2
$\begingroup$

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 ?

Thanks in advance for your help !

$\endgroup$
  • $\begingroup$ salut Brian, alors cette thèse ? $\endgroup$ – ychemama Dec 28 '18 at 10:19
  • $\begingroup$ Monsieur Yvan ! $\endgroup$ – BrianT Dec 28 '18 at 10:41
2
$\begingroup$

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.

Aristide, Tsemo. "Gerbes, 2-gerbes and symplectic fibrations." The Rocky Mountain Journal of Mathematics (2008): 727-777.

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.