# Normalizable and non normalizable modes of gauge fields in AdS/CFT

In Lorentzian AdS space there are both normalizable and non normalizable solutions and we also know (at least for scalar fields in bulk) what do they correspond to in the boundary. But I saw the calculation only for scalar fields. Can someone please give me a reference where people have calculated these modes for a gauge fields, say for a graviton field? McGreevy's lecture note says the relation $\Delta(\Delta-D)=m^2L^2$ gets modified to $(\Delta+j)(\Delta+j-D)=m^2L^2$ for form $j$ fields. Does this mean for other fields too the normalizable and non normalizable behavior remains the same: namely $Z_0^{\Delta_+}$ and $Z_0^{\Delta_-}$, as $z_0\rightarrow 0$ ($\Delta_{\pm}$ are two solutions of course)? How can that be?

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## 2 Answers

At the end of Sec 3.3.1 of the MAGOO review (hep-th/9905111) you will find a useful list of the relationship between conformal dimension $\Delta$ and masses $m$ for scalars, spinors, vectors, p-forms, first order (d/2)-forms, spin $3/2$ and massless spin $2$ fields along with a list of references to the literature where these various cases were analyzed.

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From there, for example it seems for massless spin 2 particle, $\Delta=d$. What is the difference between it and putting $j=2$ in $(\Delta+j)(\Delta+j-d)=0$? Then the values of $\Delta$ are different from $d$. And also in case $\Delta=d$, does that mean it has only one mode going like $Z_0^d$? – user1349 Feb 2 '11 at 18:42
Oh..Sorry, I see..So, for example for massless spin 2, will the only one mode go like $Z_0^d$ near the boundary? In that case, is it the one which will be non-normalizable and couple to boundary fields? Also, the reference of this relation goes back to the Witten's paper (9802150, via relations in p-29 of 9904017), but I really didn't understand how to get to this relation for massless spin 2. The relation looks so like scalar case. – user1349 Feb 2 '11 at 19:32

As Jeff Harvey points out, you can find all of the original calculations for the various form fields in the references of the MAGOO article. But you are better off working the result out for yourself, and it's not a difficult calculation.

You are only concerned with the asymptotics of the fields, so you can make some simplifications. Start with the Poincaré patch of AdS (where the metric is especially easy to work with) $$ds^2 = \frac{\ell^2}{z^2}\,(dz^2 + \eta_{ab} dx^a dx^b) ~.$$ Now focus on the $z$ dependence of the field, ignoring any dependence on the boundary coordinates. Assume the field scales like $z^{\Delta}$ as $z \to 0$. Evaluate the equation of motion with this ansatz, and the condition for $\Delta$ follows. For a scalar field this calculation takes just a few lines. Working it out for massive p-forms is a bit more involved: you will need to expand the covariant derivative of a (p+1)-form field strength and then work out the z dependence of the terms involving Christoffel symbols. But the simple form of the metric in the Poincaré patch keeps this from getting too complicated, and eventually you'll get the result you quoted for $\Delta$.

Notice that this calculation works equally well for Euclidean and Lorentzian AdS. For a discussion of the differences between the two, see hep-th/9805171.

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What I said above goes for the graviton, but you'll need to linearize the Einstein equation to get the equation of motion for the field. Choosing the right gauge will help simplify the intermediate steps of this calculation. You will eventually find that the equation of motion for the spin-2 graviton on the AdS background reduces to that of a massless scalar. – Robert McNees Feb 2 '11 at 19:54
Thanks Robert. So, if I have only one solution (as I can see for massless spin 2 gravitons, $\Delta=d$), then that means the solution goes like $z^\Delta$ as $z\rightarrow 0$. Is it then the non normalizable solution which will couple to the boundary fields? I think I am confusing because it looks normalizable to me in which case I don't understand how to get the source term that will couple to the boundary field! – user1349 Feb 2 '11 at 19:57
I got your comment on graviton after I typed my comment..Can you please also suggest the reference which does that (if any)? – user1349 Feb 2 '11 at 20:00
Sorry -- I don't remember a reference off the top of my head. You can work this result out by first linearizing the Einstein equation (i.e., write the metric as $g_{\mu\nu} = \bar{g}_{\mu\nu} + h_{\mu\nu}$, where $\bar{g}_{\mu\nu}$ is the AdS metric and $h_{\mu\nu}$ is the graviton) and then solving for the leading z dependence of the graviton. Like I said before, this will be easier if you pick a convenient gauge for $h_{\mu\nu}$. – Robert McNees Feb 2 '11 at 21:30
Thanks again Robert! I will definitely work it out..But can you please explain about my confusion on whether it will provide normalizable or non normalizable mode (basically my first comment after your response) and in that case what will be the source that will couple to boundary operator? – user1349 Feb 2 '11 at 23:50
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