# Proca gauge field in AdS space

• physicus
In summary, the conversation is about deriving the equations of motion for a massive gauge field in AdS_{d+1} space and determining the index \Delta through an ansatz. The solution involves varying the action with respect to A_\mu and using integration by parts, and plugging in the ansatz A_\mu(z)=z^\Delta A_\mu(x) to obtain the desired index.
physicus

## Homework Statement

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.

## 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

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!

## 1. What is a Proca gauge field?

A Proca gauge field is a type of quantum field that describes the behavior of a massive spin-1 particle, such as a vector boson. It is a vector field that carries a conserved charge, with its dynamics governed by the Proca Lagrangian.

## 2. What is AdS space?

AdS space, or Anti-de Sitter space, is a type of curved spacetime described by the anti-de Sitter metric. It is a maximally symmetric space with a negative cosmological constant, and is an important concept in the study of string theory and holography.

## 3. What is the significance of studying Proca gauge fields in AdS space?

Studying Proca gauge fields in AdS space allows us to better understand the behavior of massive vector bosons in a curved spacetime. It also has important implications for the holographic principle and the AdS/CFT correspondence, which relate quantum gravity in AdS space to a conformal field theory on its boundary.

## 4. How is the Proca gauge field equation modified in AdS space?

In AdS space, the Proca gauge field equation is modified by the presence of the negative cosmological constant, which acts as a source term. This results in a mass term for the gauge field and modifies its behavior in the curved spacetime.

## 5. What are the current research areas involving Proca gauge fields in AdS space?

Some current research areas involving Proca gauge fields in AdS space include the study of holographic dualities, black holes and their thermodynamics, and the role of massive vector bosons in cosmology. There is also ongoing research on the behavior of Proca gauge fields in higher dimensions and in the presence of other fields, such as gravity or fermions.

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