nLab
Lagrangian correspondence

Idea

A Lagrangian correspondence is a correspondence between two symplectic manifolds given by a Lagrangian submanifold of their product.

Definition

Definition

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 $X_0^{-}\times X_1$

$$\iota :L_{0,1}\hookrightarrow X_0^{-}\times X_1$$\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

$${\iota }^{*}(-{\pi }_0^{*}{\omega }_0+{\pi }_1^{*}{\omega }_1)=0,$$\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

$$L_{01}\circ L_{12}:={\pi }_{02}(L_{01}{\times }_{X_1}{L}_{12})$$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 ${\pi }_{02}$.

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 $(X,\omega )$ 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

$$\begin{array}{ccc}& & Z\\ & {}^{i_1}\swarrow & & {\searrow }^{i_2}\\ X_1& & {}^{i_1^{*}{\omega }_1}{\downarrow }^{i_2^{*}{\omega }_2}& & X_2\\ & {}_{{\omega }_1}\searrow & & {\swarrow }_{{\omega }_2}\\ & & {\Omega }_{\mathrm{cl}}^2\end{array}.$$\array{ && Z \\ & {}^{\mathllap{i_1}}\swarrow && \searrow^{\mathrlap{i_2}} \\ X_1 && {}^{\mathllap{i_1^\ast \omega_1}}\downarrow^{\mathrlap{i_2^\ast \omega_2}} && X_2 \\ & {}_{\mathllap{\omega_1}}\searrow && \swarrow_{\mathrlap{\omega_2}} \\ && \Omega^2_{cl} } \,.

If here $(i_1,i_2):Z\to X\times Y$ is a manifold maximal with the property of fitting into the above diagram, then this is a Lagrangian correspondence.

Examples

1. For $\varphi :X_0\to X_1$ a symplectomorphism we have

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

2. 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

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

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

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

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

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

Related entries

• symplectic dual pair

• Weinstein symplectic category

• Lagrangian correspondences and category-valued TFT

• cosmic Galois group