# On the existence and classification of prequantization spaces over a closed symplectic manifold

Let $$(M,\omega)$$ be a closed symplectic manifold. If the cohomology class $$[\omega]$$ is rational, that is if it lies in the image of the natural homomorphism $$H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$$, one can construct a prequantization bundle over $$(M,\omega)$$, that is a principal $$S^1$$-bundle $$(V,\alpha) \to (M,\omega),$$ where $$\alpha$$ is an $$S^1$$-invariant one-form on $$V$$ such that $$\pi^* \omega = d \alpha$$ (it is then a contact form).

1. Why do we need this rationality condition ?
2. Is there a case where the homomorphism $$H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R})$$ is not an embedding ? Otherwise, isn't the rationality condition equivalent to $$[\omega] \in H^2(M,\mathbb{Z})$$ ?
3. Is there a proof of the existence of a prequantization space under this condition somewhere ?
4. Often the prequantization space is a complex line bundle. Is there an explicit identification between both notions somewhere ?
5. How could one classify all the prequantization spaces over a given closed symplectic manifold ?

There exists an exact sequence $$0\rightarrow \mathbb{Z}\rightarrow\mathbb{C}\rightarrow\mathbb{C}^*\rightarrow 1$$ defined by the expontial map which induces an isomorphism $$H^2(M,\mathbb{Z})\rightarrow H^1(M,\mathbb{C}^*)$$, this gives the correspondence between $$H^2(M,\mathbb{Z})$$ and line bundles which are classify by $$H^1(M,\mathbb{C}^*)$$ via the identification with Cech cohomology.
The previous correspondence identifies the element of $$H^2(M,\mathbb{Z})$$ with the Chern class of the line bundle, it is for that reason that the condition is needed. Given a line bundle define by the trivialization $$(U_i,g_{ij}$$, you can suppose that there exists a local lift $$g'_{ij}:U_i\cap U_j\rightarrow \mathbb{C}$$ and $$c_{ijk}=g'_{ij}g'_{jk}g'_{ki}\in \mathbb{Z}$$ is a way to describe the Chern class of the line bundle.
If the class of $$\omega$$ is not rational, you can associate to it a gerbe (sheaf of categories) I used this approach in this paper.