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I've started to learn a little about principal bundles (in the smooth category) and while I see how notions like connections and curvature are abstracted from the setting of vector bundles and brought into the principal bundles world, I feel highly unmotivated regarding to why this is done in the first place.

So,

  1. How do principal bundles appear "naturally"?
  2. Why did people started thinking about connections, curvature, parallel transport on principal bundles? Why would one want to put connections on them? Study their curvature? Even determine whether they are flat or not?
  3. Do we gain something by considering the frame bundle associated to a vector bundle and studying its connection and curvature and not working directly with the vector bundle itself?
  4. What kind of problems the machinery of principal bundles helps to solve? What notions does it clarifies?

Feel free to provide examples of results or ideas from any field, including physics. Explicit geometric examples are especially welcomed.

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Many interesting geometric structures can be thought of either sections of principal bundles or sections of bundles or of "twisted products" (I don't know the modern term for this) with a structure group from which a principal bundle can be recovered. –  Alexei Averchenko Nov 6 '12 at 13:16
    
I don't.${}{}{}$ –  Graphth Nov 6 '12 at 13:57

3 Answers 3

up vote 6 down vote accepted

For 1, given any free action of a compact Lie group $G$ on a manifold $M$ (or a free proper action of a noncompact Lie group), the orbit space $M/G$ naturally has the structure of a smooth manifold such that the projection $\pi:M\rightarrow M/G$ is a smooth submersion. (Free means the only group element which fixes at least one point is the identity).

It turns out that $\pi$ is actually a $G$-principal fiber bundle. So, if you care about group actions on manifolds, principal bundles arise naturally.

For 3, one of the main uses of the frame bundle I know of is the following: Suppose a compact Lie group $G$ acts effectively on a Riemannian manifold $M$. (Effective means the only group element which fixes all points is the identity). Since the action is not free, the orbit space $M/G$ isn't a manifold in any kind of natural way (though it is still not so bad as a topological space!).

On the other hand, the action induces an action on the tangent bundle $TM$ (which still might not be free), and induces and action on the frame bundle $FM$. This induced action on the frame bundle is free, so the quotient $FM/G$ is a manifold, so all the tools of differential geometry can be used to study $FM/G$, which in turn can give information about $M$.

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Has the quotient of an effective action been studied from diffeological point of view? –  Alexei Averchenko Nov 6 '12 at 13:05
    
@Alexei: I'm not sure I know what "diffeological" means, but certainly a lot is known about the geometry and toplogy of $M/G$ even when the action is not free. Key words would be "Orbifolds" or "Alexandroff spaces". –  Jason DeVito Nov 6 '12 at 13:25
    
I didn't realize the quotients are in fact orbifolds as I've never worked with them before :) Diffeological spaces is an attempt to extend the category of smooth manifolds by considering concrete sheaves on the site of open subsets of various $\mathbb{R}^n$ and smooth maps. The category of such sheaves is very well behaved. Relevant to this case, if a set $X$ has the structure of a diffeological space, you can pushforward this structure along any surjection $f: X \to Y$, thus orbifolds are naturally diffeological spaces. I briefly saw this topic mentioned in a few preprints. –  Alexei Averchenko Nov 6 '12 at 13:57
    
I see. I don't think all $M/G$ are orbifolds, though, certainly when the isotropy groups are all finite, you get an orbifold –  Jason DeVito Nov 6 '12 at 14:58

Off the top of my head I can think of a few reasons:

  1. For vector bundles there is the useful notion of a connection 1-form. This is a little messy since it is dependent on a frame. For principal bundles however, a connection 1-form is a global, well-defined object.
  2. They are useful in defining bundles. For example if you have a principal-$G$ bundle $P$ over $M$ and a representation $V$ of $G$, you get a vector bundle over $M$ and a connection on the principal bundle induces a connection on the vector bundle. These sorts of vector bundles are often easier to work with because their sections are just $G$ invariant functions from $P \to V$. These constructions are vital in things like spin geometry.
  3. It is also useful from an algebraic topological point of view. General characteristic classes can be defined for a group $G$ which then give characteristic classes for any vector bundle associated to a principal $G$-bundle.
  4. Geometric structures on a vector bundle can be thought of as reductions of the frame bundle. For example a Riemannian metric is a reduction to $O(n)$, a complex structure is a reduction to $GL(n,\mathbb C)$, etc.

EDIT:

Here's a nice concrete example combining my point 2 with Jason's answer. Let $G$ be a compact semisimple Lie group and $T$ a maximal torus. An irreducible representation of $G$ is determined by it's highest weight $\mu$ which determines a map $e^\mu: T \to \mathbb C^\times$, i.e. a one-dimensional complex representation of $T$. By Jason's point $G \to G/T$ is a principal $T$-bundle so by my point 2, you can form the vector bundle associated to $e^\mu$. The group $G$ acts on sections of this vector bundle and the Borel-Weil theorem says that the space of holomorphic sections is the representation of $G$ with highest weight $\mu$. So in using the principal bundle $G \to G/T$, we can geometrically construct all of the representations of $G$ explicitly.

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In string theory, principal bundles can arise quite naturally when considering the compactification of extra dimensions.

T-Duality in string theory is a duality which relates string theory on one spacetime to string theory on another spacetime. The simplest example of T-duality says that bosonic string theory on $\mathbb{R}^{25} \times S_{R}^1$ is equivalent (in terms of the physical observables of the theory) to bosonic string theory on $\mathbb{R}^{25} \times S_{1/R}^1$.

(The subscript on the $S^1$ refers to the radius of the compactified dimension.)

A natural generalisation of T-duality considers a spacetime with a free circle action whose orbit is the uncompactified spacetime manifold $M$. Locally, spacetime is a product $M \times S^1$, but not globally.

One is led to consider a principal $S^1$ bundle $\pi : E \to M$, where $M$ is the uncompactified part of your spacetime, and $E$ is your total spacetime. T-duality then refers to the construction of another principal bundle $\hat{\pi}: \hat{E} \to M$ on which you have an equivalent string theory.

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