nLab Lagrangian correspondence




A Lagrangian correspondence is a correspondence between two symplectic manifolds (X i,ω i)(X_i,\omega_i) given by a Lagrangian submanifold of their product (X 1×X 2,p 1 *ω 1p 2 *ω 2)(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,ω)(X,\omega) encodes the phase space of a mechanical system and a symplectomorphism

ϕ:(X 1,ω 1)(X 2,ω 2) \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,ω j)(X_j, \omega_j) two symplectic manifolds, a Lagrangian correspondence is a correspondence ZX 0 ×X 1Z \to X^-_0 \times X_1 which is a submanifold of X 0 ×X 1X^-_0 \times X_1

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

with dim(L 0,1)=12(dim(X 0)+dim(X 1))dim(L_{0,1}) = \frac{1}{2}(dim(X_0) + dim(X_1))


ι *(π 0 *ω 0+π 1 *ω 1)=0, \iota^*(-\pi_0^* \omega_0 + \pi_1^* \omega_1) = 0 \,,

where π i\pi_i are the two projections out of the product.


The composition of two Lagrangian correspondences is

L 01L 12:=π 02(L 01× X 1L 12) L_{01} \circ L_{12} := \pi_{02}(L_{01} \times_{X_1} L_{12})

which is itself a Lagrangian correspondence in X 0 ×X 2X^-_0 \times X_2 if everything is suitably smoothly embedded by π 02\pi_{02}.


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

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

Z i 1 i 2 X 1 i 1 *ω 1=i 2 *ω 2 X 2 ω 1 ω 2 Ω cl 2. \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):ZX×Y(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 ϕ:(X 0,ω 0)(X 1,ω 1)\phi : (X_0, \omega_0) \to (X_1, \omega_1) a symplectomorphism we have that its graph graph(ϕ)X 0 ×X 1graph(\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,ω)(X,\omega) of a Lie group GG given by a moment map μ\mu, the zero locus μ 1(0)\mu^{-1}(0) consitutes a Lagrangian correspondence between (X,ω)(X,\omega) and its symplectic reduction μ 1(0)/G\mu^{-1}(0)/G.


Let XX be a manifold, G=U(n)G= U(n) the unitary group, PXP \to X a GG-principal bundle and DXD \to X a U(1)U(1)-bundle with connection.

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

For example if XX has genus gg then

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

such that j=1 gA jB jA j 1B j 1=diag(e 2πid/)/G\prod_{j=1}^g A_j B_j A_j^{-1} B_j^{-1} = diag(e^{2\pi i d/})/G

Let Y 01Y_{01} be a cobordism from X 0X_0 to X 1X_1 with extension

L(Y 01)=Image(M(Y 01)restr.M(X 0) ×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 01Y_{01} is sufficiently simple. Further assuming this we have for composition that

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

Given symplectic manifolds (X 1,ω 1)(X_1, \omega_1) and (X 2,ω 2)(X_2, \omega_2) and a symplectomorphism (X 1×X 2,p 1 *ω 1p 2 *ω 2)(X,ω)(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,ω 1)(X_1, \omega_1) and (X 2,ω 2)(X_2, \omega_2) are identified with functions on XX. 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

  • 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:

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

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

See also:

Last revised on December 1, 2023 at 08:16:30. See the history of this page for a list of all contributions to it.