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I am studying abelian Chern-Simons theory on a non-trivial manifold. Could you let me know how accurate my understanding is?

Here's what I figured out:

The action of $U(1)$ leaves the action invariant but not the Lagrangian, so to compute observables I need to fix a gauge, which is equivalent to imposing a constraint on the field. If the manifold is an homology sphere (meaning the first homology group $H_1(M)$ is trivial), I can describe the field as a connection living on the fiber at each point. Since every closed path is contractible, and every base point is equivalent, I can associate a fiber to the unique element of $H_1(M)$. Therefore, I can globally describe the fibration as a product of the fiber and the base space, making it globally trivializable. Through holonomy I can move between orbits as horizontal lift of a loop. Fixing the gauge corresponds to fix a gauge orbit, akin to fix a 'point' in the fiber.

However, if the manifold is not an homology sphere, I have an equivalence class of connections (the Deligne-Beilinson classes) for each element of $H_1(M)$. This constructs a space of inequivalent fibers forming the Deligne space. This implies that the manifold is not globally trivializable since there is no way to collapse the fibers associated with each element of $H_1(M)$ onto a single fiber, describing the fibration as a BxF product.

  1. Does this simplified interpretation make sense?

  2. Does this mean that I can simply say that if $H_1(M)$ is non-trivial, then I have one DB class of connections for each class of loops? (For example, if I have a path with one non-contractible loop, I have one class of connections; if I have a path with two non-contractible loops, I have another class of connections, and so on.)

  3. In my notes I wrote:

"specifying the transition function between the connections related to two intersecting neighborhood we can eliminate the local gauge transformation of the connection only if principal bundle is trivializable".

Why is that? Is that related to the fact that a bundle admints a section iff is trivial?

Does obstruction have something to do with all of this? (I came across this concept during my searches on internet, but I have not studied it.)

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    $\begingroup$ It's not clear to me what the specific question here is, because a lot here seems confused about the basic setup, but you give no references nor even explain the idea of Chern-Simons theory, so it's really unclear what you expect answerers here to do. "Teach me about CS theory" is really too broad a question. $\endgroup$
    – ACuriousMind
    Commented Dec 23, 2023 at 13:17
  • $\begingroup$ The question is about the configuration space of connections of gauge theory in a nontrivial manifold: i know that some of the things I wrote are true (e.g. a manifold is not an homology sphere -> the configuration space is a collection of inequivalence class of connections living on fibers related to every element of $H_1$), but i'm not sure of the way i understood them and the way I related them (e.g. a manifold is not an homology sphere -> if i have some kind of "hole" and i look for possible loops, i have a class of connections (the fiber) for every class of loop (the element of $H_1$)). $\endgroup$
    – polology
    Commented Dec 23, 2023 at 16:22
  • $\begingroup$ The reference is mainly this one: hal.science/hal-00973733/document $\endgroup$
    – polology
    Commented Dec 23, 2023 at 16:24
  • $\begingroup$ I didn't explain the idea of Chern-Simons theory because i thought that it's hard to answer those question without knowing anything about it, but perhaps it's not true or it's not the right way to post questions here (?) $\endgroup$
    – polology
    Commented Dec 23, 2023 at 16:27

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