The connection as a choice of horizontal subspace?

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SUMMARY

The discussion centers on the concept of connections in fiber bundles within gauge theory, specifically addressing the partitioning of tangent spaces into vertical and horizontal components. The vertical space, denoted as Vp, consists of vectors tangent to the fiber, while the horizontal space, Hp, is defined such that Vp + Hp equals the total tangent space TpP. A key point of confusion is the misconception that finding Vp uniquely determines Hp, as there are infinitely many valid choices for Hp that satisfy the conditions of a vector subspace. The participants clarify the importance of understanding direct sums in vector spaces to resolve these issues.

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  • Understanding of fiber bundles in gauge theory
  • Familiarity with tangent spaces and vector subspaces
  • Knowledge of direct sums in linear algebra
  • Basic concepts of gauge theory and connections
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Mathematicians, theoretical physicists, and students studying gauge theory, fiber bundles, and linear algebra who seek a deeper understanding of connections and their implications in these fields.

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Hi,
I'm trying to understand the fiber bundle formulation of gauge theory at the moment, and I'm stuck on the connection. Every reference I've found introduces the idea of a connection on a principle bundle as a kind of partitioning of the tangent space at all points in the total space into a "vertical space" and a "horizontal space". The vertical space Vp consists of vectors in TpP which are also tangent to the fiber at p, and the horizontal space Hp is a set of vectors such that Vp+Hp=TpP.
What I don't understand is why finding Vp doesn't uniquely specify Hp. It should be possible to construct TpP without defining a connection, right? If so, wouldn't Hp just be every element of TpP that is not also in Vp? I don't see how we are free to make this partition ourselves. Where am I going wrong?

Thanks for reading!
 
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Take R² for instance, and for simplicity, assume that V = {(0,y) | y in R}. Then you're saying "take H:= R² - V". But that's not a subspace! (Perhaps you overlooked the fact that H is supposed to be a vector subspace?)

On the other hand, H:={(x,0) | x in R} is a natural candidate... but there are (infinitely many) other choice as H:={ (x,ax) | x in R} for any a in R would do just as well.
 
quasar987 said:
(Perhaps you overlooked the fact that H is supposed to be a vector subspace?)
Yeah, I did. I was also getting mixed up over how to take the direct sum of two vector spaces. But now I see how it all works. Cheers!
 

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