nLab symplectic Lie n-algebroid

Idea

A Lie n-algebroid is symplectic if it is equipped with a non-degenerate binary invariant polynomial. This generalizes the notion of a symplectic form on a symplectic manifold, to which it reduces for $n = 0$.

Definition

Definition

A symplectic Lie $n$-algebroid is a pair

$$(\mathfrak{a}, \langle -,- \rangle)$$

consisting of

• a Lie n-algebroid $\mathfrak{a}$;

• a binary invariant polynomial $\langle- , - \rangle$ of degree $(n+2)$

  (a closed element in the shifted elements of the Weil algebra $W(\mathfrak{a})$)

  which is non-degenerate.

Properties

The Chern-Simons element that witnesses this transgression is the Lagrangian of the corresponding AKSZ theory sigma-model with $\mathfrak{a}$ as its target space and the invariant polynomial $\langle -,- \rangle$ as the (curvature of) its background gauge field.

Examples

$n=0$: symplectic manifold

A 0-Lie algebroid is just a smooth manifold $X$.

Its Chevalley-Eilenberg algebra is the algebra of smooth functions on $X$

$$CE(X) = C^\infty(X) \,.$$

The Weil algebra of $X$ is

$$W(X) = \Omega^\bullet(X)$$

the de Rham algebra of $X$. A degree 2-invariant polynomial on $X$ is therefore a non-degenerate closed 2-form $\omega \in \Omega^2(X)$, a symplectic 2-form.

A symplectic manifold, being a pair

$$(X,\;\; \omega)$$

consisting of a smooth manifold $X$ and a symplectic 2-form $\omega$, is a symplectic Lie 0-algebroid.

$n=1$: Poisson manifold

For a Poisson manifold $X$ with Poisson bivector $\pi \in \Gamma(T X) \wedge \Gamma(T X)$ the Chevalley-Eilenberg algebra $CE(\mathfrak{a})$ of the corresponding Poisson Lie algebroid

$$\mathfrak{a} := \mathfrak{P}(X,\pi)$$

is that of multi-vector fields on $X$, equipped with the differential $d_{CE(\mathfrak{a})} = [\pi, -]_{Sch}$ given by the Schouten bracket.

If we work locally in coordinates then $CE(\mathfrak{a})$ is generated from degree 0 elements $x^i$ and degree 1 elements $\partial_i$. The differential is

$$d_{CE(\mathfrak{a})} = [\pi, -]_{Sch} \,.$$

The Poisson tensor is $\nu := \pi = \pi^{i j} \partial_i \wedge \partial_j$ and that this is a Lie algebroid cocycle is the fact that

$$d_{CE(\mathfrak{a})} \pi = [\pi,\pi]_{Sch} = 0 \,.$$

By definition the Weil algebra $W(\mathfrak{a})$ is generated from the $x^i$, the $\partial_i$ and their shifted partners $\mathbf{d}x^i$ and $\mathbf{d}\partial_i$. The differential here is

$$d_{W(\mathfrak{a})} = [\pi , - ] + \mathbf{d} \,.$$

Proposition

The invariant polynomial $\omega$ that is in transgression with the cocycle $\nu = \pi$ is

$$\omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i) \;\;\; \in inv(\mathfrak{a}) \,.$$

Proof

One checks directly that the element

$$cs_\omega = \pi^{i j} \partial_i \wedge \partial_j + x^i \wedge \mathbf{d} \partial_i$$

is a Chern-Simons transgression element for $\nu$ and $\omega$,

i.e. $d_{W(\mathfrak{a})} cs(\omega) = \omega$. The restriction of $cs_\omega$ to $CE(\mathfrak{a})$ is evidently the Poisson tensor $\pi$.

More details on this at Chern-Simons element.

For a Poisson manifold $X$ with Poisson tensor $\pi = \pi^{i j} \partial_i \wedge \partial_j$, the pair

$$(\mathfrak{P}(X,\pi), \;\;\; \omega = (\mathbf{d} x^i) \wedge (\mathbf{d} \partial_i))$$

consisting of the Poisson Lie algebroid $\mathfrak{P}(X,\pi)$ and of the invariant polynomial $\omega$ that is in transgression with its canonical 2-cocycle $\nu = \pi$ (the Poisson tensor) is a symplectic Lie algebroid.

$n=2$: Courant algebroid

A $2$-symplectic manifold encodes and is encoded by the structure of a Courant algebroid.

A Courant 2algebroid over the point if given by a semisimple Lie algebra with the symplectic form being the Killing form. The coresponding Poisson tensor is the canonical 3-cocycle $\langle -, [-,-] \rangle$ on a semisimple Lie algebra. The extension classified by this is the string Lie 2-algebra.

Relation to other concepts

To $\infty$-Chern-Simons theory

Since the symplectic form on a symplectic Lie $n$-Algebroid may be understood Lie theoretically as an invariant polynomial on an L-∞ algebroid, every symplectic Lie $n$-algebroid serves as a target space for an ∞-Chern-Simons theory: this is AKSZ theory.

We have

• for $n = 1$ the Poisson sigma-model;
• for $n = 2$ the Courant sigma model
• and so on.

To multisymplectic geometry

There is also the closely related notion of multisymplectic geometry. See

• John Baez, Chris Rogers, Categorified Symplectic Geometry and the String Lie 2-Algebra, (arXiv)

for some relations of this to the above situation for $n = 2$. Essentially multisymplectic geometry studies the higher $n$-ary brackets induced from the binary graded symplectic form discussed here. The relation between these two pictures is the same as that between as studied in the context of hemistrict Lie 2-algebras.

An article with more details on this:

• Chris Rogers, Courant algebroids from categorified symplectic geometry (pdf).

Related concepts

• L-∞ algebroid

• symplectic groupoid

∞-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

The notion originates somewhere in the school of Alan Weinstein‘s school of higher categorial symplectic geometry. The first published appearance of the notion at least for $0 \leq n \leq 3$ is

• Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds, PhD thesis [arXiv:9910078]

A good writeup of this material is in

• Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids, in: Theodore Voronov (ed.), Quantization, Poisson Brackets and Beyond, Contemp. Math. 315, Amer. Math. Soc. (2002) [arXiv:0203110, ISBN:978-0-8218-7905-4]

The idea for all $n$ was then sketched, together with many other ideas about L-infinity algebroids in the article with the nice title

• Pavol ?evera?, Some title containing the words “homotopy” and “symplectic”, e.g. this one (arXiv)

What we call $n$-symplectic manifold here is called $\Sigma_n$-manifold there.

Warning This article here uses the term “$n$-symplectic” in a related but not identical sense to the one used here:

• M. de Leon, D. Martin de Diego, M. Salgado, S. Vilariño, K-symplectic formalism on Lie algebroids (arXiv)

A discussion of aspects of how multisymplectic geometry related to $n$-symplectic manifolds is in

• Chris Rogers, Courant algebroids from categorified symplectic geometry (pdf)

arXiv:1001.0040v1 [math-ph]

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

A discussion of symplectic Lie n-algebroids from an infinity-Lie theory perspective as discussed here is in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, AKSZ Sigma-Models in Higher Chern-Weil Theory, Int. J. Geom. Methods Mod. Phys. 10 (2013) 1250078 (arXiv:1108.4378)

The H-cohomology of graded symplectic forms is considered in

• Pavol Severa, p. 1 of On the origin of the BV operator on odd symplectic supermanifolds, Lett Math Phys (2006) 78: 55. (arXiv:0506331)