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Holographic duality tells us that there is a duality between anti-deSitter space and lower dimensional conformal field theory. However, what quantum phenomenon, exactly, can we calculate using the correspondence? To find some physical observable normally found in CFT, can we do a calculation in classical gravitation and then translate the result back into CFT language? How is this done, and what are some examples/resources? Can we do this with entanglement and particle dynamics, such as periodically driven systems and anyonic braiding?

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This is a very broad question and therefore impossible to answer completely, but I will try to answer some questions and refer to some literature.

Your statement of the AdS/CFT correspondence was not quite complete:

Type IIb string theory on asymptotically $AdS_5\times S^5$ is equivalent to $\mathcal{N}=4$ super Yang-Mills theory

In the limit of large $N$ (rank of the gauge group) and large $\lambda=g_{YM}^2N$ (t'Hooft coupling) this reduces to:

Classical (super)gravity on $AdS_5\times S^5$ is equivalent to large $N$ strongly coupled $\mathcal{N}=4$ super Yang-Mills theory

What does equivalent mean?
All states, observables and dynamics are the same. The two theories are really just different descriptions of the same physics.

Practically, how does one compute field theory quantities from gravity?
Answering this question is what is called the holographic dictionnary, for obvious reason. In principle, one can compute any boundary quantity from the bulk, as the two theories are really the same. In practice, one is usually interested in correlation functions or some other quantities like thermodynamic quantities, entanglement entropy, Wilson lines etc.

In the Euclidean setting, the dictionary for correlation functions was first worked out by Gubser, Klebanov & Polyakov and Witten. Analytic continuation to Lorentzian signature is possible in certain settings. The basic idea is that the boundary values of classical bulk fields act as sources of dual operators for the CFT correlation functions, i.e. for a scalar

$$\left\langle e^{\int d^4x \phi_0\mathcal{O}}\right\rangle_{CFT} = e^{-S_{grav}\left(\left.r^\Delta\phi(r,x)\right|_{r\to\infty}=\phi_0(x)\right)}$$

where the gravity action is evaluated on-shell subject to the given boundary condition. $\Delta$ is the scaling dimension of the dual operator $\mathcal{O}$. Using functional derivatives with respect to the source $\phi_0$, one can now compute any correlation function of the operator $\mathcal{O}$ which is dual to the bulk scalar $\phi$. So one in principle gets the quantum correlation function from the classical bulk theory.

For the Lorentzian case at non-zero tempertature, the Euclidean results cannot be simply continued due to the analytic structure of the correlators. The complication in the Lorentzian case is that there is no unique solution to the bulk equations of motion only given boundary conditions on the boundary. One has to supply further information. In the case of the reatrded two-point function this has been worked out by Son & Starinets resulting in having to impose incoming boundary conditions at the horizon of the black hole metric, that is dual to the finite temperture field theory.

For situations far from thermal equilibrium, where the bulk geometry might only develop a horizon during the dynamics, the situation is much less clear and the holographic dictionary still needs to be explored.

Other observables have also been explored in the context of holography:
- entanglement entropy via minimal surfaces in the bulk due to Ryu & Takayanagi (also relevant: Nishioka, Ryu & Takayanagi review)
- Wilson loops haven been studied by Maldacena
- etc.

Since you asked about driven systems, they have recently been studied by Rangamani, Rozali & Wong, but I cannot say anything about this work. I unfortunately do not know anything about anyonic braiding.

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