nLab
n-plectic geometry

Contents

Idea

nn-plectic geometry is a generalization of symplectic geometry to higher category theory.

It is closely related to multisymplectic geometry and n-symplectic manifolds.

Definition

Definition

For nn \in \mathbb{N}, an n-plectic vector space is a vector space VV (over the real numbers) equipped with an (n+1)(n+1)-linear skew function

ω: n+1V \omega : \wedge^{n+1} V \to \mathbb{R}

such that regarded as a function

V nV * V \to \wedge^n V^*

is has trivial kernel.

Let XX be a smooth manifold, ωΩ n+1(X)\omega \in \Omega^{n+1}(X) a differential form.

Definition

We say (X,ω)(X,\omega) is a nn-plectic manifold if

  • ω\omega is closed: dω=0d \omega = 0;

  • for all xXx \in X the map

    ω^:T xXΛ nT x nX \hat \omega : T_x X \to \Lambda^n T_x^n X

    given by contraction of vectors with forms

    vι vω v \mapsto \iota_v \omega

    is injective.

See also the definition at multisymplectic geometry.

Examples

  1. For XX orientable, take ω\omega the volume form. This is (dim(X)1)(dim(X)-1)-plectic.

  2. nT *XX\wedge^n T^* X \to X

  3. GG a compact simple Lie group,

    let ν:(x,y,z)x,[y,z]\nu : (x,y,z) \mapsto \langle x, [y,z]\rangle be the canonical Lie algebra 3-cocycle and extend it left-invariantly to a 3-form ω ν\omega_\nu on GG. Then (G,ω ν)(G,\omega_\nu) is 2-plectic.

Poisson L L_\infty-algebras

To an ordinary symplectic manifold is associated its Poisson algebra. Underlying this is a Lie algebra, whose Lie bracket is the Poisson bracket.

We discuss here how to an nn-plectic manifold for n>1n \gt 1 there is correspondingly assoociated not a Lie algebra, but a Lie n-algebra: the Poisson bracket Lie n-algebra. It is natural to call this a Poisson Lie nn-algebra (see for instance at Poisson Lie 2-algebra).

(Not to be confused with the Lie algebra of a Poisson Lie group, which is a Lie group that itself is equipped with a compatible Poisson manifold structure. It is a bit unfortunate that there is no better established term for the Lie algebra underlying a Poisson algebra apart from “Poisson bracket”.)

Consider the ordinary case in n=1n=1 for how a Poisson algebra is obtained from a symplectic manifold (X,ω)(X, \omega).

Here

ω^:T xXT x *X \hat \omega : T_x X \to T^*_x X

is an isomorphism.

Given fC (X)f \in C^\infty(X), !ν fΓ(TX)\exists ! \nu_f \in \Gamma(T X) such that df=ω(v f,)d f = - \omega(v_f, -)

Define {f,g}:=ω(v f,v g)\{f,g\} := \omega(v_f, v_g). Then (C (X,),{,})(C^\infty(X,), \{-,-\}) is a Poisson algebra.

We can generalize this to nn-plectic geometry.

Let (X,ω)(X,\omega) be nn-plectic for n>1n \gt 1.

Observe that then ω^:T xX nT xX\hat \omega : T_x X \to \wedge^n T_x X is no longer an isomorphism in general.

Definition

Say

αΩ n1(X) \alpha \in \Omega^{n-1}(X)

is Hamiltonian precisely if

v αΓ(TX) \exists v_\alpha \in \Gamma(T X)

such that

dα=ω(v α,). d \alpha = - \omega(v_\alpha, -) \,.

This makes v αv_\alpha uniquely defined.

Denote the collection of Hamiltonian forms by Ω Hamilt n1(X)\Omega^{n-1}_{Hamilt}(X).

Define a bracket

{,}:Ω Hamilt n1(X) 2Ω Hamilt n1(X) \{-,-\} : \Omega^{n-1}_{Hamilt}(X)^{\otimes_2} \to \Omega^{n-1}_{Hamilt}(X)

by

{α,β}=ω(v α,v β,,,). \{\alpha, \beta\} = - \omega(v_\alpha, v_\beta, -, \cdots, -) \,.

This satisfies

  1. k

    d{α,β}=ω([v α,v β],,,). d \{\alpha, \beta\} = - \omega([v_\alpha, v_\beta], -, \cdots, -) \,.
  2. {,}\{-,-\} is skew-symmetric;

  3. {α 1,{α 2,α 3}}\{\alpha_1, \{\alpha_2, \alpha_3\}\} + cyclic permutations
    dω(v α 1,v α 2,v α 3,,)d \omega(v_{\alpha_1}, v_{\alpha_2}, v_{\alpha_3}, -, \cdots).

So the Jacobi dientity fails up to an exact term. This will yield the structure of an L-infinity algebra.

Proposition

Given an nn-plectic manifold (X,ω)(X,\omega) we get a Lie n-algebra structure on the complex

C (X)d dRΩ 1(X)d dRΩ Hamilt n1(X) C^\infty(X) \stackrel{d_{dR}}{\to} \Omega^1(X) \stackrel{d_{dR}}{\to} \to \cdots \to \Omega^{n-1}_{Hamilt}(X)

(where the rightmost term is taken to be in degree 0).

where

  • the unary bracket is d dRd_{dR};

  • the kk-ary bracket is

    [α 1,,α k]={±ω(v α 1,,v α k) ifi:α iΩ Hamilt n1(X) 0 otherwise [\alpha_1, \cdots, \alpha_k] = \left\{ \array{ \pm \omega(v_{\alpha_1}, \cdots, v_{\alpha_k}) & if \forall i : \alpha_i \in \Omega^{n-1}_{Hamilt}(X) \\ 0 & otherwise } \right.

This is the Poisson bracket Lie n-algebra.

This appears as (Rogers, theorem 3.14).

For n=1n = 1 this recovers the definition of the Lie algebra underlying a Poisson algebra.

Prequantization

Review of the symplectic situation

Recall for n=1n=1 the mechanism of geometric quantization of a symplectic manifold.

Given a 2-form ω\omega and the corresponding complex line bundle PP, consider the Atiyah Lie algebroid sequence

adPTP/U(1)TX ad P \to T P/U(1) \to T X

The smooth sections of TP/U(1)XT P/U(1) \to X are the U(1)U(1) invariant vector fields on the total space of PP.

Using a connection \nabla on PP we may write such a section as

s(v)+f t s(v) + f \partial_t

for vΓ(TX)v \in \Gamma(T X) a vector field downstairs, s(v)s(v) a horizontal lift with respect to the given connection and fC (X)f \in C^\infty(X).

Locally on a suitable patch UXU \subset X we have that s(V) U=v U+ι vθ i Us(V)|_U = v|_U + \iota_v \theta_i|_U .

We say that v˜=s(v)+f t\tilde v = s(v) + f \partial_t preserves the splitting iff uΓ(X)\forall u \in \Gamma(X) we have

[v˜,s(u)]=s([v,u]). [\tilde v, s(u)] = s([v,u]) \,.

One finds that this is the case precisely if

df=ι vω. d f = - \iota_v \omega \,.

This gives a homomorphism of Lie algebras

C (X)Γ(TP/U(1)) C^\infty(X) \to \Gamma(T P / U(1))
fs(v f)+f t. f \mapsto s(v_f) + f \partial_t \,.

2-plectic geometry and Courant algebroids

We consder now prequantization of 2-plectic manifolds.

Let (X,ω)(X,\omega) be a 2-plectic manifold such that the de Rham cohomology class [ω]/2πi[\omega]/2 \pi i is in the image of integral cohomology (Has integral periods.)

We can form a cocycle in Deligne cohomology from this, encoding a bundle gerbe with connection.

On a cover {U iX}\{U_i \to X\} of XX this is given in terms of Cech cohomology by data

  • (g ijk:U ijkU(1))C (U ijk,U(1))(g_{i j k} : U_{i j k} \to U(1)) \in C^\infty(U_{i j k}, U(1))

  • A ijΩ 1(U ij)A_{i j} \in \Omega^1(U_{i j});

  • B iΩ 2(U i)B_i \in \Omega^2(U_i)

satisfying a cocycle condition.

Now recall that an exact Courant algebroid is given by the following data:

  • a vector bundle EXE \to X;

  • an anchor morphism ρ:ETX\rho : E \to T X to the tangent bundle;

  • an inner product ,\langle -,-\rangle on the fibers of EE;

  • a bracket [,][-,-] on the sections of EE.

Satisfying some conditions.

The fact that the Courant algebroid is exact means that

0T *XETX0 0 \to T^* X \to E \to T X \to 0

is an exact sequence.

The standard Courant algebroid is the example where

  • E=TXT *XE = T X \oplus T^* X;

  • v 1+α 1,v 2+α 2=α 2(v 1)+α 1(v 2)\langle v_1 + \alpha_1, v_2 + \alpha_2\rangle = \alpha_2(v_1) + \alpha_1(v_2);

  • the bracket is the skew-symmetrization of the Dorfman bracket

    (v 1+α 1,v 2+α 2)=[v 1,v 2]𝕃 v 1α 2(dα 1)(v 2,) (v_1 + \alpha_1, v_2 + \alpha_2) = [v_1, v_2] - \mathbb{L}_{v_1}\alpha_2 - (d \alpha_1)(v_2,-)

Now with respect to the above Deligne cocycle, build a Courant algebroid as follows:

  • on each patch U iU_i is is the standard Courant algebroid E i:=TU iT *U iE_i := T U_i \oplus T^* U_i;

  • glued together on double intersections using the dA ijd A_{i j}

This gives an exact Courant algebroid EXE \to X as well as a splitting s:TXEs : T X \to E given by the {B i}\{B_i\}.

The bracket on this EE is given by the skew-symmetrization of

[[s(v 1)α 1,s(v 2)+α 2]]=s([v 1,v 2])+ v 1α 2(dα 2)(v 2,)ω(v 1,v 2,). [ [ s(v_1) \alpha_1, s(v_2) + \alpha_2 ] ] = s([v_1, v_2]) + \mathcal{L}_{v_1} \alpha_2 - (d \alpha_2)(v_2, -) - \omega(v_1, v_2, \cdots) \,.

Here a section e=s(v)+...e = s(v) + ... preserves the splitting precisely if

for all uΓ(TX)u \in \Gamma(T X) we have

[[e,s(u)]] D=s([v,u]) [ [ e, s(u)] ]_D = s([v,u])

exactly if α\alpha is Hamiltonian and v=v αv = v_\alpha.

Theorem

Recall that to every Courant algebroid EE is associated a Lie 2-algebra L (E)L_\infty(E).

Then: we have an embedding of L-infinity algebras

ϕ:L (X,ω)L (E) \phi : L_\infty(X,\omega) \to L_\infty(E)

given by ϕ(α)=s(v α)+α\phi(\alpha) = s(v_\alpha) + \alpha.

Properties

Central extensions under geometric quantization

higher and integrated Kostant-Souriau extensions

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for 𝔾\mathbb{G}-principal ∞-connection)

(Ω𝔾)FlatConn(X)QuantMorph(X,)HamSympl(X,) (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
nngeometrystructureunextended structureextension byquantum extension
\inftyhigher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of (Ω𝔾)(\Omega \mathbb{G})-flat ∞-connections on XXquantomorphism ∞-group
1symplectic geometryLie algebraHamiltonian vector fieldsreal numbersHamiltonians under Poisson bracket
1Lie groupHamiltonian symplectomorphism groupcircle groupquantomorphism group
22-plectic geometryLie 2-algebraHamiltonian vector fieldsline Lie 2-algebraPoisson Lie 2-algebra
2Lie 2-groupHamiltonian 2-plectomorphismscircle 2-groupquantomorphism 2-group
nnn-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
nnsmooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected XX)

duality between algebra and geometry in physics:

algebrageometry
Poisson algebraPoisson manifold
deformation quantizationgeometric quantization
algebra of observablesspace of states
Heisenberg pictureSchrödinger picture
AQFTFQFT
higher algebrahigher geometry
Poisson n-algebran-plectic manifold
En-algebrashigher symplectic geometry
BD-BV quantizationhigher geometric quantization
factorization algebra of observablesextended quantum field theory
factorization homologycobordism representation

∞-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

General

Discussion in the more general context of higher differential geometry/extended prequantum field theory is in

See also the references at multisymplectic geometry and n-symplectic manifold.

A higher differential geometry-generalization of contact geometry in line with multisymplectic geometry/nn-plectic geometry is discussed in

  • Luca Vitagliano, L-infinity Algebras From Multicontact Geometry (arXiv.1311.2751)

Applications

Some more references on application, on top of those mentioned in the articles above.

A survey of some (potential) applications of 2-plectic geometry in string theory and M2-brane models is in section 2 of

and in

Revised on November 13, 2013 01:57:32 by Urs Schreiber (82.169.114.243)