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

Question

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?

Answer 1

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.

Answer 2

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.