such that regarded as a function
is has trivial kernel.
We say is a -plectic manifold if
is closed: ;
for all the map
given by contraction of vectors with forms
See also the definition at multisymplectic geometry.
, this appears naturally in 1+1 dimensional quantum field theory.
For orientable, take the volume form. This is -plectic.
let be the canonical Lie algebra 3-cocycle and extend it left-invariantly to a 3-form on . Then is 2-plectic.
We discuss here how to an -plectic manifold for 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 -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”.)
is an isomorphism.
Given , such that
Define . Then is a Poisson algebra.
We can generalize this to -plectic geometry.
Let be -plectic for .
Observe that then is no longer an isomorphism in general.
is Hamiltonian precisely if
This makes uniquely defined.
Denote the collection of Hamiltonian forms by .
Define a bracket
+ cyclic permutations
So the Jacobi dientity fails up to an exact term. This will yield the structure of an L-infinity algebra.
Given an -plectic manifold we get a Lie n-algebra structure on the complex
(where the rightmost term is taken to be in degree 0).
the unary bracket is ;
the -ary bracket is
This is the Poisson bracket Lie n-algebra.
This appears as (Rogers 11, theorem 3.14).
Using a connection on we may write such a section as
for a vector field downstairs, a horizontal lift with respect to the given connection and .
Locally on a suitable patch we have that .
We say that preserves the splitting iff we have
One finds that this is the case precisely if
satisfying a cocycle condition.
Now recall that an exact Courant algebroid is given by the following data:
a vector bundle ;
an anchor morphism to the tangent bundle;
an inner product on the fibers of ;
a bracket on the sections of .
Satisfying some conditions.
The fact that the Courant algebroid is exact means that
is an exact sequence.
The standard Courant algebroid is the example where
the bracket is the skew-symmetrization of the Dorfman bracket
Now with respect to the above Deligne cocycle, build a Courant algebroid as follows:
on each patch is is the standard Courant algebroid ;
glued together on double intersections using the
This gives an exact Courant algebroid as well as a splitting given by the .
The bracket on this is given by the skew-symmetrization of
Here a section preserves the splitting precisely if
for all we have
exactly if is Hamiltonian and .
Then: we have an embedding of L-infinity algebras
given by .
higher and integrated Kostant-Souriau extensions:
|geometry||structure||unextended structure||extension by||quantum extension|
|higher prequantum geometry||cohesive ∞-group||Hamiltonian symplectomorphism ∞-group||moduli ∞-stack of -flat ∞-connections on||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-plectic geometry||Lie n-algebra||Hamiltonian vector fields||line Lie n-algebra||Poisson Lie n-algebra|
|smooth n-group||Hamiltonian n-plectomorphisms||circle n-group||quantomorphism n-group|
(extension are listed for sufficiently connected )
|Poisson algebra||Poisson manifold|
|deformation quantization||geometric quantization|
|algebra of observables||space of states|
|Heisenberg picture||Schrödinger picture|
|higher algebra||higher geometry|
|Poisson n-algebra||n-plectic manifold|
|En-algebras||higher symplectic geometry|
|BD-BV quantization||higher geometric quantization|
|factorization algebra of observables||extended quantum field theory|
|factorization homology||cobordism representation|
|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)
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
with first discussion of application to prequantization in
Some more references on application, on top of those mentioned in the articles above.