nLab
lagrangian submanifold

Contents

Idea

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= 2nX = \mathbb{R}^{2n} equipped with its canonical symplectic form ω= i=1 ndq idp i\omega = \sum_{i = 1}^n d q_i \wedge d p^i, standard Lagrangian submanifolds are the submanifolds n 2n\mathbb{R}^n \hookrightarrow \mathbb{R}^{2n} of fixed values of the {q i} i=1 n\{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.

Definition

Definition

A (Lagrangean or) lagrangian submanifold of a symplectic manifold (X,ω)(X,\omega) is a submanifold LXL \hookrightarrow X such that the following equivalent conditions hold

Remark

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 Λ\Lambda-structures in (Ševera, section 4).

Examples in higher differential geometry

We discuss classes of examples of Lagrangian dg-submanifolds, remark 1, of symplectic Lie n-algebroids.

Of a Poisson Lie algebroid

A Poisson Lie algebroid 𝔓\mathfrak{P} is a symplectic Lie n-algebroid for n=1n = 1. Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form ω\omega. One can then say

Definition

A dg-Lagrangian submanifold of (𝔓,ω)(\mathfrak{P}, \omega) is a Lagrangian dg-submanifold, also called a Λ\Lambda-structure. (Ševera, section 4).

Remark

A foliation by such leaves is a Lagrangian foliation of a Lie algebroid.

Proposition

For (X,π)(X, \pi) the Poisson manifold underlying a Poisson Lie algebroid (𝔓,ω)(\mathfrak{P}, \omega), a dg-Lagrangian submanifold of (𝔓,ω)(\mathfrak{P}, \omega) corresponds to a coisotropic submanifold of (X,π)(X, \pi).

(Ševera, section 4)

Proof

As a vector bundle with bracket structure, the Poisson Lie algebroid 𝔓\mathfrak{P} is

T *X π TX X \array{ T^* X &&\stackrel{\pi}{\to}&& T X \\ & \searrow && \swarrow \\ && X }

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= nX = \mathbb{R}^n with the invariant polynomial/graded symplectic form on CE(𝔓)CE(\mathfrak{P}) being

ω=dx idp i, \omega = \mathbf{d} x^i \wedge \mathbf{d} p_i \,,

where the {q i} i=1 n\{q_i\}_{i = 1}^n are the canonical coordinates on n\mathbb{R}^n and where the {p i}\{p_i\} are the canonical coordinates on T x * n nT^*_x \mathbb{R}^n \simeq \mathbb{R}^n, regarded as being in degree 1.

Consider then a sub-Lie algebroid of 𝔓\mathfrak{P} over a submanifold S nS \hookrightarrow \mathbb{R}^n. That the corresponding subbundle

E T *X S X \array{ E &\hookrightarrow& T^* X \\ \downarrow && \downarrow \\ S &\hookrightarrow & X }

over SS is Lagrangian with respect to the above ω\omega means that EE consists of precisely those cotangent vectors to XX which vanish when evaluated on tangent vectors of SS. Hence

E=N *S E = N^* S

is the conormal bundle to SXS \hookrightarrow X. The inclusion N *ST S *XN^* S \hookrightarrow T^*_S X of vector bundles is an inclusion of Lie algebroids over SS precisely if the anchor map restricts to an anchor on SS, hence that contraction with the Poisson tensor restricted to conormal vectors of SS lands in tangent vectors of SS:

π(N *S)TS. \pi(N^* S) \subset T S \,.

This is the standard definition for what it means for SS to be a coisotropic submanifold.

Remark

The dg-Lagrangian submanifolds also correspond to branes in the Poisson sigma-model (see there) on (𝔓,ω)(\mathfrak{P}, \omega).

Of a Courant Lie 2-algebroid

A Courant Lie algebroid \mathfrak{C} is a symplectic Lie n-algebroid for n=2n = 2. Regarding its Chevalley-Eilenberg algebra as the algebra of functions on a dg-manifold, that dg-manifold carries a graded symplectic form ω\omega. One can then say

Definition

A dg-Lagrangian submanifold of (,ω)(\mathfrak{C}, \omega) is also called a Λ\Lambda-structure. (Ševera, section 4).

Hence we might say real polarization of (,ω)(\mathfrak{C}, \omega) is a foliation by dg-Lagrangian submanifolds.

Proposition

The dg-Lagrangian submanifolds of a Courant Lie 2-algebroid (,ω)(\mathfrak{C}, \omega) correspond to Dirac structures on (,ω)(\mathfrak{C}, \omega).

(Ševera, section 4)

type of subspace WW of inner product spacecondition on orthogonal space W W^\perp
isotropic subspaceWW W \subset W^\perp
coisotropic subspaceW WW^\perp \subset W
Lagrangian subspaceW=W W = W^\perp(for symplectic form)
symplectic spaceWW ={0}W \cap W^\perp = \{0\}(for symplectic form)

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(adapted from Ševera 00)

References

The concept of lagrangian submanifold has been defined/named in

  • Victor Maslov, Perturbation Theory and Asymptotic Methods (MSU Publ., Moscow, 1965; English translation: Mir, Moscow, 1965).

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 ”Λ\Lambda-structures” in

Revised on November 10, 2013 11:41:23 by Urs Schreiber (89.204.135.42)