nLab
n-symplectic manifold

Idea

An $n$-symplectic manifold is to an ordinary symplectic manifold as an Lie n-algebroid is to an ordinary manifold.

Definition

Traditionally, $n$-symplectic manifolds are thought of as supermanifolds with extra structure. More particularly, as NQ-supermanifolds with extra structure.

One can argue that a more conceptual way to talk about them is in terms of Lie ∞-algebroids. Technically it means precisely the same thing, but amplifying the Lie theory context should be helpful.

In supermanifold language

An $n$-symplectic manifold is a NQ-supermanifold $X$ with an $\mathbb{N}$-graded symplectic form $\omega$ of grading-degree $n$.

So $\omega$ is a closed nondegenerate super differential form that respects the $\mathbb{N}$-refined grading of the underlying ${\mathbb{Z}}_2$-garded supermanifold as indicated.

In $\mathrm{\infty }$-Lie algebroid language

An $n$-symplectic manifold is a Lie ∞-algebroid $𝒶$ equipped with a choice of binary degree $(n+2)$ invariant polynomial $\omega \in \mathrm{inv}(𝒶)$. More on this is at symplectic ∞-Lie algebroid.

Examples

$n=0$: symplectic manifold

In NQ-supermanifold language:

the 2-form being of degree 0 means that only vector fields along degree 0 coordinates may be sent to non-vanishing values. Hence for the form to be non-degenerate, there may not be any higher degree coordinates, hence the NQ-supermanifold must be an ordinary manifold. On that the 2-form is then a closed non-degenerate 2-form, hence a symplectic form.

A $0$-symplectic manifold is an ordinary symplectic manifold.

$n=1$: Poisson manifold

A $1$-symplectic manifold is, as a 1-Lie algebroid, necessarily a Poisson Lie algebroid. As such it is equivalently encoded in an ordinary Poisson manifold.

Regarded as a Lie algebroid, it should by Lie integration integrate to a Lie groupoid with extra structure. These are the symplectic groupoids.

$n=2$: Courant algebroid

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

Recall from the discussion there that one incarnation of this Courant algebroid is as a Lie 3-algebra. If the base manifold is a point, then this is the String Lie 2-algebra.

Relation 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-algebra?s.

An article with more details on this:

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

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 $0\le n\le 3$ is

• Dmitry Roytenberg, Courant algebroids, derived brackets and even symplectic supermanifolds PhD thesis (arXiv)

A good writeup of this material is in

• Dmitry Roytenberg, On the structure of graded symplectic supermanifolds and Courant algebroids (arXiv)

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 Ševera, 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.

Urs Schreiber: I find ${\Sigma }_n$ too undescriptive. But logically I should then be speaking of $n$-Poisson manifolds so that for $n=1$ we get the ordinary version, not for $n=0$. But I find symplectic still much nicer descriptive than “Poisson”, so the above choice of terminology follows aesthetics a little bit more than established terminology. But only slightly. Only by a difference of 1.

Toby: What if you changed the numbering by $1$? Must your $n$ match the $n$ in ${\Sigma }_n$?

Urs: I thought of that, but then I found it weird to have a numbering system that claims that ordinary Poisson geometry is a “2-” and hence supposed a categorified thing. In ${\Sigma }_n$-terminology the $n$ is indeed the degree of the Lie n-algebroid. That seems worthwhile to keep.

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]