$ finite temperature 4D gauge theory
\newline \Blue{periodic boundary conditions for
bosons} \newline\Blue{antiperiodic boundary conditions for fermions}

zero-energy boson modes, no zero-energy
fermion modes\newline  $\rightarrow$ SUSY is broken\newline Scalars will get masses from
loop effects 
\newline gluons are protected by
gauge symmetry\newline  low-energy effective theory is 
\Red{pure non-SUSY
Yang-Mills}

\npar high-temperature limit  lose one dimension \newline $\rightarrow$ zero-temperature, non-SUSY, 
3D Yang-Mills 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{AdS Finite Temperature}
 
in  AdS  there is a
 at high $T$
partition function
dominated by a black hole metric with a horizon size $b  = \pi T$


\beq
{{ds^2} \over{R^2 }} 
=
\left(u^2 - {b^4 \over u^2}
\right)^{-1} d u^2 + 
\left(u^2 - {b^4 \over u^2}
\right) d \tau^2 + u^2 dx^i dx^i ~
\label{Chap17:blackhole}
\eeq 

\npar
\Red{blackhole horizon $\leftrightarrow$ confinement in gauge theory}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\myhead{Wilson Loops at High Temperature}
%\label{Chap17:AdSCFT:sec:wilsonloops}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Finite Temperature and Confinement}

\beq
 \left<W(C)\right> =  e^{-\alpha(D)}~
\eeq
in black hole metric 
bounded by the horizon, $u=b$
\newline minimal area of 
$D$  is area at the horizon
\beq
\alpha(D) =R^2  b^2 \,{\rm area}(C)~
\eeq
 $\leftrightarrow$ area law confinement 
 \beq
V(r) = R^2 b^2 r ~
\eeq
string tension is very large
\beq
\sigma \sim  R^2 b^2 \sim \sqrt{g^2 N}\, \alpha^\prime b^2~
\eeq

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Glueballs}
\label{Chap17:AdSCFT:sec:glueballs}

\noindent
massless
scalar field $\Phi$
in AdS$_5$,
dilaton which  couples to Tr $F^2$
\newline 
Tr $F^2$ has  nonzero overlap
with gluon states


\Red{$\Phi$ $\leftrightarrow$   $0^{++}$ glueball}

with AdS black hole metric:
\beq
\partial_{\mu} \left[ \sqrt{g}g^{\mu \nu} \partial_{\nu}\Phi
 \right] =0~,~~~~~
\Phi =f(u )e^{ik . x}~
\eeq
\beq
u^{-1} {{d}\over{d u}} \left( \left(u^4 -
b^4 \right) u {{d f}\over{d u}} \right) - k^2 f = 0~
\eeq

for large $u$,
 $ f (u)\sim u^\lambda$ where $ m^2 =0= \lambda( \lambda + 4)$
so as $u \rightarrow \infty$ either $ f (u)\sim $ constant or $ \sim u^{-4}$.
\newline second solution is normalizable solution \newline need
$f$ to  be regular at $u=b$  $\Rightarrow$ ${{df}/{du}}$ is finite
\newline wave guide problem,  bc in the direction $\perp$ to $k$ 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Glueball Mass Gap}
\npar
no normalizable solutions for $k^2 \ge 0$\newline
discrete eigenvalues solutions for $k^2 < 0$
\newline  3D
glueball masses 
\beq
M_i^2 = - k_i^2 > 0~
\eeq

 \noindent \Blue{mass gap as expected for confining gauge theory}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{4D Glueball Masses}

 M-theory 5-brane 
 wrapped on two circles\newline  one circle is small $\to$
Type IIA  D4-branes on a  circle\newline
problem is that the supergravity limit
 $g \rightarrow 0$, $g^2 N \rightarrow \infty$
$\not \leftrightarrow$ gauge theories
we usually think about.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Strong coupling problem}
QCD$_3$ intrinsic scale: 
\beq
g_3^2 N = g^2 N T~
\eeq
\Red{hold  fixed as $T \rightarrow \infty$ need  
  $g^2 N \rightarrow 0$}
  
  \npar
QCD$_4$ intrinsic scale: 
\beq
\Lambda_{\rm QCD} = \exp\left( {{-24 \pi^2}\over{11 \, g^2 N}}\right) T ~
\eeq
\Red{hold fixed as $T \rightarrow \infty$ need 
$g^2 N \rightarrow 0$}


supergravity calculation works when extra SUSY states
have masses $\sim$ glueballs 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{4D Glueball Masses}


consider M5-branes 
wrapped on two circles where the M5-branes have  some  angular momentum $a$


\beq
\label{Chap17:pocho}
ds^2_{\rm IIA}&={2\pi \lambda A \over  3u_0} u^3 \Delta ^{1/2}\bigg[& 4
\big( -dx_0^2+dx_1^2+dx_2^2+dx^2_3\big)
+ { 4A^2\over 9u_0^2}  \ (1-{u_0^6\over u^6 \Delta }) d\theta_2^2 
\nonumber \\
&&+ {4\ du^2  \over u^4 (1-{a^4\over u^4}-{u_0^6\over u^6 }) }
d\theta^2+\frac{\tilde{\Delta}}{u^2 \Delta} \sin^2\theta d\varphi^2 \\
&&+\frac{1}{u^2 \Delta } \cos^2\theta d\Omega_2^2
 -{4a^2 A u_0^2\over 3u^6\Delta } \sin^2\theta d\theta_2 d\varphi \bigg] \nonumber 
\eeq
\beq
\Delta\equiv 1-{a^4\cos^2\theta \over u^4}\ ,\ \ \ \ \tilde{\Delta}\equiv 1-{a^4\over 
u^4} \ , \nonumber \\
\ \ \ \ A\equiv {u_0^4\over u_H^4-\frac{1}{3} a^4}\ , \ \ \ \ 
u_H^6-a^4 u_H^2-u_0^6=0\ 
\eeq
horizon  $u_H$, dilaton background $e^{2\Phi }$,
temperature $T_H$
\beq
e^{2\Phi }={8\pi\over 27} {A^3\lambda^3 u^3\Delta^{1/2}\over u_0^3} 
{1\over N^2}\ , \ \ \ \  R=(2\pi T_H)^{-1}={A\over 3u_0}\ 
\label{dilz}
\eeq
when $a/u_0\gg 1$ $R\to 0$
shrinks to zero
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{4D  Glueball Masses}

 $0^{++}$ glueballs $\leftrightarrow$ ${\rm Tr}FF$, solve

\beq
\partial_{\mu} \left[ \sqrt{g} e^{-2 \Phi} 
g^{\mu \nu} \partial_{\nu} \Phi \right]=0
\eeq



\vspace{24pt}
 $0^{-+}$ glueballs $\leftrightarrow$ ${\rm Tr}F\tilde{F}$, solve 

\beq
\partial_{\nu} \left[ \sqrt{g} g^{\mu\rho}g^{\nu\sigma}
(\partial_{\rho}A_{\sigma}-\partial_{\sigma}A_{\rho})\right]=0
\eeq


\npar
\Blue{ discrete sets of
eigenvalues, functions of  $a$}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{4D Glueball Masses: $a\to \infty$ }

\begin{table}[htbp]
\centering
\begin{tabular}{lccc}
\hline
state & lattice $N=3$ &
SUGRA  $a=0$ & SUGRA $a\to \infty$ \\
 \hline
 $0^{++}$ & $1.61 \pm 0.15$   & 1.61 {\rm (input)} & 1.61 {\rm (input)} \\
 $0^{++*}$ &  $2.48 \pm 0.23$  & 2.55 &  2.56 \\
 $0^{-+}$ & 2.59 $\pm$0.13   & 2.00 & 2.56 \\
 $0^{-+*}$ &  3.64 $\pm$0.18   & 2.98 &  3.49 \\
 \hline
\end{tabular}
\label{Chap17:tab:4ddila}
\end{table}






\noindent 
\Blue{circle KK modes
decouple $\Rightarrow$ 
real 4D gauge theory} \newline 
$0^{++}$ glueball mass ratios change only  slightly\newline
$S^4$ KK modes do not decouple

\npar $a/u_0\gg 1$,
 approaches a SUSY limit




%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{4D Glueball Mass}



%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{figure}[htb]
\centerline{\includegraphics[width=0.4\hsize]{slideimages/glueballspectrumnew}}
\label{AdS:fig:glueballspectrum}
\end{figure}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

 \noindent masses are within 4\%
of the lattice results\newline strong-coupling expansion 
off by between 7\% and 28\%
\newline \Red{SUGRA  results
are much better than we have any reason to expect}







%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{Breaking SUSY: Orbifolds}
\label{Chap17:AdSCFT:sec:orbifolds}



\beq \nonumber
\begin{array}{ccccc}
{\rm Type} \,\, {\rm IIB}\,\,{\rm  on}\,\, {\rm AdS}_5 \times S^5 &
\,\,\,\,& \leftrightarrow &
\,\,\,\,
& {\mathcal N}=4 \,  {\rm CFT} \\
{\rm KK}\,\,{\rm mode} &&&& {\rm operator} \\ &&&& \\
\downarrow & & 
{\rm orbifolding}\,\,S^5 &
&\downarrow \\  &&&& \\
{\rm AdS}_5 \times S^5/\Gamma &
& \leftrightarrow &
& {\mathcal N}<4  \, {\rm CFT} \\
{\rm invariant} \,{\rm KK}\,\,{\rm mode}&&&& {\rm invariant}\,\,{\rm operator}~
\end{array}
\eeq


construct ${\mathcal N}=1$ SUSY CFTs by orbifolding  ${\mathcal N}=4$with discrete group $\Gamma$  embedded in $SU(N)$ using
an  $N$-fold copy of the regular representation

$\leftrightarrow$ Type IIB string
theory on
 orbifold AdS$_5 \times S^5/\Gamma$\newline 
For  ${\mathcal N}=1$, the $SO(6) \simeq SU(4)_R$ isometry of $S^5$  is broken to 
$U(1)_R \times \Gamma$

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$Z_3$ Orbifold}
\beq
X^{1,2,3} \rightarrow e^{{2 \pi i}/{3}} X^{1,2,3}~,
\label{z3}
\eeq
 $X^i$ parameterize the $R^6$$\perp$ to the D3-branes

\beq
\begin{tabular}{cccc|c}
 & $SU(N)$ & $SU(N)$  & $SU(N)$ & $U(1)_R$

\\
\hline
$U^i$ & \Yfund & $\overline{\Yfund}$ & {\bf 1}  &  ${{2}\over{3}} \vbr$ \\
$V^i$ & {\bf 1} & \Yfund &  $\overline{\Yfund}$ &  ${{2}\over{3}} \vbr$ \\
$W^i$ & $\overline{\Yfund}$ & {\bf 1} &\Yfund &  ${{2}\over{3}} \vbr$ \\
\end{tabular}~,
\label{N=1}
\eeq
where $i=1,2,3$,  $SU(3)$ global symmetry is broken by the superpotential\newline 
orbifold fixed point  $X^i=0$\newline  volume of
$S^5$ is nonzero,
manifold is non-singular
\newline  supergravity description still applicable

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$Z_3$ Orbifold}
  KK modes of supergravity on AdS$_5 \times S^5/Z_3$  are
$Z_3$ invariant



for example, the   KK mode


\begin{tabular}{c|ccc} 
Spin& $SU(4)_R\sim SO(6)$ & $m^2R^2$&Operator\\ \hline 
0 &  
$\Ysquare$, $\Ytwotwotwo$, $\Ytwotwotwotwo$, $\ldots$ \vphantom{\Yfoura} &$k(k-4)~,\,\,\, k \geq 2$&${\rm Tr}(\Phi^{I_1}...\Phi^{I_k})$ \\
\end{tabular}
\newline
with $k=3$, $\Ytwotwotwo={\bf 50}$ of $SU(4)_R$
couples to a dim 3  chiral primary op\newline
$SU(4)_R \to SU(3) \times U(1)_R$ gives:
\beq
{\bf 50} \to {\bf 10}_2 + \overline{\bf 10}_{-2} + {\bf 15}_{2/3} + \overline{\bf
15}_{-2/3} ~
\label{Chap17:yuk1}
\eeq
 $Z_3$
on ${\bf 3}$ of $SU(3)$:
$(x^1,x^2,x^3) \rightarrow$ ($e^{{2 \pi i}/{3}} x^1$, $ e^{{2 \pi i}/{3}} x^2$, 
$e^{{-4 \pi i}/{3}}x^3$) \newline  ${\bf 10}$ is contained in ${\bf 3} \times {\bf 3} \times {\bf 3}$
$\Rightarrow$ ${\bf 10}$  is invariant under the $Z_3$
projection, ${\bf 10}$ has correct
$R$-charge \newline 
$\leftrightarrow$10 chiral primary operators 
${\rm Tr} \,U^{i_1}V^{i_2}W^{i_3}$  symmetric in  $i_k$ 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\myhead{$Z_3$ Orbifold}
 \begin{tabular}{c|ccc} 
Spin& $SU(4)_R\sim SO(6)$ & $m^2R^2$&Operator\\ \hline 
 0 & 
${\bf 1}$, \asymm, \Ysquare, $\ldots$ & \vphantom{\Yfoura} $k(k+4)~,\,\,\, k \geq 0$&${\rm Tr}\, \phi^k F^{\mu\nu}F_{\mu\nu}+...$
\end{tabular}

\npar
$k=0$,  dilaton transforms as  ${\bf 1}$ invariant
under the $Z_3$ projection\newline
couples to the marginal primary operator $\sum_{i=1}^3 {\rm Tr}\, F_i^2$


\npar  result is independent of 
$\Gamma$\newline ${\rm Tr} \, F^2$
is  marginal in any theory obtained by $\Gamma$
projection on  ${\mathcal N}=4$ 





\end{document}
