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Homework Help: Help differntial forms

  1. Dec 15, 2004 #1
    I need to compute the following thing

    trace ( (dg + g^g)^2) = d (tr (g ^ dg + 2/3 g ^ g ^ g))

    im messing up completely b/c I dont know how to use trace with differential forms

    tr (dg + g^g) ^ (dg + g^g))
     
  2. jcsd
  3. Dec 16, 2004 #2

    Haelfix

    User Avatar
    Science Advisor

    Just expand it out, theres only a few tricky problems

    1) This is the Chern-Simons form, so you are dealing with Matrix valued forms.. So in general

    tr (A ^ B) = -(1)^(rs) tr (B ^ A) (err 2nd wedge is a power)

    Use the Bianchi identities

    and of course

    d (A ^ B) = dA^B + (-1)^rs (A^dB)

    Note that
    tr (A^A^A^A) = 0 b/c of the last identity

    Expand out both sides, it should only take you a few lines.
     
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