Do thermodynamic quantities in CFT correspond to something different in AdS/CFT?

Question

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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Answer 1

Not so simple. The equation $Z_{QG}=Z_{CFT}$ has to be interpreted correctly. AdS has a (conformal) boundary at space-like infinity and in order to define Quantum Gravity in AdS one has to supply boundary conditions on this conformal boundary.

$Z_{QG}=Z_{CFT}$ is really a dictionary telling us which boundary conditions to choose at space-like infinity in order to calculate a given quantity in field theory. For instance, in order to calculate correlation functions (say at separated points) of the EM tensor in the CFT one has to introduce some boundary conditions for the metric field in the QG theory on AdS. Then one has to solve for this boundary problem and one would generally obtain a solution for the metric field different than AdS itself. The action of this solution is the correlation function in the CFT (this is when the QG can be approximated by a classical theory, otherwise one would need the full QG partition function with these boundary conditions).

There is a theorem that the solution to such boundary problems is unique.

To incorporate finite temperature physics one does something slightly different. Finite temperature in the field theory can be understood as a compact imaginary time direction. One then compactifies the corresponding direction in AdS near the boundary and then the idea is to search for solutions in AdS which approach $R^{d-1}xS^1$ near the boundary. Sometimes the leading solution is a black hole. Hence, roughly, you can think of the finite temperature ensemble on the field theory side as putting a black hole in QG.

Then various thermodynamic quantities in the field theory (such as entropy and energy density) can be calculated from the black hole geometry.

One may also be interested in real time transport properties (beyond thermodynamics), and this is a different story altogether.

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Comments

Thanks for clarifying those points. But thermodynamic quantities don't necessarily need to be at finite temperature.. what happens then? Going back to finite temperature, can the CFT's magnetization and internal energy be calculated from the black hole geometry? On the AdS side, do they then correspond to the black hole's magnetization and internal energy? What about the correlation functions - are they still correlation functions on the AdS side?

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If you are at zero temperature then you can try to turn on other chemical potentials, such as particle number density etc. Some of the corresponding calculations in AdS are well-understood and can be carried out. Yes, at finite temperature you can get all the quantities you are interested in from the black hole geometry (amended, perhaps, by additional fugacities) Correlation functions at zero temperature map to the partition function of QG in AdS with prescribed boundary conditions.

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