Scaling dimensions of operators in AdS/CFT

1. Feb 20, 2009

jdstokes

http://arxiv.org/abs/hep-th/9802150

Since the AdS metric diverges at the boundary, the boundary metric is only defined up to a conformal class Eq. (2.2),

$ds^2 \to d\widetilde{s}^2 = f^2 ds^2$

Similarly, the solution for a massive scalar $\varphi$ is divergent. In order to define the scaling dimension of $\varphi$, Witten writes it as Eq. (2.36)

$\varphi \sim f^{-\lambda}\varphi_0$

where $f$ is the same function used to give a finite metric in (2.2) and $\varphi_0$ depends only on the boundary.

Thus if we fix some appropriate $f$ in (2.2) then we can easily determine the scaling dimension $\lambda$ from (2.36).

What concerns me is why Witten is justified in using the same function $f$ in both (2.2) and (2.36). One would have thought that it is possible to be more general by defining say

$\varphi \sim g^{-\lambda}\varphi_0$

where in general $g\neq f$.

Suppose, for example, that $f = e^{-y}$ and $g = e^{-2y}$, then all of the scaling dimensions will be shifted from their values had we chosen $g = f$. How can the scaling dimension depend on the choice of an arbitrary function?