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Symplectic geometry



A Poisson manifold may be thought of as a Poisson Lie algebroid, a Lie algebroid with extra structure: called an n-symplectic manifold for n=1n = 1.

By Lie integration this Lie algebroid should integrate to a Lie groupoid with extra structure. Symplectic groupoids are supposed to be those objects that integrate n-symplectic manifold aka Poisson manifolds in this sense.

The groupoid algebra of these symplectic groupoids are C-star algebras that may be regarded as the quantization of the original Poisson manifold. This is described in the references below.


The original definition of Weinstein 1987 is this:


A symplectic Lie groupoid is a Lie groupoid X \mathbf{X}_\bullet whose manifold of morphisms X 1\mathbf{X}_1 is equipped with a symplectic structure whose symplectic form ωΩ closed 2(X 1)\omega \in \Omega^2_{closed}(\mathbf{X}_1) is multiplicative in that the alternating sum of its canonical pullbacks to the space X 2\mathbf{X}_2 of composable morphisms vanishes:

0=δω=pr 1 *ωcompose *ω+pr 2 *ω. 0 = \delta \omega = pr_1^* \omega - compose^* \omega + pr_2^* \omega \,.

The manifold of objects X 0\mathbf{X}_0 of a symplectic Lie groupoid X \mathbf{X}_\bullet, def. , carries the structure of a Poisson manifold which is unique, up to isomorphism, with the property that the target map t:X 1X 0t \colon \mathbf{X}_1 \to \mathbf{X}_0 is a homomorphism of Poisson manifolds (canonically regarding the symplectic manifold (X 1,ω)(\mathbf{X}_1, \omega) as a Poisson manifold).

The Poisson manifolds that arise this way as X 0\mathbf{X}_0 of a symplectic Lie groupoid are called integrable Poisson manifolds.


Reformulated more abstractly, def. says that the differential form ω\omega, when extended to a triple

(0,ω,0) k=0,1,2Ω 3k(X k) (0, \omega, 0) \in \oplus_{k = 0,1,2} \Omega^{3-k}(\mathbf{X}_{k})

is a cocycle of degree 3 in the de Rham complex of X\mathbf{X}, identified with the simplicial de Rham complex of the nerve X X_\bullet of XX.

This observation leads to the following generalization


A pre-quasi symplectic groupoid is a Lie groupoid X\mathbf{X} equipped with a differential 2-form ω 2Ω 2(X 1)\omega_2 \in \Omega^2(\mathbf{X}_1) and a differential 3-form ω 3Ω 3(X 0)\omega_3 \in \Omega^3(\mathbf{X}_0) such that

(0,ω 2,ω 3) k=0,1,2Ω 3k(X k) (0, \omega_2, \omega_3) \in \oplus_{k = 0,1,2} \Omega^{3-k}(\mathbf{X}_{k})

is a cocycle in the simplicial de Rham complex of X \mathbf{X}_\bullet, hence such that

δω 2=0 \delta \omega_2 = 0
dω 2+δω 3=0, d\omega_2 + \delta \omega_3 = 0 \,,

where δ= k(1) k k *\delta = \sum_{k} (-1)^k \partial_k^* is the alternating sum of the pullbacks along the face maps of the nerve X \mathbf{X}_\bullet.

This appears as (Xu, def. 2.1, LG-Xu, def. 2.1). This structure is called a twisted presymplectic groupoid in (BCWZ, def. 2.1).


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.


Lie integration and Poisson manifolds

Every Lie groupoid integrating a Poisson Lie algebroid is naturally a symplectic Lie groupoid. Picking always the unique source-simply connected integrating Lie groupoid produces a functor

Σ:PoissonManifoldsSymplecticGroupoids. \Sigma : PoissonManifolds \to SymplecticGroupoids \,.

When the Poisson manifold we start with happens to be a symplectic manifold, then its symplectic Lie groupoid is always the fundamental groupoid of XX:

(X,π)symplecticΣ(X,π) isoΠ(X). (X,\pi)\;\; symplectic \;\;\Rightarrow\;\; \Sigma(X,\pi) \simeq_{iso} \Pi(X) \,.

When XX is simply connected such that Π(X)\Pi(X) is the codiscrete groupoid Pair(X)Pair(X) we have that the symplectic form on Mor(Π(X))=X×XMor(\Pi(X)) = X \times X is ω(ω)\omega \otimes (-\omega), for ω\omega the symplectic form on XX.

Conversely, for every symplectic groupoid X\mathbf{X} there is a unique Poisson manifold structure on its manifold X 0\mathbf{X}_0 of objects such that the codomain map t:X 1X 0t \colon \mathbf{X}_1 \to \mathbf{X}_0 is a homomorphism of Poisson manifolds. (For instance Racaniere, theorem 6.3) One says also that X\mathbf{X} integrates the Poisson manifold X 0\mathbf{X}_0.

Symplectic realization

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.

In geometric quantization of Poisson manifolds

In the groupoid approach to quantization symplectic groupoids are used to discuss geometric quantization not just of symplectic manifolds but more generally of Poisson manifolds.

See geometric quantization of symplectic groupoids.

As Lie integration of Poisson Lie algebroid


A reduced phase space of open Poisson sigma-model

The symplectic groupoid of a Poisson manifold is also the reduced phase space of the open sector of the corresponding Poisson sigma-model. (Cattaneo-Felder 01)


Of Lie-Poisson structure


Let GG be a Lie group with Lie algebra 𝔤\mathfrak{g} and consider the dual vector space 𝔤 *\mathfrak{g}^* equipped with its Lie-Poisson structure. Then the action groupoid 𝔤 *G\mathfrak{g}^* \sslash G of the coadjoint action carries a multiplicative symplectic form ω\omega induced by the identification of the manifold of morphisms with the cotangent bundle of the group, G×𝔤 *T *GG \times \mathfrak{g}^* \simeq T^* G, induced by right translation from the Poincare form on the cotangent bundle. This makes (𝔤 *//G,ω)(\mathfrak{g}^* //G, \omega) a symplectic groupoid which Lie integrates the Lie-Poisson structure on 𝔤 *\mathfrak{g}^*.

This seems to be due to Weinstein 1991, ex. 3.2, see Bursztyn & Crainic 2005, ex. 4.3, Nuiten 2013, p. 111.

∞-Chern-Simons theory from binary and non-degenerate invariant polynomial

nn \in \mathbb{N}symplectic Lie n-algebroidLie integrated smooth ∞-groupoid = moduli ∞-stack of fields of (n+1)(n+1)-d sigma-modelhigher symplectic geometry(n+1)(n+1)d sigma-modeldg-Lagrangian submanifold/ real polarization leaf= brane(n+1)-module of quantum states in codimension (n+1)(n+1)discussed in:
0symplectic manifoldsymplectic manifoldsymplectic geometryLagrangian submanifoldordinary space of states (in geometric quantization)geometric quantization
1Poisson Lie algebroidsymplectic groupoid2-plectic geometryPoisson sigma-modelcoisotropic submanifold (of underlying Poisson manifold)brane of Poisson sigma-model2-module = category of modules over strict deformation quantiized algebra of observablesextended geometric quantization of 2d Chern-Simons theory
2Courant Lie 2-algebroidsymplectic 2-groupoid3-plectic geometryCourant sigma-modelDirac structureD-brane in type II geometry
nnsymplectic Lie n-algebroidsymplectic n-groupoid(n+1)-plectic geometryd=n+1d = n+1 AKSZ sigma-model

(adapted from Ševera 00)


The notion of symplectic groupoids was apparently proposed independently by Karasëv, Weinstein, and Zakrzewski, all motiviated from the problem of quantization.

  • Alan Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987) 101-104 [euclid:bams/1183553676]

  • Alan Weinstein, Symplectic groupoids, geometric quantization, and irrational rotation algebras, in:

    Symplectic geometry, groupoids, and integrable systems (Berkeley, CA, 1989), Springer (1991) 281-290 [doi:10.1007/978-1-4613-9719-9_19, MR1104934]

  • Alan Weinstein, Tangential deformation quantization and polarized symplectic groupoids, in: Deformation theory and symplectic geometry (Ascona, 1996), 301-314, Kluwer (1997) [ISBN:9780792345251, MR1480730]

  • M. V. Karas"ev, The Maslov quantization conditions in higher cohomology and analogs of notions developed in Lie theory for canonical fibre bundles of symplectic manifolds II, Selecta Mathematica Sovietica 8, pp. 235–257, 1989.

  • S. Zakrzewski, Quantum and classical pseudogroups I, II, Commun. Math. Phys. 134 (1990)

See also the references at geometric quantization of symplectic groupoids .

Lecture notes include

  • Sébastien Racanière, Lie algebroids, Lie groupoids and Poisson geometry (pdf)

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:

The formal groupoid version of symplectic groupoids is discussed in

Last revised on November 30, 2023 at 13:48:08. See the history of this page for a list of all contributions to it.