A Lagrangian submanifold of a symplectic manifold is a submanifold which is a maximal isotropic submanifold, hence a submanifold on which the symplectic form vanishes, and which is maximal with this property.
In the archetypical example of an even dimensional Cartesian space equipped with its canonical symplectic form , standard Lagrangian submanifolds are the submanifolds of fixed values of the coordinates. Indeed locally, every Lagrangian submanifold looks like this.
If one thinks of a symplectic manifold as a phase space of a physical system, then a Lagrangian submanifold may be thought of (locally) as the space of “all canonical momenta (= parameterization of a leaf) at fixed canonical coordinate (= parameterization of leaf space)”.
A Lagrangian submanifold equipped with a half-density is a model for a state of the physical system in semiclassical approximation (see e.g. Bates-Weinstein, p. 14). A quantum state given by a wave function (see there) is a refinement of this concept.
More generally one can consider Lagrangian submanifolds of symplectic structures in higher geometry, such as symplectic Lie n-algebroids equipped with their canonical invariant polynomials, thought of as dg-manifolds (via their Chevalley-Eilenberg algebra) and equipped with graded symplectic forms. Lagrangian dg-submanifolds of such symplectic dg-manifolds have been called -structures in (Ševera, section 4).
A Poisson Lie algebroid is a symplectic Lie n-algebroid for . Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form . One can then say
where the horizontal morphism is given by contraction/pairing with the Poisson tensor.
where the are the canonical coordinates on and where the are the canonical coordinates on , regarded as being in degree 1.
Consider then a sub-Lie algebroid of over a submanifold . That the corresponding subbundle
is the conormal bundle to . The inclusion of vector bundles is an inclusion of Lie algebroids over precisely if the anchor map restricts to an anchor on , hence that contraction with the Poisson tensor restricted to conormal vectors of lands in tangent vectors of :
This is the standard definition for what it means for to be a coisotropic submanifold.
A Courant Lie algebroid is a symplectic Lie n-algebroid for . Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form . One can then say
Hence we might say real polarization of is a foliation by dg-Lagrangian submanifolds.
The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid correspond to Dirac structures on .
|type of subspace of inner product space||condition on orthogonal space|
|Lagrangian subspace||(for symplectic form)|
|symplectic space||(for symplectic form)|
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||discussed in:|
|0||symplectic manifold||symplectic manifold||symplectic geometry||Lagrangian submanifold||–||ordinary space of states (in geometric quantization)||geometric quantization|
|1||Poisson Lie algebroid||symplectic groupoid||2-plectic geometry||Poisson sigma-model||coisotropic submanifold (of underlying Poisson manifold)||brane of Poisson sigma-model||2-module = category of modules over strict deformation quantiized algebra of observables||extended geometric quantization of 2d Chern-Simons theory|
|2||Courant Lie 2-algebroid||symplectic 2-groupoid||3-plectic geometry||Courant sigma-model||Dirac structure||D-brane in type II geometry|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||AKSZ sigma-model|
(adapted from Ševera 00)
The concept of lagrangian submanifold has been defined/named in
An introduction with an eye towards geometric quantization is for instance in
(pages 10 and onward and then section 4.3).
Lagrangian submanfolds of symplectic dg-manifolds are called “-structures” in