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symplectic manifold

Contents

Definition

Definition

A symplectic manifold is

  • a smooth manifold XX of even dimension dimX=2ndim X = 2 n;

  • equipped with a symplectic form:

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

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

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

      • ( 2T p *X,ω p)(\wedge^2 T^*_p X,\omega_p) is a symplectic vector space for every point pXp\in X.

Definition

A 2n2n-dimensional topological manifold XX is

  • a real symplectic manifold

  • equipped with a symplectic atlas:

    • an atlas consisting of smooth charts ϕ i:U iX\phi_i:U_i\to X as usual,

    • such that the transition functions ϕ j 1ϕ i:ϕ i 1(ϕ i(U i)ϕ j(U j))ϕ j 1(ϕ i(U i)ϕ j(U j))\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 ω 0= i=1 ndx idp i\omega_0=\sum_{i=1}^n dx_i\wedge dp_i on 2n\mathbb{R}^{2n} with the basis (x 1,,x n,p 1,,p n)(x_1,\ldots,x_n,p_1,\ldots,p_n).

Remark

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

ω(,):Γ(TX)Γ(T *X) \omega(-,-) : \Gamma(T X) \to \Gamma(T^* X)

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

(vT xX)(ω(v,)T x *X). (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 HC (X)H \in C^\infty(X) a smooth function and dHd H its differential 1-form, the corresponding Hamiltonian vector field v HΓ(TX)v_H \in \Gamma(T X) is the unique vector field such that

dH=ω(v H,) d H = \omega(v_H, -) \,

Equivalently, for ϕ 2nX\phi \mathbb{R}^{2n} \to X a coordinate chart of XX and ϕ *ω=ω ijdx idx j\phi^*\omega = \omega_{i j} d x^i \wedge d x^j the symplectic form on this patch, the Hamiltonian vector field v Hv_H is

v H=Hx i(ω 1) ij j. 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 2nUX\mathbb{R}^{2n} \simeq U \hookrightarrow X on which the symplectic form takes the canonical form ω| U= k=1 ndx 2kdx 2k+1\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 ωΩ 2(X)\omega \in \Omega^2(X) which is non-degenerate (but not necessarily closed) is equivalent to the existence of a Sp-structure on XX, a reduction of the structure group of the tangent bundle along the inclusion of the symplectic group into the general linear group

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

Such an Sp(2n)-structure is also called an almost symplectic structure on XX. Adding the extra condition that dω=0d \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,)GL(2n,)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)hookrightarowSp(2n,)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,ω)(X, \omega) a symplectic manifold, the vector fields vΓ(TX)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 vω=0 L_v \omega = 0

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

dι vω=0. d \iota_v \omega = 0 \,.

By the Poincare lemma it follows that there is locally a function HH with dH=ι vωd H = \iota_v \omega.

Poisson structure

Definition

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

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

by

{F,H}:=ι v Fι v Hω. \{F,H\} := \iota_{v_F} \iota_{v_H} \omega \,.

In a coordinate chart this says that

{F,H}=(Fx i)(ω 1) ij(Hx j). \{F,H\} = (\frac{\partial F}{\partial x^i}) (\omega^{-1})^{i j} (\frac{\partial H}{\partial x^j}) \,.
Proposition

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

Examples

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

type of subspace WW of inner product spacecondition on orthogonal space W W^\perp
isotropic subspaceWW W \subset W^\perp
coisotropic subspaceW WW^\perp \subset W
Lagrangian subspaceW=W W = W^\perp(for symplectic form)
symplectic spaceWW ={0}W \cap W^\perp = \{0\}(for symplectic form)

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

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

Revised on January 22, 2015 10:37:29 by Urs Schreiber (88.100.66.95)