Timeline for On prequantization bundles over integral symplectic manifolds
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| Dec 18, 2020 at 15:52 | comment | added | BrianT | Thank you for your answer @Bertram Arnold. Regarding the argument about the contactomorphism between the different choices of connection 1-forms, are you saying that this contactomorphism descends or gives rise somehow to an element of $H^1(M; U(1))$ ? | |
| Nov 26, 2020 at 12:06 | comment | added | Bertram Arnold | For section 1, the point is that $\mathrm d\alpha_t = \pi^*\omega$ and $\alpha_t|_{\operatorname{ker}(\mathrm d\alpha_t)}$ are independent of $t$, so that $\alpha_t$ is a contact form. Your argument shows that there is a contact isomorphism between the different choices of connection $1$-forms, but this will cover a nontrivial symplectomorphism of $M$. All of your other questions are answered in Chapter 8 of Woodhouse's book "Geometric quantization", if not in the language of contact geometry. | |
| Nov 26, 2020 at 11:10 | history | edited | BrianT | CC BY-SA 4.0 |
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| Nov 25, 2020 at 16:16 | comment | added | BrianT | Thank you @Bertram Arnold. Regarding your first remark, I am afraid I am not familiar enough with the concepts you use to understand your argument. Are you saying that there can be several contact forms on a given prequantization bundle over a given symplectic manifold $(M, \omega)$ ? Regarding your second remark, isn't the period entirely determined by the symplectic form ? | |
| Nov 25, 2020 at 11:20 | comment | added | Bertram Arnold | Regarding your last section, you have to be more explicit with what you mean by $S^1$: Is it a fixed quotient $\mathbb R/\mathbb Z$ (or $\mathbb R/2\pi\mathbb Z$) or general notation for a compact $1$-dimensional Lie group? In other words, when you require the Reeb flow to give a $S^1$-action, do you fix the period beforehand? I personally find it easier to fix one model for $S^1$ once and for all, which in particular will get rid of the normalization $\int_{S^1}\alpha$, and then rescale the symplectic form. | |
| Nov 25, 2020 at 11:14 | comment | added | Bertram Arnold | The line bundle with connection $(V,\alpha)$ is a lift of $\omega\in\Omega^2_{cl}(M)$ to differential integral cohomology $\check H^2(M;\mathbb Z)$, see e.g. ime.usp.br/~2wspjm/slides/2wspjm-koszul-fernandes.pdf In particular, such a lift exists if the deRham class of $\omega$ lies in $H^2(M\mathbb Z)\otimes\mathbb R\subset H^2(M;\mathbb R)$, and in that case lifts are a torsor over $H^1(M;U(1))$ (i.e. flat line bundles). For $M = S^1\times S^1$, this gives a counterexample to your first section since there are flat connections with nontrivial holonomy on the trivial circle bundle. | |
| Nov 24, 2020 at 20:47 | history | asked | BrianT | CC BY-SA 4.0 |