Lagrangian correspondence

A *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 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 *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

$\phi \;\colon\; (X_1,\omega_1) \to (X_2, \omega_2)$

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 *canonical transformation* in physics. Therefore Lagrangian correspondence have also been called *canonical relations* (Weinstein 83, p. 5).

For $(X_j, \omega_j)$ two symplectic manifolds, a **Lagrangian correspondence** is a correspondence $Z \to X^-_0 \times X_1$ which is a submanifold of $X^-_0 \times X_1$

$\iota : L_{0,1} \hookrightarrow X^-_0 \times X_1$

with $dim(L_{0,1}) = \frac{1}{2}(dim(X_0) + dim(X_1))$

and

$\iota^*(-\pi_0^* \omega_0 + \pi_1^* \omega_1) = 0
\,,$

where $\pi_i$ are the two projections out of the product.

The **composition** of two Lagrangian correspondences is

$L_{01} \circ L_{12} :=
\pi_{02}(L_{01} \times_{X_1} L_{12})$

which is itself a Lagrangian correspondence in $X^-_0 \times X_2$ if everything is suitably smoothly embedded by $\pi_{02}$.

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 SmoothSpaces of the form

$\array{
&& Z
\\
& {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}}
\\
X_1 && {i_1^\ast \omega_1 = i_2^\ast \omega_2} && X_2
\\
& {}_{\mathllap{\omega_1}}\searrow && \swarrow_{\mathrlap{\omega_2}}
\\
&& \Omega^2_{cl}
}
\,.$

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 *prequantized Lagrangian correspondence*. See there for more details.

For $\phi : (X_0, \omega_0) \to (X_1, \omega_1)$ a symplectomorphism we have that its graph $graph(\phi) \subset X_0^- \times X_1$ is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.

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 $\mu^{-1}(0)/G$.

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.

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

$M(X) = \{ (A,B, \cdots, A_g, B_g) \in G^{2g}\}$

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

$L(Y_{01}) = Image(M(Y_{01}) \stackrel{restr.}{\to} M(X_0)^- \times M(X_1) )$

is a Lagrangian correspondence if $Y_{01}$ is sufficiently simple. Further assuming this we have for composition that

$L(Y_{01} \circ Y_{12}) = L(Y_{01}) \circ L(Y_{12})
\,.$

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 $(X_2, \omega_2)$ ar identified with functions on $X$. This is identified with the calculus of generating functions for canonical transformations as used in classical mechanics. (Weinstein 83, p. 5)

The notion originates somewhere around

- Lars Hörmander,
*Fourier Integral Operators I.*, Acta Math.**127**(1971)

- Alan Weinstein,
*Symplectic manifolds and their lagrangian submanifolds*, Advances in Math. 6 (1971)

The use of Lagrangian correspondences for encoding symplectomorphisms was further highlighted in

- Alan Weinstein,
*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.

and on p. 5 and then section 3

- Alan Weinstein,
*Lectures on Symplectic Manifolds*, volume 29 of CBMS Regional Conf. Series in Math. Amer. Math. Soc., 1983. third printing.

Further developments include for instance

- Katrin Wehrheim, Chris Woodward,
*Floer Cohomology and Geometric Composition of Lagrangian Correspondences*(arXiv:0905.1368)

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

- Alberto Cattaneo, Pavel Mnev, Nicolai Reshetikhin,
*Classical and quantum Lagrangian field theories with boundary*(arXiv:1207.0239)

Revised on November 11, 2013 06:55:42
by Zoran Škoda
(161.53.130.104)