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 $X = \mathbb{R}^{2n}$ equipped with its canonical symplectic form $\omega = \sum_{i = 1}^n d q_i \wedge d p^i$, standard Lagrangian submanifolds are the submanifolds $\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n}$ of fixed values of the $\{q_i\}_{i = 1}^n$ coordinates. Indeed locally, every Lagrangian submanifold looks like this.
Lagrangian submanifolds are of central importance in symplectic geometry where they constitute leaves of real polarizations and are closely related to quantum states:
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.
A (Lagrangean or) lagrangian submanifold of a symplectic manifold $(X,\omega)$ is a submanifold $L \hookrightarrow X$ such that the following equivalent conditions hold
at each point $\ell \in L the$tangent space $T_\ell L \hookrightarrow T_\ell X$ is a Lagrangian subspace (hence a simultanously isotropic subspace and cosisotropic subspace?) of $T_\ell X$ equiiped with the symplectic form $\omega_x$;
$L \hookrightarrow X$ is a maximal isotropic submanifold.
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 $\Lambda$-structures in (Ševera, section 4).
We discuss classes of examples of Lagrangian dg-submanifolds, remark 1, of symplectic Lie n-algebroids.
A Poisson Lie algebroid $\mathfrak{P}$ is a symplectic Lie n-algebroid for $n = 1$. Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form $\omega$. One can then say
A dg-Lagrangian submanifold of $(\mathfrak{P}, \omega)$ is a Lagrangian dg-submanifold, also called a $\Lambda$-structure. (Ševera, section 4).
A foliation by such leaves is a Lagrangian foliation of a Lie algebroid.
For $(X, \pi)$ the Poisson manifold underlying a Poisson Lie algebroid $(\mathfrak{P}, \omega)$, a dg-Lagrangian submanifold of $(\mathfrak{P}, \omega)$ corresponds to a coisotropic submanifold of $(X, \pi)$.
As a vector bundle with bracket structure, the Poisson Lie algebroid $\mathfrak{P}$ is
where the horizontal morphism is given by contraction/pairing with the Poisson tensor.
It is sufficient to consider this locally over a coordinate chart and hence we set without essential restriction of generality $X = \mathbb{R}^n$ with the invariant polynomial/graded symplectic form on $CE(\mathfrak{P})$ being
where the $\{q_i\}_{i = 1}^n$ are the canonical coordinates on $\mathbb{R}^n$ and where the $\{p_i\}$ are the canonical coordinates on $T^*_x \mathbb{R}^n \simeq \mathbb{R}^n$, regarded as being in degree 1.
Consider then a sub-Lie algebroid of $\mathfrak{P}$ over a submanifold $S \hookrightarrow \mathbb{R}^n$. That the corresponding subbundle
over $S$ is Lagrangian with respect to the above $\omega$ means that $E$ consists of precisely those cotangent vectors to $X$ which vanish when evaluated on tangent vectors of $S$. Hence
is the conormal bundle to $S \hookrightarrow X$. The inclusion $N^* S \hookrightarrow T^*_S X$ of vector bundles is an inclusion of Lie algebroids over $S$ precisely if the anchor map restricts to an anchor on $S$, hence that contraction with the Poisson tensor restricted to conormal vectors of $S$ lands in tangent vectors of $S$:
This is the standard definition for what it means for $S$ to be a coisotropic submanifold.
The dg-Lagrangian submanifolds also correspond to branes in the Poisson sigma-model (see there) on $(\mathfrak{P}, \omega)$.
A Courant Lie algebroid $\mathfrak{C}$ is a symplectic Lie n-algebroid for $n = 2$. Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form $\omega$. One can then say
A dg-Lagrangian submanifold of $(\mathfrak{C}, \omega)$ is also called a $\Lambda$-structure. (Ševera, section 4).
Hence we might say real polarization of $(\mathfrak{C}, \omega)$ is a foliation by dg-Lagrangian submanifolds.
The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid $(\mathfrak{C}, \omega)$ correspond to Dirac structures on $(\mathfrak{C}, \omega)$.
type of subspace $W$ of inner product space | condition on orthogonal space $W^\perp$ | |
---|---|---|
isotropic subspace | $W \subset W^\perp$ | |
coisotropic subspace | $W^\perp \subset W$ | |
Lagrangian subspace | $W = W^\perp$ | (for symplectic form) |
symplectic space | $W \cap W^\perp = \{0\}$ | (for symplectic form) |
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(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 “$\Lambda$-structures” in