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

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(Hawkins, section 4.2)

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

(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

$$\mathrm{curv}:H_{\mathrm{diff}}^2(X,U(1))\to H_{\mathrm{dR}}^3(X)$$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 $H_{\mathrm{diff}}(X,B^{2}U(1))$, 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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$$\sqrt{\Omega }≔\sqrt{{\wedge }^{\max }\Gamma (T_{dt=0}^{*}{𝒢}_1)}\otimes \sqrt{{\wedge }^{\max }\Gamma (T_{ds=0}^{*}{𝒢}_1)}$$\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) }

$$\begin{array}{rl}{\mathrm{pr}}_1^{*}\sqrt{\Omega }\otimes {\mathrm{pr}}_2^{*}\sqrt{\Omega }& \simeq {\mathrm{pr}}_1^{*}\sqrt{{\wedge }^{\max }\Gamma (T_{ds=0}^{*}{𝒢}_1)}\otimes {\mathrm{pr}}_2^{*}\sqrt{{\wedge }^{\max }\Gamma (T_{dt=0}^{*}{𝒢}_1)}\otimes {\mathrm{pr}}_1^{*}\sqrt{{\wedge }^{\max }\Gamma (T_{dt=0}^{*}{𝒢}_1)}\otimes {\mathrm{pr}}_2^{*}\sqrt{{\wedge }^{\max }\Gamma (T_{ds=0}^{*}{𝒢}_1)}\\ & \simeq \sqrt{{\wedge }^{\max }\Gamma (T_{d\circ =0}^{*}{𝒢}_2)}\otimes \sqrt{{\wedge }^{\max }\Gamma (T_{d\circ =0}^{*}{𝒢}_2)}\otimes {\mathrm{pr}}_1^{*}\sqrt{{\wedge }^{\max }\Gamma (T_{dt=0}^{*}{𝒢}_1)}\otimes {\mathrm{pr}}_2^{*}\sqrt{{\wedge }^{\max }\Gamma (T_{ds=0}^{*}{𝒢}_1)}\\ & \simeq {\wedge }^{\max }\Gamma (T_{d\circ =0}^{*}{𝒢}_2)\otimes {\circ }^{*}\sqrt{\Omega }\end{array}$$\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 literazure 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 $s^{*}\omega +t^{*}\overline{\omega }$. Any ordinary prequantum line bundle $P$ and polarization $ℱ$ of $(X,\omega )$ induces a prequantization $s^{*}L+t^{*}\overline{L}$ and coresponding polarization of the symplectic groupoid. The resulting twisted convolution algebra? is that of compact operators on $X/ℱ$.

(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 $\stackrel{\mathrm{central}\mathrm{extension}}{\mapsto }$ geometric quantization of symplectic groupoids

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

References

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

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

Their prequantization are 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)

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

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

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