# AdS/CFT dual of $N$ D$p$-branes at finite temperature

The gravity dual of $N$ D$p$-branes at zero temperature is

$$ds^2= H^{-1/2}(r)(-dt^2+dx_p^2) + H^{1/2}(r)(dr^2 + r^2d\Omega_{8-p}^2)$$

with

$$H(r) = 1 + \left(\frac{R}{r}\right)^{7-p}$$

what is (tell me if I'm wrong) an extremal black $p$-brane.

When we consider that the boundary system is at temperature $T$, the dual metric then is

$$ds^2= H^{-1/2}(-h(r)dt^2+dx_p^2) + H^{1/2}\left(\frac{dr^2}{h(r)} + r^2d\Omega_{8-p}^2\right) \qquad(*)$$

with

$$h(r) = 1 + \left(\frac{r_0}{r}\right)^{7-p}$$

(for example, this is written here, in section 7.5, for $p=3$), but I don't know to what system corresponds this metric, except for $p=3$, which is an AdS black hole (for $r$ near the throat).

So the question is, what system has the metric $(*)$?

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AdS black holes exist in various dimensions, $p=3$ is not the only choice. The parameter can take on values above or below $3$. One famous example is the three-dimensional BTZ black hole, and higher dimensional ones are also frequently used in the correspondence.
Furthermore, I think there is a misunderstanding on the concept of a "gravity dual". The metric you wrote down is not the the gravity dual of a $\mathrm{D}p$-brane at zero temperature, it is a solution of classical gravity that corresponds to the geometry of such a brane at low energies. In this sense, it is the D-brane in a particular limit. The word "gravity dual" refers to precisely this geometry, and it is the dual of a quantum field theory living on its boundary. The AdS/CFT correspondence is about formulating higher-dimensional gravity duals (D-Branes and their low energy limits) of quantum field theories in lower dimensions.
But for $p\neq 3$ the first metric is not AdS. If I understood correctly your answer, the metric (*) corresponds to a black hole in the spacetime of the first metric? – David Pravos Oct 1 '14 at 10:59
Your statement is not true, AdS is not restricted to $p=3$. – Frederic Brünner Oct 1 '14 at 12:50