# nLab symplectic manifold

Contents

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## Definition

###### Definition

A symplectic manifold is

• a smooth manifold $X$ of even dimension $dim X = 2 n$;

• equipped with a symplectic form:

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

• 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 $p\in X$;

• $(\wedge^2 T^*_p X,\omega_p)$ 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 $\phi_i:U_i\to X$ as usual,

• such that the transition functions $\phi_j^{-1}\circ\phi_i:\phi_i^{-1}(\phi_i(U_i)\cap\phi_j(U_j))\to \phi_j^{-1}(\phi_i(U_i)\cap\phi_j(U_j))$ preserve the standard symplectic form $\omega_0=\sum_{i=1}^n dx_i\wedge dp_i$ on $\mathbb{R}^{2n}$ with the basis $(x_1,\ldots,x_n,p_1,\ldots,p_n)$.

###### Remark

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

$\omega(-,-) : \Gamma(T X) \to \Gamma(T^* X)$

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

$(v \in T_x X) \mapsto (\omega(v,-) \in T^*_x X) \,.$
###### Definition

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

This means that for $H \in C^\infty(X)$ a smooth function and $d H$ its differential 1-form, the corresponding Hamiltonian vector field $v_H \in \Gamma(T X)$ is the unique vector field such that

$d H = \omega(v_H, -) \,$

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

$v_H = \frac{\partial H}{\partial x^i} (\omega^{-1})^{i j} \partial_j \,.$

## Properties

### Darboux coordinates

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

### Relation to almost symplectic structure

The existence of a 2-form $\omega \in \Omega^2(X)$ 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

$Sp(2n) \hookrightarrow GL(2n) \,.$

Such an Sp(2n)-structure is also called an almost symplectic structure on $X$. Adding the extra condition that $d \omega = 0$ – the condition for integrability of G-structures – makes it a genuine symplectic structure. See at integrability of G-structures – Examples – Symplectic structure.

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

### Relation to almost Hermitian and Kähler structure

By the above, a symplectic manifold structure is an integrable $Sp(2n,\mathbb{R}) \hookrightarrow GL(2n,\mathbb{R})$-structure. Further reduction of the structure group along the maximal compact subgroup inclusion of the unitary group $U(n) \hookrightarow Sp(2n,\mathbb{R})$ yields is an almost Hermitian structure. If that is again first order integrable then it is Kähler structure.

Such a refinement from symplectic to Kähler structure is also called a choice of Kähler polarization.

### Symplectomorphisms

###### Proposition

For $(X, \omega)$ a symplectic manifold, the vector fields $v \in \Gamma(T X)$ 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

$L_v \omega = 0$

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

$d \iota_v \omega = 0 \,.$

By the Poincare lemma it follows that there is locally a function $H$ with $d H = \iota_v \omega$.

### Poisson structure

###### Definition

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

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

by

$\{F,H\} := \iota_{v_F} \iota_{v_H} \omega \,.$

In a coordinate chart this says that

$\{F,H\} = (\frac{\partial F}{\partial x^i}) (\omega^{-1})^{i j} (\frac{\partial H}{\partial x^j}) \,.$
###### Proposition

The bracket $\{-,-\}$ makes $C^\infty(X)$ 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$W \cap W^\perp = \{0\}$(for symplectic form)

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

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