∞-Lie theory (higher geometry)
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 $n = 0$.
A symplectic Lie $n$-algebroid is a pair
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.
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.
A 0-Lie algebroid is just a smooth manifold $X$.
Its Chevalley-Eilenberg algebra is the algebra of smooth functions on $X$
The Weil algebra of $X$ is
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
consisting of a smooth manifold $X$ and a symplectic 2-form $\omega$, is a symplectic Lie 0-algebroid.
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
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
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
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
The invariant polynomial $\omega$ that is in transgression with the cocycle $\nu = \pi$ is
One checks directly that the element
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
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.
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.
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
There is also the closely related notion of multisymplectic geometry. See
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:
∞-Chern-Simons theory from binary and non-degenerate invariant polynomial
(adapted from Ševera 00)
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
A good writeup of this material is in
The idea for all $n$ was then sketched, together with many other ideas about L-infinity algebroids in the article with the nice title
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:
A discussion of aspects of how multisymplectic geometry related to $n$-symplectic manifolds is in
A discussion of symplectic Lie n-algebroids from an infinity-Lie theory perspective as discussed here is in
The H-cohomology of graded symplectic forms is considered in