# nLab Lagrangian correspondence

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Idea

A Lagrangian correspondence is a correspondence between two symplectic manifolds $\left({X}_{i},{\omega }_{i}\right)$ given by a Lagrangian submanifold of their product $\left({X}_{1}×{X}_{2},{p}_{1}^{*}{\omega }_{1}-{p}_{2}^{*}{\omega }_{2}\right)$. 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 $\left(X,\omega \right)$ encodes the phase space of a mechanical system and a symplectomorphism

$\varphi \phantom{\rule{thickmathspace}{0ex}}:\phantom{\rule{thickmathspace}{0ex}}\left({X}_{1},{\omega }_{1}\right)\to \left({X}_{2},{\omega }_{2}\right)$\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).

## Definition

###### Definition

For $\left({X}_{j},{\omega }_{j}\right)$ two symplectic manifolds, a Lagrangian correspondence is a correspondence $Z\to {X}_{0}^{-}×{X}_{1}$ which is a submanifold of ${X}_{0}^{-}×{X}_{1}$

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

with $\mathrm{dim}\left({L}_{0,1}\right)=\frac{1}{2}\left(\mathrm{dim}\left({X}_{0}\right)+\mathrm{dim}\left({X}_{1}\right)\right)$

and

${\iota }^{*}\left(-{\pi }_{0}^{*}{\omega }_{0}+{\pi }_{1}^{*}{\omega }_{1}\right)=0\phantom{\rule{thinmathspace}{0ex}},$\iota^*(-\pi_0^* \omega_0 + \pi_1^* \omega_1) = 0 \,,

where ${\pi }_{i}$ are the two projections out of the product.

###### Definition

The composition of two Lagrangian correspondences is

${L}_{01}\circ {L}_{12}:={\pi }_{02}\left({L}_{01}{×}_{{X}_{1}}{L}_{12}\right)$L_{01} \circ L_{12} := \pi_{02}(L_{01} \times_{X_1} L_{12})

which is itself a Lagrangian correspondence in ${X}_{0}^{-}×{X}_{2}$ if everything is suitably smoothly embedded by ${\pi }_{02}$.

###### Remark

The category of Lagrangian correspondences is a full subcategory of that of correspondence of the slice topos ${\mathrm{SmoothSpaces}}_{/{\Omega }_{\mathrm{cl}}^{2}}$ of smooth spaces over the moduli space ${\Omega }_{\mathrm{cl}}^{2}$ of closed differential 2-forms:

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

$\begin{array}{ccc}& & Z\\ & {}^{{i}_{1}}↙& & {↘}^{{i}_{2}}\\ {X}_{1}& & {i}_{1}^{*}{\omega }_{1}={i}_{2}^{*}{\omega }_{2}& & {X}_{2}\\ & {}_{{\omega }_{1}}↘& & {↙}_{{\omega }_{2}}\\ & & {\Omega }_{\mathrm{cl}}^{2}\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\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 $\left({i}_{1},{i}_{2}\right):Z\to X×Y$ 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.

## Examples

###### Example

For $\varphi :\left({X}_{0},{\omega }_{0}\right)\to \left({X}_{1},{\omega }_{1}\right)$ a symplectomorphism we have that its graph $\mathrm{graph}\left(\varphi \right)\subset {X}_{0}^{-}×{X}_{1}$ is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.

###### Example

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

###### Example

Let $X$ be a manifold, $G=U\left(n\right)$ the unitary group, $P\to X$ a $G$-principal bundle and $D\to X$ a $U\left(1\right)$-bundle with connection.

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

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

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

such that ${\prod }_{j=1}^{g}{A}_{j}{B}_{j}{A}_{j}^{-1}{B}_{j}^{-1}=\mathrm{diag}\left({e}^{2\pi id/}\right)/G$

Let ${Y}_{01}$ be a cobordism from ${X}_{0}$ to ${X}_{1}$ with extension

$L\left({Y}_{01}\right)=\mathrm{Image}\left(M\left({Y}_{01}\right)\stackrel{\mathrm{restr}.}{\to }M\left({X}_{0}{\right)}^{-}×M\left({X}_{1}\right)\right)$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

$L\left({Y}_{01}\circ {Y}_{12}\right)=L\left({Y}_{01}\right)\circ L\left({Y}_{12}\right)\phantom{\rule{thinmathspace}{0ex}}.$L(Y_{01} \circ Y_{12}) = L(Y_{01}) \circ L(Y_{12}) \,.
###### Example

Given symplectic manifolds $\left({X}_{1},{\omega }_{1}\right)$ and $\left({X}_{2},{\omega }_{2}\right)$ and a symplectomorphism $\left({X}_{1}×{X}_{2},{p}_{1}^{*}{\omega }_{1}-{p}_{2}^{*}{\omega }_{2}\right)\stackrel{\simeq }{⟶}\left(X,\omega \right)$, then certain Lagrangian correspondences between $\left({X}_{1},{\omega }_{1}\right)$ and $\left({X}_{2},{\omega }_{2}\right)$ 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)

## References

The notion originates somewhere around

• 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

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

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