1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: 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
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook

Can you offer guidance or do you also need help?
Draft saved Draft deleted