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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 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

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To increase the odds of a useful answer, you might want to include a little explanation of what a "prequantization in Hitchin's sense" is. –  André Henriques Jul 5 '13 at 21:03
André Henriques@ here I mean normal definition of pre-quantization on a manifold. –  Hassan Jolany Jul 5 '13 at 21:30
Is your first sentence a question or a statement? The word order suggests that it is a question, but it lacks a question mark. Really, putting a bit more effort in typing and wording your request is much more likely to result in any answers at all. –  Vladimir Dotsenko Jul 5 '13 at 21:35
@HassanJolany, it would not hurt to give some context and/or a reference where "prequantization" or the "Hitchin sense" is explained —even if all this is somewhat standard in the subject. One of the things that keeps MO working is that we all get a chance to learn from it: frankly, your question, as it is, has not taught me anything! –  Mariano Suárez-Alvarez Jul 5 '13 at 21:59
Your question is somewhat better worded now. As for its previous state, let me put it bluntly: I strongly believe that the rule of a thumb should be that you put in writing your question at least half of the effort you expect someone to put into answering you. If you look at MO questions that get good detailed answers, and also at questions that are upvoted, you will notice that those questions are mostly quite different in style, wording, and level of detail from what you wrote in the first place. Enough said, really. –  Vladimir Dotsenko Jul 6 '13 at 7:25
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3 Answers

up vote 4 down vote accepted

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.

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What does this approach have to to say about functoriality? I am asking because in geometric quantization of Poisson manifolds one may have the same C^* algebra even if the Poisson manifold one starts with are not even Poisson-Morita equivalent. (Let me add that geometric quantization of Poisson manifolds is due, independently, to Weinstein, Karasev and Zakrzewski - and that the prequantization part was done by Weinstein-Xu and in a different approach by Huebschmann; Hawkins contribution was to explain which are the right conditions on polarizations) –  Nicola Ciccoli Jul 6 '13 at 13:39
Yes, I had the same question –  Hassan Jolany Jul 6 '13 at 14:04
@Hassan, maybe we should read the paper... ;) –  Nicola Ciccoli Jul 6 '13 at 15:13
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You may be interested in the preprint "Geometric quantization of generalized complex manifolds" by Alexander Cardona (available here), in which generalized complex manifolds are quantized by "Dolbeault quantization" (which is itself a generalization of Kostant-Souriau geometric quantization).

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Chris R@ Thanks for your comment, –  Hassan Jolany Jul 6 '13 at 10:14
Maybe one should point to the original article where the prequantization of, in particular, generalized complex structures in this sense was defined, which is Weinstein-Zambon, "Variations on Prequantization" arxiv.org/abs/math/0412502 –  Urs Schreiber Jul 6 '13 at 11:11
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Notice that the standard notion of prequantization is not of a bare manifold, but of a manifold equipped with a (pre-)symplectic form.

It seems you are asking what happens to the notion of prequantization as we pass from ordinary symplectic geometry to generalized complex geometry. But notice that you can't just ask "what's the prequantization of $T X \oplus T^\ast X$" (that's why people are complaining in the comment section...) you need to specify which (pre-)symplectic form you would like to prequantize, or else what aspect of the usual process you would want to generalize.

There is indeed a good story here. This may not be what you had in mind, but let me mention it anyway: the generalized tangent bundle underlies (locally) a Courant Lie 2-algebroid (and Courant Lie 2-algebroids is really what generalized complex geometry is about). Now, Courant Lie 2-algebroids are themselves already prequantized structures in a way. For

  • if there is a (pre-)2-plectic form (a closed differential 3-form);

  • it may be higher prequantized by a prequantum 2-bundle

  • and the 2-Atiyah Lie 2-algebroid of that prequantum 2-bundle is the corresponding Courant Lie 2-algebroid, the one that locally looks like the bundle $T X \oplus T^\ast X$ that you wrote down.

This and a bit more is discussed in some detail in our article

  • Domenico Fiorenza, Chris Rogers, Urs Schreiber, Higher geometric prequantum theory (arXiv:1304.0236)
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Your answer was very useful. I am beginner in this field and will try to follow your answer –  Hassan Jolany Jul 5 '13 at 22:44
You might want to start with the companion article arxiv.org/abs/1304.6292 which is more elementary. Section 5 discusses the relation of 2-geometric quantization to Courant Lie 2-algebroids. –  Urs Schreiber Jul 5 '13 at 23:12
In fact this is not completely disjoint from the answer by Chris. In the paper he refers to what is prequantized is a Dirac structure on the generalized tangent space $TM\oplus T^*M$, which defines a Courant algebra on the space of sections. Emphasis is put on the underlying Poisson bracket (true Poisson bracket) that one can thus define on a smaller algebra of "admissible" functions. It seems to me in Cardona's terms not every generalized comple manifold is "pre-quantizable", while you seem to say that you have no obstructions in your approach: correct? –  Nicola Ciccoli Jul 6 '13 at 10:35
No, there is an obstruction in higher geometric prequantization, the obstruction to the lift from closed higher forms to higher differential cocycles, just as in ordinary geometric quantization, but in higher generalization of that. –  Urs Schreiber Jul 6 '13 at 11:14
By the way, geometric quantization of Poisson manifolds was beautifully solved by Eli Hawkins in terms of symplectic groupoids (following Weinstein's famous proposal), see ncatlab.org/nlab/show/… . We discuss at the end of "Higher geometric prequantum theory" arxiv.org/abs/1304.0236 how that is secretly a higher geometric quantization in the above sense. –  Urs Schreiber Jul 6 '13 at 11:16
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