Proca gauge field in AdS space

[physicus]

Homework Statement

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.

Homework Equations

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

 

[mark726]

Dear physicus,

Your solution for part a) is correct. For part b), there are a few issues with your approach. First, the ansatz for A_\mu should be A_\mu(z)=z^\Delta A_\mu(x), where A_\mu(x) is a function of the boundary coordinates x^\mu. This is because the field A_\mu is defined on the boundary of AdS_{d+1}, not in the bulk. Therefore, the derivatives \partial_\mu A_\nu should be replaced by \partial_\mu A_\nu(x), and similarly for F_{\mu\nu}. Also, the expression for F_{\mu\nu} should be corrected as F_{\mu\nu}=(\partial_\mu A_\nu-\partial_\nu A_\mu). Finally, after plugging in the ansatz, the equation of motion becomes

0 = -(d+1)\frac{1}{z}F^{z\nu}+\partial_{\mu}F^{\mu\nu}+m^2z^{-2\Delta}A^\mu(x).

By equating the coefficients of z^{-2\Delta} and z^{-1}, you can solve for the values of \Delta and m^2. This will give you the desired index \Delta for the gauge field in AdS_{d+1}. I hope this helps. Good luck with your calculations!


Your colleague