Hawking-Page Transition and Confinement

The main idea

The previous page introduced AdS-Schwarzschild black holes and planar black branes as thermal states of the dual field theory. We now study a sharper question: which Euclidean bulk geometry dominates the canonical ensemble at a fixed boundary temperature?

For a CFT on the spatial sphere $S^{d-1}$, the Euclidean thermal boundary is

$$\partial M = S^1_\beta \times S^{d-1}, \qquad \beta = \frac{1}{T}.$$

There are two especially important smooth bulk fillings of this same boundary:

1. Thermal AdS, which is Euclidean global AdS with periodically identified time. It has no horizon. The thermal circle $S^1_\beta$ is noncontractible in the bulk.
2. Euclidean AdS-Schwarzschild, where the thermal circle smoothly shrinks at the horizon. Its Euclidean geometry has a cigar-like $(r,\tau)$ part.

The Hawking-Page transition is the transition between these two saddles. In the dual large-$N$ gauge theory it is interpreted as a confinement/deconfinement transition, or more precisely as a transition from a phase with only $O(1)$ thermal entropy above the vacuum to a phase with $O(N^2)$ entropy.

In classical gravity, the rule is simple:

$$Z(\beta) \approx \sum_{\text{saddles }i} e^{-I_{E,i}^{\mathrm{ren}}}, \qquad F_i = \frac{I_{E,i}^{\mathrm{ren}}}{\beta}.$$

The dominant phase is the saddle with smaller renormalized Euclidean action.

The Hawking-Page transition compares two bulk fillings of the same boundary $S^1_\beta\times S^{d-1}$. Thermal AdS has topology $S^1_\beta\times B^d$, while the Euclidean black hole has topology $D^2\times S^{d-1}$. The transition occurs at $r_h=L$, or $T_{\mathrm{HP}}=(d-1)/(2\pi L)$ for spherical AdS$_{d+1}$ Schwarzschild black holes.

This page has one central lesson: a black hole is not automatically the dominant thermal saddle just because it exists. One must compare free energies.

The canonical ensemble in global AdS

Consider Einstein gravity with negative cosmological constant in $d+1$ bulk dimensions,

$$I_E = -\frac{1}{16\pi G_{d+1}}\int_M d^{d+1}x\sqrt g\left(R+\frac{d(d-1)}{L^2}\right) -\frac{1}{8\pi G_{d+1}}\int_{\partial M}d^d x\sqrt h\,K + I_{\mathrm{ct}}.$$

Here $L$ is the AdS radius, $h_{ij}$ is the induced metric at the cutoff boundary, $K$ is the trace of the extrinsic curvature, and $I_{\mathrm{ct}}$ denotes the standard local counterterms. The Euclidean path integral with boundary metric $S^1_\beta\times S^{d-1}$ computes the thermal partition function of the boundary theory on $S^{d-1}$,

$$Z_{\mathrm{CFT}}(\beta) = \mathrm{Tr}_{\mathcal H(S^{d-1})} e^{-\beta H} \approx Z_{\mathrm{grav}}[S^1_\beta\times S^{d-1}].$$

At large $N$ and large ‘t Hooft coupling, the gravitational path integral is approximated by classical saddles,

$$\log Z \approx - I_E^{\mathrm{ren}}[g_{\mathrm{cl}}].$$

The boundary data specify the temperature and the spatial geometry, but they do not uniquely specify the bulk topology. Different smooth fillings can contribute to the same ensemble. This is why the Hawking-Page transition is possible.

The physical interpretation depends on the boundary spatial manifold. For global AdS, the dual theory lives on $S^{d-1}$. This is a compact space, so strictly speaking one should not expect ordinary finite-volume phase transitions at finite $N$. At $N=\infty$, however, the number of degrees of freedom scales like $N^2$, and the large-$N$ limit supplies a genuine thermodynamic limit even on a compact sphere.

Thermal AdS

Thermal AdS is obtained by taking Euclidean global AdS and identifying the Euclidean time coordinate,

$$\tau \sim \tau+\beta.$$

The Lorentzian global AdS metric is

$$ds^2 = -\left(1+\frac{r^2}{L^2}\right)dt^2 + \frac{dr^2}{1+r^2/L^2} +r^2 d\Omega_{d-1}^2.$$

After $t=-i\tau$, the Euclidean metric is

$$ds_E^2 = \left(1+\frac{r^2}{L^2}\right)d\tau^2 + \frac{dr^2}{1+r^2/L^2} +r^2 d\Omega_{d-1}^2.$$

There is no horizon, so smoothness imposes no special value of $\beta$. Any temperature is allowed. Topologically,

$$M_{\mathrm{thermal\ AdS}} \simeq S^1_\beta\times B^d.$$

The Euclidean time circle is noncontractible. The sphere $S^{d-1}$ shrinks smoothly in the center of AdS, just as the boundary sphere of a ball shrinks in the interior of the ball.

From the CFT point of view, thermal AdS describes a low-temperature phase in which the leading large-$N$ entropy is absent:

$$S_{\mathrm{thermal\ AdS}} = O(1) \quad \text{above the vacuum contribution.}$$

The phrase “above the vacuum contribution” matters. A CFT on $S^{d-1}$ can have a Casimir energy, and in holographic CFTs that vacuum energy may scale like $N^2$. But the thermal entropy of the thermal-AdS saddle is not $O(N^2)$ at leading classical order. The first thermal corrections come from a gas of bulk supergravity or string modes and are suppressed relative to a classical black-hole entropy.

The spherical AdS-Schwarzschild black hole

The Euclidean AdS-Schwarzschild metric with spherical horizon is

$$ds_E^2 = f(r)d\tau^2 +\frac{dr^2}{f(r)} +r^2 d\Omega_{d-1}^2,$$

with

$$f(r)=1+\frac{r^2}{L^2}-\frac{\mu}{r^{d-2}}.$$

The horizon radius $r_h$ is the largest positive root of $f(r_h)=0$, so

$$\mu = r_h^{d-2}\left(1+\frac{r_h^2}{L^2}\right).$$

Near $r=r_h$, the Euclidean $(r,\tau)$ geometry is smooth only if $\tau$ has the correct period. The general formula is

$$T = \frac{f'(r_h)}{4\pi}.$$

Using the expression for $f(r)$ gives

$$T(r_h) = \frac{1}{4\pi r_h}\left(d\frac{r_h^2}{L^2}+d-2\right).$$

The Euclidean geometry is a cigar in the $(r,\tau)$ directions: the thermal circle shrinks smoothly at $r=r_h$. Topologically,

$$M_{\mathrm{BH}} \simeq D^2\times S^{d-1}.$$

This topology difference is not decorative. It controls the Polyakov loop, the dominant saddle, and the deconfinement interpretation.

Temperature branches and local stability

The temperature function

$$T(r_h) = \frac{1}{4\pi r_h}\left(d\frac{r_h^2}{L^2}+d-2\right)$$

has a minimum for $d>2$. Differentiating with respect to $r_h$ gives

$$\frac{dT}{dr_h}=0 \quad\Longrightarrow\quad r_h^2=\frac{d-2}{d}L^2.$$

Thus

$$r_{\min}=L\sqrt{\frac{d-2}{d}}, \qquad T_{\min}=\frac{\sqrt{d(d-2)}}{2\pi L}.$$

For $T<T_{\min}$, no smooth spherical AdS-Schwarzschild black hole with that boundary temperature exists. For $T>T_{\min}$, there are two black-hole branches:

• a small black hole with $r_h<r_{\min}$, negative specific heat, and local thermodynamic instability;
• a large black hole with $r_h>r_{\min}$, positive specific heat, and local thermodynamic stability.

This is already different from asymptotically flat Schwarzschild thermodynamics. AdS acts like a gravitational box. Large AdS black holes can be in stable equilibrium with their Hawking radiation because the AdS boundary reflects radiation back into the bulk.

Local stability, however, is not the same as dominance. A locally stable black hole can still have larger free energy than thermal AdS.

Free energy and the Hawking-Page point

The ADM mass of the spherical AdS-Schwarzschild solution is

$$M = \frac{(d-1)\Omega_{d-1}}{16\pi G_{d+1}} r_h^{d-2}\left(1+\frac{r_h^2}{L^2}\right),$$

where $\Omega_{d-1}$ is the volume of the unit $S^{d-1}$. The Bekenstein-Hawking entropy is

$$S = \frac{A_h}{4G_{d+1}} = \frac{\Omega_{d-1}r_h^{d-1}}{4G_{d+1}}.$$

In the canonical ensemble the relevant thermodynamic potential is

$$F=M-TS.$$

Substituting the formulas above gives the remarkably simple result

$$F_{\mathrm{BH}} = \frac{\Omega_{d-1}}{16\pi G_{d+1}} r_h^{d-2} \left(1-\frac{r_h^2}{L^2}\right),$$

where the zero of free energy is chosen so that thermal AdS has vanishing thermal free energy after the vacuum subtraction.

Therefore

$$F_{\mathrm{BH}}>0 \quad \text{for} \quad r_h<L,$$

and

$$F_{\mathrm{BH}}<0 \quad \text{for} \quad r_h>L.$$

The Hawking-Page transition occurs at

$$r_h=L.$$

The corresponding temperature is

$$T_{\mathrm{HP}} = T(L) = \frac{d-1}{2\pi L}.$$

For $T<T_{\mathrm{HP}}$, thermal AdS dominates the canonical ensemble. For $T>T_{\mathrm{HP}}$, the large AdS black hole dominates.

The transition is first order for $d>2$ because the entropy jumps at leading large $N$. At the transition,

$$S_{\mathrm{BH}}(r_h=L) = \frac{\Omega_{d-1}L^{d-1}}{4G_{d+1}},$$

which is of order $N^2$ in a matrix large-$N$ holographic CFT. Thermal AdS has no such classical horizon entropy.

The CFT interpretation: confinement and deconfinement

The thermal phases are most cleanly distinguished by the large-$N$ scaling of the free energy and entropy.

In a matrix large-$N$ gauge theory, the deconfined plasma has order $N^2$ gluonic degrees of freedom:

$$F_{\mathrm{deconfined}} \sim N^2, \qquad S_{\mathrm{deconfined}} \sim N^2.$$

A confined phase has only color-singlet excitations, so its thermal entropy is order one at fixed temperature:

$$F_{\mathrm{confined}}-F_{\mathrm{vac}} \sim O(1), \qquad S_{\mathrm{confined}}\sim O(1).$$

The Hawking-Page dictionary is then

Bulk saddle	Boundary phase	Leading thermal entropy	Geometry
thermal AdS	confined-like phase	$O(1)$	no horizon
large AdS black hole	deconfined phase	$O(N^2)$	horizon
small AdS black hole	unstable saddle	$O(N^2)$ but negative specific heat	horizon

This is the gravitational origin of the statement that black-hole formation in global AdS is dual to deconfinement in the large-$N$ gauge theory.

One must read this statement carefully. The canonical example, $\mathcal N=4$ SYM on $S^3$, is not QCD on flat space. It is a conformal gauge theory on a compact sphere. The word “confinement” here means that the low-temperature phase has the large-$N$ counting, center-symmetry behavior, and color-singlet spectrum characteristic of confinement. It does not mean that the conformal theory on flat $\mathbb R^3$ has a mass gap or a linear quark potential.

The Polyakov loop and the contractible thermal circle

A useful diagnostic of deconfinement in gauge theory is the Polyakov loop,

$$P(\vec x) = \frac{1}{N}\mathrm{Tr}\,\mathcal P \exp\left(i\int_0^\beta d\tau\, A_\tau(\tau,\vec x)\right).$$

In a pure gauge theory with center symmetry, $\langle P\rangle=0$ in the confined phase and $\langle P\rangle\neq 0$ in the deconfined phase. With adjoint matter, the center symmetry is not explicitly broken by dynamical fundamentals, so this remains a natural large-$N$ diagnostic.

In holography, a Polyakov loop is computed by a Euclidean fundamental string worldsheet whose boundary wraps the thermal circle at the AdS boundary:

$$\langle P\rangle \sim \exp(-S_{\mathrm{F1}}^{\mathrm{ren}}).$$

Now the topology matters.

In thermal AdS, the thermal circle is noncontractible. A disk-shaped fundamental string worldsheet ending on a single thermal circle at the boundary cannot smoothly fill the loop in the bulk. At leading classical order this gives

$$\langle P\rangle = 0 \qquad \text{thermal AdS.}$$

In Euclidean AdS-Schwarzschild, the thermal circle contracts at the horizon. A disk worldsheet can end on the boundary thermal circle and cap off smoothly in the interior. This gives

$$\langle P\rangle \neq 0 \qquad \text{black-hole phase.}$$

This is one of the cleanest geometric translations of deconfinement: the thermal circle becomes contractible in the dominant Euclidean geometry.

There is a small caveat. In a finite-volume system at finite $N$, symmetries are not truly spontaneously broken. The sharp order parameter appears after taking $N\to\infty$ first.

AdS$_5$/CFT$_4$ specialization

For the canonical AdS$_5$/CFT$_4$ example, set $d=4$. The spherical AdS$_5$ black hole has

$$T(r_h) = \frac{1}{4\pi r_h}\left(4\frac{r_h^2}{L^2}+2\right) = \frac{1}{2\pi r_h}\left(1+2\frac{r_h^2}{L^2}\right).$$

The minimum temperature is

$$T_{\min} = \frac{\sqrt{8}}{2\pi L} = \frac{\sqrt 2}{\pi L},$$

while the Hawking-Page temperature is

$$T_{\mathrm{HP}} = \frac{3}{2\pi L}.$$

The free energy is

$$F_{\mathrm{BH}} = \frac{\pi}{8G_5}r_h^2\left(1-\frac{r_h^2}{L^2}\right),$$

since $\Omega_3=2\pi^2$. It changes sign at $r_h=L$.

Using

$$\frac{L^3}{G_5}\sim N^2,$$

we see that the black-hole entropy scales as

$$S_{\mathrm{BH}} \sim \frac{r_h^3}{G_5} \sim N^2 \left(\frac{r_h}{L}\right)^3.$$

At the transition this is already $O(N^2)$. The dual transition is therefore a large-$N$ deconfinement transition of $\mathcal N=4$ SYM on $S^3$ at strong coupling.

Why there is no Hawking-Page transition for the planar black brane

The spherical Hawking-Page transition is often confused with the planar black-brane thermodynamics used for $\mathcal N=4$ SYM on $\mathbb R^3$. These are different ensembles.

The planar black brane has metric

$$ds^2 = \frac{r^2}{L^2}\left(-f(r)dt^2+d\vec x^{\,2}\right) +\frac{L^2}{r^2}\frac{dr^2}{f(r)}, \qquad f(r)=1-\left(\frac{r_h}{r}\right)^d.$$

Its boundary is

$$S^1_\beta\times \mathbb R^{d-1}$$

or a large torus approximation to it. For a conformal theory in infinite flat space, there is no dimensionless parameter like $TL$ built from a sphere radius. Once $T>0$, the black brane gives the deconfined plasma saddle with free-energy density

$$f \sim - N^2 T^d.$$

There is no finite-temperature Hawking-Page transition in the strict planar conformal case. The black brane dominates over the corresponding thermal Poincare AdS saddle for any nonzero temperature, after comparing densities.

This is not a contradiction. The global transition uses the finite sphere radius $L$ as an additional scale. The planar plasma is the high-temperature, large-volume limit of the global black hole.

Confining geometries and Hawking-Page-like transitions

The global Hawking-Page transition is not the only holographic route to confinement. In QCD-like holographic models, the low-temperature geometry often has an IR cap. A common Euclidean topology story is this:

• In the confined phase, a spatial circle contracts smoothly in the bulk. The thermal circle is noncontractible.
• In the deconfined phase, the thermal circle contracts at a horizon. The spatial circle remains finite.

This exchange of contractible cycles is the geometric core of many confinement/deconfinement transitions in holography.

For example, in Witten’s model of a confining large-$N$ gauge theory, one compactifies a spatial direction with supersymmetry-breaking boundary conditions. At low temperature, the spatial circle caps off smoothly; at high temperature, a black-hole geometry dominates. The phase transition is Hawking-Page-like, but it is not the same as the spherical AdS$_{d+1}$ transition above.

The diagnostic Wilson loops also know about the geometry. In a confining background, a spatial Wilson loop can have an area law because the string worldsheet is forced to sit near the IR cap. In a deconfined black-hole background, temporal Wilson loops are allowed because the thermal circle caps off at the horizon.

What exactly jumps?

At leading classical order, the transition is a jump between two geometries with different topology. The most important quantities behave as follows.

Entropy

Thermal AdS has no horizon, so

$$S_{\mathrm{thermal\ AdS}}=O(1)$$

above the vacuum piece. A large AdS black hole has

$$S_{\mathrm{BH}}=\frac{A_h}{4G_{d+1}}=O(N^2).$$

Thus the entropy jumps by $O(N^2)$ at $N=\infty$.

Energy

The energy also jumps by $O(N^2)$. In the CFT this is the latent heat of the first-order deconfinement transition.

Polyakov loop

At leading large $N$,

$$\langle P\rangle = 0 \quad \text{in thermal AdS}, \qquad \langle P\rangle \neq 0 \quad \text{in the black-hole phase}.$$

Spectrum

Thermal AdS corresponds to a gas of color-singlet states. The black-hole phase corresponds to a deconfined plasma with a continuous-looking spectrum in the large-volume or high-temperature limit. On the gravity side, the difference is between normal modes in a horizonless spacetime and dissipative quasinormal modes in a black-hole spacetime.

Relation to large-$N$ matrix models

The same physics can be described in the CFT using the holonomy around the thermal circle. Define the thermal holonomy

$$U = \mathcal P\exp\left(i\int_0^\beta d\tau\,A_\tau\right).$$

The Polyakov loops are moments of the eigenvalue distribution,

$$u_n = \frac{1}{N}\mathrm{Tr}\,U^n.$$

A center-symmetric confined phase has an approximately uniform eigenvalue distribution on the unit circle, so

$$\langle \nu_1\rangle=0.$$

A deconfined phase has a gapped or clumped eigenvalue distribution, giving

$$\langle \nu_1\rangle\neq 0.$$

At weak coupling on $S^3$, this matrix-model language gives a useful perturbative picture of deconfinement. At strong coupling and large $N$, the gravitational Hawking-Page transition is the dual description.

The details of the weak-coupling and strong-coupling transitions need not match quantitatively. Protected quantities are special; thermal free energies and deconfinement temperatures are generally not protected.

Common mistakes

Mistake 1: “Every AdS black hole dominates at its temperature.”

No. The black-hole solution may exist but fail to dominate the canonical ensemble. For $T_{\min}<T<T_{\mathrm{HP}}$, the large black hole is locally stable but still has higher free energy than thermal AdS.

Mistake 2: “The small black hole is the confined phase.”

No. The small black hole has a horizon and $O(N^2)$ entropy, but it is thermodynamically unstable in the canonical ensemble. The confined-like phase is thermal AdS, not the small black hole.

Mistake 3: “The Hawking-Page transition is the same as the black-brane plasma at any $T$.”

No. The Hawking-Page transition for global AdS depends on the boundary sphere radius. The planar black brane describes a CFT on flat space and has no finite-temperature confinement/deconfinement transition in the conformal case.

Mistake 4: “Confinement means the same thing in every holographic model.”

No. In $\mathcal N=4$ SYM on $S^3$, the confined-like phase is defined by large-$N$ counting and center-symmetry diagnostics on a compact space. In QCD-like holographic models, confinement may also mean a mass gap, area-law spatial Wilson loops, and an IR cap in the geometry.

Mistake 5: “The free energy of thermal AdS is literally zero.”

Only after a choice of subtraction. The physical comparison is between renormalized Euclidean actions with the same boundary geometry. The CFT on a sphere can have a vacuum Casimir energy; the key point is the absence of leading $O(N^2)$ thermal entropy in the low-temperature saddle.

Exercises

Exercise 1: Derive the black-hole temperature

For the spherical AdS-Schwarzschild metric

$$f(r)=1+\frac{r^2}{L^2}-\frac{\mu}{r^{d-2}},$$

use $f(r_h)=0$ to show that

$$T=\frac{1}{4\pi r_h}\left(d\frac{r_h^2}{L^2}+d-2\right).$$

Solution

The horizon condition gives

$$\mu=r_h^{d-2}\left(1+\frac{r_h^2}{L^2}\right).$$

Differentiate $f(r)$:

$$f'(r)=\frac{2r}{L^2}+(d-2)\frac{\mu}{r^{d-1}}.$$

At $r=r_h$,

$$f'(r_h) =\frac{2r_h}{L^2} +(d-2)\frac{1}{r_h}\left(1+\frac{r_h^2}{L^2}\right).$$

Therefore

$$f'(r_h) =\frac{d r_h}{L^2}+\frac{d-2}{r_h},$$

and Euclidean smoothness gives

$$T=\frac{f'(r_h)}{4\pi} =\frac{1}{4\pi r_h}\left(d\frac{r_h^2}{L^2}+d-2\right).$$

Exercise 2: Find $T_{\min}$ and $T_{\mathrm{HP}}$

For $d>2$, show that the black-hole temperature has a minimum at

$$r_{\min}=L\sqrt{\frac{d-2}{d}},$$

and find $T_{\min}$. Then show that the Hawking-Page transition occurs at

$$T_{\mathrm{HP}}=\frac{d-1}{2\pi L}.$$

Solution

Write

$$T(r_h)=\frac{1}{4\pi}\left(\frac{d r_h}{L^2}+\frac{d-2}{r_h}\right).$$

Then

$$\frac{dT}{dr_h} =\frac{1}{4\pi}\left(\frac{d}{L^2}-\frac{d-2}{r_h^2}\right).$$

Setting this to zero gives

$$r_h^2=\frac{d-2}{d}L^2.$$

Thus

$$r_{\min}=L\sqrt{\frac{d-2}{d}}.$$

Substituting into $T(r_h)$ gives

$$T_{\min} =\frac{\sqrt{d(d-2)}}{2\pi L}.$$

The Hawking-Page transition is where the black-hole free energy changes sign,

$$F_{\mathrm{BH}} \propto 1-\frac{r_h^2}{L^2}=0.$$

Therefore $r_h=L$. Substituting into $T(r_h)$ gives

$$T_{\mathrm{HP}} =\frac{1}{4\pi L}(d+d-2) =\frac{d-1}{2\pi L}.$$

Exercise 3: Derive the free energy from $M-TS$

Using

$$M = \frac{(d-1)\Omega_{d-1}}{16\pi G_{d+1}} r_h^{d-2}\left(1+\frac{r_h^2}{L^2}\right),$$ $$S = \frac{\Omega_{d-1}r_h^{d-1}}{4G_{d+1}},$$

and

$$T = \frac{1}{4\pi r_h}\left(d\frac{r_h^2}{L^2}+d-2\right),$$

show that

$$F=M-TS = \frac{\Omega_{d-1}}{16\pi G_{d+1}} r_h^{d-2}\left(1-\frac{r_h^2}{L^2}\right).$$

Solution

First compute

$$TS = \frac{\Omega_{d-1}r_h^{d-1}}{4G_{d+1}} \frac{1}{4\pi r_h}\left(d\frac{r_h^2}{L^2}+d-2\right).$$

Thus

$$TS = \frac{\Omega_{d-1}}{16\pi G_{d+1}} r_h^{d-2} \left(d\frac{r_h^2}{L^2}+d-2\right).$$

Subtracting from $M$ gives

$$F = \frac{\Omega_{d-1}}{16\pi G_{d+1}} r_h^{d-2} \left[(d-1)\left(1+\frac{r_h^2}{L^2}\right)-d\frac{r_h^2}{L^2}-(d-2)\right].$$

The bracket simplifies to

$$(d-1)-(d-2)+\left(d-1-d\right)\frac{r_h^2}{L^2} =1-\frac{r_h^2}{L^2}.$$

Therefore

$$F = \frac{\Omega_{d-1}}{16\pi G_{d+1}} r_h^{d-2}\left(1-\frac{r_h^2}{L^2}\right).$$

Exercise 4: Polyakov loops and topology

Explain why the topology of the Euclidean thermal circle implies $\langle P\rangle=0$ in thermal AdS but allows $\langle P\rangle\neq 0$ in the AdS black-hole phase.

Solution

In holography, a Polyakov loop is represented by a Euclidean fundamental string worldsheet ending on the thermal circle at the boundary.

In thermal AdS, the boundary thermal circle is noncontractible in the bulk. A disk-shaped string worldsheet cannot end on a single noncontractible loop and close smoothly in the interior. Thus there is no classical disk saddle for a single Polyakov loop, giving $\langle P\rangle=0$ at leading large $N$.

In the Euclidean black-hole geometry, the thermal circle shrinks smoothly at the horizon. The string worldsheet can form a disk: its boundary is the thermal circle at infinity, and its interior caps off at the horizon. Thus a finite-action worldsheet exists and $\langle P\rangle$ can be nonzero.

This is the geometric version of deconfinement.

Exercise 5: Why the planar brane has no finite-temperature Hawking-Page transition

Use dimensional analysis to explain why a conformal theory on $\mathbb R^{d-1}$ should have free-energy density

$$f(T)\propto -N^2 T^d$$

in the deconfined plasma phase, and why there is no special finite temperature analogous to $T_{\mathrm{HP}}$.

Solution

On $\mathbb R^{d-1}$, a CFT at temperature $T$ has no scale other than $T$. The free-energy density has dimension $d$, so dimensional analysis gives

$$f(T)= -c N^2 T^d$$

for some positive constant $c$ in the deconfined phase. There is no sphere radius $L$ or confinement scale $\Lambda$ with which to form a dimensionless quantity such as $TL$ or $T/\Lambda$.

The global Hawking-Page transition occurs because the boundary sphere supplies a scale $L$, and the spherical black-hole free energy can change sign as $r_h/L$ varies. For the planar brane, the corresponding black-brane free-energy density is negative for every $T>0$. Thus the conformal plasma on flat space has no finite-temperature Hawking-Page transition.

Further reading

• S. W. Hawking and D. N. Page, “Thermodynamics of Black Holes in Anti-de Sitter Space,” Communications in Mathematical Physics 87 (1983) 577.
• E. Witten, “Anti-de Sitter Space, Thermal Phase Transition, And Confinement In Gauge Theories,” arXiv:hep-th/9803131.
• O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri, and Y. Oz, “Large N Field Theories, String Theory and Gravity,” arXiv:hep-th/9905111.
• B. Sundborg, “The Hagedorn Transition, Deconfinement and $\mathcal N=4$ SYM Theory,” arXiv:hep-th/9908001.
• O. Aharony, J. Marsano, S. Minwalla, K. Papadodimas, and M. Van Raamsdonk, “The Hagedorn/Deconfinement Phase Transition in Weakly Coupled Large N Gauge Theories,” arXiv:hep-th/0310285.