A Lagrangian correspondence is a correspondence between two symplectic manifolds given by a Lagrangian submanifold of their product . 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 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 two symplectic manifolds, a Lagrangian correspondence is a correspondence which is a submanifold of
where are the two projections out of the product.
The composition of two Lagrangian correspondences is
which is itself a Lagrangian correspondence in if everything is suitably smoothly embedded by .
For a symplectomorphism we have that its graph is a Lagrangian correspondence and composition of syplectomorphisms corresponds to composition of Lagrangian correspondences.
Let be a manifold, the unitary group, a -principal bundle and a -bundle with connection.
Then there is the moduli space of connections on with central curvature and given determinant.
For example if has genus then
Let be a cobordism from to with extension
is a Lagrangian correspondence if is sufficiently simple. Further assuming this we have for composition that
Given symplectic manifolds and and a symplectomorphism , then certain Lagrangian correspondences between and ar identified with functions on . 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.
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