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lagrangian submanifold in nLab

Contents

Context

Symplectic geometry

symplectic geometry

higher symplectic geometry

Background

Basic concepts

Classical mechanics and quantization

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=ℝ 2nX = \mathbb{R}^{2n} equipped with its canonical symplectic form ω=∑ i=1 ndq i∧dp i\omega = \sum_{i = 1}^n d q_i \wedge d p^i, standard Lagrangian submanifolds are the submanifolds ℝ n↪ℝ 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

Examples in higher differential geometry

We discuss classes of examples of Lagrangian dg-submanifolds, remark , 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

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

ω=dx i∧dp 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 ℝ n\mathbb{R}^n and where the {p i}\{p_i\} are the canonical coordinates on T x *ℝ n≃ℝ 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↪ℝ 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 S↪XS \hookrightarrow X. The inclusion N *S↪T 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.

Of a Courant Lie 2-algebroid

A Courant Lie algebroid ℭ\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 (ℭ,ω)(\mathfrak{C}, \omega) is also called a Λ\Lambda-structure. (Ševera, section 4).

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

Proposition

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

(Ševera, section 4)

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).