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\title{Complex Matrix Model Duality}
\author{T.W. Brown}


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

\institute[DESY]
{
  DESY, Hamburg, Germany
}

\date[Johannesburg April 2010]
{DESY Theory Workshop\newline September 2010}



\begin{document}

\begin{frame}
\titlepage

\begin{center}
  based on \href{http://www.arxiv.org/abs/1009.0674}{1009.0674} [hep-th] and forthcoming by Gopakumar
\end{center}

\end{frame}





%\begin{frame}{From Feynman graphs to closed string correlation functions}

%\end{frame}



\begin{frame}{Moduli space of punctured Riemann surfaces from graphs}

Theorem thanks to Mumford, Strebel, Harer and Penner in 80's:
\vspace{0.2cm} 

\alert{Moduli space} of genus $g$ Reimann surfaces with $s$ punctures
decorated by boundary lengths $\cM_{g,s} \times \R_+^s$ has a
\alert{cell decomposition} given by inequivalent ribbon graphs with
\alert{$s$ faces} and lengths assigned to each edge.
\vspace{0.2cm}

Top-dim cells $\dim_\R = 6g-6+3s$ come from edge-lengths of graphs
with 3-valent vertices; lower-dim cells from collapsing edges to get
higher-valency vertices.
\vspace{0.2cm}

E.g. for the thrice-punctured sphere $\cM_{0,3}\times \R_+^3$ have graphs with 3 faces:

  % 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
  % Actually for some reason this didn't work.  I ended up printing
  % the file.eps to pdf with evince and then doing pdfcrop
  \begin{center}
  \resizebox{!}{2.2cm}{\includegraphics{cell_decomp-crop}}
  \end{center}




\end{frame}



\begin{frame}{Including closed string operators: the Kontsevich model}

Generating function
of correlation functions in 2d topological
gravity given by Kontsevich matrix model
  \begin{equation*}
 \exp \sum_{g=0}^\infty \corr{\exp \sum_k t_k \cO_k}_g =  \rho(\L)^{-1} \int [dM]^H_{n\times n}\;\; e^{-\tfrac{1}{2}\tr(\L M^2) + \tfrac{i}{6}\tr(M^3)}
  \end{equation*}
  The coupling coefficients are encoded in the constant matrix $\L$
  \begin{equation*}
    t_k = -\sum_{i=1}^n \frac{1}{k\l_i^k}=-\frac{1}{k}\tr(\L^{-k})
  \end{equation*}

  The colour index for each \alert{face} gets associated to eigenvalues $\l_i$
  and hence to the couplings $t_k$.  The propagator can be transformed
  into an integral over the corresponding edge length $p$ using the
  Schwinger trick
  \vspace{0.1cm}
\begin{equation*}
 \corr{M^i_j\; M^k_l}\;\; =\;\; \delta^i_l \delta^k_j \frac{2}{\l_i + \l_j}\;\; =\;\; 2\delta^i_l \delta^k_j\;\int_0^\infty dp\; e^{-p\,(\l_i+\l_j)}
\end{equation*}



\end{frame}


\begin{frame}{The $\C$ $Z$ matrix model}

The $Z$ model for an $N\times N$ $\C$ matrix ($N$ large but finite) is
\begin{equation*}
  \cZ(\{t\},\{{\ov t}\}) 
 =\int [dZ]^{\C}_{N\times N}\;\; e^{-\tr(ZZ^\dagger) + \sum_{k=1}^\infty t_k \tr(Z^k) +\sum_{k=1}^\infty{\ov t}_k\tr( Z^{\dagger k})}
\end{equation*}
It generates the full genus expansions of
\begin{itemize}
\item Extremal correlation functions of certain half-BPS local
  operators for free $4d, \cN=4$ super Yang-Mills with gauge group
  $U(N)$.
\item Scattering of integer-momentum tachyons in $c=1$ string at
  self-dual radius, cosmological constant $\m = iN$. The map to
  tachyons is $\cT_k \to \tr(Z^k)$ and $\cT_{-k} \to \tr(Z^{\dagger
    k})$.
\end{itemize}

\vspace{0.5cm}
The individual correlation functions are
\begin{equation*}
 \corr{ \tr(Z^{k_1}) \cdots \tr(Z^{k_p})\;\; \tr(Z^{\dagger{\ov k}_1}) \cdots \tr(Z^{\dagger{\ov k}_q}) } \label{eq:tachycorr}
\end{equation*}


How do we rewrite this model so that closed string insertions associate with \alert{faces} rather than \alert{vertices}?

\end{frame}





\begin{frame}{The dual $\C$ $F$ matrix model}


The dual $F$ model is the \alert{same} function of the couplings $\{t\},\{{\ov t}\}$
\begin{align*}
\cZ(\{t\},\{{\ov t}\}) =  \int [dF]^\C_{n\times n}\;\;e^{-\tr(AFBF^\dagger) +N \sum_{k=1}^\infty \frac{1}{k} \tr\left[\left(FF^\dagger \right)^{k} \right]} 
\end{align*}
The couplings $\{t\}$ and $\{{\ov t}\}$ are encoded in constant matrices $A,B$
\begin{equation*}
   t_k = \sum_{i=1}^n \frac{1}{ka_i^k} = \frac{1}{k} \tr A^{-k} \hspace{1cm} {\ov t}_k = \sum_{j=1}^n \frac{1}{kb_j^k} = \frac{1}{k} \tr B^{-k}
\end{equation*}

The colour index for each face of the $F$ model Feynman diagrams
comes with either $a_i$'s or $b_j$'s, so the couplings $\{t\}$ and
$\{{\ov t}\}$ are associated to \alert{faces} (dual to vertices of $Z$ model)
\begin{equation*}
  \corr{F^i_j \; F^\dagger{}^k_l } = \frac{\delta^i_l \delta^k_j}{(a_i b_j-N)} 
\end{equation*}



\end{frame}

\begin{frame}

\begin{figure}[h]
\begin{center}
\begin{picture}(150,150)
\put(-70,160){$\int [dZ]\;e^{-\tr(ZZ^\dagger)}\frac{1}{\det( A \otimes \bI_N - \bI_n \otimes Z)}\frac{1}{\det( B \otimes \bI_N - \bI_n \otimes Z^\dagger)}$}
\put(-70,125){$ = \int [dZ,C,D]\; e^{-\tr\left[ZZ^\dagger + C^\dagger A C -  \alert{C^\dagger ZC} + D^\dagger BD -\alert{D^\dagger Z^\dagger D }\right]}$}
\put(150,105){\resizebox{!}{2.9cm}{\includegraphics{oomm-inCD-crop}}}
\put(-70,75){$ = \int [dC,D]\; e^{-\tr\left[ C^\dagger A C   + D^\dagger BD -\alert{CC^\dagger DD^\dagger } \right]}$}
\put(150,67){\resizebox{!}{0.7cm}{\includegraphics{oomm-outZ-crop}}}
\put(-70,35){$ = \int [dF,C,D]\; e^{-\tr\left[ FF^\dagger -\alert{D^\dagger F^\dagger C}  -\alert{C^\dagger F D} + C^\dagger A C   + D^\dagger BD \right]}$}

\put(150,25){\resizebox{!}{1.1cm}{\includegraphics{oomm-inF-crop}}}

\put(-70,-20){$=  \int [dF]\;e^{-\tr(AFBF^\dagger) +N \sum_{k=1}^\infty \frac{1}{k}\alert{ \tr\left[\left(FF^\dagger \right)^{k} \right]}}$}

\put(150,-60){\resizebox{!}{2.7cm}{\includegraphics{oomm-outCD-crop}}}

\end{picture}
\end{center}
\end{figure}

\end{frame}



\begin{frame}{Example in action}

  \begin{center}
  \resizebox{!}{8cm}{\includegraphics{oomm-graphd-crop}}
  \end{center}


\end{frame}



\begin{frame}{Calculation: two-point function on the torus}
 $Z$ model graph is solid line;  dual $F$ model graph is dashed line
  % 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
  % Actually for some reason this didn't work.  I ended up printing
  % the file.eps to pdf with evince and then doing pdfcrop
  \begin{center}
  \resizebox{!}{3.2cm}{\includegraphics{evince-crop}}
  \end{center}

Each $F$ model graph captures \alert{infinitely many} bunched $Z$ graphs
\begin{align*}
&  \corr{\tr(FF^\dagger FF^\dagger FF^\dagger)}_{\textrm{torus}} +   \corr{\tr(FF^\dagger FF^\dagger)\;\;\tr(FF^\dagger FF^\dagger)}_{\textrm{torus}} \\
& \hspace{3cm} = \sum_{k=3}^\infty t_k {\ov t}_kk \left[\binom{k}{3} + \binom{k}{4}  \right]N^{k-2}  \nn\\
& \hspace{3cm} = \sum_{k=3}^\infty t_k {\ov t}_k  \corr{\tr(Z^k)\;\;\tr(Z^{\dagger k})}_{\textrm{torus}}
\end{align*}

\end{frame}


\begin{frame}{Discrete Schwinger parameterisation of $F$ matrix model}


Following analysis of Chekhov-Makeenko model (the Hermitian version of
the $F$ model), rewrite $a_i = \sqrt{N} e^{\ve l_i}$ and $b_j =
\sqrt{N} e^{\ve m_j}$ with discretisation parameter $\ve$.  The
propagator becomes
\begin{equation*}
  \frac{1}{a_ib_j - N} = \frac{1}{Ne^{\ve l_i}e^{\ve m_j} - N} = \frac{1}{N}\sum_{p=1}^\infty e^{-p\ve( l_i+ m_j)}
\end{equation*}
The sum is a \alert{discrete} Schwinger parameterisation of the edge
length for the propagator.  Each summand comes from an edge of
\alert{integer length} $p\ve$.

\vspace{0.2cm}

Edge lengths correspond to co\"ordinates on (subspaces of) the moduli
space of Riemann surfaces, so $F$ model correlation functions
\alert{localise} on \alert{discrete points} in the moduli space.  For
each dual $Z$ correlation function there are furthermore only a 
\alert{finite number} of points.

\vspace{0.2cm}

[This is no surprise since the correlation functions count very
  particular holomorphic Belyi maps from algebraic Riemann surfaces to
  $\CP^1$, whose Strebel differentials have integer-length critical
  graphs.]


\end{frame}



\begin{frame}{Conclusions}

  \begin{itemize}
  \item Same complex matrix model ($Z$ model) generates tachyon
    scattering for $c=1,R=1$ string and computes correlation functions
    of half-BPS operators in free $d=4,\cN=4$ SYM.  Closed string
    insertions appear as \alert{vertices} of the Feynman diagrams.
  \item There is a \alert{dual} complex matrix model ($F$ model)
    where now closed string insertions are associated to \alert{faces}
    of the dual diagrams.
    \item Using a discrete Schwinger parameterisation of the $F$ model
      propagators, the correlation functions localise on
      \alert{discrete} points in (subspaces of) the moduli space of
      Riemann surfaces.
  \end{itemize}
\vspace{0.1cm}
Questions:
\begin{itemize}
    \item What is the geometric relation between the $c=1,R=1$ string
      and this sector of (small radius) $AdS_5/CFT_4$?
    \item How does the dual worldsheet theory localise on the moduli space?
    \item Can correlation functions of other operators of (free)
      $\cN=4$ SYM be treated this way to gain a microscopic understanding of AdS/CFT?

  \end{itemize}

\end{frame}






\end{document}
