Let $M$ be a pre-symplectic manifold.In recent years several Geometrists are working on $TM \bigoplus T^\*M$ which has fascinated the complex and Poisson geometry. In recent decade also Nigel Hitchin introduced [generalized complex manifold](http://en.wikipedia.org/wiki/Generalized_complex_structure) on this manifold and found several applications in physics. I have a question about geometric quantization on $TM \bigoplus T^\*M$ . It is well-known that we have at least one pre-quantization on cotangent bundle $T^\*M$.
 Is there any pre-quantization on generalized tangent bundle $TM \bigoplus T^\*M$ with Generalized complex structure (in Hitchin sence)?. The prequantization in the case of existence is unique.?


PS:I edited my question after some comments



prequantization on $TM \bigoplus T^*M$
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](http://ncatlab.org/nlab/show/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](http://ncatlab.org/nlab/show/Poisson%20sigma-model). This curious "[holographic](http://ncatlab.org/nlab/show/holographic%20principle)" 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](http://ncatlab.org/nlab/show/geometric%20quantization) of Poisson manifolds should/would proceed via some kind of geometric quantization of their induced [symplectic groupoids](http://ncatlab.org/nlab/show/symplectic%20groupoid), which are the Lie groupoids [that Lie integrate](http://ncatlab.org/nlab/show/Lie%20integration) the [Poisson Lie algebroid](http://ncatlab.org/nlab/show/Poisson%20Lie%20algebroid) of the Poisson manifold. This program of [geometric quantization of symplectic groupoids](http://ncatlab.org/nlab/show/geometric%20quantization%20of%20symplectic%20groupoids) 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 


* Domenico Fiorenza, Chris Rogers, Urs Schreiber, *[Higher geometric prequantum theory](http://ncatlab.org/schreiber/show/Higher%20geometric%20prequantum%20theory)* ([arXiv:1304.0236](http://arxiv.org/abs/1304.0236))


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](http://ncatlab.org/nlab/show/prequantum+circle+n-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](http://ncatlab.org/nlab/show/higher%20differential%20geometry): 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](http://ncatlab.org/schreiber/show/Higher%20local%20prequantum%20field%20theory))
* 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](http://ncatlab.org/schreiber/show/Higher%20local%20prequantum%20field%20theory)) 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](http://arxiv.org/abs/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.


