nLab geometric quantization of symplectic groupoids

under construction

Idea

Traditional geometric quantization applies to symplectic manifolds but not to Poisson manifolds. However, every Poisson manifold can be regarded as a symplectic Lie n-algebroid: a Poisson Lie algebroid. This is symplectic, in higher symplectic geometry. Its Lie integration is a symplectic groupoid.

There is an generalization of the machinery of geometric quantization to symplectic groupoids which hence provides a geometric quantization of Poisson manifolds.

Definition

Geometric prequantization of a symplectic groupoid

Formulated in the language of higher differential geometry, what is traditionally called a prequantization of a symplectic groupoid $SG(X,\pi) = \exp(\mathfrak{P}(X,\pi))$ (see Hawkins, section 4.2) is a morphism of smooth 2-groupoids

$$SG(X,\pi) \longrightarrow \mathbf{B}^2 U(1)_{conn^1}$$

to the moduli 2-stack of circle 2-bundles with 1-form connection, such that this 2-connection is trivial on the underlying Poisson manifold

$$\array{ && X \\ & \swarrow && \searrow \\ \ast && \swArrow && SG(X,\pi) \\ & \searrow && \swarrow_{\mathrlap{\widehat{\chi}}} \\ && \mathbf{B}^2 U(1)_{conn^1} }$$

(hgp 13, Bongers 13 Nuiten 13) and such that the image of $\chi$ in Lie algebroid cohomology (hence in Poisson cohomology) is the given Poisson bitensor $\pi \in CE^\bullet(\mathfrak{P}(X,\pi))$.

By (FRS 11, based on FSS 10) the Lie algebroid cocycle

$$\pi \;\colon\; \mathfrak{P}(X,\pi) \longrightarrow \mathbb{R}[2]$$

directly Lie integrates to a morphism of smooth 2-groupoids of the form

$$\exp(\pi)_{conn} \;\colon\; SG(X,\pi)_{conn} = \tau_1 \exp(\mathfrak{P}(X,\pi))_{conn} \longrightarrow \mathbf{B}(\mathbb{R}/\Gamma)_{conn} \,,$$

which is a cocycle in degree-3 ordinary differential cohomology on the differential refinement $SG(X,\pi)_{conn}$ of the symplectic groupoid by Lie algebroid valued differential forms (the moduli stack of the 2d Chern-Simons theory induced by the cocycle), with coefficients in the quotient group $\mathbb{R}/\Gamma$, where $\Gamma \hookrightarrow \mathbb{R}$ is the group of periods of

$$\left( T S^2 \longrightarrow \mathfrak{P}(X,\pi) \stackrel{\pi}{\longrightarrow} \mathbb{R}[2] \right) \in \Omega^2_{cl}(S^2) \,.$$

For $\Gamma = \mathbb{Z}$ this hence yields

$$SG(X,\pi) = \tau_1 \exp(\mathfrak{P}(X,\pi))_{conn} \longrightarrow \mathbf{B}U(1)_{conn} \,.$$

This integrality condition is the one that appears in the traditional literature as (Bonechi-Cattaneo-Zabzine 05, (1.7)-(1.11)) based on (Crainic-Zhu 04, theorem 3).

Forgetting the differential refinement and the principal 2-connection structure we get an underlying circle 2-group-principal 2-bundle on the symplectic groupoid, modulated by

$$\exp(\pi) \;\colon\; SH(X,\pi) \longrightarrow \mathbf{B}^2 U(1) \,.$$

That this coincides with the one in the traditional literature can be seen fairly explicitly for instance from (Bonechi-Cattaneo-Zabzine 05, remark 3).

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Notice that given a symplectic groupoid $(X,\omega)$, the symplectic form defines a class in degree-3 de Rham cohomology $H_{dR}^3(X)$.

(Notice that, while this $\omega$ is typically expressed as a 2-form on $X_1$, this represents indeed a degree-3 cocycle in the simplicial de Rham complex of the nerve of $X$).

We say that $\omega$ is integral if it is in the image of the curvature map

$$curv : H^2_{diff}(X,U(1)) \to H^3_{dR}(X)$$

from the ordinary differential cohomology of $X$. If this is the case, we say that a lift $(\hat X, \nabla)$ of $\omega$ to $\mathbf{H}_{diff}(X, \mathbf{B}^2 U(1))$, hence to the 2-groupoid of circle 2-bundles with connection over $X$, is a prequantum line bundle for $(X,\omega)$.

Notice that this traditional terminology is off by one: the underlying $\hat X \to X$ is a circle 2-group-principal 2-bundle on $X$.

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Polarization of a symplectic groupoid

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(Hawkins, section 4.3)

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Geometric quantization of a symplectic groupoid

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(Hawkins, section 5)

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$$\sqrt{\Omega} \coloneqq \sqrt{\wedge^{max} \Gamma (T_{d t = 0}^\ast \mathcal{G}_1) } \otimes \sqrt{\wedge^{max} \Gamma (T_{d s = 0}^\ast \mathcal{G}_1) }$$

$$\begin{aligned} pr_1^\ast \sqrt{\Omega} \otimes pr_2^\ast \sqrt{\Omega} & \simeq pr_1^\ast \sqrt{\wedge^{max} \Gamma (T_{d s = 0}^\ast \mathcal{G}_1) } \otimes pr_2^\ast \sqrt{\wedge^{max} \Gamma (T_{d t = 0}^\ast \mathcal{G}_1) } \otimes pr_1^\ast \sqrt{\wedge^{max} \Gamma (T_{d t = 0}^\ast\mathcal{G}_1) } \otimes pr_2^\ast \sqrt{\wedge^{max} \Gamma (T_{d s = 0}^\ast \mathcal{G}_1) } \\ & \simeq \sqrt{\wedge^{max} \Gamma (T_{d \circ = 0}^\ast \mathcal{G}_2) } \otimes \sqrt{\wedge^{max} \Gamma (T_{d \circ = 0}^\ast \mathcal{G}_2) } \otimes pr_1^\ast \sqrt{\wedge^{max} \Gamma (T_{d t = 0}^\ast\mathcal{G}_1) } \otimes pr_2^\ast \sqrt{\wedge^{max} \Gamma (T_{d s = 0}^\ast \mathcal{G}_1) } \\ & \simeq \wedge^{max} \Gamma(T_{d \circ = 0}^\ast \mathcal{G}_2) \otimes \circ^\ast \sqrt{\Omega} \end{aligned}$$

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Properties

Relation to deformation quantization

There does not seem to be in the literature a precise relation between the methods of geometric quantization discussed here and methods of deformation quantization. But the following similarity might be relevant:

If the task is to quantize a Poisson manifold, then both methods, Maxim Kontsevich‘s construction of deformation quantization as well as Eli Hawkins’ geometric quantization pass through a 2-plectic geometry on the Poisson Lie algebroid which is induced by the Poisson manifold; Kontsevich’s construction of the star product, as clarified by Cattaneo and Felder, is really that of the 3-point function in the 2-dimension sigma-model QFT whose target space is that Poisson Lie algebroid – the Poisson sigma-model –, and the symplectic 2-groupoid that Hawkins et al consider is the “extended” geometric quantization over the as in extended prequantum field theory associated with this theory.

For more on this see at extended geometric quantization of 2d Chern-Simons theory.

Examples

Ordinary geometric quantization of a symplectic manifold

For $(X,\pi = \omega^{-1})$ an ordinary symplectic manifold the symplectic groupoid is just the pair groupoid equipped with the multiplicative form $s^* \omega + t^* \bar \omega$. Any ordinary prequantum line bundle $P$ and polarization $\mathcal{F}$ of $(X,\omega)$ induces a prequantization $s^* L + t^* \bar L$ and coresponding polarization of the symplectic groupoid. The resulting twisted convolution algebra? is that of compact operators on $X/\mathcal{F}$.

(EH, example 6.1)

Moyal quantization of Poisson vector space

For $(X,\pi)$ a Poisson vector space, hence a vector space $X = V$ equipped with a constant (translating invariant) Poisson bivector, the geometric quantization of the corresponding symplectic groupoid yields the Moyal quantization of $(V, \pi)$.

(GBV 94, EH 06, example 6.2)

Related concepts

• symplectic manifold, geometric quantization

• C-star algebraic deformation quantization

• Poisson Lie algebroid$\stackrel{\exp(-)}{\mapsto}$ symplectic groupoid $\stackrel{central extension}{\mapsto}$ geometric quantization of symplectic groupoids

• symplectic Lie n-algebroid$\stackrel{\exp(-)}{\mapsto}$ symplectic ∞-groupoid $\stackrel{central extension}{\mapsto}$ geometric quantization of symplectic ∞-groupoids

References

Symplectic groupoids were introduced as intended tools for the quantization of Poisson manifolds in

• Alan Weinstein, Symplectic groupoids and Poisson manifolds, Bull. Amer. Math. Soc. (N.S.) 16 (1987), 101–104; 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; Tangential deformation quantization and polarized symplectic groupoids, in Deformation theory and symplectic geometry (Ascona, 1996), 301–314, Kluwer (1997) MR1480730

• Alan Weinstein, Ping Xu, Extensions of symplectic groupoids and quantization, Journal für die reine und angewandte Mathematik (1991) Volume 417 (pdf)

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

Their prequantization is developed in

• 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, 2005, Volume 232, 423-454 (arXiv:math/0311154)

• Marius Crainic, Prequantization and Lie brackets (arXiv:0403269)

• Marius Crainic, Chenchang Zhu, Integrability of Jacobi structures (arXiv:math/0403268)

• Francesco Bonechi, Alberto Cattaneo, Maxim Zabzine, Geometric quantization and non-perturbative Poisson sigma model, Adv. Theor. Math. Phys. 10:683-712, 2006 (arXiv:math/0507223)

A notion of polarization and of actual geometric quantization of symplectic groupoids, yielding a strict deformation quantization of the underlying Poisson manifold, originates in

• Alan Weinstein, Noncommutative geometry and geometric quantization in P. Donato et al. (eds.) Symplectic geometry and Mathematical physics, Birkhäuser 1991

and is further developed in

• Eli Hawkins, A groupoid approach to quantization, J. Symplectic Geom. Volume 6, Number 1 (2008), 61-125. (arXiv:math.SG/0612363)

The case over Poisson vector spaces leading to Moyal quantization was proven earlier in

• José Gracia-Bondia, Joseph Varilly, From geometric quantization to Moyal quantization, J. Math. Phys. 36 (1995) 2691-2701 (arXiv:hep-th/9406170)

and had been claimed without proof in

• Alan Weinstein, in P. Donato et al. (eds.) Symplectic Geometry and Mathematical Physics, (Birkhäuser, Basel, 1991); p. 446.

The interpretation of symplectic groupoids in higher geometry is made fairly explicit in (LGX) above. This is further expanded on in

• Domenico Fiorenza, Chris Rogers, Urs Schreiber, last section of Higher geometric prequantum theory 2013

• Stefan Bongers, section 4 of Geometric quantization of symplectic and Poisson manifolds, MSc thesis 2014

The cohomological quantization of symplectic groupoids in this sense, making the construction in (Weinstein 91, Hawkins 08) Morita invariant is in

• Joost Nuiten, section 5.2.2 of Cohomological quantization of local boundary prequantum field theory, MSc thesis, August 2013