# “Dictionary” between QFTs in D and D-1 dimensions?

Consider the Einstein field equations. Suppose, for instance, that the RHS, the stress-energy tensor, is uniquely determined by the electromagnetic field. Now, if we imagine a quantized version of these Einstein equations, we have a quantum field theory in $D$ dimensions (here the quantum electromagnetic field) as "source" of a "quantum gravity" theory in $D$ dimensions, so we may consider these equations as a kind of "dictionary" between these two theories.

On the other hand, there are some holographic dualities which relates some quantum theory of gravity in a $D$ dimensional space with boundary, with some quantum field theory on the boundary ($D-1$) dimensions

So, the question is: Is it posible to "skip gravity" and to directly relate the QFT in $D$ dimensions (source of gravity), and the other QFT (on the boundary) in $D-1$ dimensions.

Holographic dualities like the ones you mentioned are dynamic in nature i.e. you can determine the dynamics of the quantum field theory in terms of the dynamics of the gravity theory. The most famous one is the duality between the $\mathcal{N} = 4$ super Yang-Mill theory in 4 dimensions with a gauge group of $SU(N)$ with large $N$ living on the world volume of a stack of D3 branes and the type IIB supergravity compactified on $AdS_5 \times S_5$. Given that such dualities are dynamical in nature, you would need to consider the full Einstein equation and not just focus on the stress energy tensor. The geometric component is required to understand the dynamic quantities in gravity which is in turn required to make the duality valid.
To offer an example as to why this is not right, let us consider a $D-1$ dimensional dual to some $D$ dimensional gravity theory with vanishing stress energy tensor, for which you have nothing on the RHS of the Einstein field equation. In this case the dynamics is determined solely by the vacuum solution to Einstein's equations. What now?