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Yang Mills Stress Tensor

  1. Mar 17, 2007 #1
    I am working through a set of notes on conformal field theory by Schellekens and want to show the conformal invariance of N=4 SYM theory in four dimensions. I start with the action
    [tex]S=\frac{1}{4g}\int d^Dx \sqrt{g}Tr\left(F_{\mu\nu}F^{\mu \nu})[/tex]
    There's only the metric in the action to worry about, in the Jacobian. (Is this wrong?)

    But then the stress tensor I get is this (Abelian case):
    [tex]T^{\alpha\beta} = g^{\alpha\beta}F_{\mu\nu}F^{\mu \nu}[/tex].

    I'm pretty sure that this isn't right because I was assuming I'd use the SYM equation of motion to show the divergence condition on the stress tensor. Instead, I get something like (Abelian case):
    [tex]\partial_{\alpha}T^{\alpha\beta} = \partial^{\beta}F_{\mu\nu}F^{\mu \nu}[/tex].

    Can anyone point me in the right directions? Am I missing something in the actoin (i.e. a hiding metric)?
    Last edited: Mar 17, 2007
  2. jcsd
  3. Mar 18, 2007 #2


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    F^{\mu \nu} = g^{\mu \rho} g^{\nu \sigma} F_{\rho \sigma}
  4. Mar 18, 2007 #3
    Sigh. Then I definitely see how to use the eom to get the divergence condition.

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