Geometric quantization associates to a symplectic manifold $(M,\omega)$ a hermitian line bundle $L \to M$ with connection $\nabla$ whose curvature is $\omega$ (up to some constant).
Without talking about curvatures of connections and hermitian line bundles, one can also define a prequantization space over $(M,\omega)$ as a principal $S^1$-bundle $\pi : (V,\alpha) \to (M,\omega)$, where $\alpha$ is an $S^1$-invariant $1$-form on $V$ satisfying $d \alpha = \pi^* \omega$. These two conditions imply that $\alpha$ is a contact form.
I am looking for explanations or references regarding the following questions:
- it is said everywhere that the existence of such prequantization bundle is equivalent to the cohomology class of $\omega$ lying in the image of the natural homomorphism $$H^2(M,\mathbb{Z}) \to H^2(M,\mathbb{R}).$$ I know that it has to do with some identification involving the first Chern class, but I can't find any good reference or detailed proof about this anywhere;
- I would also like to know what classifies such bundles over a given symplectic manifold;
- I would like to find a detailed proof of the equivalence of these two definitions;
- Is there a physical meaning to the fact that $\alpha$ is a contact form ?
