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Kahler Metric

  1. Dec 12, 2008 #1
    If you want some background on this question (i.e. why I'm asking), let me know.

    But either way, this is a basic (stupid) question: what happens if you calculate the inverse Kahler metric and it's singular?
  2. jcsd
  3. Dec 12, 2008 #2
    Ok, a bit more details. And I think I see the problem.

    Seiberg and Nelson claim that U(1)R -> broken SUSY, except in some special cases. This proof hinges on a redefinition of fields so that you can factor the superpotential in to a part that has R charge and one that doesn't:

    W = T f(a,b,c,...)

    where T has r charge 2, and all of the other fields have r charge 0. What I didn't realize an hour ago is that this redefinition probably means that your fields aren't canonically normalized---i.e. at least ONE of them looks like A/X, where A and X are two chiral superfields.

    You run into problems when T = 0...specifically, when T = 0, it could be that some other field, say a, can be zero, eg a = 0, and you can satisfy the F=0 constraints with a good U(1)R, so you might be fooled and think that you have a SUSY ground state. In the specific example I'm looking at (and probably more generally), a = A/x and T = X (this is a simple ORaifeartaigh model):

    [tex]\mathcal{W} =\frac{1}{2} h X A^2 + mAB + gX[/tex].

    In terms of the new fields:

    [tex]\mathcal{W} = T \{\frac{1}{2}h a^2 + mab + g \}, [/tex]

    where g is the dimensionful constant, not a field!

    Anyway, this means that when you now compute the scalar potential, you have to write

    [tex]V \sim F_i \bar{F}_{\bar{j}} K^{i\bar{j}}[/tex]

    i.e. the Kahler metric is no longer flat. In the case I am describing, when T = 0, the new Kahler metric (in terms of non-canonical fields) is now singular.

    So it seems I have a bit of a hole: I know that F = 0, but K = infinity. Obviously something weird is happening. In terms of the original OR model, SUSY is clearly broken (SUSY is broken everywhere). But how do I see that SUSY breaking in terms of the new fields?

    I suspect that there's something to do with a singular Kahler matrix, but I'd like a reference to a paper, or a gentle pat on the head with a "Good boy", or something.
    Last edited: Dec 12, 2008
  4. Dec 14, 2008 #3

    Is it a bad question, or does no one know the answer? Feel free to tell me I'm a dumbass :)
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