In AdS/CFT a charged Black hole is probably someway equivalent to introducing a chemical potential (Chemical potential) at the boundary theory. Is there a quick way to see how it is or how does this correspondence work?

I am sure it's well argued, but I haven't found a satisfactory explanation in any paper. Specially I will like to understand by connecting it with the thermodynamic case. In addition, it will be very helpful, if you can suggest me any introductory references or something with a nice explanation. Thanks!

• Are you referring to this arxiv.org/abs/hep-th/9908109 ? Feb 25 '13 at 3:52
• A chemical potential is the thermodynamic dual of the number of particles $N$ of a certain kind. That's why $\mu N$ will be inevitably mathematically analogous to many things found in AdS/CFT and physics in general expect that 1) this analogy is a particular case of the mathematical similarities between classical statistical physics and the path integral in quantum mechanics, 2) the objects in AdS/CFT aren't really "number of particles" and "chemical potential" so it's just a vague mathematical similarity. Feb 25 '13 at 10:30
• Let me mention that if you want to learn AdS/CFT or any advanced high-energy physics, you shouldn't try to organize this knowledge as exercises in thermodynamics because they're not exercises in thermodynamics. Some analogies work but some analogies don't work. If you want to assume that everything is just simple 19th century thermodynamics, well, then you're just wrong. Feb 25 '13 at 10:32
• @Frederick: Yes, that's where (one of many actually) I saw the statement of chemical potential. But do you have any other references which review it? Feb 25 '13 at 14:52
• @Motl: Okay.. I will not try to make contact with thermodynamics. But is there any intuitive way to see why charged BH might be dual to CFT with chemical potential? May be I should go through some references first, but your answer would be helpful. Feb 25 '13 at 14:55

One can get an intuitive picture of the relation by realizing the following: in statistical physics, conserved quantities are implemented by Lagrange multipliers, see for example these lecture notes (pdf). Chemical potential is defined as the Lagrange multiplier corresponding to the conservation of charge. Now the assumption that a charged black hole might somehow be related to chemical potential is not too far-fetched. The chemical potential is identified with the $U(1)$ gauge field generated by the charge, directly affecting the boundary theory.