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Connections and Curvature

  1. Feb 7, 2007 #1
    Hi folks. I am a mathematician and my research
    is on the curvature equation

    D(\gamma) = F

    where \gamma is a Lie-algebra valued one-form and F is a Lie-algebra-valued 2-form.

    I want a very
    rough idea how fiber bundles and associated vector
    bundles are used in physics. I've tried to read up on it, but most
    of the stuff is too technical and I
    just want sort of a bird's eye view of how they are used.

    More specifically, I'd like to know if you guys ever look at vector bundles whose structure group is Nilpotent. The problem I am working on is when the base manifold is dimension 3 and the structure group is nilpotent.
  2. jcsd
  3. Feb 8, 2007 #2
    Do you mean 'Nakahara : Geometry, Topology, and Physics' instead? I'm guessing the likelyhood of two such similarly named authors and books is unlikely ;) Excellent recommendation though. I'm trying to work through the bundle chapters at the moment :cry:
  4. Feb 11, 2007 #3
    Hi Chris.

    I have read Frankel's book (or rather, parts of it). It's not too technical, but it is confusing and still doesn't really answer my question about how nilpotent structure groups are used.

    I got your email that you replied to this posting, but I could not read most of it - there was just a lot of junk with html code in it. Could you reply in the forum here or email me directly?

    Thanks a lot for replying - nobody else has :cry:
  5. Feb 11, 2007 #4
    Not sure if this is of any help, but this is a brief summary of my understanding of how things work in one application of fibre bundles:

    Physics makes extensive use of a type of model called a gauge theory. Such a theory is "invariant" (i.e. the field equations retain their form) under "gauge transformations". A simple example is a system consisting of charged scalar fields, and electromagnetic interactions. The charged scalar fields (think of wave functions in quantum mechanics) are modelled as sections of a (complex) line bundle. We want the theory to be invariant under phase transformations (wave functions are only significant up to a phase factor). It would be nice if the theory was also invariant up to a phase factor which could be defined independently at each point. In order to do this, we need to take our field equations for our complex field and replace any derivatives in them by covariant derivatives, i.e. we introduce a connection on our bundle. This connection corresponds to the electromagnetic potential and its curvature to the electromagnetic field. The structure group of the principal bundle to which our complex line bundle is associated is U(1), which is abelian, and therefore nilpotent and so may be of some interest in your case?
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