# Proca gauge field in AdS space

1. Jan 20, 2013

### physicus

1. The problem statement, all variables and given/known data

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

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

b) Determine the index $\Delta$ by inserting the ansatz $A_\mu(z)=z^\Delta$ 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. $\mu,\nu = z,0,\ldots,d-1$ take d+1 values including the z-direction. Is that right? In the metric however, $x^\mu$ has only d components $\mu=0,\ldots,d-1$.

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

The determinant of the metric is $g=z^{-2(d+1)}$

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

Is that right?

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

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

Cheers, physicus