nLab Lagrangian correspondence

Idea

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 Lagrangian 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).

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 $X^-_0 \times X_1$

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

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.

Definition

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 $\pi_{02}$.

Remark

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.

Examples

Example

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.

Example

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

Example

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

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

Example

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. (Weinstein 83, p. 5)

Related entries

• symplectic dual pair

• Weinstein symplectic category

• prequantized Lagrangian correspondence

• Lagrangian correspondences and category-valued TFT

• cosmic Galois group

References

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.

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:

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

More on the construction of symplectic categories with Lagrangian correspondences as morphisms:

• Katrin Wehrheim, Chris T. Woodward, Functoriality for Lagrangian correspondences in Floer theory, Quantum Topology 1 2 (2010) 129-170 [doi:10.4171/qt/4, arXiv:0708.2851]

and on its (∞,1)-category-theoretic version:

• Damien Calaque, Three lectures on derived symplectic geometry and topological field theories, Indagationes Mathematicae 25 5 (2014) 926–947 [doi:10.1016/j.indag.2014.07.005]

• Rune Haugseng, Iterated spans and classical topological field theories, Mathematische Zeitschrift 289 3 (2018) 1427–1488 [arXiv, doi:10.1007/s00209-017-2005-x]

See also:

• Katrin Wehrheim, Chris Woodward, Floer Cohomology and Geometric Composition of Lagrangian Correspondences [arXiv:0905.1368]