nLab symplectic manifold

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 real symplectic manifold $X$ is

• a $2n$-dimensional topological 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-degeneracy of the symplectic form implies that it defines an isomorphism

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

between sections of the tangent bundle – vector fields – and sections of the cotangent bundle – differential 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 $U(n) \hookrightarrow Sp(2n,\mathbb{R})$ 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

• 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

• bilinear form, quadratic form, sesquilinear form

• Kähler form, Hermitian form

• MSp

• symplectic orbifold

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

• holomorphic symplectic manifold

• 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.

Discussion of the torsion-invariants of almost symplectic structures includes

• Rui Albuquerque, Roger Picken, On invariants of almost symplectic connections (arXiv:1107.1860)

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)

On symplectic orbifolds:

• Misha Verbitsky, Holomorphic symplectic geometry and orbifold singularities, Asian J. Math. 4 (2000), no. 3, 553-563 (arXiv:math/9903175)

On a proposal for homotopy theory of symplectic manifolds:

• Vardan Oganesyan, The first step towards symplectic homotopy theory [arXiv:2304.10529]