% NB: use pdflatex to compile NOT pdftex.  Also make sure youngtab is
% there...

% converting eps graphics to pdf with ps2pdf generates way too much
% whitespace in the resulting pdf, so crop with pdfcrop
% cf. http://www.cora.nwra.com/~stockwel/rgspages/pdftips/pdftips.shtml




\documentclass{beamer}


\usepackage[vcentermath]{youngtab}

\usepackage{hyperref}
%\usepackage{beamerthemesplit}


\usetheme{default}

\def\nn{\nonumber}

\def\d{\partial}
\def\tr{\operatorname{tr}}
\def\Dim{\operatorname{Dim}}
\def\ket#1{\left| #1\right\rangle}
\def\L{\Lambda}

\def\cO{{\cal O}}

\def\corr#1{\left\langle #1 \right\rangle}
\def\cN{{\cal N}}

\def\a{\alpha}
\def\m{\mu}
\def\n{\nu}

\def\nat{\textrm{nat}}


% from Sean Murray
\def\tyng(#1){\hbox{\tiny$\yng(#1)$}\,}
\def\tyoung(#1){\hbox{\tiny$\young(#1)$}}
\newcommand{\svdots}{\scriptstyle{\vdots}}%for youngtab
\newcommand{\scdots}{\scriptstyle{\cdots}}%for youngtab
\newcommand{\weevdots}{\scriptstyle{\vbox{\hbox{.}\vskip 1pt\hbox{.}\vskip 1pt\hbox{.}}}}%for youngtab




\title{AdS/CFT Beyond the Planar Limit}
\author{T.W. Brown}


\author[T.W.~Brown]
{
  T.W.~Brown
}

%\institute[DESY]
%{
%  DESY
%}

\date[DESY November 2009]
{DESY, November 2009}



\begin{document}

\begin{frame}
\titlepage

{\scriptsize
\begin{itemize}
\item Diagonal multi-matrix correlators and BPS operators in N=4 SYM (\href{http://www.arxiv.org/abs/0711.0176}{0711.0176} [hep-th]) TWB, Paul Heslop and Sanjaye Ramgoolam
\item Permutations and the Loop (\href{http://www.arxiv.org/abs/0801.2094}{0801.2094} [hep-th]) TWB
\item Diagonal free field matrix correlators, global symmetries and giant gravitons (\href{http://www.arxiv.org/abs/0806.1911}{0806.1911} [hep-th]) TWB, PJH, SR
\item Thesis and unpublished
\end{itemize}
}
\end{frame}






\begin{frame}{IIB superstrings on $AdS_5 \times S^5$}

  % see http://amath.colorado.edu/documentation/LaTeX/reference/figures.html
  \begin{columns}
    \column{.45\textwidth}
  \includegraphics[width=50mm]{manifold.jpg}

    \column{.55\textwidth}
  \begin{equation*}
 \alert{  \frac{1}{\a'}} \int d\tau d\sigma \left[ \textrm{ non-linear } \sigma \textrm{-model } \right]
  \end{equation*}

  \end{columns}
    \vskip.4in
    Perturbative expansion in the string coupling $\alert{g_s}$
    \vskip.2in
    \centering
 \includegraphics[width=110mm]{string_expansion.jpg}

\end{frame}





\begin{frame}{$\cN=4$ SUSY Yang-Mills: a Conformal Field Theory}
  \begin{equation*}
    \alert{\frac{N}{\lambda} }\int d^4 x \tr\Big[F_{\m\n} F^{\m\n} + D^\m\phi_i D_\m\phi_i - [\phi_i, \phi_j][\phi_i, \phi_j]  
  \end{equation*}
  \begin{equation*}
       + \psi \sigma^\m D_\m \psi - \psi \phi \psi \Big]\;\; +\;\; \theta \int d^4 x \tr[ F_{\m\n} \tilde  F^{\m\n}]
  \end{equation*}
  \vskip.1in
 Gauge group $\alert{U(N)}$; fields in adjoint.  Compute
 correlation functions of gauge-invariant local operators.
 
  \centering
  
  \vskip.2in
 \includegraphics[width=100mm]{feynman2.jpg}



\end{frame}



\begin{frame}{The AdS/CFT correspondence}



\begin{align*}
  \left\{\begin{array}{c} \textrm{IIB superstrings on} \\ AdS_5\times
  S^5 \end{array}\right\}  & =   \left\{\begin{array}{c} \cN=4 \textrm{ SUSY} \\ \textrm{ Yang-Mills in 4d } 
  \end{array} \right\} \\
& \\
    \frac{R^2}{\alert{\alpha'}} & = \alert{\sqrt{\lambda}} \\
& \\
   \alert{ g_s} & =\alert{ \frac{\lambda}{N}}
\end{align*}
\centering
\vskip.2in
bosonic symmetries $\alert{SO(2,4)\times SO(6)}$  match

\end{frame}




\begin{frame}{The Planar Limit}
  
  AdS/CFT has been successfully studied in the \alert{planar} limit.\newline
  $\lambda$ fixed, $N \to \infty$ $\Rightarrow$ $g_s \to 0$.
  \vskip.1in
  \begin{itemize}
  \item \alert{Single-trace} local operators in $\cN=4$ SYM.
  \vskip.1in
  \item \alert{Classical} string theory in bulk for strict $N \to \infty$
    limit.
  \vskip.1in
    \item Beautiful story of spinning strings, spin chains and
      integrability.
  \end{itemize}
  
  \centering
  \vskip.3in
 \includegraphics[width=90mm]{planar2.jpg}



\end{frame}



\begin{frame}{Parameter space}
  % see http://amath.colorado.edu/documentation/LaTeX/reference/figures.html
  \includegraphics[width=105mm]{parameters.jpg}

\end{frame}




\begin{frame}{General Programme}
  We want to study AdS/CFT at \alert{finite N}.
  \vskip.2in
    \begin{itemize}
    \item \alert{Multi-trace} and \alert{determinant}-type operators
      in CFT.
      \vskip.1in
    \item Must deal with \alert{Stringy Exclusion Principle}.
      \vskip.1in
    \item Correlation functions involve complicated combinatorics.
      \vskip.1in
    \item \alert{Non-perturbative} quantum gravity effects in the
      bulk, giant graviton branes, black holes.
    \end{itemize}

\end{frame}









\begin{frame}{Beasts of the field}
  In  $\cN=4$ super Yang-Mills the fields are
  \begin{equation*}
    X,Y,Z,X^\dagger, Y^\dagger, Z^\dagger; \lambda^A_\alpha, \bar{ \lambda}^A_{\dot{\alpha}}; F_{\mu\nu}\quad \textrm{plus derivatives} \quad D_{\mu}
  \end{equation*}
  
  Each field is in the \alert{adjoint} of the gauge group \alert{$U(N)$}
  \begin{equation*}
    (W_m)^i_j
  \end{equation*}
  $i,j = 1,2 \dots N$.
  
  $m$ runs over different fields.
  \vskip.15in
  To get gauge-invariant operators, usual route is to multiply these
  $N \times N$ matrices together and take \alert{traces}
  \begin{align*}
    :\tr(XYX^\dagger)\tr(YZ)\tr(Y^\dagger): \;\; = \;\; X^{i_1}_{i_2}\, Y^{i_2}_{i_3}\, X^{\dagger}{}^{i_3}_{i_1} \;\; Y^{i_4}_{i_5}\,Z^{i_5}_{i_4}\;\; Y^{\dagger}{}^{i_6}_{i_6}\; - \; \dots
  \end{align*}
  

\end{frame}



\begin{frame}{Correlation functions}
 
Wick contract with, e.g., 
\begin{equation*}
  \left\langle X^{i}_j(x) \;\; X^{\dagger}{}^k_l(y) \right\rangle \;\; = \;\; \delta^i_l\delta^k_j \; \frac{1}{(x-y)^2}
\end{equation*}
\vskip.1in
Even at tree level this gives a complicated \alert{$\frac{1}{N}$ expansion}
\begin{equation*}
  \left\langle \tr(XXXX)[x] \;\; \tr(X^\dagger X^\dagger X^\dagger X^\dagger)[y]\right\rangle =  \frac{4N^4}{(x-y)^{8}}\; \alert{\left(1 + \frac{5}{N^2}\right)}
\end{equation*}
\begin{equation*}
  \left\langle \tr(XXXX)[x] \;\; \tr(X^\dagger X^\dagger ) \tr(X^\dagger X^\dagger)[y]\right\rangle = \frac{4N^4}{(x-y)^{8}} \alert{\left(\frac{4}{N} + \frac{2}{N^3}\right)}
\end{equation*}
\vskip.2in
Mixing between different trace structures only suppressed when the
op. length $n < N$. [For giant graviton $n \sim N$, black hole $n \sim N^2$.]

\end{frame}







\begin{frame}{Stringy Exclusion Principle}

  For an $N \times N$ matrix $A$, traces of powers \alert{bigger than $N$} can
  always be written in terms of traces of powers \alert{$\leq N$}.

  \vskip.2in
  For example, if \alert{$N=2$} for the $2\times 2$ matrix $A = \left(\begin{array}{cc}
 a&b \\ c&d \end{array} \right)  $
  \begin{equation*}
  \alert{\tr(A^{3})} = \frac{3}{2} \tr(A^2)\tr(A) -\frac{1}{2} \tr(A)\tr(A)\tr(A) 
  \end{equation*}
  \vskip.2in
  So working with traces is problematic for operators with $\Delta \geq N$ ...
\end{frame}




\begin{frame}{Operators with multiple fields}

  Trace the same field content (e.g. for $U(2)$ rep $\tyng(2,2)$) and you get
  \vskip.2in
  \includegraphics{22trace-crop}
  \vskip.2in
  where $\Phi^p\Phi_p = \epsilon^{pq}\Phi_p \Phi_q = [X,Y]$.
\end{frame}



\begin{frame}{Solution: separation}

  Separate:

  \begin{itemize}
    \item \alert{Representation} of \alert{global symmetry group}
      $PSU(2,2|4)$, which organises field content and its symmetrisation
  \end{itemize}

  from

  \begin{itemize}
    \item \alert{Trace structure}, which it turns out will involve the
      representation theory of the \alert{gauge group} $U(N)$
  \end{itemize}

  \vskip.3in

  The permutation group \alert{$S_n$} plays a vital role.
  \vskip.2in
  Simplest to see for half-BPS case, where the representation of the
  global symmetry group is trivial.

\end{frame}


\begin{frame}{Half BPS operators: only trace structure}

  Here we have only \alert{one} type of field: $X$.  Multi-trace ops
  labelled by (conjugacy classes of) elements of the symmetric group
  \alert{$S_n$}.
  \vskip.2in 
  E.g. $\tr(XX)\tr(XX)$ can be written using $\a =
  \alert{(12)(34)} \in S_4$
  \vskip.04in
\begin{equation*}
  \tr(XX)\tr(XX) = X^{i_1}_{i_{\a(1)}}\;X^{i_2}_{i_{\a(2)}}\;X^{i_3}_{i_{\a(3)}}\;X^{i_4}_{i_{\a(4)}} = X^{i_1}_{i_2}\;X^{i_2}_{i_{1}}\;X^{i_3}_{i_{4}}\;X^{i_4}_{i_{3}}
\end{equation*}
\vskip.25in 
%\center
%\begin{figure}[t]
%\begin{center}
%left, bottom, right, top
%\includegraphics[width=150mm,trim=100 0 -10 -20 ]{tracealpha.pdf}
%\end{center}
%\end{figure}
%use pdfcrop http://www.maths.manchester.ac.uk/~kd/latextut/pdfbyex.htm
%pdf tips: http://www.cora.nwra.com/~stockwel/rgspages/pdftips/pdftips.shtml
%\begin{picture}(x_dim,y_dim)
\resizebox{10cm}{!}{\includegraphics{tracealpha-crop}}

\end{frame}

\begin{frame}{The Schur polynomials}
  Define linear \alert{change of basis} to Schur polynomials
  \begin{equation*}
    \tr_{\alert{R}}(X^{\otimes n}) \equiv
 \frac{1}{n!} \sum_{\a \in S_n}  \chi_{\alert{R}}(\a) \;\;X^{i_1}_{i_{\a(1)}}  X^{i_2}_{i_{\a(2)}} \cdots  X^{i_n}_{i_{\a(n)}}
  \end{equation*}

  \vskip.1in
  \alert{$R$} is Young diagram of \alert{$n$} boxes: rep
  \alert{both} of \alert{$U(N)$} and \alert{$S_n$}, sorts
  multi-trace structure (cf. Wilson loop). 2-pt function
  \alert{diagonal}
      \begin{equation*}
        \corr{\chi_R(X{\scriptstyle (x)})\;\chi_S(X^\dagger{\scriptstyle (y)})}  =  \; \alert{\delta_{RS}}\; \alert{\Dim_N\,R}\;  \nn%\frac{1}{(x-y)^{2n}} \nn   %\; \frac{n!}{d_R} \frac{1}{(x-y)^{2n}} \nn
      \end{equation*}
    
     $ \alert{\Dim_N\,R}$ is $U(N)$ dimension of $R$; it capture the $N$ expansion, e.g.
      \begin{equation*}
        Dim_N \;\tyng(3,2,1) = \frac{N^2(N+1)(N+2)(N-1)N(N-2)}{45}
      \end{equation*}
  

      \vskip.15in
      (The half-BPS sector is not renormalised, so this holds for all
      values of the coupling $\lambda$.  This will not be true in
      general...)
\end{frame}


\begin{frame}{Physical meaning of Schur polynomials}
  

  \begin{itemize}
    \item Encode finite $N$ \alert{stringy exclusion principle}, since
      reps of $U(N)$ have Young diagrams with at most $N$ rows.
    \vskip.1in
    \item Row-lengths $\sim$ $N$ occupied energy levels of \alert{free fermions} from complex matrix model.
    \vskip.1in
    \item For $n \sim N$ map to \alert{giant gravitons}, single column $[1^N]$ to giant in $S^5$, single row $[N]$ in  $AdS_5$.  General:
      \alert{LLM-type geometries}.
    \vskip.1in
    \item Can gain qualitative understanding of black hole microstates.
  \end{itemize}
      \vskip.2in

      \center
      \resizebox{8cm}{!}{\includegraphics{droplets-crop}}


   %N occupied energy levels $E_i$
   %*Replaces* trace basis for large $\D = R = n\sim N$

   % These are correct ops at finite N
\end{frame}




\begin{frame}{Outline of method for multiple  non-commuting fields}
  Solution: \alert{group theory}.

  \vskip.2in
  Organise \alert{multi-trace operators} of $\cN=4$ SYM at
  \alert{finite $N$} into reps of the \alert{global} symmetry group
  and reps of the \alert{local} gauge group (which will control
  multi-trace structure \`a la Wilson loop).

  \vskip.2in
  \begin{enumerate}
  \item Start with $n$ fields with none of their indices contracted
  \begin{equation*}
    (W_{m_1})^{i_1}_{j_1}\;   (W_{m_2})^{i_2}_{j_2} \;  \cdots \;     (W_{m_n})^{i_n}_{j_n} 
  \end{equation*}
  \vskip.1in
  \item Build into reps of $G$ and $U(N)$.
  \vskip.1in
  \item Enforce gauge invariance.
  \end{enumerate}


\end{frame}







\begin{frame}{Technical slide 1/4: Example of $U(2)$}
  Take the fundamental representation $V_F$ of $U(2)$
  \begin{equation*}
    V_F = \left(  
    \begin{array}{c}
      X \\ Y
    \end{array} \right)
  \end{equation*}
  \pause
  and then consider the simplest tensor product
  \begin{equation*}
  V_F \otimes V_F
  \end{equation*}
  We can re-arrange into irreducible reps of $U(2)$
  \begin{align*}
    \left(  
    \begin{array}{c}
      X \\ Y
    \end{array} \right) \otimes \left(  
    \begin{array}{c}
      X \\ Y
    \end{array} \right) & = \left(  
    \begin{array}{c}
      X\otimes X \\
      X\otimes Y + Y \otimes X \\
      Y \otimes Y
    \end{array} \right) \oplus  \left(  
    \begin{array}{c}
      X \otimes Y - Y \otimes X
    \end{array} \right)  \\
& \\
    \tyng(1)\hspace{4mm} \otimes\hspace{4mm} \tyng(1)\hspace{5mm} & = \hspace{16mm}\tyng(2) \hspace{15mm}\oplus \hspace{15mm}\tyng(1,1)
  \end{align*}
\end{frame}

\begin{frame}{Technical slide 2/4}
  We hit a problem with three copies of the fundamental
  \begin{align*}
 V_F^{\otimes 3}  = &\left(  
    \begin{array}{c}
      X\otimes X \otimes X \\
      \vdots
    \end{array} \right) 
    \oplus
    \left(
    \begin{array}{c}
      X\otimes X \otimes \alert{Y} - X\otimes \alert{Y} \otimes X \\
      \vdots
    \end{array} \right)\\
  &  \oplus
    \left(
    \begin{array}{c}
      X\otimes X \otimes \alert{Y} - \alert{Y}\otimes X \otimes X \\
      \vdots
    \end{array} \right)
  \end{align*}

  In terms of Young diagrams
  \begin{equation*}
    \tyng(1)\hspace{4mm} \otimes\hspace{4mm} \tyng(1)\hspace{4mm} \otimes\hspace{4mm} \tyng(1)\hspace{4mm}  = \hspace{6mm}\tyng(3) \hspace{4mm}\oplus \hspace{4mm} \alert{2} \;\;\; \tyng(2,1)
  \end{equation*}
  \vskip6mm
  How do we account for this multiplicity?
\end{frame}




\begin{frame}{Technical slide 3/4: Schur-Weyl duality}

  For $V_F^{\otimes n}$, an $\alert{n}$-tensor products of the
  fundamental of $U(\alert{K})$, there are \alert{two} commuting
  group actions:
    \vskip.1in
  \begin{itemize}
  \item \alert{$U(K)$}: the action of $U(K)$ on its
    fundamental rep
    \vskip.15in
  \item \alert{$S_n$}: permutes the $n$ different copies of $V_F$
  \end{itemize}
  \pause
  \vskip.2in
  So organise $V_F^{\otimes n}$ in terms of representations of the two groups:
  \begin{equation*}
    V_F^{\otimes n}\; \equiv\;\; \stackrel{n}{\overbrace{ \tyng(1) \otimes \tyng(1) \otimes \cdots \otimes \tyng(1)}}\;\;=\;\; \bigoplus_{\L}\;  V_\L^{\alert{U(K)}} \otimes V_{\L}^{\alert{S_n}}
  \end{equation*}
  where $\L$ runs over Young diagrams with $\alert{n}$ boxes and at
  most $\alert{K}$ rows.
  \vskip.2in
  [To answer question: $\dim  V_{\tyng(2,1)}^{\alert{S_3}} = \alert{2}$. ]
\end{frame}




\begin{frame}{Technical slide 4/4: Clebsch-Gordan coefficients}

  We can express this map from $V_F^{\otimes n}$ to reps of $U(K)$ and
  $S_n$ using Clebsch-Gordan coefficients $C$.

  \begin{eqnarray*}
    C:& V_F^{\otimes n}&\to \;\;  V_\Lambda^{\alert{U(K)}} \otimes V_{\Lambda}^{\alert{S_n}} \\
    & \\
 C^{m_1m_2 \dots m_n}_{\Lambda,M_\Lambda,a_\Lambda} &  W_{m_1} \otimes   W_{m_2} \otimes\cdots \otimes W_{m_n}   &= \;\; \ket{\Lambda,M_\Lambda,a_\Lambda}
  \end{eqnarray*}
  \vskip.2in
  \begin{itemize}
  \item $m_k = \{1, 2, \dots K\}$\quad\quad (for $U(2)$, $W_1 = X$, $W_2 = Y$)
    \vskip.14in
  \item $M_\L$ labels $U(K)$ state in $V_\Lambda^{\alert{U(K)}}$
    \vskip.1in
  \item $a_\L$ labels $S_n$ state in $V_{\Lambda}^{\alert{S_n}}$
    \vskip.1in
    \item Clebsch map is invertible
  \end{itemize}

  

\end{frame}




\begin{frame}{The solution: use C-G coefficients}

  Consider operators with $\alert{n}$ fields, a generic example being
  \begin{equation*}
    (W_{m_1})^{i_1}_{j_1}\;   (W_{m_2})^{i_2}_{j_2} \;  \cdots \;     (W_{m_n})^{i_n}_{j_n} 
  \end{equation*}
  where $\{ W_m\}$ are the fields of a subsector $G \subset
  PSU(2,2|4)$.
  \vskip.1in
 
  Combine indices into rep of $G \times S_n$ and two of $U(N) \times
  S_n$
%  \begin{equation*}
%\ket{\Lambda(\alert{G}),M_\Lambda,m_\Lambda} \otimes \ket{R(\alert{U(N)}),M_R,m_R}\otimes \ket{\bar{S}(\alert{U(N)}),M_S,m_S}
%  \end{equation*}

  \begin{equation*}
    C^{m_1 \dots m_n}_{\Lambda(\alert{G}),M_\Lambda,a_\Lambda} \;   C^{i_1 \dots i_n}_{R(\alert{U(N)}),M_R,a_R}\;    C^{j_1 \dots j_n}_{\bar{S}(\alert{U(N)}),M_S,a_S}\;\; (W_{m_1})^{i_1}_{j_1}  \cdots    (W_{m_n})^{i_n}_{j_n} 
  \end{equation*}
  \vskip.15in
  \begin{itemize}
  \item Enforce gauge invariance: pick singlet ${\bf 1} \in R \otimes
    \bar{S}$\newline
    (implies $R = S$, sum over $M_R = M_S$)
  \item Impose overall $S_n$ invariance
  \end{itemize}
\end{frame}




\begin{frame}{Operator for general $G$}
%  Trace with a ${\alert{U(N)}}$ representation $R$ to get the final operator

\begin{figure}[h]
\begin{center}
\begin{picture}(150,65)
\put(2,60){rep and state of ${\alert{G}}$}
\put(2,35){ $
    {\cal O}\big[\overbrace{\L\,({\alert{G}}),\,M_{\L}}, \,R\,({\alert{U(N)}}),\,\tau \big]$}
\put(80,18){\vector(1,2){5}}
\put(-70,8){$R$ of ${\alert{U(N)}}$ gives multi-trace structure}
\put(134,18){\vector(-1,2){5}}
\put(135,8){(multiplicity)}
\end{picture}
\end{center}
\end{figure}
\begin{itemize}
  \item \alert{Complete} basis on space of multi-trace operators at
    finite $N$ built out of fundamental fields of $G$.
    \vskip.15in
  \item \alert{Free} 2-point function totally \alert{diagonal} on all labels,
    proportional to $\Dim_N\,R$.
    \vskip.15in
  \item Operators given in detail for $G = U(3),
    SL(2)$,$O(2)$,$SO(2,4)$, prescription given for $SO(6)$.
    \vskip.15in
  \item For $SL(2)$, in regime of large quantum numbers, spectrum of
    our basis matches excitations of giant gravitons.
\end{itemize}
\end{frame}



\begin{frame}{Partition function at finite $N$}
At finite $N$ we have for the \alert{free} theory
\begin{equation*}
  \mathcal{Z}_N = \int_{U(N)} [dU]
\, \exp\left\{ \sum_{m=1}^\infty {1 \over m} f({\bf x}^m) {\rm
tr}(U^{\dagger})^m {\rm tr} U^m \right\}
\end{equation*}
where  $f({\bf x})$ is the character for the
fundamental fields; for $U(K)$ this is just the trace of the matrix 
\begin{equation*}
  f({\bf x}) = \chi_F^{U(K)} ({\bf x}) = x_1 + x_2+\cdots +  x_K
\end{equation*}

Expanding we match \alert{exactly} the coefficient of each irrep of $G$

\begin{equation*}
    \mathcal{Z}_N =  \sum_n \sum_{\L(U(K))} \alert{ \sum_{R(U(N))} C(R,R,\L)}\;\;   \chi_{\L}({\bf x})
\end{equation*}
where $C(R,R,\L)$ is the number of times $\L$ appears in the
symmetric group tensor product $R \otimes R$.
\end{frame}


\begin{frame}{Subsectors}
 
  We can do this classification for the following sub-sectors
  $\alert{G} \subset PSU(2,2|4)$ of the global superconformal symmetry
  group (and product groups $G_1 \times G_2$) :
  \vskip.2in
  \begin{eqnarray}
    \textrm{half BPS } \alert{U(1)} : & \{W_m\} & = \{X\} \nn \\
    \alert{U(3)}: & \{W_m\} & = \{X,Y,Z\} \nn \\
    \alert{U(3|2)}: & \{W_m\} & = \{X,Y,Z;\psi_1,\psi_2\} \nn \\
    \alert{O(2)}: & \{W_m\} & = \{X,X^\dagger\} \nn \\
    \alert{SO(6)}: & \{W_m\} & = \{X,Y,Z,X^\dagger,Y^\dagger,Z^\dagger\} \nn \\
    \alert{SL(2,\mathbb{R})}: & \{W_m\} & = \{X,\d X,\d^2X,\d^3X, \dots \} \nn \\
    \alert{SO(2,4)}: & \{W_m\} & =  \{ X,\d_\mu X, \d_\mu\d_\nu X, \dots \} \nn 
  \end{eqnarray}
 
\end{frame}





\begin{frame}{More complicated example: $SL(2,\mathbb{R})$}

Take arbitrarily many derivatives of a field $\{W_{m}\} =
\{\d^m X\}$
\begin{equation*}
  \d^{m_1} X \otimes   \d^{m_2} X \otimes \cdots  \otimes  \d^{m_n} X
\end{equation*}
In $V_F^{\otimes n}$ sort into ops with $k = m_1 + \cdots + m_n$ derivatives
(spread out across the $n$ sites) and remove all descendants of form $\alert{D}^p(\cdots)$

\begin{picture}(150,100)
\put(40,50){$\left(\begin{array}{c}\d^{(1)}\\\d^{(2)}\\\vdots \\\d^{(n)}\end{array}  \right)$}
\put(100,60){\vector(3,1){40}}
\put(150,75){$\alert{D} = \d^{(1)}+ \d^{(2)}+\cdots + \d^{(n)}$}
\put(100,50){\vector(3,-1){40}}
\put(150,20){$\left(\begin{array}{c}\d^{(1)} -\d^{(2)} \\\d^{(2)}-\d^{(3)}\\\vdots \\\d^{(n-1)} -\d^{(n)}\end{array}  \right)$}
\end{picture}
\vskip.2in This is split of the canonical permutation rep $V_{\nat}$
of $S_n$ into the trivial and the `standard' rep $V_{\nat} =
V_{[n]}\oplus V_{[n-1,1]}$.
\vskip.1in
Build HWS of $SL(2,\mathbb{R})$ with $V_{[n-1,1]}$ (i.e. the differences).
\end{frame}





\begin{frame}{Free three-point function}
  We can also use this formalism to work out the free non-extremal
  three-point function
  % see http://amath.colorado.edu/documentation/LaTeX/reference/figures.html
  \begin{equation*}
    \Big\langle  \cO[{\bf \L}_1, R_1](x_1)\;\;  \cO[{\bf \L}_2, R_2](x_2)\;\;  \cO[{\bf \L}_3, R_3](x_3)\Big\rangle
  \end{equation*}
  
  \vskip.2in
  \center
  \includegraphics{muscle-crop}

\end{frame}

\begin{frame}{Three-point gauge spin network}
  On the legs between the operators the gauge group representations
  need not form a singlet.  The three-point function becomes a $G
  \times U(N)$ spin network.  Also extends to one-loop...
  \vskip.2in
  % see http://amath.colorado.edu/documentation/LaTeX/reference/figures.html
  % converted with epstopdf unspin.eps
  \includegraphics{unspin-crop}


\end{frame}




\begin{frame}{One loop two-point function}
  At one loop this basis is no longer diagonal.  Operators mix and we
  must rediagonalise.  Multiplets also re-organise in a highly
  non-trivial way. Take for example the $U(2)$ sector, $\L =
  \tyng(2,2)$.
  \vskip.2in 


  Some operators are no longer protected and join long multiplets
  \begin{align*}
    \tr([X,Y][X,Y]) &\quad \quad \Delta = 4 + \alert{\frac{3\lambda}{4\pi^2} + \cO(\lambda^2)}
  \end{align*}
  (This is a descendant of the Konishi at weak coupling.)
  \vskip.2in
  The $\frac{1}{4}$-BPS operators, which are
  protected, are in 1-to-1 correspondence with the chiral ring and
  receive $\frac{1}{N}$ corrections, e.g.
  \begin{align*}
   \tr(\Phi^r \Phi^s)\tr(\Phi_r \Phi_s)  + \frac{2}{N} \tr([X,Y][X,Y])
  \end{align*}
\end{frame}

\begin{frame}{Action of dilatation operator}

 Analyse mixing with one-loop \alert{dilatation operator}, e.g. $U(2)$
 sector
  \begin{equation*}
    :\tr( [X,Y][\tilde X,\tilde Y]) :
  \end{equation*}
  $\tilde X \sim \frac{\partial}{\partial X}$.  This gives matrix of
  anomalous dimensions.
  \vskip.2in 
  Write its action on two sites by introducing \alert{extra $(n+1)$th index}
\center
\resizebox{!}{20cm}{\includegraphics[trim=-5 0 -20 0 ]{dil2site.pdf}}




\end{frame}





\begin{frame}{Constrained mixing  at one loop}

  Operators now mix via \alert{$(n+1)$}-box $U(N)$ reps $T$.
  \vskip.15in

  The ${\alert{U(N)}}$ $n$-box representations $R$ and $S$,
  controlling multi-trace structure, then \alert{only} mix if they
  both fit into the same \alert{$(n+1)$}-box rep $T$, i.e. $R$ and $S$ must be
  related by \alert{repositioning} a \alert{single} box.
    \vskip.3in


\begin{figure}[h]
\begin{center}
\begin{picture}(220,80)
\put(-40,70){$R = $ \tyoung(\ \ \ \ ,\ \ \times,\ \ )}
\put(25,70){mixes with}
\put(90,70){$S_1 = $ \tyoung(\ \ \ \ \times,\ \ ,\ \ )}
\put(165,70){via}
\put(190,70){$T = $ \tyoung(\ \ \ \ \times,\ \ \times,\ \ )}
\put(25,25){but \alert{not} with}
\put(90,25){$S_2 = $ \tyoung(\ \ \ \ \ ,\ ,\ ,\ ,\ )}
\end{picture}
\end{center}
\end{figure}
\end{frame}

\begin{frame}{Solution for commuting matrices}

$\frac{1}{4}$ and $\frac{1}{8}$-BPS ops at weak coupling in chiral
  ring built from symmetrised traces, i.e. commuting matrices.
  Characterise in terms of symmetric functions of eigenvalues.
  Still transform under $S_{\alert{N}}\subset U(\alert{N})$; want invariants of this
  group from $\left(V_{\nat}^{S_N}\right)^{\otimes n}$.
  \vskip.1in
\begin{equation*}
    C^{m_1 \dots m_n}_{\Lambda(\alert{G}),M_\Lambda,a_\Lambda} \;   C^{e_1 \dots e_n}_{[N](\alert{S_N}),\Lambda(\alert{S_n}),a_\Lambda}\quad x_{m_1}^{e_1}\; x_{m_2}^{e_2}\; \cdots \;x_{m_n}^{e_n}
\end{equation*}
\vskip.1in
$e_i \in \{1, 2,\dots N\}$. Generating function for multiplicity at finite $N$


\begin{equation*}
  \prod_{m,n=0}^\infty  \frac{1}{1- \alert{\nu} x^m y^n} = \sum_{N=0}^{\infty} \alert{\nu}^N \mathcal{Z}^{\frac{1}{4}\textrm-BPS}_N(x,y)
\end{equation*}

\begin{equation*}
  \mathcal{Z}_N^{\frac{1}{4}\textrm-BPS}(x,y) = \sum_{\L \textrm{ of } U(2) } \alert{\textrm{dim}_{[N],\Lambda}^{S_N \times S_n}} \;\;\chi_\L(x,y)
\end{equation*}

Map combinatorics to supergravity geometries \`a la LLM?
\end{frame}




\begin{frame}{Conclusions}

\begin{itemize}
  \item To study many phenomena in AdS/CFT need $N$ \alert{finite}.
  \item Organised multi-trace ops into \alert{complete basis}
    that \alert{transforms in irreps of $G$}$\subset PSU(2,2|4)$,
    traces sorted by $U(N)$.
  \item This basis \alert{diagonalises} the free two-point function,
    including all finite $N$ corrections.  Higher-point functions also simple.
  \item One-loop mixing \alert{highly constrained}.
  \item $\frac{1}{4}$ and $\frac{1}{8}$-BPS ops in chiral ring characterised in terms of functions of \alert{eigenvalues} of fields.
   \item Focus in future:
     \begin{itemize}
     \item Clarify field versus bulk description of $\frac{1}{4}$ and $\frac{1}{8}$-BPS states.
     \item String theory dual to zero/weakly-coupled field theory.
     \item Diagonalise spectrum at 1-loop.
     \item Sixteenth-BPS states: how do they furnish black hole
       entropy?
     \item Understand information loss.
     \end{itemize}
\end{itemize}



\end{frame}



\end{document}
