#
nLab

symplectic leaf

### Context

#### Symplectic geometry

**symplectic geometry**

higher symplectic geometry

## Background

geometry

differential geometry

## Basic concepts

almost symplectic structure, metaplectic structure, metalinear structure

symplectic form, n-plectic form

symplectic Lie n-algebroid

symplectic infinity-groupoid

symplectomorphism, symplectomorphism group

Hamiltonian action, moment map

symplectic reduction, BRST-BV formalism

isotropic submanifold, Lagrangian submanifold, polarization

## Classical mechanics and quantization

Hamiltonian mechanics

quantization

deformation quantization,

**geometric quantization**, higher geometric quantization

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# Contents

## Idea

For $(X, \{-,-\})$ a Poisson manifold, a *symplectic leaf* is a maximal submanifold $Y \hookrightarrow X$ on which the Poisson bracket restricts to a symplectic manifold structure.

$X$ is foliated by its symplectic leaves.

## References

Regular foliations by symplectic leafs have originally been found and studied in

- F. Bayen, M. Flato, C. Fronsdal, A. Lichnerovicz & D. Sternheimer,
*Deformation theory and quantization*, Ann. Phys. I l l (1978) 61-151.

A detailed technical review is in the notes

- Jordan Watts,
*An introduction to Poisson manifolds* (2007) (pdf)

Last revised on April 23, 2013 at 20:50:26.
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