# Role of the canonical ensemble and electric charge in AdS/CFT

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If we consider a charged black hole in AdS spacetime, we can either do thermodynamics in the grand canonical or the canonical ensemble. In the former, we fix the electrostatic potential $\Phi=A_t(r=\infty)$ at the boundary of the bulk such that $\left<Q\right>=-\frac{1}{\beta}\left(\frac{\partial S}{\partial\beta}\right)_{\beta}$, where $S$ is the Euclidean action. In the latter, we fix the charge $Q$ of the black hole and we do not consider $\Phi$ at all. The phase diagram of the black hole is highly dependent on the choice of ensemble, see for example this paper by Chamblin, Emparan, Johnson and Myers. One could therefore expect that this choice also has an influence on the CFT side.

In the AdS/CFT dictionary, the charged black hole gives a global $U(1)$ symmetry on the CFT side. Here $\Phi$ in the bulk corresponds to a chemical potential $\mu$ on the CFT, so that we usually consider the grand canonical ensemble when using the correspondence.

My questions are as follows:

• Do we ever consider the canonical ensemble in AdS/CFT?
• If so, what would $Q$ determine on the CFT side (in the same way $\Phi$ determines $\mu$)?
• If we work in the grand canonical ensemble, does $\left<Q\right>$ play any role on the CFT side, or do we only need $\Phi$?
This post imported from StackExchange Physics at 2014-07-09 07:41 (UCT), posted by SE-user ScroogeMcDuck
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