# nLab n-plectic geometry

### Context

#### Symplectic geometry

symplectic geometry

higher symplectic geometry

# Contents

## 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 $n\in ℕ$, an n-plectic vector space is a vector space $V$ (over the real numbers) equipped with an $\left(n+1\right)$-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, $\omega \in {\Omega }^{n+1}\left(X\right)$ a differential form.

###### Definition

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

• $\omega$ is closed: $d\omega =0$;

• for all $x\in X$ the map

$\stackrel{^}{\omega }:{T}_{x}X\to {\Lambda }^{n}{T}_{x}^{n}X$\hat \omega : T_x X \to \Lambda^n T_x^n X

given by contraction of vectors with forms

$v↦{\iota }_{v}\omega$v \mapsto \iota_v \omega

is injective.

## Examples

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

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

3. $G$ a compact simple Lie group,

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

## Poisson ${L}_{\infty }$-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 $n$-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 $\left(X,\omega \right)$.

Here

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

is an isomorphism.

Given $f\in {C}^{\infty }\left(X\right)$, $\exists !{\nu }_{f}\in \Gamma \left(TX\right)$ such that $df=-\omega \left({v}_{f},-\right)$

Define $\left\{f,g\right\}:=\omega \left({v}_{f},{v}_{g}\right)$. Then $\left({C}^{\infty }\left(X,\right),\left\{-,-\right\}\right)$ is a Poisson algebra.

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

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

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

Definition

Say

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

is Hamiltonian precisely if

$\exists {v}_{\alpha }\in \Gamma \left(TX\right)$\exists v_\alpha \in \Gamma(T X)

such that

$d\alpha =-\omega \left({v}_{\alpha },-\right)\phantom{\rule{thinmathspace}{0ex}}.$d \alpha = - \omega(v_\alpha, -) \,.

This makes ${v}_{\alpha }$ uniquely defined.

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

Define a bracket

$\left\{-,-\right\}:{\Omega }_{\mathrm{Hamilt}}^{n-1}\left(X{\right)}^{{\otimes }_{2}}\to {\Omega }_{\mathrm{Hamilt}}^{n-1}\left(X\right)$\{-,-\} : \Omega^{n-1}_{Hamilt}(X)^{\otimes_2} \to \Omega^{n-1}_{Hamilt}(X)

by

$\left\{\alpha ,\beta \right\}=-\omega \left({v}_{\alpha },{v}_{\beta },-,\cdots ,-\right)\phantom{\rule{thinmathspace}{0ex}}.$\{\alpha, \beta\} = - \omega(v_\alpha, v_\beta, -, \cdots, -) \,.

This satisfies

1. k

$d\left\{\alpha ,\beta \right\}=-\omega \left(\left[{v}_{\alpha },{v}_{\beta }\right],-,\cdots ,-\right)\phantom{\rule{thinmathspace}{0ex}}.$d \{\alpha, \beta\} = - \omega([v_\alpha, v_\beta], -, \cdots, -) \,.
2. $\left\{-,-\right\}$ is skew-symmetric;

3. $\left\{{\alpha }_{1},\left\{{\alpha }_{2},{\alpha }_{3}\right\}\right\}$ + cyclic permutations
$d\omega \left({v}_{{\alpha }_{1}},{v}_{{\alpha }_{2}},{v}_{{\alpha }_{3}},-,\cdots \right)$.

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 $\left(X,\omega \right)$ we get a Lie n-algebra structure on the complex

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

$\left[{\alpha }_{1},\cdots ,{\alpha }_{k}\right]=\left\{\begin{array}{cc}±\omega \left({v}_{{\alpha }_{1}},\cdots ,{v}_{{\alpha }_{k}}\right)& \mathrm{if}\forall i:{\alpha }_{i}\in {\Omega }_{\mathrm{Hamilt}}^{n-1}\left(X\right)\\ 0& \mathrm{otherwise}\end{array}$[\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

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

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

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

$s\left(v\right)+f{\partial }_{t}$s(v) + f \partial_t

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

Locally on a suitable patch $U\subset X$ we have that $s\left(V\right){\mid }_{U}=v{\mid }_{U}+{\iota }_{v}{\theta }_{i}{\mid }_{U}$ .

We say that $\stackrel{˜}{v}=s\left(v\right)+f{\partial }_{t}$ preserves the splitting iff $\forall u\in \Gamma \left(X\right)$ we have

$\left[\stackrel{˜}{v},s\left(u\right)\right]=s\left(\left[v,u\right]\right)\phantom{\rule{thinmathspace}{0ex}}.$[\tilde v, s(u)] = s([v,u]) \,.

One finds that this is the case precisely if

$df=-{\iota }_{v}\omega \phantom{\rule{thinmathspace}{0ex}}.$d f = - \iota_v \omega \,.

This gives a homomorphism of Lie algebras

${C}^{\infty }\left(X\right)\to \Gamma \left(TP/U\left(1\right)\right)$C^\infty(X) \to \Gamma(T P / U(1))
$f↦s\left({v}_{f}\right)+f{\partial }_{t}\phantom{\rule{thinmathspace}{0ex}}.$f \mapsto s(v_f) + f \partial_t \,.

### 2-plectic geometry and Courant algebroids

We consder now prequantization of 2-plectic manifolds.

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

• $\left({g}_{ijk}:{U}_{ijk}\to U\left(1\right)\right)\in {C}^{\infty }\left({U}_{ijk},U\left(1\right)\right)$

• ${A}_{ij}\in {\Omega }^{1}\left({U}_{ij}\right)$;

• ${B}_{i}\in {\Omega }^{2}\left({U}_{i}\right)$

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 $\left[-,-\right]$ 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

• $E=TX\oplus {T}^{*}X$;

• $⟨{v}_{1}+{\alpha }_{1},{v}_{2}+{\alpha }_{2}⟩={\alpha }_{2}\left({v}_{1}\right)+{\alpha }_{1}\left({v}_{2}\right)$;

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

$\left({v}_{1}+{\alpha }_{1},{v}_{2}+{\alpha }_{2}\right)=\left[{v}_{1},{v}_{2}\right]-{𝕃}_{{v}_{1}}{\alpha }_{2}-\left(d{\alpha }_{1}\right)\left({v}_{2},-\right)$(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 $\left\{{B}_{i}\right\}$.

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

$\left[\left[s\left({v}_{1}\right){\alpha }_{1},s\left({v}_{2}\right)+{\alpha }_{2}\right]\right]=s\left(\left[{v}_{1},{v}_{2}\right]\right)+{ℒ}_{{v}_{1}}{\alpha }_{2}-\left(d{\alpha }_{2}\right)\left({v}_{2},-\right)-\omega \left({v}_{1},{v}_{2},\cdots \right)\phantom{\rule{thinmathspace}{0ex}}.$[ [ 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 $e=s\left(v\right)+...$ preserves the splitting precisely if

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

$\left[\left[e,s\left(u\right)\right]{\right]}_{D}=s\left(\left[v,u\right]\right)$[ [ 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 ${L}_{\infty }\left(E\right)$.

Then: we have an embedding of L-infinity algebras

$\varphi :{L}_{\infty }\left(X,\omega \right)\to {L}_{\infty }\left(E\right)$\phi : L_\infty(X,\omega) \to L_\infty(E)

given by $\varphi \left(\alpha \right)=s\left({v}_{\alpha }\right)+\alpha$.

## 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)

$\left(\Omega 𝔾\right)\mathrm{FlatConn}\left(X\right)\to \mathrm{QuantMorph}\left(X,\nabla \right)\to \mathrm{HamSympl}\left(X,\nabla \right)$(\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla)
$n$geometrystructureunextended structureextension byquantum extension
$\infty$higher prequantum geometrycohesive ∞-groupHamiltonian symplectomorphism ∞-groupmoduli ∞-stack of $\left(\Omega 𝔾\right)$-flat ∞-connections on $X$quantomorphism ∞-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
$n$n-plectic geometryLie n-algebraHamiltonian vector fieldsline Lie n-algebraPoisson Lie n-algebra
$n$smooth n-groupHamiltonian n-plectomorphismscircle n-groupquantomorphism n-group

(extension are listed for sufficiently connected $X$)

duality between algebra and geometry in physics:

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

$n\in ℕ$symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $\left(n+1\right)$-d sigma-modelhigher symplectic geometry$\left(n+1\right)$d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension $\left(n+1\right)$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
$n$symplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometry$d=n+1$ AKSZ sigma-model

## References

### General

• Chris Rogers, ${L}_{\infty }$ 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