# nLab symplectic manifold

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Definition

###### Definition

A symplectic manifold is

• a smooth manifold $X$ of even dimension $\mathrm{dim}X=2n$;

• equipped with a symplectic form:

• a closed smooth 2-form $\omega \in {\Omega }_{\mathrm{cl}}^{2}\left(X\right)$;

• such that $\omega$ is non-degenerate, which means equivalently that

• ${\omega }^{\wedge n}=\omega \wedge \omega \wedge \cdots \wedge \omega$ has the maximal rank at every point $p\in X$;

• $\left({\wedge }^{2}{T}_{p}^{*}X,{\omega }_{p}\right)$ is a symplectic vector space for every point $p\in X$.

###### Definition

A $2n$-dimensional topological manifold $X$ is

• a real symplectic manifold

• equipped with a symplectic atlas:

• an atlas consisting of smooth charts ${\varphi }_{i}:{U}_{i}\to X$ as usual,

• such that the transition functions ${\varphi }_{j}^{-1}\circ {\varphi }_{i}:{\varphi }_{i}^{-1}\left({\varphi }_{i}\left({U}_{i}\right)\cap {\varphi }_{j}\left({U}_{j}\right)\right)\to {\varphi }_{j}^{-1}\left({\varphi }_{i}\left({U}_{i}\right)\cap {\varphi }_{j}\left({U}_{j}\right)\right)$ preserve the standard symplectic form ${\omega }_{0}={\sum }_{i=1}^{n}{\mathrm{dx}}_{i}\wedge {\mathrm{dp}}_{i}$ on ${ℝ}^{2n}$ with the basis $\left({x}_{1},\dots ,{x}_{n},{p}_{1},\dots ,{p}_{n}\right)$.

###### Remark

The non-degenracy of the symplectic form implies that it defines an isomorphism

$\omega \left(-,-\right):\Gamma \left(TX\right)\to \Gamma \left({T}^{*}X\right)$\omega(-,-) : \Gamma(T X) \to \Gamma(T^* X)

between sections of the tangent bundlevector fields – and sections of the cotangent bundledifferential 1-forms – on $X$ by the map

$\left(v\in {T}_{x}X\right)↦\left(\omega \left(v,-\right)\in {T}_{x}^{*}X\right)\phantom{\rule{thinmathspace}{0ex}}.$(v \in T_x X) \mapsto (\omega(v,-) \in T^*_x X) \,.
###### Definition

The vector fields in the image of the exact 1-forms under the isomorphism, remark 1, are called Hamiltonian vector fields.

This means that for $H\in {C}^{\infty }\left(X\right)$ a smooth function and $dH$ its differential 1-form, the corresponding Hamiltonian vector field ${v}_{H}\in \Gamma \left(TX\right)$ is the unique vector field such that

$dH=\omega \left({v}_{H},-\right)\phantom{\rule{thinmathspace}{0ex}}$d H = \omega(v_H, -) \,

Equivalently, for $\varphi {ℝ}^{2n}\to X$ a coordinate chart of $X$ and ${\varphi }^{*}\omega ={\omega }_{ij}d{x}^{i}\wedge d{x}^{j}$ the symplectic form on this patch, the Hamiltonian vector field ${v}_{H}$ is

${v}_{H}=\frac{\partial H}{\partial {x}^{i}}\left({\omega }^{-1}{\right)}^{ij}{\partial }_{j}\phantom{\rule{thinmathspace}{0ex}}.$v_H = \frac{\partial H}{\partial x^i} (\omega^{-1})^{i j} \partial_j \,.

## Properties

### Darboux coordinates

By Darboux's theorem every symplectic manifold has an atlas by coordinate charts ${ℝ}^{2n}\simeq U↪X$ on which the symplectic form takes the canonical form $\omega {\mid }_{U}={\sum }_{k=1}^{n}d{x}^{2k}\wedge d{x}^{2k+1}$.

### Symplectic and almost symplectic structure

The existence of a 2-form $\omega \in {\Omega }^{2}\left(X\right)$ which is non-degenerate (but not necessarily closed) is equivalent to the existence of a Sp-structure on $X$, a reduction of the structure group of the tangent bundle along the inclusion of the symplectic group into the general linear group

$\mathrm{Sp}\left(2n\right)↪\mathrm{GL}\left(2n\right)\phantom{\rule{thinmathspace}{0ex}}.$Sp(2n) \hookrightarrow GL(2n) \,.

Such an Sp(2n)-structure is also called an almost symplectic structure on $X$. Adding the extra condition that $d\omega =0$ makes it a genuine symplectic structure.

A metaplectic structure on a symplectic or almost symplectic manifold is in turn lift of the structure group to the metaplectic group.

### Symplectomorphisms

###### Proposition

For $\left(X,\omega \right)$ a symplectic manifold, the vector fields $v\in \Gamma \left(TX\right)$ that generate diffeomorphisms that preserve the symplectic structure are precisely the locally Hamiltonian vector fields.

###### Proof

The condition in question is that the Lie derivative

${L}_{v}\omega =0$L_v \omega = 0

vanishes. By Cartan's magic formula and using that $d\omega =0$ this is equivalently

$d{\iota }_{v}\omega =0\phantom{\rule{thinmathspace}{0ex}}.$d \iota_v \omega = 0 \,.

By the Poincare lemma it follows that there is locally a function $H$ with $dH={\iota }_{v}\omega$.

### Poisson structure

###### Definition

For $\left(X,\omega \right)$ a symplectic manifold, define a bilinear skew-symmetric map

$\left\{-,-\right\}:{C}^{\infty }\left(X\right)\otimes {C}^{\infty }\left(X\right)\to {C}^{\infty }\left(X\right)$\{-,-\} : C^\infty(X) \otimes C^\infty(X) \to C^\infty(X)

by

$\left\{F,H\right\}:={\iota }_{{v}_{F}}{\iota }_{{v}_{H}}\omega \phantom{\rule{thinmathspace}{0ex}}.$\{F,H\} := \iota_{v_F} \iota_{v_H} \omega \,.

In a coordinate chart this says that

$\left\{F,H\right\}=\left(\frac{\partial F}{\partial {x}^{i}}\right)\left({\omega }^{-1}{\right)}^{ij}\left(\frac{\partial H}{\partial {x}^{j}}\right)\phantom{\rule{thinmathspace}{0ex}}.$\{F,H\} = (\frac{\partial F}{\partial x^i}) (\omega^{-1})^{i j} (\frac{\partial H}{\partial x^j}) \,.
###### Proposition

The bracket $\left\{-,-\right\}$ makes ${C}^{\infty }\left(X\right)$ a Poisson algebra.

## Examples

The notion of symplectic manifold is equivalent to that of symplectic Lie n-algebroid for $n=0$. (See there.)

type of subspace $W$ of inner product spacecondition 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$W\cap {W}^{\perp }=\left\{0\right\}$(for symplectic form)

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

$n\in ℕ$symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $\left(n+1\right)$-d sigma-modelhigher symplectic geometry$\left(n+1\right)$d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension $\left(n+1\right)$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
$n$symplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometry$d=n+1$ AKSZ sigma-model