By Lie integration this Lie algebroid should integrate to a Lie groupoid with extra structure. Symplectic groupoids are supposed to be these objects that integrate n-symplectic manifold aka Poisson manifolds in this sense.
The original definition of (Weinstein) is this:
A symplectic Lie groupoid is a Lie groupoid whose manifold of morphisms is equipped with a symplectic structure whose symplectic form is multiplicative in that the alternating sum of its canonical pullbacks to the space of composable morphisms vanishes:
The manifold of objects of a symplectic Lie groupoid , def. 1, carries the structure of a Poisson manifold which is unique, up to isomorphism, with the property that the target map is a homomorphism of Poisson manifolds (canonically regarding the symplectic manifold as a Poisson manifold).
The Poisson manifolds that arise this way as of a symplectic Lie groupoid are called integrable Poisson manifolds.
Reformulated more abstractly, def. 1 says that the differential form , when extended to a triple
This observation leads to the following generalization
Since therefore a (pre-)symplectic groupoid is really a Lie groupoid equipped with a cocycle in degree-3 de Rham cohomology (instead of degree 2 as for a symplectic manifold), it is really rather an object in 2-plectic geometry.
Conversely, for every symplectic groupoid there is a unique Poisson manifold structure on its manifold of objects such that the codomain map is a homomorphism of Poisson manifolds. (For instance Racaniere, theorem 6.3) One says also that integrates the Poisson manifold .
The source map of a symplectic groupoid over a Poisson manifold constitutes a symplectic realization of this Poisson manifold, hence its canonical desingularization via Lie integration. See at symplectic realization for more.
Let be a Lie group with Lie algebra and consider the dual vector space equipped with its Lie-Poisson structure. Then the action groupoid of the coadjoint action carries a multiplicative symplectic form induced by the identification of the manifold of morphisms with the cotangent bundle of the group, , induced by right translation from the Poincare form on the cotangent bundle. This makes a symplectic groupoid which Lie integrates the Lie-Poisson structure on .
This appears as (Bursztyn-Crainic, example 4.3).
|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)
Alan Weinstein, Symplectic groupoids, geometric quantization, and irrational rotation algebras in Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), 281–290, Springer, New York, (1991) MR1104934.
See also the references at geometric quantization of symplectic groupoids .
Lecture notes include
The notion of pre-quasi-symplectic groupoids is introduced and the intepretation of symplectic groupoids in higher geometry is made fairly explicit in
These “pre-quasi-symplectic groupoids” had been called “twisted presymplectic groupoids” in
The identification with reduced phase spaces of the open Poisson sigma-model is in
Further developments include
The formal groupoid version of symplectic groupoids is discussed in