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People generalize the statements of AdS/CFT correspondence by adding black hole (charged black hole) in the gravity theory to provide the dual gauge theory finite temperature (finite density). I have some problems (may be very basic) understanding these concepts with charged BH intuitively.

  1. Is it true that whenever one adds a charged BH in the bulk one introduces conserved current (matter degrees of freedom) in the boundary? For N=4 SYM what are they?

  2. What does the chemical potential do in dual field theory side? Actually I don't really have clear intuitive picture of chemical potential unlike pressure, temperature etc.

  3. When one adds charge to a BH the temperature of the BH depends on both mass and charge. Now what are the independent parameters one can tune to make the BH extremal? What do these changes mean in the dual field theory picture?

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I'm also new to the subject, but I'll try to share what I've learned so far if you haven't found the answers already. First, since we want to introduce a charged Black Hole in the bulk ($Reissner$-$Nordstr\ddot{o}m$-$AdS$ Black Hole), it'll be dual to a field theory at finite temperature and charge density with the partition function of the grand canonical ensemble: $$\mathcal{Z}=tr\Big[\exp \Big(-\beta \,(\hat{H}-\sum\limits_{i}{{{\mu }_{i}}{{{\hat{Q}}}_{i}}})\Big)\Big]$$ where $\beta$ is the inverse temperature and ${\mu}_{i}$ are the chemical potentials associated with the charges ${\hat{Q}}_{i}$ which themselves are the Noether charges associated with conserved currents $J_{(i)}^{\nu }$ under global $U(1)$ symmetry. For $\mathcal{N}=4\,$ SYM, these currents are conserved under the $U(1)$ subgroup of the global $SO(6)$ R-symmetry [1][2].

Now for simplicity, first let's consider the pure $Ad{{S}_{5}}$ with a non-zero electric local gauge field ${{A}_{0}}(z)$ , where the index is for its temporal component. Solving the equations of motion will yield an asymptotic solution of the form: $${{A}_{0}}\simeq A_{0}^{(0)}(1+A_{0}^{(1)}{{z}^{2}}) \quad as \quad z\to0$$ By the AdS/CFT dictionary, since the partition function of the field theory will have the additional source term of the form $\int{{{d}^{4}}x}\,{{J}^{\mu}}{{A}_{\mu}}$ , therefore the boundary value of the bulk gauge field (i.e. $A_{0}^{(0)}$) will be the source (i.e. chemical potential) conjugate to the conserved charge density $J^0$ [3].

Now by keeping in mind the above reasoning, in the case of $Reissner$-$Nordstr\ddot{o}m$-$AdS$ BH we will have the following form for the bulk gauge field: $${{A}_{0}}\propto Q\,(z_{h}^{d-2}-{{z}^{d-2}})\quad for \quad d\ge 3$$ where $z_{h}$ is the location of the event horizon obtained by $f(z_{h})=0$ . You can find the explicit form of $f(z)$ in [4]. Now recall that the chemical potential is the boundary value of the gauge field, so we'll have: $$\mu =\frac{1}{L}\underset{z\to 0}{\mathop{\lim }}\,{{A}_{0}}(z)$$ where $\frac{1}{L}$ is introduced to ensure that $\mu$ has the correct dimension of energy in the boundary theory. For your 3rd question, by computing the Hawking temperature (T), the ratio $\frac{\mu}{T}$ will give us a good transition from zero charge where $\frac{\mu}{T}=0$ to the extremal case where $\frac{\mu}{T} \to \infty$ for the fixed $z_{h}$. You can find its explicit form and the parameters it depends, with some good discussions in [4]. I hope this answer will be helpful.

[1] J. Kapusta, et al., Phys. Rev. D 28, 3093 (1983)
[2] H. Nastase, Introduction to the AdS/CFT Correspondence, CUP, 2015
[3] M. Natsuume, AdS/CFT Duality User Guide, Springer, 2015
[4] D. Galante, M. Schvellinger, JHEP 1207 (2012) 096, https://arxiv.org/abs/1205.1548 (2012)

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