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Proca gauge field in AdS space

  1. Jan 20, 2013 #1
    1. The problem statement, all variables and given/known data

    Consider a massivegauge field in [itex]AdS_{d+1}[/itex] space given by the action
    [itex] S=\int_{AdS} d^{d+1}x\sqrt{g}\left(\frac{1}{4}F_{\mu\nu}F^{\mu \nu}+\frac{m^2}{2}A_\mu A^\mu \right)[/itex]

    a) Derive the equations of motion for [itex]A_\mu[/itex] in the Poincaré patch of [itex]AdS_{d+1}[/itex]. The metric is given by [itex]ds^2=\frac{1}{z^2}(dz^2+\delta_{\mu\nu}dx^\mu dx^\nu)[/itex].

    b) Determine the index [itex]\Delta[/itex] by inserting the ansatz [itex]A_\mu(z)=z^\Delta[/itex] into the equations of motion.


    2. Relevant equations



    3. The attempt at a solution

    First of all I think that the indices used in the action and the metric are misleading. [itex]\mu,\nu = z,0,\ldots,d-1[/itex] take d+1 values including the z-direction. Is that right? In the metric however, [itex]x^\mu[/itex] has only d components [itex]\mu=0,\ldots,d-1[/itex].

    By varying the action with respect to [itex]A_\mu[/itex] and using integration by parts one obtains the equation
    [itex] \partial_\mu(\sqrt{g}F^{\mu\nu})+m^2\sqrt{g}A^\nu = 0[/itex]

    The determinant of the metric is [itex]g=z^{-2(d+1)}[/itex]

    I plug this in
    [itex]0 = \partial_\mu(z^{-d-1}F^{\mu\nu})+m^2z^{-d-1}A^\nu = -(d+1)z^{-d-2}F^{z\nu}+z^{-d-1}\partial_\mu F^{\mu\nu}+m^2z^{-d-1}A^\mu[/itex]
    [itex]\Rightarrow 0 = -(d+1)\frac{1}{z}F^{z\nu}+\partial_{\mu}F^{\mu\nu}+m^2A^\mu[/itex]

    Is that right?

    b) I have to plug in [itex]A_\mu(z)=z^\Delta[/itex]
    Therefore [itex]\partial_\nu A_\mu = \delta_{\nu z}\Delta z^{\Delta-1}[/itex].
    [itex]F_{\mu\nu}=(\delta_{\mu z}-\delta_{\nu z})\Delta z^{\Delta-1}[/itex]

    Somehow, I get problems with the indices. Does someone know how to do this more elegantly?

    Cheers, physicus
     
  2. jcsd
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