Hassan asks me in the comment section of my other reply to post the following as a separate reply, too.

The issue of geometric quantization of Poisson manifolds came up in the discussion section, and its potential relation to higher and/or generalized complex geometry. Indeed, one can find the following nice story here:

First recall that Kontsevich fomously gave a general formula for formal deformation quantization of Poisson manifolds, which later Cattaneo and Felder realized as being the 3-point function of the open string in the perturbatively quantized Poisson-sigma-model. This curious "holographic" quantization of particle mechanics as open string endpoints remained conceptually mysterious since, I would say. It just so happened.

Independent of that, Weinstein suggested that the geometric quantization of Poisson manifolds should/would proceed via some kind of geometric quantization of their induced symplectic groupoids, which are the Lie groupoids that Lie integrate the Poisson Lie algebroid of the Poisson manifold. This program of geometric quantization of symplectic groupoids was finally completed rather beautifully by Eli Hawkins, who thereby gives *strict* $C^\ast$-algebraic deformation quantization of Poisson manifolds.

One may observe though, and this we indicate in the Examples-section 2.6.3 of

that what Eli Hawkins does is *secretly* really a 2-geometric quantization: in symplectic groupoid theory it is traditional to speak of a *multiplicatice* prequantum bundle on the space of morphisms of the symplectic groupoid... but of course this is equivalently and more intrinsically a prequantum 2-bundle (a bundle gerbe) on the whole groupoid. So this is as in generalized complex geometry (the general version "twisted by a 3-form") only that it is more genuinely higher differential geometric: here the base space is not a smooth manifold but a genuine Lie groupoid not equivalent to a manifold.

Moreover one can observe that:

the symplectic groupoid is the moduli stack of (instanton sectors) of the 2d Chern-Simons field theory which is the non-perturbative version of the Poisson sigma-model;

the above prequantum 2-bundle is the corresponding local action functional (in the general sense of Higher local prequantum field theory)

its specific incarnation as a multiplicative line bundle on the space of morphisms implies that the inclusion of the original Poisson manifold into the symplectic groupoid is (in the formally precise sense, as explained at Higher local prequantum field theory) a *boundary condition* for the Poisson sigma model.

lastly, that from this perspective the geometric quantization that Eli Hawkins writes out can be seen to to be equivalently the 2-geometric holographic boundary field quantization of the prequantum 2-bundle on the moduli stack of fields of the 2d Chern-Simons theory which is the non-perturbatively integrated Poisson sigma-model.

All this is in a way a higher geometric definition and refinement of gemetric quantization of generalized complex geometry in the sense that if we would replace the moduli stack of fields here with a plain manifold, then the corresponding Atiyah-2-groupoid of the prequantum 2-bundle is the Lie integration of the Courant Lie 2-algebroid which is "twisted" by that prequantum 2-bundle.

As I said, an indication of this story is in the examples section 2.6.3 of arXiv:1304.0236. More details will appear, as noted there, in a two Master theses of two Master students in Utrecht, Joost Nuiten and Stefan Bongers, that will be made public end of August 2013, hence in a bit less than months from now.