nLab
n-plectic geometry

Idea

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

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

such that regarded as a function

$$V\to {\wedge }^n{V}^{*}$$V \to \wedge^n V^*

is has trivial kernel.

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

Definition

We say $(X,\omega )$ is a $n$-plectic manifold if

• $\omega$ is closed: $d\omega =0$;

• for all $x\in X$ the map

  $$\hat{\omega }:T_{x}X\to {\Lambda }^n{T}_x^{n}X$$\hat \omega : T_x X \to \Lambda^n T_x^n X

  given by contraction of vectors with forms

  $$v\mapsto {\iota }_v\omega$$v \mapsto \iota_v \omega

  is injective.

See also the definition at multisymplectic geometry.

Examples

• $n=1$ – this is the case of an ordinary symplectic manifold this appears in Hamiltonian mechanics;

• $n=2$, this appears naturally in 1+1 dimensional quantum field theory.

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

2. ${\wedge }^n{T}^{*}X\to X$

3. $G$ a compact simple Lie group,

   let $\nu :(x,y,z)\mapsto ⟨x,[y,z]⟩$ be the canonical Lie algebra 3-cocycle and extend it left-invariantly to a 3-form ${\omega }_{\nu }$ on $G$. Then $(G,{\omega }_{\nu })$ is 2-plectic.

Poisson $L_{\mathrm{\infty }}$-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 $n$-plectic manifold for $n>1$ 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 $n$-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 $n=1$ for how a Poisson algebra is obtained from a symplectic manifold $(X,\omega )$.

Here

$$\hat{\omega }:T_{x}X\to T_x^{*}X$$\hat \omega : T_x X \to T^*_x X

is an isomorphism.

Given $f\in C^{\mathrm{\infty }}(X)$, $\exists !{\nu }_f\in \Gamma (TX)$ such that $df=-\omega (v_f,-)$

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

We can generalize this to $n$-plectic geometry.

Let $(X,\omega )$ be $n$-plectic for $n>1$.

Observe that then $\hat{\omega }:T_{x}X\to {\wedge }^n{T}_{x}X$ is no longer an isomorphism in general.

Definition

Say

$$\alpha \in {\Omega }^{n-1}(X)$$\alpha \in \Omega^{n-1}(X)

is Hamiltonian precisely if

$$\exists v_{\alpha }\in \Gamma (TX)$$\exists v_\alpha \in \Gamma(T X)

such that

$$d\alpha =-\omega (v_{\alpha },-).$$d \alpha = - \omega(v_\alpha, -) \,.

This makes $v_{\alpha }$ uniquely defined.

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

Define a bracket

$$\{-,-\}:{\Omega }_{\mathrm{Hamilt}}^{n-1}(X{)}^{{\otimes }_2}\to {\Omega }_{\mathrm{Hamilt}}^{n-1}(X)$$\{-,-\} : \Omega^{n-1}_{Hamilt}(X)^{\otimes_2} \to \Omega^{n-1}_{Hamilt}(X)

by

$$\{\alpha ,\beta \}=-\omega (v_{\alpha },v_{\beta },-,\cdots ,-).$$\{\alpha, \beta\} = - \omega(v_\alpha, v_\beta, -, \cdots, -) \,.

This satisfies

1. k

   $$d\{\alpha ,\beta \}=-\omega ([v_{\alpha },v_{\beta }],-,\cdots ,-).$$d \{\alpha, \beta\} = - \omega([v_\alpha, v_\beta], -, \cdots, -) \,.

2. $\{-,-\}$ is skew-symmetric;

3. $\{{\alpha }_1,\{{\alpha }_2,{\alpha }_3\}\}$ + cyclic permutations
   $d\omega (v_{{\alpha }_1},v_{{\alpha }_2},v_{{\alpha }_3},-,\cdots )$.

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

Proposition

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

$$C^{\mathrm{\infty }}(X)\stackrel{d_{\mathrm{dR}}}{\to }{\Omega }^1(X)\stackrel{d_{\mathrm{dR}}}{\to }\to \cdots \to {\Omega }_{\mathrm{Hamilt}}^{n-1}(X)$$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 $d_{\mathrm{dR}}$;

• the $k$-ary bracket is

  $$[{\alpha }_1,\cdots ,{\alpha }_k]=\{\begin{array}{cc}\pm \omega (v_{{\alpha }_1},\cdots ,v_{{\alpha }_k})& \mathrm{if}\forall i:{\alpha }_i\in {\Omega }_{\mathrm{Hamilt}}^{n-1}(X)\\ 0& \mathrm{otherwise}\end{array}$$[\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 $n=1$ this recovers the definition of the Lie algebra underlying a Poisson algebra.

Prequantization

Review of the symplectic situation

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

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

$$\mathrm{ad}P\to TP/U(1)\to TX$$ad P \to T P/U(1) \to T X

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

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

$$s(v)+f{\partial }_t$$s(v) + f \partial_t

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

Locally on a suitable patch $U\subset X$ we have that $s(V){\mid }_U=v{\mid }_U+{\iota }_v{\theta }_i{\mid }_U$ .

We say that $\tilde{v}=s(v)+f{\partial }_t$ preserves the splitting iff $\forall u\in \Gamma (X)$ we have

$$[\tilde{v},s(u)]=s([v,u]).$$[\tilde v, s(u)] = s([v,u]) \,.

One finds that this is the case precisely if

$$df=-{\iota }_v\omega .$$d f = - \iota_v \omega \,.

This gives a homomorphism of Lie algebras

$$C^{\mathrm{\infty }}(X)\to \Gamma (TP/U(1))$$C^\infty(X) \to \Gamma(T P / U(1))

$$f\mapsto s(v_f)+f{\partial }_t.$$f \mapsto s(v_f) + f \partial_t \,.

2-plectic geometry and Courant algebroids

We consder now prequantization of 2-plectic manifolds.

Let $(X,\omega )$ be a 2-plectic manifold such that the de Rham cohomology class $[\omega ]/2\pi i$ 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 $\{U_i\to X\}$ of $X$ this is given in terms of Cech cohomology by data

• $(g_{ijk}:U_{ijk}\to U(1))\in C^{\mathrm{\infty }}(U_{ijk},U(1))$

• $A_{ij}\in {\Omega }^1(U_{ij})$;

• $B_i\in {\Omega }^2(U_i)$

satisfying a cocycle condition.

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

• a vector bundle $E\to X$;

• an anchor morphism $\rho :E\to TX$ to the tangent bundle;

• an inner product $⟨-,-⟩$ on the fibers of $E$;

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

Satisfying some conditions.

The fact that the Courant algebroid is exact means that

$$0\to T^{*}X\to E\to TX\to 0$$0 \to T^* X \to E \to T X \to 0

is an exact sequence.

The standard Courant algebroid is the example where

• $E=TX\oplus T^{*}X$;

• $⟨v_1+{\alpha }_1,v_2+{\alpha }_2⟩={\alpha }_2(v_1)+{\alpha }_1(v_2)$;

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

  $$(v_1+{\alpha }_1,v_2+{\alpha }_2)=[v_1,v_2]-{𝕃}_{v_1}{\alpha }_2-(d{\alpha }_1)(v_2,-)$$(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 $U_i$ is is the standard Courant algebroid $E_i:=T{U}_i\oplus T^{*}{U}_i$;

• glued together on double intersections using the $d{A}_{ij}$

This gives an exact Courant algebroid $E\to X$ as well as a splitting $s:TX\to E$ given by the $\{B_i\}$.

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

$$[[s(v_1){\alpha }_1,s(v_2)+{\alpha }_2]]=s([v_1,v_2])+{ℒ}_{v_1}{\alpha }_2-(d{\alpha }_2)(v_2,-)-\omega (v_1,v_2,\cdots ).$$[ [ 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 $e=s(v)+...$ preserves the splitting precisely if

for all $u\in \Gamma (TX)$ we have

$$[[e,s(u)]{]}_D=s([v,u])$$[ [ e, s(u)] ]_D = s([v,u])

exactly if $\alpha$ is Hamiltonian and $v=v_{\alpha }$.

Theorem

Recall that to every Courant algebroid $E$ is associated a Lie 2-algebra $L_{\mathrm{\infty }}(E)$.

Then: we have an embedding of L-infinity algebras

$$\varphi :L_{\mathrm{\infty }}(X,\omega )\to L_{\mathrm{\infty }}(E)$$\phi : L_\infty(X,\omega) \to L_\infty(E)

given by $\varphi (\alpha )=s(v_{\alpha })+\alpha$.

Properties

Central extensions under geometric quantization

higher and integrated Kostant-Souriau extensions

(∞-group extension of ∞-group of bisections of higher Atiyah groupoid for $𝔾$-principal ∞-connection)

$$(\Omega 𝔾)\mathrm{FlatConn}(X)\to \mathrm{QuantMorph}(X,\nabla )\to \mathrm{HamSympl}(X,\nabla )$$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\mathrm{\infty }$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega 𝔾)$-flat ∞-connections on $X$	quantomorphism ∞-group
1	symplectic geometry	Lie algebra	Hamiltonian vector fields	real numbers	Hamiltonians under Poisson bracket
1		Lie group	Hamiltonian symplectomorphism group	circle group	quantomorphism group
2	2-plectic geometry	Lie 2-algebra	Hamiltonian vector fields	line Lie 2-algebra	Poisson Lie 2-algebra
2		Lie 2-group	Hamiltonian 2-plectomorphisms	circle 2-group	quantomorphism 2-group
$n$	n-plectic geometry	Lie n-algebra	Hamiltonian vector fields	line Lie n-algebra	Poisson Lie n-algebra
$n$		smooth n-group	Hamiltonian n-plectomorphisms	circle n-group	quantomorphism n-group

(extension are listed for sufficiently connected $X$)

Related concepts

• 2-plectic geometry

• symplectic infinity-groupoid

• higher geometric quantization

duality between algebra and geometry in physics:

algebra	geometry
Poisson algebra	Poisson manifold
deformation quantization	geometric quantization
algebra of observables	space of states
Heisenberg picture	Schrödinger picture
AQFT	FQFT
higher algebra	higher geometry
Poisson n-algebra	n-plectic manifold
En-algebras	higher symplectic geometry
BD-BV quantization	higher geometric quantization
factorization algebra of observables	extended quantum field theory
factorization homology	cobordism representation

∞-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

General

• Chris Rogers, Higher symplectic geometry PhD thesis (2011) (arXiv:1106.4068)

• Chris Rogers, $L_{\mathrm{\infty }}$ algebras from multisymplectic geometry , Letters in Mathematical Physics April 2012, Volume 100, Issue 1, pp 29-50 (arXiv:1005.2230, journal).

• Chris Rogers, 2-plectic geometry, Courant algebroids, and categorified prequantization , arXiv:1009.2975.

• Chris Rogers, Higher geometric quantization, talk at Higher Structures 2011 in Göttingen (pdf slides)

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

• Domenico Fiorenza, Chris Rogers, Urs Schreiber,

  Higher geometric prequantum theory,

  L-∞ algebras of local observables from higher prequantum bundles

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

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

• Christian Saemann, Richard Szabo, Groupoid quantization of loop spaces (arXiv:1203.5921)

and in

• Christian Saemann, Richard Szabo, Quantization of 2-Plectic Manifolds (arXiv:1106.1890)