From what I've (hopefully) understood from the AdS/CFT correspondence, physical quantities have a dual version. For example, the position in the bulk is the scale size (in renormalization), and waves in a curved gravitational background are the dynamics of quantum criticality.

But the partition functions of both the gravitational and the conformal field theory sides are equal (in the right limits):

\begin{equation}
Z_{QG} = Z_{CFT}
\end{equation}

Which allows things like correlation functions to be calculated on one side, for 'use' on the other.

From this, it would seem like thermodynamic quantities like magnetization and internal energy have the same meaning on each side of the correspondence. Is that correct? Or does, for example internal energy, have a different meaning on the gravitational side?

If it remains an internal energy, what exactly is it an internal energy of?

Similarly, if the correlation functions in the CFT are, for example, static and relate to different positions, what do they correspond to in the gravitational side?

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