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\begin{document}

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\section*{Lagrangian correspondence}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{symplectic_geometry}{}\paragraph*{{Symplectic geometry}}\label{symplectic_geometry}

[[!include symplectic geometry - contents]]

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

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{Definition}{Definition}\dotfill \pageref*{Definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_entries}{Related entries}\dotfill \pageref*{related_entries} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{Lagrangian correspondence} is a [[correspondence]] between two [[symplectic manifolds]] $(X_i,\omega_i)$ given by a [[Lagrangian submanifold]] of their [[product]] $(X_1 \times X_2, p_1^\ast \omega_1 - p_2^\ast \omega_2)$. The [[graph of a function|graph]] of any [[symplectomorphism]] induces a Lagragian correspondence.

Lagrangian correspondences are supposed to form, subject to some technicalities, the [[morphisms]] of a [[category]] to be called the \emph{[[symplectic category]]}.

When [[symplectic geometry]] is used to model [[mechanics]] in [[physics]], then a [[symplectic manifold]] $(X,\omega)$ encodes the [[phase space]] of a [[mechanical system]] and a [[symplectomorphism]]

\begin{displaymath}
\phi \;\colon\; (X_1,\omega_1) \to (X_2, \omega_2)
\end{displaymath}
encodes a process undergone by this system, for instance the time evolution induced by a [[Hamiltonian vector field]]. In particular if this is a [[Hamiltonian symplectomorphism]] then this is traditionally called a \emph{[[canonical transformation]]} in physics. Therefore Lagrangian correspondence have also been called \emph{canonical relations} (\hyperlink{Weinstein83}{Weinstein 83, p. 5}).

\hypertarget{Definition}{}\subsection*{{Definition}}\label{Definition}

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

\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]].

\end{defn}
\begin{defn}
\label{}\hypertarget{}{}
The \textbf{composition} of two Lagrangian correspondences is

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

\end{defn}
\begin{remark}
\label{}\hypertarget{}{}
The category of Lagrangian correspondences is a [[full subcategory]] of that of correspondence of the [[slice topos]] $SmoothSpaces_{/\Omega^2_{cl}}$ of [[smooth spaces]] over the [[moduli space]] $\Omega^2_{cl}$ of closed [[differential 2-forms]]:

a [[symplectic manifold]] $(X,\omega)$ is given by a map of [[smooth spaces]] $\omega \colon X \to \Omega^2_{cl}$ (generally this is a [[presymplectic manifold]]) and a correspondence in $SmoothSpaces_{/\Omega^2_{cl}}$ is a [[commuting diagram]] in [[smooth spaces|SmoothSpaces]] of the form

\begin{displaymath}
\itexarray{
    && Z
    \\
    & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}}
    \\
    X_1 && {i_1^\ast \omega_1 = i_2^\ast \omega_2} && X_2
    \\
    & {}_{\mathllap{\omega_1}}\searrow && \swarrow_{\mathrlap{\omega_2}}
    \\
    && \Omega^2_{cl}
  }
  \,.
\end{displaymath}
If here $(i_1, i_2) \colon Z \to X \times Y$ is a manifold maximal with the property of fitting into the above diagram, then this is a Lagrangian correspondence.

From this is naturally induced the notion of a \emph{[[prequantized Lagrangian correspondence]]}. See there for more details.

\end{remark}
\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{example}
\label{}\hypertarget{}{}
For $\phi : (X_0, \omega_0) \to (X_1, \omega_1)$ a [[symplectomorphism]] we have that its [[graph of a function|graph]] $graph(\phi) \subset X_0^- \times X_1$ is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.

\end{example}
\begin{example}
\label{}\hypertarget{}{}
For a [[Hamiltonian action]] on a [[symplectic manifold]] $(X,\omega)$ of a [[Lie group]] $G$ given by a [[moment map]] $\mu$, the [[zero locus]] $\mu^{-1}(0)$ consitutes a Lagrangian correspondence between $(X,\omega)$ and its [[symplectic reduction]] $\mu^{-1}(0)/G$.

\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}
\begin{example}
\label{}\hypertarget{}{}
Given symplectic manifolds $(X_1, \omega_1)$ and $(X_2, \omega_2)$ and a [[symplectomorphism]] $(X_1 \times X_2 , p_1^\ast \omega_1 - p_2^\ast \omega_2) \stackrel{\simeq}{\longrightarrow} (X,\omega)$, then certain Lagrangian correspondences between $(X_1, \omega_1)$ and $(X_2, \omega_2)$ are identified with functions on $X$. This is identified with the calculus of generating functions for [[canonical transformations]] as used in [[classical mechanics]]. (\hyperlink{Weinstein83}{Weinstein 83, p. 5})

\end{example}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[symplectic dual pair]]


\item [[Weinstein symplectic category]]


\item [[prequantized Lagrangian correspondence]]


\item [[Lagrangian correspondences and category-valued TFT]]


\item [[cosmic Galois group]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

The notion originates somewhere around

\begin{itemize}%
\item [[Lars Hörmander]], \emph{Fourier Integral Operators I.}, Acta Math. \textbf{127} (1971)

\end{itemize}
\begin{itemize}%
\item [[Alan Weinstein]], \emph{Symplectic manifolds and their lagrangian submanifolds}, Advances in Math. 6 (1971) 

\end{itemize}
The use of Lagrangian correspondences for encoding [[symplectomorphisms]] was further highlighted in

\begin{itemize}%
\item [[Alan Weinstein]], \emph{The symplectic ``category''}, In Differential geometric methods in mathematical physics (Clausthal, 1980), volume 905 of Lecture Notes in Math., pages 45--51. Springer, Berlin, 1982.

\end{itemize}
and on p. 5 and then section 3

\begin{itemize}%
\item [[Alan Weinstein]], \emph{Lectures on Symplectic Manifolds}, volume 29 of CBMS Regional Conf. Series in Math. Amer. Math. Soc., 1983. third printing.

\end{itemize}
Further developments include for instance

\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}
A review of the way Lagrangian correspondences encode symplectomorphisms induced by Hamiltonian time evolution in the context of [[field theory]] and generalization to the broader context of [[BV-BRST formalism]] is in

\begin{itemize}%
\item [[Alberto Cattaneo]], [[Pavel Mnev]], [[Nicolai Reshetikhin]], \emph{Classical and quantum Lagrangian field theories with boundary} (\href{http://arxiv.org/abs/1207.0239}{arXiv:1207.0239})

\end{itemize}
[[!redirects Lagrangian correspondences]]

[[!redirects canonical relation]] [[!redirects canonical relations]]



\end{document}