nLab symplectic groupoid

Idea

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

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.

Definition

The original definition of Weinstein 1987 is this:

Definition

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

$$0 = \delta \omega = pr_1^* \omega - compose^* \omega + pr_2^* \omega \,.$$

Remark

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

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

Remark

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

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

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

This observation leads to the following generalization

Definition

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

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

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

$$\delta \omega_2 = 0$$

$$d\omega_2 + \delta \omega_3 = 0 \,,$$

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

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).

Remark

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.

Properties

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

$$\Sigma : PoissonManifolds \to SymplecticGroupoids \,.$$

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

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

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

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

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)

Examples

Of Lie-Poisson structure

Example

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

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

Related concepts

• symplectic manifold

• symplectic Lie n-algebroid

• symplectic infinity-groupoid

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

$n \in \mathbb{N}$	symplectic Lie n-algebroid	Lie integrated smooth ∞-groupoid = moduli ∞-stack of fields of $(n+1)$-d sigma-model	higher symplectic geometry	$(n+1)$d sigma-model	dg-Lagrangian submanifold/ real polarization leaf	= brane	(n+1)-module of quantum states in codimension $(n+1)$	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
$n$	symplectic Lie n-algebroid	symplectic n-groupoid	(n+1)-plectic geometry	$d = n+1$ AKSZ sigma-model

(adapted from Ševera 00)

References

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

• Ping Xu, Momentum Maps and Morita Equivalence, J. Diff. Geom (arXiv:math/0307319)

• Camille Laurent-Gengoux, Ping Xu, Quantization of pre-quasi-symplectic groupoids and their Hamiltonian spaces in The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics 232 (2005) 423-454 [arXiv:math/0311154, doi:10.1007/0-8176-4419-9_14]

These “pre-quasi-symplectic groupoids” had been called “twisted presymplectic groupoids” in

• Henrique Bursztyn, Marius Crainic, Alan Weinstein, Chenchang Zhu, Integration of twisted Dirac brackets (arXiv:math/0303180)

The identification with reduced phase spaces of the open Poisson sigma-model is in

• Alberto Cattaneo, Giovanni Felder, Poisson sigma models and symplectic groupoids, in Quantization of Singular Symplectic Quotients, (ed. Klaas Landsman, M. Pflaum, M. Schlichenmeier), Progress in Mathematics 198 (Birkhäuser,

  2001), 61–93. (arXiv:math/0003023)

Further developments:

• Alan Weinstein, Noncommutative geometry and geometric quantization, in P. Donato et al. (eds.) Symplectic geometry and Mathematical physics, Progr. Math 99 Birkhäuser (1991) 446-461. doi:10.1007/978-1-4757-2140-9_23.

• Ping Xu, Morita equivalence and symplectic realizations of Poisson manifolds, Annales scientifiques de l’École Normale Supérieure, Sér. 4, 25 no. 3 (1992) (NUMDAM)

• Henrique Bursztyn, Marius Crainic, Dirac structures, momentum maps and quasi-Poisson manifolds, in: The Breadth of Symplectic and Poisson Geometry, Progress in Mathematics 232, Birkhäuser (2005) 1-40 [doi:10.1007/0-8176-4419-9_1]

• F. Bonechi, N. Ciccoli, N. Staffolani, M. Tarlini, The quantization of the symplectic groupoid of the standard Podles sphere (arXiv:1004.3163)

• Joost Nuiten, pp. 108 in: Cohomological quantization of local prequantum boundary field theory, MSc thesis, Utrecht (August 2013) [pdf, dspace:1874/282756, talk slides]

The formal groupoid version of symplectic groupoids is discussed in

• Alberto Cattaneo, Benoit Dherin, Giovanni Felder, Formal symplectic groupoid, Comm. Math. Phys. 253 (2005), no. 3, 645–674 math.SG/0312380 doi; Formal Lagrangian operad, math.SG/0505051