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What is the difference between Principle G bundles v.s. Flat G connection?

I heard that for a discrete group $G$ (in physics, or a finite group $G$ in math), the principle G bundles is the same as the flat G connection.

I know the definitions, but I wish to know explicit cases where I can see their differences. Like this MO post.

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Is this a real question? Can we then ask whether there is any difference between vector bundles and flat Koszul connections? Of course, there is: these are just notions of different order! –  Yuri Vyatkin Jun 16 at 2:59
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Of course, the point of this question is to provide some good examples saying that why and how they are different? –  mystery Jun 16 at 3:04
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@mystery: this question is like asking the difference between a horse and a saddle - while related, they are not comparable objects. A G-connection is an additional object defined "on top of" a G-bundle. –  Anthony Carapetis Jun 16 at 3:36
    
@ Anthony, that was a good analogy... If that was an correct answer. –  mystery Jun 16 at 3:52
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It seems that what you're trying to ask is whether every $G$-connection is flat when $G$ is discrete. This seems true to me - unless I'm mistaken there is only a single choice of $G$-connection because the vertical dimension is zero. –  Anthony Carapetis Jun 16 at 3:56

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up vote 7 down vote accepted

For a topological group $G$ and a path-connected base space $X$, a principal $G$-bundle is classified by a map $X \to BG$ where $BG$ is the classifying space, whereas a principal $G$-bundle with flat connection (which presumably is what you meant to ask about) is classified by a map $\pi_1(X) \to G$ (its monodromy), or equivalently by a map $X \to BG_{\delta}$, where $G_{\delta}$ is $G$ equipped with the discrete topology.

The two spaces above are the same when $G$ is discrete but different in general. For example,

$$\pi_1(BG_{\delta}) \cong G$$

while

$$\pi_1(BG) \cong \pi_0(G).$$

So already on a circle the classifications of principal $G$-bundles vs. principal $G$-bundles with flat connection are very different, and the difference persists for higher-dimensional spheres: $BG_{\delta}$ has no higher homotopy, but the higher homotopy groups of $BG$ are the higher homotopy groups of $G$ shifted down one index.

If your question was on a more basic level than this, the point is that a connection is extra structure you can put on a principal bundle that equips it with a notion of parallel transport, and flatness is an extra property guaranteeing that parallel transport only depends on the homotopy class of the path. Without a connection you don't even get a notion of parallel transport.

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