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 .
A symplectic Lie -algebroid is a pair
The Chern-Simons element that witnesses this transgression is the Lagrangian of the corresponding AKSZ theory sigma-model with as its target space and the invariant polynomial as the (curvature of) its background gauge field.
A 0-Lie algebroid is just a smooth manifold .
Its Chevalley-Eilenberg algebra is the algebra of smooth functions on
The Weil algebra of is
A symplectic manifold, being a pair
consisting of a smooth manifold and a symplectic 2-form , is a symplectic Lie 0-algebroid.
is that of multi-vector fields on , equipped with the differential given by the Schouten bracket.
If we work locally in coordinates then is generated from degree 0 elements and degree 1 elements . The differential is
The Poisson tensor is and that this is a Lie algebroid cocycle is the fact that
By definition the Weil algebra is generated from the , the and their shifted partners and . The differential here is
The invariant polynomial that is in transgression with the cocycle is
One checks directly that the element
is a Chern-Simons transgression element for and ,
i.e. . The restriction of to is evidently the Poisson tensor .
More details on this at Chern-Simons element.
For a Poisson manifold with Poisson tensor , the pair
A -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 on a semisimple Lie algebra. The extension classified by this is the string Lie 2-algebra.
Since the symplectic form on a symplectic Lie -Algebroid may be understood Lie theoretically as an invariant polynomial on an L-∞ algebroid, every symplectic Lie -algebroid serves as a target space for an ∞-Chern-Simons theory: this is AKSZ theory.
There is also the closely related notion of multisymplectic geometry. See
for some relations of this to the above situation for . Essentially multisymplectic geometry studies the higher -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:
|symplectic Lie n-algebroid||Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of -d sigma-model||higher symplectic geometry||d sigma-model||dg-Lagrangian submanifold/ real polarization leaf||= brane||(n+1)-module of quantum states in codimension||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|
|symplectic Lie n-algebroid||symplectic n-groupoid||(n+1)-plectic geometry||AKSZ sigma-model|
(adapted from Ševera 00)
A good writeup of this material is in
The idea for all was then sketched, together with many other ideas about L-infinity algebroids in the article with the nice title
What we call -symplectic manifold here is called -manifold there.
Warning This article here uses the term ”-symplectic” in a related but not identical sense to the one used here:
A discussion of aspects of how multisymplectic geometry related to -symplectic manifolds is in
A discussion of symplectic Lie n-algebroids from an infinity-Lie theory perspective as discussed here is in