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
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 $X^-_0 \times X_1$
with $dim(L_{0,1}) = \frac{1}{2}(dim(X_0) + dim(X_1))$
and
where $\pi_i$ are the two projections out of the product.
The composition of two Lagrangian correspondences is
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
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
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
is a Lagrangian correspondence if $Y_{01}$ is sufficiently simple. Further assuming this we have for composition that
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)
The notion originates somewhere around
The use of Lagrangian correspondences for encoding symplectomorphisms was further highlighted in
and on p. 5 and then section 3
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:
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:
Last revised on December 1, 2023 at 08:16:30. See the history of this page for a list of all contributions to it.