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WZW model

  1. Aug 27, 2009 #1
    Hi I'm trying to understand very basic aspects of the WZW model, and would appreciate any help. The main thing i don't get is how you replace

    [tex]
    I = \frac{1}
    {2}\int\limits_0^1 {\gamma _{\alpha \beta } } \frac{{dx^\alpha }}
    {{d\tau }}\frac{{dx^\beta }}
    {{d\tau }}d\tau
    [/tex]

    with
    [tex]
    I = \frac{{ - k}}
    {4}\int\limits_0^1 {tr} (g^{ - 1} \frac{d}
    {{d\tau }}g)^2 d\tau
    [/tex]


    for some group g

    My thoughts are I don't know how the metric enters and the connection doesn't. For example consider

    [tex]
    {\mathbf{g}} = P\exp \int\limits_0^1 {\frac{{dx^\mu }}
    {{d\tau }}} {\mathbf{A}}_\mu d\tau
    [/tex]

    then

    [tex]
    {\mathbf{g}}^{ - {\mathbf{1}}} \frac{d}
    {{d\tau }}{\mathbf{g}} = - \frac{{dx^\mu }}
    {{d\tau }}{\mathbf{A}}_\mu
    [/tex]

    so the connection enters but not sure how the metric does cos I'm not familiar with any identity that says

    [tex]
    tr\left( {{\mathbf{A}}_\mu {\mathbf{A}}_\tau } \right) = \gamma _{\mu \tau }
    [/tex]
     
  2. jcsd
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