Other AdS Spacetimes

Abstract

We discuss various spacetimes which often appear in AdS/CFT. We describe charged AdS black holes, the Schwarzschild-AdS black hole in the other dimensions, various branes (M-branes and D$p$-branes), and some other examples. Many of them are asymptotically AdS spacetimes, but some are not.

Appendix: Explicit form of Other AdS Spacetimes \({}^\blacklozenge \)

11.1.1 SAdS$_{p+2}$ Black Hole

In this section, we consider black holes with planar horizon only. The SAdS$_{p+2}$ black hole is given by

$$\begin{aligned} ds_{p+2}^2&= - f dt^2 + \frac{dr^2}{f} + \left( \frac{r}{L}\right) ^2 d{\varvec{x}}_{p}^2, \end{aligned}$$

(11.8)

$$\begin{aligned} f&= \left( \frac{r}{L}\right) ^2 \left\{ 1 - \left( \frac{r_0}{r} \right) ^{p+1} \right\} . \end{aligned}$$

(11.9)

Thermodynamic quantities are given by

$$\begin{aligned} T&= \frac{p+1}{4\pi L^2} r_0~,\end{aligned}$$

(11.10)

$$\begin{aligned} s&= \frac{1}{4 G_{p+2}} \left( \frac{r_0}{L}\right) ^{p}~,\end{aligned}$$

(11.11)

$$\begin{aligned} \varepsilon&= \frac{p}{16\pi G_{p+2} L} \left( \frac{r_0}{L} \right) ^{p+1}~,\end{aligned}$$

(11.12)

$$\begin{aligned} P&= \frac{1}{16\pi G_{p+2} L} \left( \frac{r_0}{L} \right) ^{p+1} = \frac{1}{p}\varepsilon ~. \end{aligned}$$

(11.13)

11.1.2 M-branes

Zero-temperature case   The M2-brane at zero temperature is given by

$$\begin{aligned} ds_{11}^2&= f_2^{-2/3} (-dt^2 + dx_1^2 + dx_2^2)+ f_2^{1/3} (dr^2 + r^2 d\varOmega _{7}^2), \end{aligned}$$

(11.14)

$$\begin{aligned} f_2&= 1+\left( \frac{r_2}{r}\right) ^{6}. \end{aligned}$$

(11.15)

In the near-horizon limit $r \ll r_2$, the metric becomes

$$\begin{aligned} ds_{11}^2&\rightarrow \left( \frac{r}{r_2} \right) ^4 (-dt^2 + dx_1^2 + dx_2^2) + \left( \frac{r_2}{r} \right) ^2 (dr^2 + r^2 d\varOmega _{7}^2) \end{aligned}$$

(11.16)

$$\begin{aligned}&= \left( \frac{2{\tilde{r}}}{r_2} \right) ^2 (-dt^2 + dx_1^2 + dx_2^2) + \left( \frac{r_2}{2} \right) ^2 \frac{d{\tilde{r}}^2}{{\tilde{r}}^2} + r_2^2 d\varOmega _{7}^2, \end{aligned}$$

(11.17)

where $r =(2{\tilde{r}} r_2)^{1/2}$. The geometry reduces to the AdS$_4 \times S^7$ spacetime with AdS radius $L = r_2/2$ and the $S^7$ radius $L_{S^7} = 2L = r_2$. Note that the AdS radius differs from the sphere radius unlike the D3-brane case.

The M5-brane at zero temperature is given by

$$\begin{aligned} ds_{11}^2&= f_5^{-1/3} (-dt^2 + d{\varvec{x}}_5^2) + f_5^{2/3} (dr^2 + r^2 d\varOmega _{4}^2), \end{aligned}$$

(11.18)

$$\begin{aligned} f_5&= 1+\left( \frac{r_5}{r}\right) ^{3}. \end{aligned}$$

(11.19)

In the near-horizon limit $r \ll r_5$, the metric becomes

$$\begin{aligned} ds_{11}^2&\rightarrow \left( \frac{r}{r_5} \right) (-dt^2 + d{\varvec{x}}_5^2) + \left( \frac{r_5}{r} \right) ^2 (dr^2 + r^2 d\varOmega _{4}^2) \end{aligned}$$

(11.20)

$$\begin{aligned}&= \left( \frac{{\tilde{r}}}{2r_5} \right) ^2 (-dt^2 + d{\varvec{x}}_5^2) + (2r_5)^2 \frac{d{\tilde{r}}^2}{{\tilde{r}}^2} + r_5^2 d\varOmega _{4}^2, \end{aligned}$$

(11.21)

where $r = {\tilde{r}}^2/(4r_5)$. The geometry reduces to the AdS$_7 \times S^4$ spacetime with AdS radius $L = 2r_5$ and the $S^4$ radius $L_{S^4} = L/2 = r_5$.

AdS/CFT dictionary   One can obtain the AdS/CFT dictionary for M-branes as in the D3-brane in Chap. 5 App. 2. For the M2-brane, the number of the spatial dimensions transverse to the brane is eight. Thus, the Newtonian potential is $G_{11}\mathsf T _2/r^6$ instead of $GM/r$ in three spatial dimensions. Using the dimensional analysis and the fact that there are $N_c$ branes, one gets

$$\begin{aligned} r_2^6&\simeq G_{11}\mathsf T _2 \simeq N_c l_{11}^6, \end{aligned}$$

(11.22)

$$\begin{aligned} G_{11}&=: l_{11}^9, \end{aligned}$$

(11.23)

where $l_{11}$ is the 11-dimensional Planck length. In string theory, there is the fundamental length scale, the string scale $l_s$, and the Planck length $l_{10}$ is not fundamental. They are related to each other by $l_{10}^8 \simeq g_s^2 l_s^8$. However, in the 11-dimensions, it is not yet clear if there is a fundamental length scale like $l_s$, so here we use $l_{11}$ instead. From Eqs. (11.22) and (11.23), we get

$$\begin{aligned} \frac{r_2^9}{G_{11}} \simeq N_c^{3/2}. \end{aligned}$$

(11.24)

This is the M2 version of the first equation of Eq. (5.74).

For the M5-brane, the number of the spatial dimensions transverse to the brane is five. Thus, the Newtonian potential is $G_{11}\mathsf T _5/r^3$, so

$$\begin{aligned} r_5^3 \simeq G_{11}\mathsf T _5 \simeq N_c l_{11}^3. \end{aligned}$$

(11.25)

Thus,

$$\begin{aligned} \frac{r_5^9}{G_{11}} \simeq N_c^{3}. \end{aligned}$$

(11.26)

If one works out numerical coefficients, the results are

$$\begin{aligned} N_c^{3/2}&= \frac{\sqrt{2} \pi ^5 (2L)^9}{16\pi G_{11}} = 12\sqrt{2} \pi \frac{L^2}{16\pi G_{4}} \quad (\text {M2}),\end{aligned}$$

(11.27)

$$\begin{aligned} N_c^3&= \frac{\pi ^5L^9}{2\cdot 16\pi G_{11}} = 3\pi ^3\frac{L^5}{16\pi G_{7}} \quad (\text {M5}). \end{aligned}$$

(11.28)

Finite-temperature case   The M2-brane at finite temperature is given by

$$\begin{aligned} ds_{11}^2&= f_2^{-2/3} (-hdt^2 + dx_1^2 + dx_2^2) + f_2^{1/3} (h^{-1}dr^2 + r^2 d\varOmega _{7}^2), \end{aligned}$$

(11.29)

$$\begin{aligned} h&= 1-\left( \frac{r_0}{r}\right) ^{6} = 1-\left( \frac{{\tilde{r}}_0}{{\tilde{r}}}\right) ^{3}, \end{aligned}$$

(11.30)

where $r_0 =(2{\tilde{r}}_0 r_2)^{1/2}$. The M5-brane at finite temperature is given by

$$\begin{aligned} ds_{11}^2&= f_5^{-1/3} (-hdt^2 + d{\varvec{x}}_5^2) + f_5^{2/3} (h^{-1}dr^2 + r^2 d\varOmega _{4}^2), \end{aligned}$$

(11.31)

$$\begin{aligned} h&= 1-\left( \frac{r_0}{r}\right) ^{3} = 1-\left( \frac{{\tilde{r}}_0}{{\tilde{r}}}\right) ^{6}, \end{aligned}$$

(11.32)

where $r_0 = {\tilde{r}}_0^2/(4r_5)$. In the near-horizon limit ${\tilde{r}}_0 < {\tilde{r}} \ll L$, the M2 and M5-branes reduce to the SAdS$_4$ and SAdS$_7$ black holes. One can then use the results of thermodynamic quantities for the SAdS$_{p+2}$ black hole. For example, the energy density for the M2-brane is given by

$$\begin{aligned} \varepsilon \simeq \frac{1}{G_4 L} \left( \frac{r_0}{L} \right) ^{3} \simeq \frac{L^9}{G_{11}} \left( \frac{r_0}{L^2} \right) ^{3} \simeq N_c^{3/2} T^3, \end{aligned}$$

(11.33)

where $1/G_4 \simeq L^7/G_{11}$. For the M5-brane,

$$\begin{aligned} \varepsilon \simeq \frac{1}{G_7 L} \left( \frac{r_0}{L} \right) ^{6} \simeq \frac{L^9}{G_{11}} \left( \frac{r_0}{L^2} \right) ^{6} \simeq N_c^3 T^6, \end{aligned}$$

(11.34)

where $1/G_7 \simeq L^4/G_{11}$. If one works out numerical coefficients, the results are

$$\begin{aligned} s&= \sqrt{2} \pi ^2 \left( \frac{2}{3}\right) ^3 N_c^{3/2} T^2, \end{aligned}$$

(11.35)

$$\begin{aligned} \varepsilon&= \sqrt{2} \pi ^2 \left( \frac{2}{3}\right) ^4 N_c^{3/2} T^3, \end{aligned}$$

(11.36)

for the M2-brane, and

$$\begin{aligned} s&= 3\pi ^3 \left( \frac{2}{3}\right) ^7 N_c^{3} T^5~, \end{aligned}$$

(11.37)

$$\begin{aligned} \varepsilon&= \frac{5\pi ^3}{2} \left( \frac{2}{3}\right) ^7 N_c^{3} T^6~, \end{aligned}$$

(11.38)

for the M5-brane.

11.1.3 D$p$-brane

Zero-temperature case   The D$p$-brane at zero temperature is given by

$$\begin{aligned} ds_{10}^2&= Z_p^{-1/2} (-dt^2 + d{\varvec{x}}_p^2) + Z_p^{1/2} (dr^2 + r^2 d\varOmega _{8-p}^2), \end{aligned}$$

(11.39)

$$\begin{aligned} e^{-2\phi }&= Z_p^{(p-3)/2}, \end{aligned}$$

(11.40)

$$\begin{aligned} Z_p&= 1+ \left( \frac{r_p}{r} \right) ^{7-p}. \end{aligned}$$

(11.41)

When $p\ne 3$, the dilaton $\phi$ has a nontrivial behavior. In the near-horizon limit $r \ll r_p$, the metric becomes

$$\begin{aligned} ds_{10}^2&\rightarrow \left( \frac{r}{r_p}\right) ^{(7-p)/2} ( -dt^2 + d{\varvec{x}}_p^2 ) + \frac{dr^2}{(\frac{r}{r_p})^{(7-p)/2}} + r_p^2 \left( \frac{r}{r_p} \right) ^{(p-3)/2} d\varOmega _{8-p}^2 \end{aligned}$$

(11.42)

$$\begin{aligned}&= \left( \frac{r}{r_p}\right) ^{(p-3)/2} \left\{ \left( \frac{r}{r_p}\right) ^{5-p} ( - dt^2 + d{\varvec{x}}_p^2 ) + \frac{dr^2}{(\frac{r}{r_p})^2} + r_p^2 d\varOmega _{8-p}^2 \right\} . \end{aligned}$$

(11.43)

The metric is invariant up to an overall coefficient under a scaling:

$$\begin{aligned} x^\mu \rightarrow a x^\mu , \quad r \rightarrow a^{-2/(5-p)} r, \quad ds^2 \rightarrow a^{-(p-3)/(5-p)} ds^2. \end{aligned}$$

(11.44)

AdS/CFT dictionary   One can obtain the AdS/CFT dictionary for the D$p$-branes as in the D3-brane in Chap. 5 App. 2. For the D$p$-brane, the number of the spatial dimensions transverse to the brane is $(9-p)$. Thus, the Newtonian potential is $G_{10}\mathsf T _p/r^{7-p}$. One thus obtains

$$\begin{aligned} r_p^{7-p}&\simeq g_s N_c l_s^{7-p}, \end{aligned}$$

(11.45)

$$\begin{aligned} G_{10}&\simeq g_s^2 l_s^8, \end{aligned}$$

(11.46)

$$\begin{aligned} g_s\, l_s^{p-3}&\simeq g_\text {YM}^2, \end{aligned}$$

(11.47)

where the last two expressions are Eq. (5.10). If one works out numerical coefficients, the results are

$$\begin{aligned} r_p^{7-p}&= \frac{(2\pi )^{7-p}}{(7-p) \varOmega _{8-p}} g_s N_c l_s^{7-p}, \end{aligned}$$

(11.48)

$$\begin{aligned} 16\pi G_{10}&= (2\pi )^7 g_s^2 l_s^8, \end{aligned}$$

(11.49)

$$\begin{aligned} 2(2\pi )^{p-2}\, g_s\, l_s^{p-3}&= g_\text {YM}^2. \end{aligned}$$

(11.50)

When $p=3$, Eqs. (11.48)–(11.50) reduces to Eq. (5.74) for the D3-brane.

Finite-temperature case   The D$p$-brane at finite temperature is given by

$$\begin{aligned} ds_{10}^2&= Z_p^{-1/2} (-h dt^2 + d{\varvec{x}}_p^2) + Z_p^{1/2} (h^{-1}dr^2 + r^2 d\varOmega _{8-p}^2), \end{aligned}$$

(11.51)

$$\begin{aligned} h&= 1-\left( \frac{r_0}{r}\right) ^{7-p}, \end{aligned}$$

(11.52)

(in the so-called string metric). Thermodynamic quantities in the near-horizon limit are given by

$$\begin{aligned} T&= \frac{7-p}{4\pi }\frac{r_0^{(5-p)/2}}{r_p^{(7-p)/2}},\end{aligned}$$

(11.53)

$$\begin{aligned} s&= e^{-2\phi } \frac{a}{4G_{10}} = \frac{1}{4 G_{10}} \varOmega _{8-p} (r_p r_0)^{(7-p)/2} r_0,\end{aligned}$$

(11.54)

$$\begin{aligned} \varepsilon&= \frac{9-p}{32\pi G_{10}} \varOmega _{8-p} r_0^{7-p},\end{aligned}$$

(11.55)

$$\begin{aligned} P&= \frac{5-p}{9-p} \varepsilon ,\end{aligned}$$

(11.56)

$$\begin{aligned} C&:= \frac{\partial \varepsilon }{\partial T} = \frac{9-p}{4 G_{10}(5-p)} \varOmega _{8-p} (r_p r_0)^{(7-p)/2} r_0. \end{aligned}$$

(11.57)

Here, the area law of the black hole entropy is modified as

$$\begin{aligned} s = \left. e^{-2\phi } \frac{a}{4G_{10}} \right| _{r=r_0} \end{aligned}$$

(11.58)

since the dilaton plays the role of the effective Newton’s constant (Sect. 5.2.5).

11.1.4 RN-AdS Black Holes

The RN-AdS$_5$ black hole is given by

$$\begin{aligned} ds_5^2&= - f dt^2 + \frac{dr^2}{f} + \left( \frac{r}{L}\right) ^2 d{\varvec{x}}_3^2, \end{aligned}$$

(11.59)

$$\begin{aligned} f&= \left( \frac{r}{L}\right) ^2 \left\{ 1 - \left( \frac{r_+}{r} \right) ^2 \right\} \left\{ 1 - \left( \frac{r_-}{r} \right) ^2 \right\} \left\{ 1 + \frac{r_+^2+r_-^2}{r^2} \right\} , \end{aligned}$$

(11.60)

$$\begin{aligned} A_0&= - \frac{r_+}{L^2} \sqrt{3\alpha ^2(1+\alpha ^2)} \left( \frac{r_+^2}{r^2} - 1\right) , \end{aligned}$$

(11.61)

($\alpha :=r_-/r_+$). The horizons are located at $r=r_+, r_-$.

Thermodynamic quantities are given by

$$\begin{aligned} T&= \frac{r_+}{2\pi L^2}(2 - \alpha ^2 - \alpha ^4),\end{aligned}$$

(11.62)

$$\begin{aligned} \varOmega&= - \frac{V_3}{16\pi G_5 L} \left( \frac{r_+}{L} \right) ^4 (1 + \alpha ^2 + \alpha ^4),\end{aligned}$$

(11.63)

$$\begin{aligned} s&= - \frac{1}{V_3} \left( \frac{\partial \varOmega }{\partial T} \right) _{V_3, \mu } = \frac{1}{4 G_5}\left( \frac{r_+}{L}\right) ^3,\end{aligned}$$

(11.64)

$$\begin{aligned} \mu&= \left. A_0 \right| _{r\rightarrow \infty } = \frac{r_+}{L^2} \sqrt{ 3\alpha ^2(1+ \alpha ^2) },\end{aligned}$$

(11.65)

$$\begin{aligned} \rho&= - \frac{1}{V_3} \left( \frac{\partial \varOmega }{\partial \mu } \right) _{T, V_3} = \frac{1}{8\pi G_5} \left( \frac{r_+}{L} \right) ^3 \sqrt{ 3\alpha ^2(1+ \alpha ^2) },\end{aligned}$$

(11.66)

$$\begin{aligned} \varepsilon&= \frac{\varOmega }{V_3} + Ts + \mu \rho = \frac{3}{16\pi G_5 L} \left( \frac{r_+}{L} \right) ^4 (1 + \alpha ^2 + \alpha ^4),\end{aligned}$$

(11.67)

$$\begin{aligned} P&= - \frac{\varOmega }{V_3} = \frac{1}{3} \varepsilon . \end{aligned}$$

(11.68)

To compute thermodynamic quantities, one has to fix, e.g., $\mu$. To do so, rewrite $d\mu =0$ as the condition for $r_+$ and $\alpha$:

$$\begin{aligned} d\mu = \frac{\partial \mu }{\partial r_+} dr_+ + \frac{\partial \mu }{\partial \alpha } d\alpha = 0 \rightarrow d\alpha = - \frac{\alpha (1+\alpha ^2)}{1+2\alpha ^2} \frac{dr_+}{r_+}. \end{aligned}$$

(11.69)

Imposing this condition, one gets

$$\begin{aligned} d\varOmega&= \frac{\partial \varOmega }{\partial r_+} dr_+ + \frac{\partial \varOmega }{\partial \alpha } d\alpha = - \frac{V_3}{8\pi G_5 L^2} \left( \frac{r_+}{L} \right) ^3 (2 + \alpha ^2 + \alpha ^4) dr_+,\end{aligned}$$

(11.70)

$$\begin{aligned} dT&= \frac{\partial T}{\partial r_+} dr_+ + \frac{\partial T}{\partial \alpha } d\alpha = \frac{1}{2\pi L^2} (2 + \alpha ^2 + \alpha ^4) dr_+. \end{aligned}$$

(11.71)

Thus,

$$\begin{aligned} s = - \frac{1}{V_3} \left( \frac{\partial \varOmega }{\partial T} \right) _{V_3, \mu } = \frac{1}{4 G_5}\left( \frac{r_+}{L}\right) ^3. \end{aligned}$$

(11.72)

Similarly, the RN-AdS$_4$ black hole is given by

$$\begin{aligned} ds_4^2&= - f dt^2 + \frac{dr^2}{f} + \left( \frac{r}{L}\right) ^2 d{\varvec{x}}_2^2, \end{aligned}$$

(11.73)

$$\begin{aligned} f&= \left( \frac{r}{L}\right) ^2 \left( 1 - \frac{r_+}{r} \right) \left( 1 - \frac{r_-}{r} \right) \left( 1 + \frac{r_++r_-}{r} + \frac{r_+^2+r_+ r_-+r_-^2}{r^2} \right) , \end{aligned}$$

(11.74)

$$\begin{aligned} A_0&= - \frac{2r_+}{L^2} \sqrt{ \alpha (1+\alpha +\alpha ^2) } \left( \frac{r_+}{r}-1\right) , \end{aligned}$$

(11.75)

($\alpha :=r_-/r_+$). Thermodynamic quantities are given by

$$\begin{aligned} T&= \frac{r_+}{4\pi L^2}(3 - \alpha - \alpha ^2 - \alpha ^3),\end{aligned}$$

(11.76)

$$\begin{aligned} \varOmega&= - \frac{V_2}{16\pi G_4 L} \left( \frac{r_+}{L} \right) ^3 (1 + \alpha + \alpha ^2 + \alpha ^3),\end{aligned}$$

(11.77)

$$\begin{aligned} s&= - \frac{1}{V_2} \left( \frac{\partial \varOmega }{\partial T} \right) _{V_2, \mu } = \frac{1}{4 G_4}\left( \frac{r_+}{L}\right) ^2,\end{aligned}$$

(11.78)

$$\begin{aligned} \mu&= \left. A_0 \right| _{r\rightarrow \infty } = \frac{2r_+}{L^2} \sqrt{ \alpha (1+ \alpha +\alpha ^2) },\end{aligned}$$

(11.79)

$$\begin{aligned} \rho&= - \frac{1}{V_2} \left( \frac{\partial \varOmega }{\partial \mu } \right) _{T, V_2} = \frac{1}{8\pi G_4} \left( \frac{r_+}{L} \right) ^2 \sqrt{ \alpha (1+ \alpha +\alpha ^2) },\end{aligned}$$

(11.80)

$$\begin{aligned} \varepsilon&= \frac{\varOmega }{V_2} + Ts + \mu \rho = \frac{1}{8\pi G_4 L} \left( \frac{r_+}{L} \right) ^3 (1 + \alpha + \alpha ^2 + \alpha ^3),\end{aligned}$$

(11.81)

$$\begin{aligned} P&= - \frac{\varOmega }{V_2} = \frac{1}{2} \varepsilon . \end{aligned}$$

(11.82)

11.1.5 $d=5$ R-charged Black Hole and Holographic Current Anomaly

The $d=5$ R-charged black hole is the solution of the five-dimensional $\fancyscript{N}=2$ gauged $U(1)^3$ supergravity:

$$\begin{aligned} \mathsf S&= \frac{1}{16\pi G_5} \int d^{5}x\, \sqrt{-g} \left( R - \frac{L^2}{2}G_{ij} F^i_{MN}F^{MN~j} - G_{ij} \partial _M X^i \partial ^M X^j + \frac{4}{L^2}\, \sum _{i=1}^3 \frac{1}{X^i} \right. \nonumber \\&\qquad \qquad \qquad \qquad \left. + \frac{L^3}{24\sqrt{-g}}\, \varepsilon ^{ABCDE}\,\varepsilon _{ijk}\, F^i_{AB} F^j_{CD} A^k_{E} \right) . \end{aligned}$$

(11.83)

Here, $F^i_{MN}$ are the field strength of the three $U(1)$ gauge fields $A^i_M$ ($i= 1, 2, 3$). The fields $X^i$ ($i=1, 2, 3$) represent three real scalar fields which are not independent but are subject to the constraint $X^1\, X^2\, X^3 = 1$, and their “moduli space metric” $G_{ij}$ is given by

$$\begin{aligned} G_{ij} = \frac{1}{2} \text {diag}[(X^1)^{-2},\, (X^2)^{-2},\, (X^3)^{-2}]. \end{aligned}$$

(11.84)

Note that $\varepsilon ^{ABCDE}$ is the Levi-Civita symbol not the Levi-Civita tensor: it is purely numerical and takes values $\pm 1$.

The solution is given by

$$\begin{aligned} ds_5^2&= - \fancyscript{H}^{-2/3}f dt^2 + \fancyscript{H}^{1/3} f^{-1} dr^2 + \fancyscript{H}^{1/3} \left( \frac{r}{L}\right) ^2 d{\varvec{x}}_3^2, \end{aligned}$$

(11.85)

$$\begin{aligned} A_0^i&= - \frac{\tilde{\kappa _i}}{L} \left[ \left( \frac{r_+}{r}\right) ^2\frac{1}{H_i} - \frac{1}{1+\kappa _i} \right] ,\end{aligned}$$

(11.86)

$$\begin{aligned} X_i&= \frac{\fancyscript{H}^{1/3}}{H_i}, \end{aligned}$$

(11.87)

where the outer horizon is located at $r=r_+$, and

$$\begin{aligned} H_i&:= 1+ \kappa _i \left( \frac{r_+}{r}\right) ^2, \quad \fancyscript{H} := H_1 H_2 H_3, \end{aligned}$$

(11.88)

$$\begin{aligned} f&:= \left( \frac{r}{L}\right) ^2 \left[ \fancyscript{H} - \left( \frac{r_+}{r}\right) ^4 \prod _{i=1}^3 (1+\kappa _i) \right] ,\end{aligned}$$

(11.89)

$$\begin{aligned} \tilde{\kappa }_i&:= \frac{r_+}{L} \sqrt{\kappa _i} \prod _{l=1}^3 (1+\kappa _l)^{1/2}. \end{aligned}$$

(11.90)

Thermodynamic quantities are given by⁵

$$\begin{aligned} T&= \frac{2 + \kappa _1 + \kappa _2 + \kappa _3 - \kappa _1\kappa _2\kappa _3}{2\sqrt{(1+\kappa _1)(1+\kappa _2)(1+\kappa _3)}}T_0, \end{aligned}$$

(11.91)

$$\begin{aligned} s&= \frac{\pi ^2 N_c^2 T_0^3}{2} \prod _{i=1}^3 (1+\kappa _i)^{1/2},\end{aligned}$$

(11.92)

$$\begin{aligned} \mu _i&= \left. A_0^i \right| _{r\rightarrow \infty } =\pi T_0 \frac{\sqrt{\kappa _i} }{1 + \kappa _i} \prod _{l=1}^3 (1+\kappa _l)^{1/2}, \end{aligned}$$

(11.93)

$$\begin{aligned} \rho _i&= \frac{\pi N_c^2 T_0^3}{4}~\sqrt{\kappa _i} \prod _{l=1}^3 (1+\kappa _l)^{1/2},\end{aligned}$$

(11.94)

$$\begin{aligned} \varepsilon&= \frac{3 \pi ^2 N_c^2 T_0^4}{8} \prod _{i=1}^3 (1+\kappa _i),\end{aligned}$$

(11.95)

$$\begin{aligned} P&= \frac{1}{3} \varepsilon , \end{aligned}$$

(11.96)

where $T_0:= r_+/(\pi L^2)$.

Equal-charge case   The scalar fields are constant $X^i=1$ when $F^i_{MN} = F_{MN}/\sqrt{3}$. Then, the action becomes

$$\begin{aligned} \mathsf S&= \frac{1}{16\pi G_5} \int d^{5}x\, \left\{ \sqrt{-g} \left( R + \frac{12}{L^2}-\frac{L^2}{4} F_{MN}F^{MN} \right) + \frac{L^3}{12\sqrt{3}}\, \varepsilon ^{ABCDE}\, F_{AB} F_{CD} A_{E} \right\} . \end{aligned}$$

(11.97)

The action (11.97) contains the Chern-Simons term . We consider the electric solution, so the Chern-Simons term does not affect the solution. Then, the R-charged black hole reduces to the RN-AdS$_5$ black hole.

However, the Chern-Simons term has an important consequence. Because the Chern-Simons term explicitly contains the gauge potential $A_E$, the bulk action is not gauge invariant at the AdS boundary. From the boundary point of view, this gives the current anomaly .

The $\fancyscript{N}=4\ \mathrm{SYM}$ has the global $\textit{SO}(6)$ R-symmetry, so the theory has conserved currents. When the SYM is coupled with the external gauge fields for the currents, it is known that the R-symmetry becomes anomalous and the currents are not conserved. In the present case, we pick up a $U(1)$ subgroup of the R-symmetry, and we have a $U(1)$ current. Since we have the bulk Maxwell field, the SYM has the external $U(1)$ gauge field for the current. But the bulk Chern-Simons term spoils the bulk gauge invariance and the current is not conserved as we see below.

The equation of motion for the Maxwell field is given by

$$\begin{aligned} \partial _N \left( \sqrt{-g} F^{MN} \right) + \frac{L}{4\sqrt{3}} \varepsilon ^{MABCD}\,F_{AB} F_{CD} = 0. \end{aligned}$$

(11.98)

Using the GKP-Witten relation, one can show

$$\begin{aligned} \langle J^\mu \rangle _s&= \frac{\delta \underline{\mathsf{S }}}{\delta A_\mu ^{(0)}} \end{aligned}$$

(11.99)

$$\begin{aligned}&= \left. \frac{L^2}{16\pi G_5} \left[ \sqrt{-g} F^{\mu r} + \frac{L}{3\sqrt{3}} \varepsilon ^{r\mu \nu \rho \sigma }\,A_{\nu }F_{\rho \sigma } \right] \right| _{r\rightarrow \infty }. \end{aligned}$$

(11.100)

The first term is essentially the same as Eqs. (10.33) and (10.34). Substituting Eq. (11.100) into the $r$-component of Eq. (11.98), one gets

$$\begin{aligned} \partial _\mu \langle J^\mu \rangle _s = -\frac{1}{16\pi G_5} \frac{L^3}{12\sqrt{3}} \varepsilon ^{\mu \nu \rho \sigma }\,F_{\mu \nu }F_{\rho \sigma }, \end{aligned}$$

(11.101)

where $\varepsilon ^{\mu \nu \rho \sigma }=\varepsilon ^{r\mu \nu \rho \sigma }$. Thus, the current is not conserved.

See, e.g., Refs. [22, 23] for phenomenological applications of the anomaly.