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% from Sean Murray
\def\tyng(#1){\hbox{\tiny$\yng(#1)$}\,}
\def\tyoung(#1){\hbox{\tiny$\young(#1)$}}
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\title{Cut-and-join operators and $\cN=4$ SYM}
\author{T.W. Brown}


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

\institute[DESY]
{
  DESY
}

\date[Hannover February 2010]
{Nordic String Meeting, Hannover, February 2010}



\begin{document}

\begin{frame}
\titlepage

\begin{center}
  \href{http://www.arxiv.org/abs/1002.2099}{1002.2099} [hep-th]
\end{center}

\end{frame}






\begin{frame}{General Programme}
    \begin{itemize}
    \item Study $\frac{1}{N}$ corrections to $\cN=4,d=4$ super Yang-Mills with guage group $U(N)$.
      \vskip.2in
    \item Multi-trace operators with $\Delta_0 \equiv n < N^{\frac{1}{2}}$.  Organise into:
      \vskip.05in
      \begin{itemize}
      \item Representations of the global symmetry group;
      \item Operators with fixed trace structure, e.g. single/double trace.
      \end{itemize}
      \vskip.2in
    \item Focus on theory at tree level and one loop.
      \vskip.05in
      \begin{itemize}
      \item Messy mixing problem;
      \item Want to find operators with well-defined conformal dimensions;
      \item Is there a string dual to the free gauge theory?
      \end{itemize}      
    \end{itemize}

\end{frame}



\begin{frame}{Two different attitudes}

\alert{Two} different attitudes to $\frac{1}{N}$ corrections, depending on coupling.
\vskip.2in
\begin{itemize}
\item For \alert{free} theory, $\lambda=0$, treat $\frac{1}{N}$ as a string
  coupling ordering the non-planar expansion of correlation functions.
  Multi-trace operators identified with multi-string states.
\vskip.2in
\item For $\alert{\lambda > 0}$ the correct string expansion
  is in $g_s = \frac{\lambda}{N}$.  Treat $\frac{1}{N}$ corrections as a
  modification to the gauge theory/string theory state
  identification.
\end{itemize}
\end{frame}






\begin{frame}{Review of half-BPS sector}

{\scriptsize Based on Vaman and Verline \href{http://arxiv.org/abs/hep-th/0209215}{0209215}; Corley, Jevicki and Ramgoolam \href{http://arxiv.org/abs/hep-th/0111222}{0111222}.}
\vskip.1in
Trace structures of operators map to \alert{conjugacy classes} of $S_n$.

E.g. for $\a = (123)(45)(6) \in S_6$
\begin{align*}
  \tr(X^3) \tr(X^2) \tr(X) & = X^{i_1}_{i_{2}}X^{i_2}_{i_{3}}X^{i_3}_{i_{1}}X^{i_4}_{i_{5}}X^{i_5}_{i_{4}}X^{i_6}_{i_{6}}\nn \\
& = 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_5}_{i_{\a(5)}} X^{i_6}_{i_{\a(6)}} 
\end{align*}
\vskip.02in
Conjugacy classes labelled by \alert{partitions} of $n$, e.g. $[3,2,1]$ here.
\vskip.2in
Two-point function given by cut-and-join operators
\begin{equation*}
  \corr{ \tr(\a'\, X^{\dagger n}) \;\tr(\a\, X^n)}_{\textrm{non-planar}} = N^n\bra{\a'}\O_n\ket{\a}
\end{equation*}
{\scriptsize (We're dropping the spacetime dependence here and onwards.)}
\end{frame}



\begin{frame}{Cut-and-join operators}
Basic cut-and-join operator is a sum over the transpositions in $S_n$
\begin{equation*}
  \S_{[2]} = \sum_{i < j} (ij)
\end{equation*}

It cuts a single trace/cycle $[n] = (123 \cdots n)$ into two
\begin{equation*}
  \S_{[2]} \ket{n} \sim \ket{n_1,n_2}
\end{equation*}

It both joins a double trace and cuts it into three
\begin{equation*}
  \S_{[2]} \ket{n_1,n_2} \sim \ket{n} + \ket{n_1,n_2,n_3}
\end{equation*}

Tree-level mixing given by
\begin{align*}
  \O_n & =  \sum_{\s \in S_n} \frac{1}{N^{T(\s)}} \;\s\nn \\
& = 1 + \frac{1}{N} \S_{[2]}+ \frac{1}{N^2} \left(  \S_{[3]}+\S_{[2,2]} \right) + \cO\left(\frac{1}{N^3}\right)  
\end{align*}
\end{frame}





\begin{frame}{Inner product and full non-planar correlation function}

The inner product is given by the leading \emph{planar} two-point function
\begin{equation*}
  \braket{\a'}{\a} \sim \delta_{\a' \in [\a]}
\end{equation*}

The leading term of the (extremal) three-point function
\begin{equation*}
  \bra{n_1,n_2} \left( \frac{1}{N} \S_{[2]} \right) \ket{n} = \frac{nn_1n_2}{N}
\end{equation*}

The first correction to the single-trace 2-p't f'n from the torus
\begin{equation*}
  \bra{n}\left(  \frac{1}{N^2}\left[ \S_{[3]}+\S_{[2,2]}\right]\right) \ket{n} = \frac{n}{N^2}\left[ \binom{n}{3} +\binom{n}{4} \right]
\end{equation*}

What do these numbers mean in a putative worldsheet theory?
\end{frame}



\begin{frame}{Bunching of homotopic propagators}
  The $\S_{[3]}$ term gives propagators on the torus bunched into 3 groups; $\S_{[2,2]}$ gives propagators bunched into 4 groups.
  % remarkably ps2pdf torworld.eps cuts off part of the diagram because
  % it doesn't fit inside A4, so you need to do
  % ps2pdf -sPAPERSIZE=ledger torworld.eps  
  % and then pdfcrop torworld.pdf
  \begin{center}
  \resizebox{!}{3.2cm}{\includegraphics{torworld-crop}}
  \end{center}
In Gopakumar's model, each $\S_C$ gives a different skeleton graph of homotopically-bunched propagators for the relevant genus $g$.
  \vskip.1in
Suggestively, these are Hurwitz numbers counting $n$-branched covers of
$\C P^1$ by surfaces of genus $g$ with three branch points, two
labelled by the operators and the third by the cut-and-join $\S_C$.

\end{frame}



\begin{frame}{Two-dimensional factorisation of correlation functions}

Another feature is that for large $n$ the higher genus correlation
functions factorise into planar 3-point functions, e.g. for torus
\begin{equation*}
  \frac{1}{N^2} \left(  \S_{[3]}+\S_{[2,2]} \right)  \to \frac{1}{2} \left( \frac{1}{N} \S_{[2]} \right)^2
\end{equation*}

  \vskip.05in
  % remarkably ps2pdf torus.eps cuts off part of the diagram because
  % it doesn't fit inside A4, so you need to do
  % ps2pdf -sPAPERSIZE=ledger torus.eps  
  % and then pdfcrop torus.pdf
  \resizebox{!}{1.7cm}{\includegraphics{torus-crop}}
  \vskip.1in

This is the result of the exponentiation of the tree-level mixer
\begin{align*}
  \O_n &  = \exp\left(\frac{1}{N} \S_{[2]} - \frac{1}{2N^2}\left[\binom{n}{2} + \S_{[3]}\right] + \cO \left(\frac{1}{N^3}\right) \right) \nn \\
& \to \exp\left(\frac{1}{N} \S_{[2]} \right)
\end{align*}
NB: additional terms subleading in $\frac{n^2}{N}$.


\end{frame}


\begin{frame}{Multiple fields: a few simple examples I}



  Tracing the same field content for $U(2)\subset SU(4)_R$ rep $\Lambda =\tyng(2,2)$ we sometimes have to `twist' the trace to get a non-vanishing operator
  \vskip.2in
  \includegraphics{22trace-crop}
  \vskip.2in
  where $\Phi^p\Phi_p = \epsilon^{pq}\Phi_p \Phi_q = [X,Y]$.

\end{frame}



\begin{frame}{Multiple fields: a few simple examples II}
  
  Things also get complicated when for a given representation and
  trace structure there is \alert{more than one} operator, e.g. for the $U(2)$
  rep $\tyng(4,2) \sim [X,Y][X,Y]\,XX$ with trace structure $\alert{[4,2]}$
  \vskip.2in
  \begin{equation*}
    \tr([X,Y][X,Y])\tr(XX)
  \end{equation*}
  \begin{equation*}
    \tr(XX\Phi^r\Phi^s)\tr(\Phi_r\Phi_s)
  \end{equation*}
  \vskip.2in
  (remembering that $\Phi^p\Phi_p = \epsilon^{pq}\Phi_p \Phi_q = [X,Y]$).

\end{frame}




\begin{frame}{Solution for multiple fields}
For $U(2)$
sector organise $n$ copies of fields  $\{X,Y\}$ into reps
\begin{equation*}
  V_{\bf 2}^{\otimes n} = \bigoplus_{|\L| = n}  V_{\L}^{U(2)} \otimes V_{\L}^{S_n}
\end{equation*}

Can then write all multitrace operators as
\begin{equation*}
  \ket{\L,M;\a,\g} \equiv\frac{1}{n!} \sum_{\s \in S_n} S^{\L,\a}_{a\g}\;B_{ b\b}^{\L,\vec{\m}}\; D^{\L}_{ab}(\s) \;\tr(\s^{-1}\a\s\; \overbrace{X \cdots X}^{\m_1}\; \overbrace{Y  \cdots Y}^{\m_2})
\end{equation*}
\begin{itemize}
\item $\alert{\L}$ tells us the rep. of $U(2)$ (a two-row $n$-box Young diagram)
\item $\alert{M}$ tells us the state within that rep.
\item $\alert{\a}$ is a partition of $n$ giving the trace structure
\item $\alert{\g}$ labels the multiplicity for this $\L$ and $\a$;  no. of values is
  \begin{equation*}
     \frac{1}{|\Sym(\a)|} \sum_{\rho \in \Sym(\a)} \chi_\L(\rho)
  \end{equation*}
\end{itemize}

\end{frame}

\begin{frame}{Example operators}

\begin{equation*}
  \ket{\L =  \tyng(2,2), M = HWS; \a = [4], \g=1}  = \tr([X,Y][X,Y])
\end{equation*}

\begin{equation*}
  \ket{\L = \tyng(2,2), M = HWS; \a = [2,2], \g=1} = \tr(\Phi^r\Phi^s)
  \tr(\Phi_r\Phi_s)
\end{equation*}

\vskip.1in
\begin{align*}
  \ket{\tyng(4,2),HWS;[4,2],1} & = \tr([X,Y][X,Y])\tr(XX) \nn \\
& \nn \\
\ket{\tyng(4,2),HWS;[4,2],2} 
 & =
    \tr(XX\Phi^r\Phi^s)\tr(\Phi_r\Phi_s) \nn \\
& \quad\quad \quad+ \frac{1}{6}
    \tr([X,Y][X,Y])\tr(XX)
\end{align*}

\end{frame}


\begin{frame}{Inner product and non-planar 2-point function}

The inner product (i.e. \emph{planar} two-point function) is diagonal
\begin{equation*}
  \braket{\L',M';\a',\g'}{\L,M;\a,\g} \propto \delta^{\L\L'}\delta^{MM'} \delta^{\a\a'}\delta^{\g\g'}
\end{equation*}
\vskip.2in

As for the half-BPS sector, the cut-and-join operators give the full non-planar free two-point function
\begin{align*}
&\corr{\cO^\dagger[\L',M';\a',\g']\;\;\cO[\L,M;\a,\g]}_{\textrm{non-planar}}  \nn \\
&\hskip 2.3cm = \delta^{\L\L'}\delta^{MM'}\;N^n \,\bra{\L,M;\a',\g'}\, \O_n \,\ket{\L,M;\a,\g} 
\end{align*}

\end{frame}


\begin{frame}{From $U(2)$ to $PSU(2,2|4)$}

  This works automatically for $U(2) \to U(K_1|K_2)$.
  To extend these results for the \alert{free} theory to the other
  fields of $\cN=4$ SYM treat the infinite-dimensional singleton
  rep. of $PSU(2,2|4)$ as the fundamental of $U(\infty|\infty)$.  (The $\L$ are now unrestricted $S_n$ reps, also known as the higher spin YT-pletons.)
\vskip.2in However as soon as we turn on the coupling the $PSU(2,2|4)$
group structure asserts itself.  Each rep $\L$ breaks down into an
infinite number of $PSU(2,2|4)$ reps.  This decomposition is tricky
and not known in general.  Using the technology of Schur-Weyl duality
we \emph{can} do this for e.g. $SO(6)$ and $SO(2,4)$.


\end{frame}

\begin{frame}{One-loop}

 Analyse mixing with one-loop \alert{dilatation operator}, e.g. $U(2)$
 sector
  \begin{equation*}
    :\tr( [X,Y][\tfrac{\partial}{\partial X},\tfrac{\partial}{\partial Y}]) :
  \end{equation*}

\vskip.1in Operators with \alert{anomalous} dimensions have
commutators $[X,Y]$ within a trace. Label them $\ket{\L,M;\a^a,\g^a}$,
e.g.
\begin{equation*}
  \ket{\tyng(2,2),HWS;[4]^a,1^a} = \tr([X,Y][X,Y])
\end{equation*}
\begin{equation*}
  \ket{\tyng(4,2),HWS;[4,2]^a,1^a} = \tr([X,Y][X,Y])\tr(XX)
\end{equation*}


\end{frame}


\begin{frame}{How do we find the quarter-BPS operators?}

  On general grounds the protected BPS operators must be orthogonal to those operators with anomalous dimensions in the full non-planar two-point function.  So choose $\a^q, \g^q$ such that

  \begin{equation*}
    \braket{\L,M;\a^a,\g^a}{\L,M;\a^q,\g^q} = 0 \quad \quad \forall a,q
  \end{equation*}

  The $\frac{1}{4}$-BPS ops. are defined with the \alert{inverse} of the tree-level mixer

  \begin{equation*}
   \boxed{ \tfrac{1}{4}\mbox{-BPS } = \O_n^{-1}\;\ket{\L,M;\a^q,\g^q}}
  \end{equation*}

  \begin{equation*}
    \mbox{for } \quad \O_n^{-1} = 1 -\tfrac{1}{N} \S_{[2]} +
    \tfrac{1}{N^2} \left[\tfrac{n(n-1)}{2} + 2\S_{[3]} + \S_{[2,2]} \right]
 + \cO\left(\tfrac{1}{N^3}\right)
  \end{equation*}

\end{frame}


\begin{frame}{Quarter-BPS examples}

  \scriptsize
  \begin{align*}
\O_n^{-1}\;\ket{\tyng(2,2),HWS;[2,2]^q,1^q} & =
    \tr(\Phi^r\Phi^s) \tr(\Phi_r\Phi_s) \nn \\&\hskip0.3cm+ \frac{2}{N}
    \tr([X,Y][X,Y])\hskip1.2cm\nn \\& \hskip0.3cm -\frac{2}{N^2} \tr(\Phi^r\Phi^s)
    \tr(\Phi_r)\tr(\Phi_s)
  \end{align*}

  \vskip.1in
  \begin{align*}
& \O_n^{-1}\;\ket{\tyng(4,2),HWS;[4,2]^q,2^q} \nn \\
 &\hskip.5cm =
    \tr(XX\Phi^r\Phi^s)\tr(\Phi_r\Phi_s) + \frac{1}{6}
    \tr([X,Y][X,Y])\tr(XX)\nn \\
 &\hskip.8cm  + 
    \frac{8}{3N}\tr(\Phi^r\Phi_r \Phi^s \Phi_sXX) -\frac{16}{3N}
    \tr(\Phi^r \Phi^s\Phi_r \Phi_sXX) \nn \\
&\hskip.8cm- \frac{4}{3N} \tr(\Phi^r \Phi^s)\tr(\Phi_r \Phi_s)\tr(XX) \nn \\
&\hskip.8cm - \frac{1}{N} \tr(\Phi^r \Phi^sXX)\tr(\Phi_r)\tr( \Phi_s) - \frac{1}{6N} \tr(\Phi^r\Phi_r \Phi^s \Phi_s)\tr(X)\tr(X) \nn \\
&\hskip.8cm -\frac{4}{N} \tr(\Phi^r \Phi^sX)\tr(\Phi_r \Phi_s)\tr(X)       +\frac{2}{N} \tr(\Phi^r \Phi^sX)\tr(\Phi_r X)\tr(\Phi_s)      +\cO\left(\frac{1}{N^2}\right) 
  \end{align*}
  \normalsize

\end{frame}
\begin{frame}{Conclusions}

  \begin{itemize}
  \item Full non-planar free theory has a universal structure given by
    cut-and-join operators, with many stringy features.
    \vskip.1in
    \begin{itemize}
    \item Can we turn this into a concrete description of the dual string?
    \end{itemize}

    \vskip.4in
  \item Some features also appear in the weak coupling regime, at
    least in identifying the quarter-BPS operators.
    \vskip.1in
    \begin{itemize}
    \item Does any of this apply to ops with anomalous dimensions?
    \end{itemize}



  \end{itemize}

\end{frame}



\end{document}
