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\begin{document}

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\section*{Lagrangian correspondences and category-valued TFT}

This entry describes classes of examples of [[A-∞ category]]-valued [[FQFTs]] defined on a version of the [[symplectic category]].

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{overview}{Overview}\dotfill \pageref*{overview} \linebreak
\noindent\hyperlink{lagrangian_correspondences}{Lagrangian correspondences}\dotfill \pageref*{lagrangian_correspondences} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{overview}{}\subsection*{{Overview}}\label{overview}

Let $(X.\omega)$ be a [[compact space|compact]] [[symplectic manifold]]. At least in good cases to this is associated a [[Fukaya category]] $Fuk(X)$ of [[Lagrangian submanifold]]s and an enlarged version $Fuk^#(X)$.

Write $X^-$ for the symplectiv manifold $(X,-\omega)$.

Now if $(X_j, \omega_j)$ for $j = 0,1$ are two Lagrangian submanifolds and $L_{01} \subset  X^-_0 \times X_1$ a Lagrangian correspondence then we get an [[A-∞ category|A-∞ functor]] $\phi(L_{01}) : Fuk^#(X_0) \to Fuk^#(X_1)$

\begin{theorem}
\label{}\hypertarget{}{}
\textbf{(Wehrheim, Woodward)}

For $L_{01} \subset X^{-}_0 \times X_1$ and $L_{12} \subset X^{-}_1 \times X_2$ Lagrangian submanifolds, assuming monotonicity and Maslov conditions we have an $A_\infty$-homotopy

\begin{displaymath}
\Phi(L_{01}) \circ \Phi(L_{12}) \simeq \Phi(L_{01} \circ L_{12})
  \,,
\end{displaymath}
where on the right we have a natural notion of composition of Lagrangian submanifolds.

\end{theorem}
(\href{WehrheimWoodward07}{Wehrheim-Woodward 07, section 3.2})

This is the symplectic version of [[Mukai functors]].

\begin{example}
\label{}\hypertarget{}{}
For $X_0$ and $X_1$ a compact [[Riemann surfaces]] and $M(X_0), M(X_1)$

their [[moduli spaces]] of fixed determinant rank $n$-bundles, and for $Y_{01}$ a [[cobordism]] (compact, oriented) from $X_0$ to $X_1$ then consider

\begin{displaymath}
L(Y_{01}) := Image(
      M(Y_{01}) \stackrel{restriction}{\to} M(X_0)^- \times M(X_1)
  )
\end{displaymath}
If $Y_{01}$ is \emph{elementary} in that there exists a [[Morse function]] $Y \to \mathbb{R}$ with $\leq 1$ critical points then $L(Y_{01})$ is a Lagrangian correspondence.

\end{example}
\begin{cor}
\label{}\hypertarget{}{}
The assignment

\begin{displaymath}
Y_{01} \mapsto \Phi(L(Y_{01}))
\end{displaymath}
defines a 2+1-dimensional [[FQFT]] for connected cobordisms with values in [[A-∞ categories]].

\end{cor}
This is supposed to be the 2+1-dimensional part of [[Donaldson theory]].

Other theories that fit into this framework:

\begin{enumerate}%
\item symplectic [[Khovanov theory]] (Seidel-Smith and Rezazodegab)


\item [[Heegard-Floer theory]]

$X$ a surface $\mapsto$ $Fuk^# sym X$

elementary cobordisms $\mapsto$ $\Phi($vanishing cycle$)$



\end{enumerate}
\hypertarget{lagrangian_correspondences}{}\subsection*{{Lagrangian correspondences}}\label{lagrangian_correspondences}

Write $X^-j = (X_j , -\omega_j)$ for a [[symplectic manifold]] with its symplectic form reversed.

\begin{defn}
\label{}\hypertarget{}{}
For $(X_j, \omega_j)$ two [[symplectic manifold]]s, a \textbf{[[Lagrangian correspondence]]} is a Lagrangian submanifold of $X^-_0 \times X_1$, that is

\begin{displaymath}
\iota : L_{0,1} \hookrightarrow X^-_0 \times X_1
\end{displaymath}
with $dim(L_{0,1}) = \frac{1}{2}(dim(x_0) + dim(X_1))$

and

\begin{displaymath}
\iota^*(-\pi_0^* \omega_0 + \pi_1^* \omega_1) = 0
  \,,
\end{displaymath}
where $\pi_i$ are the two projections out of the [[product]].

The \textbf{composition} of two Lagrangian submanifolds is

\begin{displaymath}
L_{01} \circ L_{12} := 
    \pi_{02}(L_{01} \times_{X_1} L_{12})
\end{displaymath}
which is a Lagrangian correspondence in $X^-_0 \times X_2$ if everything is suitably smoothly embedded by $\pi_{02}$.

\end{defn}
\begin{example}
\label{}\hypertarget{}{}
For $\phi : X_0 \to X_1$ a [[symplectomorphism]] we have

$graph(\phi) \subset X_0^- \times X_1$ is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
Let $X$ be a [[manifold]], $G= U(n)$ the [[unitary group]], $P \to X$ a $G$-[[principal bundle]] and $D \to X$ a $U(1)$-bundle with [[connection on a bundle|connection]].

Then there is the [[moduli space]] $M(X) = M(P,D)$ of connections on $P$ with central curvature and given determinant.

For example if $X$ has [[genus]] $g$ then

\begin{displaymath}
M(X) = \{ (A,B, \cdots, A_g, B_g) \in G^{2g}\}
\end{displaymath}
such that $\prod_{j=1}^g A_j B_j A_j^{-1} B_j^{-1} = diag(e^{2\pi i d/})/G$

Let $Y_{01}$ be a [[cobordism]] from $X_0$ to $X_1$ with extension

\begin{displaymath}
L(Y_{01}) = Image(M(Y_{01}) \stackrel{restr.}{\to} 
     M(X_0)^- \times M(X_1) )
\end{displaymath}
is a Lagrangian correspondence if $Y_{01}$ is sufficiently simple. Further assuming this we have for composition that

\begin{displaymath}
L(Y_{01} \circ Y_{12}) = L(Y_{01}) \circ L(Y_{12})
      \,.
\end{displaymath}
\end{example}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Katrin Wehrheim]], [[Chris Woodward]], \emph{Functoriality for Lagrangian correspondences in Floer theory} (\href{http://arxiv.org/abs/0708.2851}{arXiv:0708.2851})

\end{itemize}
\begin{itemize}%
\item [[Katrin Wehrheim]], [[Chris Woodward]], \emph{Floer Cohomology and Geometric Composition of Lagrangian Correspondences} (\href{http://arxiv.org/abs/0905.1368}{arXiv:0905.1368})

\end{itemize}


\end{document}