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

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

such that regarded as a function

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

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

  $$\hat \omega : T_x X \to \Lambda^n T_x^* X$$

  given by contraction of vectors with forms

  $$v \mapsto \iota_v \omega$$

  is injective.

Definition

For $(X,\omega)$ a $n$-plectic manifold and $U\subset X$ a submanifold, we say $U$ is a $k$-isotropic, $k$-Lagrangian, and $k$-coisotropic submanifold is for every $p\in U$, the tangent space $T_p U\subset T_p X$ is a $k$-isotropic, $k$-Lagrangian, and $k$-coisotropic subspace, respectively (see n-plectic vector space for the definition).

Proposition

Given a diffeomorphism $f:X\to X$ for $(X,\omega)$ a $n$-plectic manifold, its graph $\Gamma(f)\subset X\times X$ is a $n$-Lagrangian submanifold of the $n$-plectic manifold $(X\times X, \pi_1 ^*\omega - \pi_2 ^* \omega)$ if and only if $f$ is an $n$-plectomorphism, meaning

$$f^*(\omega) = \omega$$

See e.g. Proposition 3.7 in de León & Vilariño 2012.

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 $\nu : (x,y,z) \mapsto \langle x, [y,z]\rangle$ 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 $L_\infty$-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 $n \gt 1$ 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 $n$-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

$$\hat \omega : T_x X \to T^*_x X$$

is an isomorphism.

Given $f \in C^\infty(X)$, $\exists ! \nu_f \in \Gamma(T X)$ such that $d f = - \omega(v_f, -)$

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

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

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

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

Definition

Say

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

is Hamiltonian precisely if

$$\exists v_\alpha \in \Gamma(T X)$$

such that

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

This makes $v_\alpha$ uniquely defined.

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

Define a bracket

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

by

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

This satisfies

1. k

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

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

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

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

where

• the unary bracket is $d_{dR}$;

• the $k$-ary bracket is

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

This is the Poisson bracket Lie n-algebra.

This appears as (Rogers 11, 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

$$ad P \to T P/U(1) \to T X$$

The smooth sections of $T P/U(1) \to X$ 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

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

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

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

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

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

One finds that this is the case precisely if

$$d f = - \iota_v \omega \,.$$

This gives a homomorphism of Lie algebras

$$C^\infty(X) \to \Gamma(T P / U(1))$$

$$f \mapsto s(v_f) + f \partial_t \,.$$

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

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

• $A_{i j} \in \Omega^1(U_{i j})$;

• $B_i \in \Omega^2(U_i)$

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 $\rho : E \to T X$ to the tangent bundle;

• an inner product $\langle -,-\rangle$ 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

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

is an exact sequence.

The standard Courant algebroid is the example where

• $E = T X \oplus T^* X$;

• $\langle v_1 + \alpha_1, v_2 + \alpha_2\rangle = \alpha_2(v_1) + \alpha_1(v_2)$;

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

  $$(v_1 + \alpha_1, v_2 + \alpha_2) = [v_1, v_2] - \mathbb{L}_{v_1}\alpha_2 - (d \alpha_1)(v_2,-)$$

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 $d A_{i j}$

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

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

$$[ [ s(v_1) \alpha_1, s(v_2) + \alpha_2 ] ] = s([v_1, v_2]) + \mathcal{L}_{v_1} \alpha_2 - (d \alpha_2)(v_2, -) - \omega(v_1, v_2, \cdots) \,.$$

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

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

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

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

Theorem

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

Then: we have an embedding of L-infinity algebras

$$\phi : L_\infty(X,\omega) \to L_\infty(E)$$

given by $\phi(\alpha) = s(v_\alpha) + \alpha$.

Related entries

• geometry of physics – prequantum geometry

Properties

Central extensions under geometric quantization

higher and integrated Kostant-Souriau extensions:

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

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

$n$	geometry	structure	unextended structure	extension by	quantum extension
$\infty$	higher prequantum geometry	cohesive ∞-group	Hamiltonian symplectomorphism ∞-group	moduli ∞-stack of $(\Omega \mathbb{G})$-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

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

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

∞-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

The observation that the would-be Poisson bracket induced by a higher degree closed form extends to the Poisson bracket Lie n-algebra is due to

• Chris Rogers, $L_\infty$ algebras from multisymplectic geometry , Letters in Mathematical Physics 100 1 (2012) 29-50 [arXiv:1005.2230, journal]

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

with first discussion of application to prequantization in

• 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

• M. de León, S. Vilariño. Lagrangian submanifolds in k-symplectic settings (2012). (arXiv:1202.3964).

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

A higher differential geometry-generalization of contact geometry in line with multisymplectic geometry/$n$-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

• 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)

Examples

On $n$-plectic formulation of gravity, Maxwell theory and Einstein-Maxwell theory:

• Petr Hořava, On a covariant Hamilton-Jacobi framework for the Einstein-Maxwell theory Classical and Quantum Gravity 8 11 (1991) 2069 [doi:10.1088/0264-9381/8/11/016]

• Dmitri Vey, $n$-Plectic Maxwell Theory [arxiv:1303.2192]

• Dmitri Vey, Multisymplectic formulation of vielbein gravity. De Donder-Weyl formulation, Hamiltonian $(n-1)$-forms, Classical and Quantum Gravity 32 9 (2015) [arxiv:1404.3546, doi:10.1088/0264-9381/32/9/095005[

• Patrick Cabau, Fernand Pelletier. Partial Nambu-Poisson structures on a convenient manifold (2024). (arXiv:2404.08688).

For teleparallel gravity

• Igor V. Kanatchikov. The De Donder-Weyl Hamiltonian formulation of TEGR and its quantization (2023) [arXiv:2308.10052]