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Going from a complex rep to a real rep

  1. Jan 17, 2007 #1
    Not sure if this is appropriate here or the differential geometry forum, but anyway....

    I've got a Kahler potential and from that can compute it's metric and then move onto the dynamics of the fields which the potential is a function of. One of the terms I consider is [tex]K_{a\bar{b}}\dot{\Phi}^{a}\dot{\Phi}^{\bar{b}}[/tex] where [tex]\Phi[/tex] is a complex field.

    Now instead of considering the complex metric [tex]K_{a\bar{b}}[/tex] I want to go a point of view involving just real fields, though obviously twice as many. Papers I've read and books I've flicked through just seem to say "And [tex]G_{ij}[/tex] is the metric associated with the real representation of the Kahler potential" or words to that effect. Problem is, I'm not sure how to compute it, but have a guess by the following logic :

    Let K be a function of n complex fields [tex]\Phi[/tex] and so by definition of metric :

    [tex]ds^{2} = K_{a\bar{b}}d\Phi^{a} d\Phi^{\bar{b}} = G_{ij}d\phi^{i}d\phi^{j}[/tex]

    where [tex]\Phi^{a} = \phi^{a} + i\phi^{a+n}[/tex]. Putting that into the Kahler form gives

    [tex]K_{a\bar{b}}(d\phi^{a} + id\phi^{a+n})(d\phi^{a} - id\phi^{a+n}) = K_{ab}d\phi^{a}d\phi^{b}+K_{ab}d\phi^{a+n}d\phi^{b+n} = G_{ij}d\phi^{i}d\phi^{j}[/tex]

    Matching efficents of [tex]d\phi^{i}d\phi^{j}[/tex] gives [tex]G_{ab} = K_{ab} = G_{a+n \, b+n}[/tex]

    This then gives the general form

    [tex]G_{ij} = \left( \begin{array}{cc}K_{ab} & 0 \\ 0 & K_{ab} \end{array} \right) [/tex]

    This seems to make sense, is it correct?
     
    Last edited: Jan 17, 2007
  2. jcsd
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