On a principal bundle, at each point you have a tangent vector space. At a given point, the vectors tangent to the fiber form the vertical vector space. Then the vector space at that point is a direct sum of the vertical vector space and what's called the horizontal vector space.

This is all standard stuff in gauge theory. What I don't intuitively understand is why is the horizontal vector space not unique?


It's not unique simply because its definition - it's another vector space in the direct sum that must be equal to the total vector space - doesn't determine it uniquely.

While the vertical vector space is uniquely determined - you may perform a simple test whether a particular vector from the total vector space belongs to the vertical vector space or not - a similar test doesn't exist for the horizontal vector space.

So it's like writing the two-dimensional plane as a direct sum of two vector spaces. The vertical vector space may be determined - for example, it may contain all vertical vectors of the form $(0,y)$ for a real $y$. However, the horizontal vector space may be chosen to be the space of all vectors of the form $x(1,a)$ where $a$ is any fixed number (labeling different choices of the horizontal vector space) and $x$ takes any real value.

You could think that the vectors $(x,0)$ are preferred - i.e. the choice $a=0$ is privileged. But that's only true if you want to construct an orthogonal basis - or if you had a reason to take the adjective "horizontal" literally. In general, the two vector spaces don't have to be orthogonal, and in some cases, an inner product isn't even available. And there's no God-given definition of "horizontality" (unlike my Cartesian example). The two-dimensional plane may be generated from pretty much any basis with 2 vectors, so even if you choose one of them, the other vector may be anything (except for multiples of the first one) to get the right direct sum. And all these solutions are equally good.

The example above easily generalizes to the case when both spaces are multi-dimensional.

  • $\begingroup$ This makes sense. The vector spaces are not necessarily orthogonal, in fact "orthogonal" or "not orthogonal" are not even defined without an inner product. Even if there is an inner product you can have non-orthoganal bases and so infinitely many possible "horizontal" vector spaces. $\endgroup$ – yalis Feb 15 '11 at 23:36

Giving a horizontal bundle is equivalent as giving a connection. It is one of the definitions actually. Connections are not unique. They are extra structure on the bundle.



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