Contents

Definition

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

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

Properties

Darboux coordinates

By Darboux's theorem every symplectic manifold has an atlas by coordinate charts ${ℝ}^{2n}\simeq U\hookrightarrow X$ 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 $\omega \in {\Omega }^2(X)$ 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

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

Poisson structure

In a coordinate chart this says that

Examples

• every Kähler manifold is canonically also a symplectic manifold;

• the critical locus over any local action functional becomes a symplectic manifold after dividing out symmetries: the reduced phase space.

Related concepts

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

• isotropic submanifold, coisotropic submanifold

• symplectic singularity, symplectic resolution

• symplectic reduction

• symplectic connection

• geometric quantization, canonical momentum

• contact manifold

• G2-manifold

type of subspace $W$ of inner product space	condition 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-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	discussed in:
0	symplectic manifold	symplectic manifold	symplectic geometry		Lagrangian submanifold	–	ordinary space of states (in geometric quantization)	geometric quantization
1	Poisson Lie algebroid	symplectic groupoid	2-plectic geometry	Poisson sigma-model	coisotropic submanifold (of underlying Poisson manifold)	brane of Poisson sigma-model	2-module = category of modules over strict deformation quantiized algebra of observables	extended geometric quantization of 2d Chern-Simons theory
2	Courant Lie 2-algebroid	symplectic 2-groupoid	3-plectic geometry	Courant sigma-model	Dirac structure	D-brane in type II geometry
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d=n+1$ AKSZ sigma-model

(adapted from Ševera 00)

References

See the references at symplectic geometry.

The generalization of the notion of symplectic manifolds to dg-manifolds is sometimes known as PQ-supermanifolds , due to

• M. Alexandrov, M. Kontsevich, A. Schwarz, O. Zaboronsky, The geometry of the master equation and topological quantum field theory, Int. J. Modern Phys. A 12(7):1405–1429, 1997 (arXiv:hep-th/9502010)