2.3 Generalisations of AdS/CFT
2.3.2 Finite temperature
entropy associated with a given the state operator reads
S(⇢) = Tr (⇢log⇢), (2.60) where log⇢ is well-defined due to the state matrix being diagonalisable.
In the canonical ensemble, similar to the probability of a configuration of energy E in the classical context, the state matrix is now given by
⇢= 1
Zcane Hˆ , (2.61)
whereZcan is the canonical partition function Zcan = Tr⇣
e Hˆ⌘
, (2.62)
which normalises the state operator such that Tr⇢= 1.
Finally, in the same fashion, the grand canonical ensemble is defined by its partition function
Zgrand = Tr exp Hˆ X
k
µkQˆk
!!
, (2.63)
where we promoted the particle numbers Nk to conserved charge operators Qˆk.5 The state operator is then likewise given by
⇢= 1 Zgrand
e (Hˆ PkµkQˆk), (2.64) which concludes our recapitulation of both classical and quantum statistical mechanics.
imaginary time over a compact region of size , which turns out to be the in-verse temperature, = 1/kBT, withkB the Boltzmann constant. Depending on the nature of the fields involved, we need to set periodic or antiperiodic boundary conditions for bosons and fermions, respectively. For zero temper-ature systems, a generic partition function is given by
Z = Z
D[ ]e+i S[ ], (2.65)
where denotes the entity of field and S denotes the action of the system, given by
S[ ] = Z
dt Z
ddxL( ), (2.66)
with L some Lagrangian defining the theory. By exchanging t ! i⌧, we end up with the Euclidean path integral
Z = Z
D[ ]e SE, (2.67)
which we already encountered in (2.46) and SE is the Euclidean action. So far, we did nothing but a Wick rotation, which is always possible for time independent boundary conditions. However, if we compactify the integra-tion regime of imaginary time to ⌧ 2 [0, ) by identifying ⌧ ⇠ ⌧ + , we need to determine boundary conditions for the fields involved. It turns out that bosonic fields need periodic boundary conditions (0) = + ( ), while fermionic fields require anti-periodic boundary conditions (0) = ( ) for this compactification to make sense. The path integral becomes
Z = Z
D[ ]
(0)=± ( )
e SE[ ], (2.68)
and compactifying the integration regime to ⌧ 2 [0, ], where we imposed (anti-)periodic boundary conditions in the path integral as discussed above.
SE denotes the euclidean action, given by SE[ ] =
Z
0
d⌧
Z
ddx L( ). (2.69)
It’s called Euclidean, because in the process of exchanging real time for imaginary time, we e↵ectively also changed the signature of the metric from { ,+,+, . . .} to {+,+,+, . . .}. The Euclidean action still contains tempo-ral derivatives of the fields. If we require the fields and operators to be stationary, as they should be in thermal equilibrium, we see can trivially
integrate over the temporal direction, which yields a factor of . The other factor reduces to an integration of the Hamiltonian density over all spatial dimensions, which yields the Hamiltonian ˆH of the system. Hence, we see that (2.67) defines the canonical partition function in the sense of statistical mechanics,
Zcan = Z
D[ ]
(0)=± ( )
e H[ ] = Tr⇣
e H[ ]ˆ ⌘
. (2.70)
With the state operator defined in the usual way
⇢= 1
Zcane H[ ]ˆ , (2.71)
the free energy of the system is given by
F =T log (Zcan)⇡T log(SE), (2.72) where we performed a saddle point approximation for the second equality.
The latter, of course, is only possible for weakly coupled systems, which will not be the case in the weak form of the gauge/gravity duality.
Gravity side
On the gravity side of the duality, the incorporation of temperature appear both naturally and astonishing. Since the seminal paper by Hawking [10], it is known that we can assign a temperature to black holes in the semiclas-sical regime. More precisely, it was shown that black holes emit particles with an emission spectrum matching that of a perfect black body at a cer-tain temperature, calledHawking temperature TH. In an asymptotically flat Schwarzschild geometry, the Hawking temperature is given in terms of the mass of the black hole by
TH = ~c3
8⇡kBGNM (2.73)
where we used SI units to show the equation in its full glory, combining constants from quantum mechanics, gravitation and thermodynamics alto-gether. Most remarkable, the temperature of the black hole decreases with its mass, imprinting itself in negative heat capacity leading to instability of the black hole. While evaporating due to the mentioned Hawking radiation, black holes in asymptotically flat geometries are getting hotter and hotter.
Their ultimate faith is subject to speculation. Only a full theory of quan-tum gravity will be able to solve the dynamics as the curvature at the event
horizon enters the regime in which Hawking’s semiclassical analysis is not reliable anymore.
For asymptotically curved backgrounds, the situation is di↵erent. Espe-cially in asymptotically AdS spaces, the temperature of black holes grows with their mass, so that for negative cosmological constants, black holes can be thermodynamically stable. For the finite temperature generalisation of the AdS/CFT conjecture, the objects responsible for the temperature on the gravity side are actually near-extremal Dp-branes, which can be used in the same fashion as the Dp-branes for the derivation of the original conjecture.
These are non-BPS solutions to type IIB supergravity with metric ds2 =H(r) 1/2 f(r) dt2+ dx2 +H(r)1/2
✓ dr2
f(r) +r2 ds2 S5
◆
(2.74) withH(r) = 1+L4/r4 andf(r) = 1 r4H/r4. The solution looks familiar from equation (2.33), but has the additional blackening factorf(r) in itsttand rr-components. This yields an event horizon at r = rH, where gtt(rH) = 0. In the same near horizon limitr ⌧Land by changing coordinates to z ⌘L2/r as in the BPS case, the background metric on which perturbative superstring theory is defined is now given by
ds2 = L2 z2
✓
h(z) dt2+ dx2+ dz2 h(z)
◆
+L2 ds2 S5 , (2.75) where we defined h(z) = f(L2/z). This is a black hole or black brane in asymptotically AdS space, respectively, depending on whether or not we compactify the x-directions.
One way to see why black holes are thermal objects is by performing the same mathematical procedure as for quantum field theories at finite tem-perature: We perform a Wick rotation t ! i⌧ and, in addition, have a closer look at what happens near the event horizon atz =zH by introducing another radial variable ⇢2 =L2⇣
1 zz
H
⌘
. The metric becomes ds2 = d⇢2 +⇢2
✓ 4 zH2 d⌧2
◆ + L2
z2H dx2+L2 ds2 S5 . (2.76) If we neglect the last two summands of this metric, it looks similar to a flat metric around ⇢ = 0, which is the location of the event horizon. Event horizons are non-local concepts of (super)gravity, meaning that in general we can only find their positions if we know the metric globally. Hence, it should not display any special local behaviour like curvature singularities or topological kinks. However, for a metric of the form ds2 = d⇢2 +⇢2d 2,
there exists the possibility of a conical singularity, at ⇢ = 0, if the angular coordinate is not 2⇡-periodic. Requiring that the metric at hand is really flat at ⇢ = 0, we find that ⌧ needs to be periodic, ⌧ ⇠⌧ + and its period is determined by
✓ 2 zH
◆2
= (2⇡)! 2, (2.77)
which yields = ⇡zH. Like in thermal field theory, we identify = 1/TH, whereTH is now the Hawking temperature associated with the black hole or brane, and obtain
TH = 1
⇡zH . (2.78)
We remember thatz =L2/r, sozH really is a length scale, where, however, largerzH means smaller black holes/branes due to its definition. Due to the relation we just derived, we see that in asymptotic AdS-space, larger black holes have higher temperature. This renders them thermodynamically stable, as their heat capacity, unlike in asymptotically flat spacetimes, is positive.
The conclusion of this section is, that the original AdS/CFT conjecture can be generalised to N = 4 SU(N) Super Yang-Mills theory at finite tem-perature T if the gravity side of the duality has a black brane with Hawk-ing temperature TH, which we identify with the field theory temperature, T ⌘TH. Just like the original AdS/CFT correspondence is really an equiv-alence of the microcanonical partition functions by (2.48), in the canonical ensemble we have to identify the free energies of both theories [16,84]. Ther-mal correlators can then be derived by functional derivatives on both sides.