Some confusion about asymptotic safety
I'm trying to understand why asymptotic safety is a reasonable approach to quantum gravity.
Here is my present understanding:
1. Asymptotic safety is roughly the statement that perturbative gravity (by which i mean a spin 2 field) has a UV fixed point where it becomes like any other conformal field theory.
2. As a conformal field theory, the UV fixed point has a particular scaling for the density of states as a function of energy which follows from conformal invariance.
3. In quantum gravity one expects (essentially because of large black holes) that the density of states at very high energy is different from that of a conformal field theory in the same dimension. For example, in anti de Sitter space the density of states is that of a conformal field theory in one lower dimension (holography).
I don't see how 1 and 2 can possibly be compatible with 3. Nevertheless, I assume people in asymptotic safety have an answer to this kind of concern. Perhaps I have misunderstood what asymptotic safety is, for example, maybe the theory is not an ordinary quantum field theory as I have assumed. Does anyone know?
Thanks for your input!