Applied holography

Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257 www.elsevier.com/locate/npbps

Applied holography Dam Thanh Sona∗ a

Institute for Nuclear Theory, University of Washington, Box 351550, Seattle, WA 98195-1550, USA

We review recent progress in applying the AdS/CFT correspondence to finite-temperature field theory. In particular, we show how the hydrodynamic behavior of field theory is reflected in the low-momentum limit of correlation functions computed through a real-time AdS/CFT prescription, which we formulate. We also show how the hydrodynamic modes in field theory correspond to the low-lying quasinormal modes of the AdS black p-brane metric. We provide a proof of the universality of the viscosity/entropy ratio within a class of theories with gravity duals and formulate a viscosity bound conjecture. We also review possible applications of gauge/gravity duality in nonrelativistic systems, and discuss the holographic realization of nonrelativistic hydrodynamics.

1. INTRODUCTION The study of quantum field theory at high temperature has a long history. It was first motivated by the Big Bang cosmology when it was hoped that early phase transitions might leave some imprints on the Universe. One of those phase transitions is the QCD phase transitions (which could actually be a crossover) which happened at a temperature around Tc ∼ 200 MeV, when matter turned from a gas of quarks and gluons (the quark-gluon plasma, or QGP) into a gas of hadrons. The only way to study QGP experimentally is by colliding two heavy atomic nuclei. Most recent experiments are conducted at the Relativistic Heavy Ion Collider (RHIC) at Brookhaven National Laboratory. Although significant circumstantial evidence for the QGP was accumulated [1], a theoretical interpretation of most of the experimental data proved difficult, because the QGP created at RHIC is far from being a weakly coupled gas of quarks and gluons. Indeed, the temperature of the plasma, as inferred from the spectrum of final particles, is only approximately 170 MeV, near the confinement scale of QCD. This is deep in the nonperturbative regime of QCD, where reliable theoretical tools are lacking. Most notably, the kinetic coefficients of the ∗ Work supported, in part, by DOE grant DE-FG0200ER41132.

0920-5632/$ – see front matter © 2009 Published by Elsevier B.V. doi:10.1016/j.nuclphysbps.2009.10.017

QGP, which enter the hydrodynamic equations (reviewed in Sec. 2), are not theoretically computable at these temperatures. The paucity of information about the kinetic coefficients of the QGP in particular and of strongly coupled thermal quantum field theories in general is one of the main reasons for our interest in their computation in a class of strongly coupled field theories, even though this class does not include QCD. The necessary technological tool is the anti–de Sitter–conformal field theory (AdS/CFT) correspondence [2–4], discovered in the investigation of D-branes in string theory. This correspondence allows one to describe the thermal plasmas in these theories in terms of black holes in AdS space. The main part of these lectures is devoted to this application of AdS/CFT correspondence. The AdS/CFT correspondence is reviewed in Sec. 3. The first calculation of this type, that of the shear viscosity in N = 4 supersymmetric YangMills (SYM) theory [5], is followed by the theoretical work to establish the rules of real-time finite-temperature AdS/CFT correspondence [6, 7]. Applications of these rules to various special cases [8–11] clearly show that even very exotic field theories, when heated up to finite temperature, behave hydrodynamically at large distances and time scales (provided that the number of space-time dimensions is 2+1 or higher). This development is reviewed in Sec. 3.3. Moreover,

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the way AdS/CFT works reveals deep connections to properties of black holes in classical gravity. For example, the hydrodynamic modes of a thermal medium are mapped, through the correspondence, to the low-lying quasi-normal modes of a black-brane metric. It seems that our understanding of the connection between hydrodynamics and black hole physics is still incomplete; we may understand more about gravity by studying thermal field theories. One idea along this direction is reviewed in Sec. 3.5. From the point of view of heavy-ion (QGP) physics, a particularly interesting finding has been the formulation of a conjecture on the lowest possible value of the ratio of viscosity and volume density of entropy. This conjecture was motivated by the universality of this ratio in theories with gravity duals. This is reviewed in Sec. 3.6. We then review an extension of hydrodynamic theory—the so-called second-order hydrodynamics, which goes one step further than the usual Navier-Stokes hydrodynamics [12,13]. This development is reviewed in Sec. 4. The final part of these lectures is devoted to the discussion of the recent proposal of a gravity dual of nonrelativistic conformal field theories, which relies on the so-called Schr¨odinger geometry [14,15]. We show how the nonrelativistic Navier-Stokes equation arises as the lowenergy dynamics of black holes in asymptotically Schr¨ odinger space [16]. In these lectures we use the “mostly plus” metric signature − + ++.

2. HYDRODYNAMICS From the modern perspective, hydrodynamics [17] is best thought of as an effective theory, describing the dynamics at large distances and time-scales. Unlike the familiar effective field theories (for example, the chiral perturbation theory), it is normally formulated in the language of equations of motion instead of an action principle. The reason for this is the presence of dissipation in thermal media. In the simplest case, the hydrodynamic equations are just the laws of conservation of energy

and momentum, ∂μ T μν = 0 .

(1)

To close the system of equations, we must reduce the number of independent elements of T μν . This is done through the assumption of local thermal equilibrium: If perturbations have long wavelengths, the state of the system, at a given time, is determined by the temperature as a function of coordinates T (x) and the local fluid velocity uμ , which is also a function of coordinates uμ (x). Because uμ uμ = −1, only three components of uμ are independent. The number of hydrodynamic variables is four, equal to the number of equations. In hydrodynamics we express T μν through T (x) and uμ (x) through the so-called constitutive equations. Following the standard procedure of effective field theories, we expand in powers of spatial derivatives. To zeroth order, T μν is given by the familiar formula for ideal fluids, T μν = ( + P )uμ uν + P g μν ,

(2)

where  is the energy density, and P is the pressure. Normally one would stop at this leading order, but qualitatively new effects necessitate going to the next order. Indeed, from Eq. (2) and the thermodynamic relations d = T dS, dP = sdT , and  + P = T s (s is the entropy per unit volume), one finds that entropy is conserved [18] ∂μ (suμ ) = 0 .

(3)

Thus, to have entropy production, one needs to go to the next order in the derivative expansion. At the next order, we write T μν = ( + P )uμ uν + P g μν − σ μν ,

(4)

where σ μν is proportional to derivatives of T (x) and uμ (x) and is termed the dissipative part of T μν . To write these terms, let us first fix a point x and go to the local rest frame where ui (x) = 0. In this frame, in principle one can have dissipative corrections to the energy-momentum density T 0μ . However, one recalls that the choice of T and uμ is arbitrary, and thus one can always redefine them

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so that these corrections vanish, σ 00 = σ 0i = 0, and so at a point x, T 00 = ,

T 0i = 0 .

(5)

The only nonzero elements of the dissipative energy-momentum tensor are σij . To the next-toleading order there are extra contributions whose forms are dictated by rotational symmetry:   2 σij = η ∂i uj + ∂j ui − δij ∂k uk + ζδij ∂k uk . (6) 3 Going back to the general frame, we can now write the dissipative part of the energymomentum tensor as σ μν

   2 μα νβ λ =P P η ∂α uβ + ∂β uα − gαβ ∂λ u 3  λ + ζgαβ ∂λ u , (7)

where P μν = g μν + uμ uν is the projection operator onto the directions perpendicular to uμ . If the system contains a conserved current, there is an additional hydrodynamic equation related to the current conservation, ∂μ j μ = 0 .

(8)

The constitutive equation contains two terms: j μ = ρuμ − DP μν ∂ν α ,

(9)

where ρ is the charge density in the fluid rest frame and D is some constant. The first term corresponds to convection, the second one to diffusion. In the fluid rest frame, j = −D∇ρ, which is Fick’s law of diffusion, with D being the diffusion constant. 2.1. Kubo’s Formula For Viscosity As mentioned above, the hydrodynamic equations can be thought of as an effective theory describing the dynamics of the system at large lengths and time scales. Therefore one should be able to use these equations to extract information about the low-momentum behavior of Green’s functions in the original theory.

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For example, let us recall how the two-point correlation functions can be extracted. If we couple sources Ja (x) to a set of (bosonic) operators Oa (x), so that the new action is  (10) S = S0 + Ja (x)Oa (x) , x

then the source will introduce a perturbation of the system. In particular, the average values of Oa will differ from the equilibrium values, which we assume to be zero. If Ja are small, the perturbations are given by the linear response theory as  (11) Oa (x) = − GR ab (x − y)Jb (y) , y

where

GR ab

is the retarded Green’s function

0 0 iGR ab (x − y) = θ(x − y )[Oa (x), Ob (y)] .

(12)

The fact that the linear response is determined by the retarded (and not by any other) Green’s function is obvious from causality: The source can influence the system only after it has been turned on. Thus, to determine the correlation functions of T μν , we need to couple a weak source to T μν and determine the average value of T μν after this source is turned on. To find these correlators at low momenta, we can use the hydrodynamic theory. So far in our treatment of hydrodynamics we have included no source coupled to T μν . This deficiency can be easily corrected, as the source of the energy-momentum tensor is the metric gμν . One must generalize the hydrodynamic equations to curved space-time and from it determine the response of the thermal medium to a weak perturbation of the metric. This procedure is rather straightforward and in the interest of space is left as an exercise to the reader. Here we concentrate on a particular case when the metric perturbation is homogeneous in space but time dependent: gij (t, x) = δij + hij (t), hij  1 g0i (t, x) = 0 . g00 (t, x) = −1,

(13) (14)

Moreover, we assume the perturbation to be traceless, hii = 0. Because the perturbation

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is spatially homogeneous, if the fluid moves, it can only move uniformly: ui = ui (t). However, this possibility can be ruled out by parity, so the fluid must remain at rest all the time: uμ = (1, 0, 0, 0). We now compute the dissipative part of the stress-energy tensor. The generalization of Eq. (7) to curved space-time is  σ

μν

=P

μα

P

νβ

η(∇α uβ + ∇β uα )    2 + ζ − η gαβ ∇ · u . 3

(15)

Substituting uμ = (1, 0, 0, 0) and gμν from Eq. (13), we find only contributions to the traceless spatial components, and these contributions come entirely from the Christoffel symbols in the covariant derivatives. For example, σxy = 2ηΓ0xy = η∂0 hxy .

(16)

By comparison with the expectation from the linear response theory, this equation means that we have found the zero spatial momentum, lowfrequency limit of the retarded Green’s function of T xy :  (ω, 0) = d4 x eiωt θ(t)[Txy (x), Txy (0)] GR xy,xy = −iηω + O(ω 2 )

(17)

(modulo contact terms). We have, in essence, derived the Kubo’s formula relating the shear viscosity and a Green’s function: η = − lim

ω→0

1 Im GR xy,xy (ω, 0) . ω

(18)

There is a similar Kubo’s relation for the charge diffusion constant D. 2.2. Hydrodynamic Modes If one is interested only in the locations of the poles of the correlators, one can simply look for the normal modes of the linearized hydrodynamic equations, that is, solutions that behave as e−iωt+ik·x . Owing to dissipation, the frequency ω(k) is complex. For example, the equation of charge diffusion, ∂t ρ − D∇2 ρ = 0,

(19)

corresponds to a pole in the current-current correlator at ω = −iDk 2 . To find the poles in the correlators between elements of the stress-energy tensor one can, without loss of generality, choose the coordinate system so that k is aligned along the x3 -axis: k = (0, 0, k). Then one can distinguish two types of normal modes: 1. Shear modes correspond to the fluctuations of pairs of components T 0a and T 3a , where a = 1, 2. The constitutive equation is T 3a = −η∂3 ua = −

η ∂3 T 0a , +P

(20)

and the equation for T 0a is ∂t T 0a −

η ∂ 2 T 0a = 0 . +P 3

(21)

That is, it has the form of a diffusion equa3 tion for T 0a . Substituting e−iωt+ikx into the equation, one finds the dispersion law ω = −i

η k2 . +P

(22)

2. Sound modes are fluctuations of T 00 , T 03 , and T 33 . There are now two conservation equations, and by diagonalizing them one finds the dispersion law   k2 i 4 ω = cs k − η+ζ , (23) 2 3 +P where cs = dP/d. This is simply the sound wave, which involves the fluctuation of the energy density. It propagates with velocity cs , and its damping is related to a linear combination of shear and bulk viscosities. In CFTs it is possible to use conformal Ward identities to show that the bulk viscosity vanishes: ζ = 0. Hence, we shall concentrate our attention on the shear viscosity η. 2.3. Viscosity In Weakly Coupled Field Theories We now briefly consider the behavior of the shear viscosity in weakly coupled field theories,

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with the λφ4 theory as a concrete example. At weak coupling, there is a separation between two length scales: The mean free path of particles is much larger than the distance scales over which scatterings occur. Each scattering event takes a time of order T −1 (which can be thought of as the time required for final particles to become onshell). The mean free path mfp can be estimated from the formula mfp ∼

1 , nσv

(24)

where n is the density of particles, σ is the typical scattering cross section, and v is the typical particle velocity. Inserting the values for thermal λφ4 theory, n ∼ T 3 , σ ∼ λ2 T −2 , and v ∼ 1, one finds mfp ∼

1 1  . λ2 T T

(25)

The viscosity can be estimated from kinetic theory to be η ∼ mfp ,

(26)

where  is the energy density. From  ∼ T 4 and the estimate of mft , one finds η∼

T3 . λ2

(27)

In particular, the weaker the coupling λ, the larger the viscosity η. This behavior is explained by the fact that the viscosity measures the rate of momentum diffusion. The smaller λ is, the longer a particle travels before colliding with another one, and the easier the momentum transfer. It may appear counterintuitive that viscosity tends to infinity in the limit of zero coupling λ → 0: At zero coupling there is no dissipation, so should the viscosity be zero? The confusion arises owing to the fact that the hydrodynamic theory, and hence the notion of viscosity, makes sense only on distances much larger than the mean free path of particles. If one takes λ → 0, then to measure the viscosity one has to do the experiment at larger and larger length scales. If one fixes the size of the experiment and takes λ → 0, dissipation disappears, but it does not tell us anything about the viscosity.

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As will become apparent below, a particularly interesting ratio to consider is the ratio of shear viscosity and entropy density s. The latter is proportional to T 3 ; thus η 1 ∼ 2. s λ

(28)

One has η/s  1 for λ  1. This is a common feature of weakly coupled field theories. Extrapolating to λ ∼ 1, one finds η/s ∼ 1. We shall see that theories with gravity duals are strongly coupled, and η/s is of order one. More surprisingly, this ratio is the same for all theories with gravity duals. To compute rather than estimate the viscosity, one can use Kubo’s formula. It turns out that one has to sum an infinite number of Feynman graphs to even find the viscosity to leading order. Another way that leads to the same result is to first formulate a kinetic Boltzmann equation for the quasi-particles as an intermediate effective description, and then derive hydrodynamics by taking the limit of very long lengths and time scales in the kinetic equation. Interested readers should consult Refs. [19,20] for more details. 3. AdS/CFT CORRESPONDENCE 3.1. Review Of AdS/CFT Correspondence At Zero Temperature This section briefly reviews the AdS/CFT correspondence at zero temperature. It contains only the minimal amount of materials required to understand the rest of the review. Further information can be found in existing reviews and lecture notes [21,22]. The original example of AdS/CFT correspondence is between N = 4 supersymmetric YangMills (SYM) theory and type IIB string theory on AdS5 ×S5 space. Let us describe the two sides of the correspondence in some more detail. The N = 4 SYM theory is a gauge theory with a gauge field, four Weyl fermions, and six real scalars, all in the adjoint representation of the color group. Its Lagrangian can be written down explicitly, but is not very important for our purposes. It has a vanishing beta function and is a conformal field theory (CFT) (thus the CFT

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in AdS/CFT). In our further discussion, we frequently use the generic terms “field theory” or CFT for the N = 4 SYM theory. On the string theory side, we have type IIB string theory, which contains a finite number of massless fields, including the graviton, the dilaton Φ, some other fields (forms) and their fermionic superpartners, and an infinite number of massive string excitations. It has two parameters: the string length ls (related to the slope parameter α by α = ls2 ) and the string coupling gs . In the long-wavelength limit, when all fields vary over length scales much larger than ls , the massive modes decouple and one is left with type IIB supergravity in 10 dimensions, which can be described by an action [23] 

√ 1 S= 2 d10 x −g e−2Φ R + 4 ∂ μ Φ∂μ Φ 2κ10  + · · · , (29) where κ10 is the 10-dimensional gravitational constant, √ κ10 = 8πG = 8π 7/2 gs ls4 , (30) and · · · stay for the contributions from fields other than the metric and the dilaton. One of these fields is the five-form F5 , which is constrained to be self-dual. The type IIB string theory lives is a 10-dimensional space-time with the following metric: ds2 =

r2 R2 2 2 2 (−dt + dx ) + dr + R2 dΩ25 . (31) R2 r2

The metric is a direct product of a fivedimensional sphere (dΩ25 ) and another fivedimensional space-time spanned by t, x, and r. An alternative form of the metric is obtained from Eq. (31) by a change of variable z = R2 /r, ds2 =

R2 (−dt2 + dx2 + dz 2 ) + R2 dΩ25 . z2

To support the metric (31) (i.e., to satisfy the Einstein equation) there must be some background matter field that gives a stress-energy tensor in the form of a negative cosmological constant in AdS5 and a positive one in S5 . Such a field is the self-dual five-form field F5 mentioned above. Field theory has two parameters: the number of colors N and the gauge coupling g. When the number of colors is large, it is the ’t Hooft coupling λ = g 2 N that controls the perturbation theory. On the string theory side, the parameters are gs , ls , and radius R of the AdS space. String theory and field theory each have two dimensionless parameters which map to each other through the following relations: g 2 = 4πgs , g 2 Nc =

R . ls4

(34)

Equation (33) tells us that, if one wants to keep string theory weakly interacting, then the gauge coupling in field theory must be small. Equation (34) is particularly interesting. It says that the large ’t Hooft coupling limit in field theory corresponds to the limit when the curvature radius of space-time is much larger than the string length ls . In this limit, one can reliably decouple the massive string modes and reduce string theory to supergravity. In the limit gs  1 and R  ls , one has classical supergravity instead of string theory. The practical utility of the AdS/CFT correspondence comes, in large part, from its ability to deal with the strong coupling limit in gauge theory. One can perform a Kaluza-Klein reduction [24] by expanding all fields in S5 harmonics. Keeping only the lowest harmonics, one finds a fivedimensional theory with the massless dilaton, SO(6) gauge bosons, and gravitons [25]:

(32)

Both coordinates r and z are known as the radial coordinate. The limiting value r = ∞ (or z = 0) is the boundary of the AdS space. It is a simple exercise to check that the (t, x, r) part of the metric is a space with constant negative curvature, or an anti de-Sitter (AdS) space.

(33)

4

S5D =

N2 8π 2 R3



 1 d5 x R5D − 2Λ − ∂ μ Φ∂μ Φ 2  R2 a aμν F F + · · · . (35) − 8 μν

In AdS/CFT, an operator O of field theory is put in a correspondence with a field φ (“bulk”

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field) in supergravity. We elaborate on this correspondence below; here we keep the operator and the field unspecified. In the supergravity approximation, the mathematical statement of the correspondence is Z4D [J] = eiS[φcl ] .

(36)

On the left is the partition function of a field theory, where the source J coupled to the operator O is included:     Z4D [J] = Dφ exp iS + i d4 x JO . (37) On the right, S[φcl ] is the classical action of the classical solution φcl to the field equation with the boundary condition: lim

z→0

φcl (z, x) = J(x) . zΔ

(38)

Here Δ is a constant that depends on the nature of the operator O (namely, on its spin and dimension). In the simplest case, Δ = 0, and the boundary condition becomes φcl (z=0) = J. Differentiating Eq. (36) with respect to J, one can find the correlation functions of O. For example, the two-point Green’s function of O is obtained by differentiating Scl [φ] twice with respect to the boundary value of φ, G(x − y) = −iT O(x)O(y)

δ 2 S[φcl ] = − . δJ(x)δJ(y) φ(z=0)=J

(39)

The AdS/CFT correspondence thus maps the problem of finding quantum correlation functions in field theory to a classical problem in gravity. Moreover, to find two-point correlation functions in field theory, one can be limited to the quadratic part of the classical action on the gravity side. The complete operator to field mapping can be found in Refs. [4,21]. For our purpose, the following is sufficient: • The dilaton Φ corresponds to O = −L = 1 2 4 Fμν + · · · , where L is the Lagrangian density.

• The gauge field Aaμ corresponds to the conserved R-charge current J aμ of field theory. • The metric tensor corresponds to the stressenergy tensor T μν . More precisely, the partition function of the four-dimensional field 0 is equal to theory in an external metric gμν 0 Z4D [gμν ] = exp(iScl [gμν ]) ,

(40)

where the five-dimensional metric gμν satisfies the Einstein’s equations and has the following asymptotics at z = 0: ds2 = gμν dxμ dxν =

R2 0 (dz 2 + gμν dxμ dxν ) . z2

(41)

From the point of view of hydrodynamics, the operator 14 F 2 is not very interesting because its correlator does not have a hydrodynamic pole. In contrast, we find the correlators of the R-charge current and the stress-energy tensor to contain hydrodynamic information. We simplify the graviton part of the action further. Our two-point functions are functions of the momentum p = (ω, k). We can choose spatial coordinates so that k points along the x3 -axis. This corresponds to perturbations that propagate along the x3 direction: hμν = hμν (t, r, x3 ). These perturbations can be classified according to the representations of the O(2) symmetry of the (x1 , x2 ) plane. Owing to that symmetry, only certain components can mix; for example, h12 does not mix with any other components, whereas components h01 and h31 mix only with each other. We assume that only these three metric components are nonzero and introduce shorthand notations φ = h12 ,

a0 = h10 ,

a3 = h13 .

(42)

The quadratic part of the graviton action acquires a very simple form in terms of these fields: N2 S2 = 8π 2 R3



4

√



1 −g − g μν ∂μ φ∂ν φ 2  1 μα νβ − 2 g g fμν fαβ , (43) 4geff

d x dr

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2 = gxx . In where fμν = ∂μ aν − ∂ν aμ , and geff deriving Eq. (43), our only assumption about the metric is that it has a diagonal form,

ds2 = gtt dt2 + grr dr2 + gxx dx2 ,

(44)

so it can also be used below for the finitetemperature metric. As a simple example, let us compute the twopoint correlation function of T xy , which corresponds to φ in gravity. The field equation for φ is √ ∂μ ( −g g μν ∂ν φ) = 0 . (45) The solution to this equation, with the boundary condition φ(p, z = 0) = φ0 (p), can be written as φ(p, z) = fp (z)φ0 (p) ,

(46)

where the mode function fp (z) satisfies the equation    fp p2 − 3 fp = 0 (47) 3 z z with the boundary condition fp (0) = 1. The mode equation (47) can be solved exactly. Assuming p is spacelike, p2 > 0, the exact solution and its expansion around z = 0 is 1 (pz)2 K2 (pz) 2 (pz)2 (pz)4 = 1− − ln pz + O((pz)4 ) . 4 16

fp (z) =

(48)

where N2 1 f−p (z)∂z fp (z) . 16π 2 z 3

Txy Txy p = −2 lim F(p, z) = z→0

(50)

N2 4 p ln p2 . (51) 64π 2

Note that we have dropped the term ∼ p4 ln z, which, although singular in the limit z → 0, is a contact term [i.e., a term proportional to a derivative of δ(x) after Fourier transform]. Removing such terms by adding local counter terms to the supergravity action is known as the holographic renormalization [26]. It is, in a sense, a holographic counterpart to the standard renormalization procedure in quantum field theory, here applied to composite operators. For time-like p, p2 < 0, there are two solutions to Eq. (47) which involve Hankel functions H (1) (z) and H (2) (z) instead of K2 (z). Neither function blows up at z → ∞, and it is not clear which should be picked. Here we encounter, for the first time, a subtlety of Minkowski-space AdS/CFT, which is discussed in great length in subsequent sections. At zero temperature this problem can be overcome by an analytic continuation from space-like p. However, this will not work at nonzero temperatures. 3.2. Black Three-Brane Metric At nonzero temperatures, the metric dual to N = 4 SYM theory is the black three-brane metric, ds2 =

The second solution to Eq. (47), (pz)2 I2 (pz), is ruled out because it blows up at z → ∞. We now substitute the solution into the quadratic action. Using the field equation, one can perform integration by parts and write the action as a boundary integral at z = 0. One finds  1 N2 d4 x 3 φ(x, z)φ (x, z)|z→0 S= 16π 2 z (49)  d4 p = φ (−p)F(p, z)φ (p)| , 0 0 z→0 (2π)4

F(p, z) =

Differentiating the action twice with respect to the boundary value φ0 one finds

r2 R2 2 2 2 dr + R2 dΩ25 , (52) (−f dt + dx ) + R2 r2 f

with f = 1 − r04 /r4 . The event horizon is located at r = r0 , where f = 0. In contrast to the usual Schwarzschild black hole, the horizon has three flat directions x. The metric (52) is thus called a black three-brane metric. We frequently use an alternative radial coordinate u, defined as u = r02 /r2 . In terms of u, the boundary is at u = 0, the horizon at u = 1, and the metric is ds2 =

(πT R)2 (−f (u)dt2 + dx2 ) u2 R2 + 2 du2 + R2 dΩ25 . 4u f (u)

(53)

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The Hawking temperature is determined completely by the behavior of the metric near the horizon. Let us concentrate on the (t, r) part of the metric, ds2 = −

4r0 R2 2 dr2 . (r − r )dt + 0 R2 4r0 (r − r0 )

(54)

Changing the radial variable from r to ρ, r = r0 +

ρ , r0

(55)

and the metric components become nonsingular:   R2 4r02 2 2 2 2 ds = 2 dρ − 2 ρ dt . (56) r0 R Note also that after a Wick rotation to Euclidean time τ , the metric has the form of the flat metric in cylindrical coordinates, ds2 ∼ dρ2 + ρ2 dϕ2 , where ϕ = 2r0 R−2 τ . To avoid a conical singularity at ρ = 0, ϕ must be a periodic variable with periodicity 2π. This fact matches with the periodicity of the Euclidean time in thermal field theory τ ∼ τ + 1/T , from which one finds the Hawking temperature: TH =

r0 . πR2

(57)

One of the first finite-temperature predictions of AdS/CFT correspondence is that of the thermodynamic potentials of the N = 4 SYM theory in the strong coupling regime. The entropy is given by the Bekenstein-Hawking formula S = A/(4G), where A is the area of the horizon of the metric (52); the result can then be converted to parameters of the gauge theory using Eqs. (30), (33), and (34). One obtains s=

π2 2 3 S = N T , V 2

space as a cigar-shaped surface, then the horizon r = r0 is the tip of the cigar. Thus, r0 is the minimal radius where the space ends, and there is no point in space with r less than r0 . The only boundary condition at r = r0 is that fields are regular at the tip of the cigar, and the AdS/CFT correspondence is formulated as Z4D [J] = Z5D [φ]|φ(z=0)→J .

2

(58)

which is 3/4 of the entropy density in N = 4 SYM theory at zero ’t Hooft coupling. We now try to generalize the AdS/CFT prescription to finite temperature. In the Euclidean formulation of finite-temperature field theory, field theory lives in a space-time with the Euclidean time direction τ compactified. The metric is regular at r = r0 : If one views the (τ, r)

225

(59)

3.3. Real-time AdS/CFT In many cases we must find real-time correlation functions not given directly by the Euclidean path-integral formulation of thermal field theory. One example is the set of kinetic coefficients expressed, through Kubo’s formulas, via a certain limit of real-time thermal Green’s functions. Another related example appears if we want to directly find the position of the poles in the correlation functions that would correspond to the hydrodynamic modes. In principle, some real-time Green’s functions can be obtained by analytic continuation of the Euclidean ones. For example, an analytic continuation of a two-point Euclidean propagator gives a retarded or advanced Green’s function, depending on the way one performs the continuation. However, it is often very difficult to directly compute a quantity of interest in that way. In particular, it is very difficult to get the information about the hydrodynamic (small ω, small k) limit of real-time correlators from Euclidean propagators. The problem here is that we need to perform an analytic continuation from a discrete set of points in Euclidean frequencies (the Matsubara frequencies) ω = 2πin, where n is an integer, to the real values of ω. In the hydrodynamic limit, we are interested in real and small ω, whereas the smallest Matsubara frequency is already 2πT . Therefore, we need a real-time AdS/CFT prescription that would allow us to directly compute the real-time correlators. However, if one tries to naively generalize the AdS/CFT prescription, one immediately faces a problem. Namely, now r = r0 is not the end of space but just the location of the horizon. Without specifying a boundary condition at r = r0 , there is an ambiguity in defining the solution to the field equations, even as the boundary condition at r = ∞ is set.

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As an example, let us consider the equation of motion of a scalar field in the black hole background, ∂μ (g μν ∂ν φ) = 0. The solution to this equation with the boundary condition φ = φ0 at u = 0 is φ(p, u) = fp φ0 (p), where fp (u) satisfies the following equation in the metric (53): fp −

1 + u2  w2 q2 fp + fp = 0 . f − p uf uf 2 uf

(60)

Here the prime denotes differentiation with respect to u, and we have defined the dimensionless frequency and momentum: w=

ω , 2πT

q=

k . 2πT

(61)

Near u = 0 the equation has two solutions, f1 ∼ 1 and f2 ∼ u2 . In the Euclidean version of thermal AdS/CFT, there is only one regular solution at the horizon u = 1, which corresponds to a particular linear combination of f1 and f2 . However, in Minkowski space there are two solutions, and both are finite near the horizon. One solution termed fp behaves as (1 − u)−iw/2 , and the other is its complex conjugate fp∗ ∼ (1 − u)iw/2 . These two solutions oscillate rapidly as u → 1, but the amplitude of the oscillations is constant. Thus, the requirement of finiteness of fp allows for any linear combination of f1 and f2 near the boundary, which means that there is no unique solution to Eq. (60). 3.3.1. Prescription For Retarded TwoPoint Functions Physically, the two solutions fp and fp∗ have very different behavior. Restoring the e−iωt phase in the wave function, one can write e−iωt fp ∼ e−iω(t+r∗ ) ,

(62)

e−iωt fp∗

(63)

iω(t−r∗ )

∼e

,

where the coordinate r∗ =

ln(1 − u) 4πT

(64)

was introduced so that Eqs. (62) and (63) looked like plane waves. In fact, Eq. (62) corresponds to a wave that moves toward the horizon (incoming

wave) and Eq. (63) to a wave that moves away from the horizon (outgoing wave). The simplest idea, which is motivated by the fact that nothing should come out of a horizon, is to impose the incoming-wave boundary condition at r = r0 and then proceed as instructed by the AdS/CFT correspondence. However, now we encounter another problem. If we write down the classical action for the bulk field, after integrating by parts we get contributions from both the boundary and the horizon:  z=zH d4 p φ (−p)F(p, z)φ (p) . (65) S= 0 0 (2π)4 z=0 If one tried to differentiate the action with respect to the boundary value φ0 , one would find G(p) = F(p, z)|z0H + F(−p, z)|z0H .

(66)

From the equation satisfied by fp and from fp∗ = f−p , it is easy to show that the imaginary part of F(p, z) does not depend on z; hence the quantity G(p) in Eq. (66) is real. This is clearly not what we want, as the retarded Green’s functions are, in general, complex. Simply throwing away the contribution from the horizon does not help because F(−p, z) = F ∗ (p, z) owing to the reality of the equation satisfied by fp . A partial solution to this problem was suggested in Ref. [6]. It was postulated that the retarded Green’s function is related to the function F by the same formula that was found at zero temperature: GR (p) = −2 lim F(p, z) . z→0

(67)

In particular, we throw away all contributions from the horizon. This prescription was established more rigorously in Ref. [7] (following an earlier suggestion in Ref. [27]) as a particular case of a general real-time AdS/CFT formulation, which establishes the connection between the close-time-path formulation of real-time quantum field theory with the dynamics of fields in the whole Penrose diagram of the AdS black brane. We briefly describe this connection in the next subsection. Here we accept Eq. (67) as a postulate and proceed to extract physical results from it.

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227

It is also easy to generalize this prescription to the case when we have more than one field. In that case, the quantity F becomes a matrix Fab , whose elements are proportional to the retarded Green’s function Gab .

function as the trace of a product of four evolution operators,

3.3.2. The close-time-path formalism and AdS/CFT The general prescription for real-time Green’s functions was developed in Ref. [7]. One needs to to consider not only the region of space outside the horizon r = r0 , but the whole Penrose diagram of the AdS black hole. In Kruskal coordinates (which will be introduced later) the Penrose diagram has 4 quadrants (see Fig. 1). The left and the right quadrants both have their own boundary. Previously, our discussion was limited to the right quadrant. Time flow forward in the right quadrant R and backward in the left quadrant time L.

and then, using the path-integral representation of the evolution operator, write the partition function as      Z = Dφ exp i dt dx L(φ) (69)

Z = Tr e−βH = Tr e−(β−σ)H ei(tf −ti )H e−σH e−i(tf −ti )H

C

where the action is evaluate along the contour C of Fig.2. The advantage of introducing the countour C is that one can introduce the sources coupled to the field φ. In particular, one can introduce as source J1 (x) that lives on the upper part of C, and another source J2 (x) that lives on the lower part of that contour. One can then define a 2 × 2 matrix of propagators, Gab (x, y) =

U

=0

F

L

R =0

V

P

δ 2 ln Z δJa (x)δJb (y)

One notices a striking similarity with the closetime-path (CTP) formulation of thermal field theory. In this formulation, one write the partition

(70)

These propagators are also called the SchwingerKeldysh propagators. In quantum field theory the choise of σ is arbitrary. Changing σ by Δσ does not affect the diagonal components of the CTP propagators, but changes the off-diagonal ones by factors of e±βΔσ There are two special choices that are convenient. The first is σ = 0, in which case the upper and lower parts of the CTP countour lies on top of each other. In this choice the retarded and advanced propagators are related in a simple way to the CTP propagators, GR = G11 − G12

Figure 1. The Penrose diagram of an AdS black hole

(68)

(71)

The other choice is σ = β/2. In this case the CTP propagator matrix is symmetric: G12 = G21 . One now note that the Penrose diagram of an AdS black hole also has two boundaries, which match with the upper and lower parts of the CTP contour. Therefore, one can tentatively formulate the following correspondence between the CTP partition function in field theory and the supergravity partition function, Z4D [J1 , J2 ] = Z5D [φcl ]

(72)

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ti A t

-C

tf



tf − iσ

? t −iβ B ti

Figure 2. The Schwinger-Keldysh contour where in the right hand side φcl is the classical solution to the equation of motion which becomes J1 on the right-handed boundary and J2 on the left-handed boundary. 3.4. Calculating hydrodynamic quantities As an illustration of the real-time AdS/CFT correspondence, we compute the correlator of Txy . First we write down the equation of motion for φ = hxy : φp −

1 + u2  w2 − q 2 f φp + φp = 0 . uf uf 2

(73)

In contrast to the zero-temperature equation, now ω and k enter the equation separately rather than through the combination ω 2 − k 2 . Thus the Green’s function will have no Lorentz invariance. The equation cannot be solved exactly for all ω and k. However, when ω and k are both much smaller than T , or w, q  1, one can develop series expansion in powers of w and q. There are two solutions that are complex conjugates of each other. The solution that is an incoming wave at u = 1 and normalized to 1 at u = 0 is fp (z) = (1 − u2 )−iw/2 + O(w2 , q 2 ) . The kinetic term in the action for φ is  f π2 N 2 T 4 du φ2 . S=− 8 u

(74)

(75)

Applying the general formula (67), one finds the retarded Green’s function of Txy , GR xy,xy (ω, k) = −

π2 N 2 T 4 iw , 4

(76)

and, using Kubo’s formula for η, the viscosity, π (77) η = N 2T 3 . 8 It is instructive to compute other correlators that have poles corresponding to hydrodynamic modes. As a warm-up, let us compute the twopoint correlators of the R-charge currents, which should have a pole at ω = −iDk2 , where D is the diffusion constant. We first write down Maxwell’s equations for the bulk gauge field. Let the spatial momentum be aligned along the x3 axis: p = (ω, 0, 0, k). Then the equations for A0 and A3 are coupled: wA0 + qf A3 = 0 , 1 2 (q A0 + wqA3 ) = 0 , A0 − uf f 1 (w2 A3 + wqA0 ) = 0 . A3 + A3 + f uf 2

(78) (79) (80)

One can eliminate A3 and write down a thirdorder equation for A0 , A 0 +

(uf )  w2 − q 2 f  A0 + A0 = 0 . uf uf 2

(81)

Near u=1 we find two independent solutions, A0 ∼ (1−u)±iw/2 , and the incoming-wave boundary condition singles out (1 − u)−iw/2 . One can substitute A0 = (1 − u)−iw/2 F (u) into Eq. (81). The resulting equation can be solved perturbatively in w and q 2 . We find A0 = C(1 − u)−iw/2 ×   2u2 1+u iw 2 ln + q ln . × 1+ 2 1+u 2u

(82)

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Using Eq. (79) one can express C through the boundary values of A0 and A3 at u = 0: q 2 A0 + wqA3 . (83) C= iw − q 2 u=0 Differentiating the action with respect to the boundary values, we find, in particular, J0 J0 p =

N 2T k2 , 16π iω − Dk 2

(84)

where ξ=

π 2 3 N T , 8

D=

1 . 4πT

(91)

Thus, we found that the correlator contains a diffusive pole ω = −iDk 2 , just as anticipated from hydrodynamics. Furthermore, the magnitude of the momentum diffusion constant D also matched our expectation. Indeed, if one recalls the value of η from Eq. (77) and the entropy density from Eq. (58), one can check that

where 1 D= . 2πT

(85)

The correlator given by Eq. (84) has the expected hydrodynamic diffusive pole, and D is the Rcharge diffusion constant. Similarly, one can observe the appearance of the shear mode in the correlators of the metric tensor. We note that the shear flow along the x1 direction with velocity gradient along the x3 direction involves T01 and T31 , hence the interesting metric components are a0 = h10 and a3 = h13 . Two of the field equations are qf  a = 0, w 3 1+u2  1 a3 − a3 + (w2 a3 + wqa0 ) = 0. uf uf 2

a0 −

(86) (87)

They can be combined into a single equation: a 0 −

2u  2uf − q 2 f + w2  a + a0 = 0 . f 0 uf 2

(88)

Again, the solution can be found perturbatively in w and q: a0

= C(1 − u)

−iw/2

 u 1+u u − iw 1 − u − ln 2 2  q2 + (1 − u) . (89) 2





Applying the prescription, one finds the retarded Green’s functions. For example, 2

Gtx,tx (ω, k) =

ξk , iω − Dk 2

(90)

D=

η . +P

(92)

3.5. The “membrane paradigm” Let us now look at the problem from a different perspective. The existence of hydrodynamic modes in thermal field theory is reflected by the existence of the poles of the retarded correlators computed from gravity. Are there direct gravity counterparts of the hydrodynamic normal modes? If the answer to this question is yes, then there must exist linear gravitational perturbations of the metric that have the dispersion relation identical to that of the shear hydrodynamic mode, ω ∼ −iq 2 , and of the sound mode, ω = cs q −iγq 2 . It turns out that one can explicitly construct the gravitational counterpart of the shear mode. (It should be possible to find a similar construction for the sound mode, but it has not been done in the literature; for a recent work on the subject, see [28].) Our discussion is physical but somewhat sketchy; for more details see Ref. [29]. First, let us construct a gravity perturbation that corresponds to a diffusion of a conserved charge (e.g., the R-charge in N = 4 SYM theory). To keep the discussion general, we use the form of the metric (44), with the metric components unspecified. Our only assumptions are that the metric is diagonal and has a horizon at r = r0 , near which g00 = −γ0 (r − r0 ),

grr =

γr . r − r0

(93)

The Hawking temperature can be computed by the method used to arrive at Eq. (57), and one finds T = (4π)−1 (γ0 /γr )1/2 .

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We also assume that the action of the gauge field dual to the conserved current is    √ 1 μν (94) Sgauge = dx −g − 2 F Fμν , 4geff where geff is an effective gauge coupling that can be a function of the radial coordinate r. For simplicity we set geff to a constant in our derivation of the formula for D; it can be restored by replac√ √ 2 in the final answer. ing −g → −g/geff The field equations are   1 √ μν ∂μ −g F = 0. (95) 2 geff We search for a solution to this equation that vanishes at the boundary and satisfies the incomingwave boundary condition at the horizon. The first indication that one can have a hydrodynamic behavior on the gravity side is that Eq. (95) implies a conservation law on a fourdimensional surface. We define the stretched horizon as a surface with constant r just outside the horizon, r = rh = r0 + ε,

ε  r0 ,

(96)

and the normal vector nμ directed along the r direction (i.e., perpendicularly to the stretched horizon). Then with any solution to Eq. (95), one can associate a current on the stretched horizon: (97) j μ = nν F μν . rh

The antisymmetry of F μν implies that j μ has no radial component, j r = 0. The field equation (95) and the constancy of nν on the stretched horizon imply that this current is conserved: ∂μ j μ = 0. To establish the diffusive nature of the solution, we must show the validity of the constitutive equation j i = −D∂i j 0 . Such constitutive equation breaks time reversal and obviously must come from the absorptive boundary condition on the horizon. The situation is analogous to the propagation of plane waves to a non-reflecting surface in classical electrodynamics. In this case, we have the relation B = −n×E between electric and magnetic fields. In our case,

the corresponding relation is

γr F0i Fir = − , γ0 r − r0

(98)

valid when r is close to r0 . This relates ji ∼ Fir to the parallel to the horizon component of the electric field F0i , which is one of the main points of the “membrane paradigm” approach to black hole physics [30,31]. We have yet to relate ji to j0 ∼ F0r , which is the component of the electric field normal to the horizon. To make the connection to F0r , we use the radial gauge Ar = 0, in which F0i ≈ −∂i A0 .

(99)

Moreover, when k is small the fields change very slowly along the horizon. Therefore, at each point on the horizon the radial dependence of the scalar potential A0 is determined by the Poisson equation, √ ∂r ( −g g rr g 00 ∂r A0 ) = 0 , (100) whose solution, which satisfies A0 (r = ∞) = 0, is ∞ g00 (r )grr (r ) . A0 (r) = C0 dr −g(r )

(101)

r

This means that the ratio of the scalar potential A0 and electric field F0r approaches a constant near the horizon: √ ∞ g00 grr A0 −g (r) . (102) = (r0 ) dr √ F0r r=r0 g00 grr −g r0

Combining the formulas j i ∼ F0i ∼ ∂i A0 , and A0 ∼ F0r ∼ j 0 , we find Fick’s law j i = −D∂i j 0 , with the diffusion constant √ ∞ 2 |g00 |grr geff −g √ . (103) (r ) dr D= 0 2 −g gxx geff |g00 |grr r0

Thus, we found that for a slowly varying solution to Maxwell’s equations, the corresponding charge on the stretched horizon evolves according to the diffusion equation. Therefore, the gravity solution must be an overdamped one, with

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ω = −iDk 2 . This is an example of a quasi-normal mode. We also found the diffusion constant D directly in terms of the metric and the gauge coupling geff . The reader may notice that our quasinormal modes satisfy a vanishing Dirichlet condition at the boundary r=∞. This is different from the boundary condition one uses to find the retarded propagators in AdS/CFT, so the relation of the quasinormal modes to AdS/CFT correspondence may be not clear. It can be shown, however, that the quasi-normal frequencies coincide with the poles of the retarded correlators [32,33]. We can now apply our general formulas to the case of N = 4 SYM theory. The metric components are given by Eq. (52). For the R-charge current geff = const, Eq. (103) gives D = 1/(2πT ), in agreement with our AdS/CFT computation. For the shear mode of the stress-energy tensor 2 = gxx , so D = 1/(4πT ), we have effectively geff which also coincides with our previous result. In both cases, the computation is much simpler than the AdS/CFT calculation.

3.6. The viscosity/entropy ratio 3.6.1. Universality In all thermal field theories in the regime described by gravity duals the ratio of shear viscosity η to (volume) density of entropy s is a universal constant equal to 1/(4π) [¯h/(4πkB ), if one restores ¯h, c and the Boltzmann constant kB ]. One proof of the universality is based on the relationship between graviton’s absorption cross section and the imaginary part of the retarded Green’s function for Txy [34]. Another way to prove the universality [35] is via the direct AdS/CFT calculation of the correlation function in Kubo’s formula (18). We, however, follow a different method. It is based on the formula for the viscosity derived from the membrane paradigm. A similar proof was given by Buchel & Liu [36]. The observation is that the shear gravitational perturbation with k = 0 can be found exactly by performing a Lorentz boost of the black-brane metric (52). Consider the coordinate transforma-

tions r, t, xi → r , t , xi of the form r = r , t + vy  t= √ ≈ t + vy  , 1 − v2 y  + vt y=√ ≈ y  + vt , 1 − v2 xi = xi ,

(104)

where v < 1 is a constant parameter and the expansion on the right corresponds to v  1. In the new coordinates, the metric becomes ds2 = g00 dt 2 + grr dr 2 + gxx (r)

p 

(dx i )2

i=1

+ 2v(g00 + gxx )dt dy  .

(105)

This is simply a shear fluctuation at k = 0. In our language, the corresponding gauge potential is a0 = vg xx (g00 + gxx ) .

(106)

This field satisfies the vanishing boundary condition a0 (r = ∞) = 0 owing to the restoration of Poincar´e invariance at the boundary: g00 /gxx → −1 when r → ∞. This clearly has a much simpler form than Eq. (101) for the solution to the generic Poisson equation. The simple form of solution (106) is valid only for the specific case of 2 the shear gravitational mode with geff = gxx . We have also implicitly used the fact that the metric satisfies the Einstein equations, with the stressenergy tensor on the right being invariant under a Lorentz boost. Equation (102) now becomes gxx (r0 ) a0 1 + g xx g00 =− = . (107) xx f0r ∂r (g g00 ) γ0 r→r0

r→r0

The shear mode diffusion constant is

√ γ0 γr 1 γr a0 = . D= = f0r gxx (r0 ) γ0 4πT

(108)

r→r0

Because D = η/( + P ), and  + P = T s in the absence of chemical potentials, we find that 1 η = . s 4π

(109)

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In fact, the constancy of this ratio has been checked directly for theories dual to Dpbrane [29], M -brane [10], Klebanov-Tseytlin and Maldacena-Nunez backgrounds [36], N = 2∗ SYM theory [37] and others. Curiously, the viscosity to entropy ratio is also equal to 1/4π in the pre-AdS/CFT “membrane paradigm” hydrodynamics [38]: there, for a four-dimensional Schwarzschild black hole one has ηm.p. = 1/16πGN , while the Bekenstein-Hawking entropy is s = 1/4GN . As remarked in Sec. 2, the ratio η/s is much larger than the one for weakly coupled theories. The fact that we found the ratio to be parametrically of order one implies that all theories with gravity duals are strongly coupled. In N = 4 SYM theory, the ratio η/s has been computed to the next order in the inverse ’t Hooft coupling expansion [39] 1 η = s 4π

  135ζ(3) 1+ . 8(g 2 N )3/2

(110)

The sign of the correction can be guessed from the fact that in the limit of zero ’t Hooft coupling g 2 N → 0, the ratio diverges, η/s → ∞. 3.6.2. A viscosity bound? From our discussion above, one can argue that h ¯ η ≥ s 4π

(111)

in all systems that can be obtained from a sensible relativistic quantum field theory by turning on temperatures and chemical potentials. The bound, if correct, implies that a liquid with a given volume density of entropy cannot be arbitrarily close to being a perfect fluid (which has zero viscosity). As such, it implies a lower bound on the viscosity of the QGP one may be creating at RHIC. Interestingly, some model calculations suggest that the viscosity at RHIC may be not too far away from the lower bound [40,41]. One place where one may think that the bound should break down is superfluids. The ability of a superfluid to flow without dissipation in a channel is sometimes described as “zero viscosity”. However, within the Landau’s two-fluid model, any

superfluid has a measurable shear viscosity (together with three bulk viscosities). For superfluid helium, the shear viscosity has been measured in a torsion-pendulum experiment by Andronikashvili [42]. If one substitutes the experimental values, the ratio η/s for helium remains larger than h ¯ /4πkB ≈ 6.08 × 10−13 K s for all ranges of temperatures and pressures, by a factor of at least 8.8. As discussed in Sec. 2.3, the ratio η/s is proportional to the ratio of the mean free path and the de Broglie wavelength of particles, mfp η ∼ . s λ

(112)

For the quasi-particle picture to be valid, the mean free path must be much larger than the de Broglie wavelength. Therefore, if the coupling is weak and the system can be described as a collection of quasi-particles, the ratio η/s is larger than 1. The situation here is similar to the so-called Mott’s minimal metallic conductivity. It was argued that the mean free path of electrons in a metal with disorders cannot be smaller than the inverse Fermi momentum. From this Mott argued that the electrical conductivity of a metal reaches a certain minimal value before the transition to insulating phase happens. However, it is now known that there exist quantum phase transitions where the conductivity goes continuously to zero. Such transitions are described by the scaling theory of localization. Mott’s minimal electric conductivity is still a useful concept, but not an absolute minumum. Could something similar happens with the viscosity/entropy ratio? Could there be a phase transition where η/s goes to zero? It is, however, difficult to imagine that one can localize all the carriers of energy and momentum. It is now known that there are examples of theories where η/s received negative 1/N corrections that make it smaller than 1/4π [43,44]. These field theories are dual to gravity with a R2 correction in the action. However, if one tries to make η/s small, one seems to encounter problems: in Gauss-Bonnet gravity one cannot make η/s smaller than 16/25 of h ¯ /4π without violat-

D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

ing causality [45]. The quesiton of what is the absolute minumum of η/s is an interesting open problem.

4. SECOND-ORDER ICS

HYDRODYNAM-

When the parameter kmfp is not too small, one may want to go beyond the first order in kmfp . This is the case, for example, in the early stages of heavy-ion collisions. There are two sources of corrections beyond the kmfp order. First, there are corrections due to thermal fluctuations of hydrodynamic variables contributing via nonlinearities of the hydrodynamic equations. The fluctuation corrections lead to nonanalytic low-momentum behavior of certain correlators [46] (similarly to the chiral logarithms that emerge from loops in chiral perturbation theory) and are, for example, essential for describing non-trivial dynamical critical behavior near phase transitions [47]. Such corrections are calculable in the framework of hydrodynamics with thermal noise. The second source of corrections are secondorder terms (order (kmfp )2 ) in the hydrodynamic equations, sometimes called the Burnett corrections [48]. These corrections come with additional transport coefficients. These second-order transport coefficients are not calculable from hydrodynamics, but have to be determined from underlying microscopic description or input phenomenologically, similarly to first-order transport coefficients such as viscosity. In gauge theories with a large number of colors Nc the corrections of the first type (fluctuation) are suppressed by 1/Nc2 [46] and therefore the corrections of the second type (Burnett) dominate in the limit of fixed k and Nc → ∞. For this reason, in this paper, we concentrate on the second type of corrections. Moreover, we shall consider the case of conformal theories, where the number of second-order transport coefficients is substantially reduced. In the real-world applications we deal with fluids which are not exactly conformal, however, e.g., QCD at sufficiently high temperatures is approximately conformal.

233

4.1. Conformal invariance in hydrodynamics To set the stage, let us emphasize again that hydrodynamics is a controlled expansion scheme ordered by the power of the parameter kmfp , or equivalently, by the number of derivatives of the hydrodynamic fields. We shall set up this expansion paying particular attention to the consequences of the conformal invariance on the equations of hydrodynamics. 4.1.1. Conformal invariance and Weyl anomalies The hydrodynamic fields are expectation values of certain quantum fields, such as e.g., components of the stress-energy tensor, averaged over small but macroscopic volumes and time intervals. Such averages can, in principle, be calculated in the close-time-path (CTP) formalism [49]. Consider a generic finite-temperature field theory in the CTP formulation. Turning on external metrics on the upper and lower contours, the partition function is 1 2 Z[gμν , gμν ]=



 1 Dφ1 Dφ2 exp iS[φ1 , gμν ]  2 − iS[φ2 , gμν ] ,

(113)

where φ1 and φ2 represent the two sets of all fields living on the upper and lower parts of the contours, and S[φ, gμν ] is the general coordinate invariant action. The one-point Green’s function of the stressenergy tensor is obtained by differentiating the partition function (the metric signature here is − + ++): 2i δ ln Z , 1 −g1 δgμν 2i δ ln Z √ , 2 −g2 δgμν

T 1μν  = − √

(114)

T 2μν  =

(115)

where . . . denote the mean  value under the path √ 1,2 integral and −g1,2 ≡ −detgμν . In this paper we consider conformally invariant theories. In such theories the action S[φ, gμν ] evaluated on classical equations of motion δS/δφ = 0 and viewed as a functional of

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the external metric gμν is invariant under local dilatations, or Weyl transformations: gμν → e−2ω gμν ,

(116)

with parameter ω a function of space-time coordinates. As a consequence, classical stressenergy tensor Tclμν ≡ δS/δgμν is traceless since gμν Tclμν = −(1/2)δS/δω = 0. In the conformal quantum theory (113) the Weyl anomaly [50,51] implies 1 1 T 1μν  = Wd [gμν ], gμν

(117a)

2 gμν T 2μν 

(117b)

=

2 Wd [gμν ],

where Wd is the Weyl anomaly in d dimensions, which is identically zero for odd d. For d = 4: W4 [gμν ] = − +

a (Rμνλρ Rμνλρ − 4Rμν Rμν + R2 ) 16π 2

c (Rμνλρ Rμνλρ − 2Rμν Rμν + 13 R2 ), 16π 2

(118)

where Rμνλρ and Rμν (R) are the Riemann tensor and Ricci tensor (scalar),

and for  SU (Nc ) N = 4 SYM theory a = c = 14 Nc2 − 1 [21]. The righthand side of Eqs. (117) contains four derivatives. In general, for even d = 2k, W2k contains 2k derivatives of the metric. Let us now explore the consequences of Weyl anomalies for hydrodynamics. The hydrodynamic equations (without noise) do not capture the whole set of CTP Green’s functions, but only the retarded ones. Hydrodynamics determines the stress-energy tensor T μν (more precisely, its slowly varying average over sufficiently long scales) in the presence of an arbitrary (also slowly varying) source gμν . The connection to the CTP partition function can be made explicit by writing 1 1 = gμν + γμν , gμν 2

1 2 gμν = gμν − γμν . (119) 2

If γμν = 0 then Z = 1 since the time evolution on the lower contour exactly cancels out the time evolution on the upper contour. When γμν is small one can expand  √ i dx −g γμν T μν + O(γ 2 ), ln Z = (120) 2

where T μν (x) depends on gμν , and is the stressenergy tensor in the presence of the source gμν . At long distance scales it should be the same as computed from hydrodynamics. Substituting Eqs. (119) and (120) into Eq. (117), the O(1) and O(γ) terms yield two equations: (121a) gμν T μν = Wd [gμν ], √ αβ δ[ −g T (x)] √ + −gT αβ δ d (x − y) gμν (x) δgμν (y) δ ( −g(x) Wd [gμν (x)]). = (121b) δgαβ (y) In odd dimensions, the right hand sides of Eqs. (121) are zero. In even dimensions, they contain d derivatives. In a hydrodynamic theory, where one keeps less than d derivatives, they can be set to zero. For example, at d = 4, the Weyl anomaly is visible in hydrodynamics only if one keeps terms to the fourth order in derivatives. This is two orders higher than in second-order hydrodynamics considered in this paper. For larger even d, one has to go to even higher orders to see the Weyl anomaly. Thus, we can neglect Wd on the right hand side: second-order hydrodynamic theory is invariant under Weyl transformations. The two conditions (121) then become gμν T μν = 0, αβ

gμν

δT (y) =− δgμν (x)





d + 1 δ d (x − y)T αβ . 2

(122) (123)

Since the r.h.s. of equation (123) is −(1/2)δT μν /δω it implies the following tranformation law for T μν under Weyl transformations (116): T μν → e(d+2)ω T μν .

(124)

Noting that ln Z is invariant under Weyl transformations this could have been gleaned from Eq. (120) already. A simple rule of thumb is that for tensors transforming homogeneously ...μm ...μm Aμν11...ν → eΔA ω Aμν11...ν , n n

(125)

the conformal weight ΔA equals the mass dimension plus the difference between the number of

D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

in derivatives,

contravariant and covariant indices: ΔA = [A] + m − n.

(126)

4.1.2. First order hydrodynamics as derivative expansion The existence of hydrodynamic description owes itself to the presence of conserved quantities, whose densities can evolve (oscillate or relax to equilibrium) at arbitrarily long times provided the fluctuations are of large spatial size. Correspondingly, the expectation values of such densities are the hydrodynamic fields. In the simplest case we shall consider here, i.e., in a theory without conserved charges, there are 4 such hydrodynamic fields: energy density T 00 and 3 components of the momentum density T 0i . It is common and convenient to use the local velocity uμ instead of the momentum density variable. It can be defined as the boost velocity needed to go from the local rest frame, where the momentum density T 0i vanishes, back to the lab frame. Similarly, it is convenient to use ε – the energy density in the local rest frame – instead of the T 00 in the lab frame. The 4 equations for thus defined variables ε and uμ are conservation equations of the energy-momentum tensor ∇μ T μν = 0. In a covariant form the above definitions of ε and uμ can be summarized as T μν = ε uμ uν + T⊥μν .

235

(127)

In hydrodynamics, the remaining components T⊥μν (spatial in the local rest frame: uμ T⊥μν = 0) of the stress-energy tensor T μν appearing in the conservation equations are not independent variables, but rather instantaneous functions of the hydrodynamic variables ε and uμ and their derivatives. In the hydrodynamic limit, this is the consequence of the fact that the hydrodynamic modes are infinitely slower than all other modes, the latter therefore can be integrated out. All quantities appearing in hydrodynamic equations are averaged over these fast modes, and are functions of the slow varying hydrodynamic variables. The functional dependence of T⊥μν (constituitive equations) can be expanded in powers of derivatives of ε and uμ . Writing the most general form of this expansion consistent with symmetries gives, up to 1st order

T⊥μν = P (ε)Δμν − η(ε)σ μν − ζ(ε)Δμν (∇·u), (128) where the symmetric, transverse tensor with no derivatives Δμν is given by Δμν = g μν + uμ uν .

(129)

In the local rest frame it is the projector on the spatial subspace. The symmetric, transverse and traceless tensor of first derivatives σ μν is defined by σ μν = 2 ∇μ uν  ,

(130)

where for a second rank tensor Aμν the tensor defined as 

Aμν  ≡

1 μα νβ Δ Δ (Aαβ + Aβα ) 2 1 Δμν Δαβ Aαβ ≡ Aμν − d−1

(131)

is transverse uμ Aμν = 0 (i.e., only spatial components in the local rest frame are nonzero) and traceless gμν Aμν = 0. In the gradient expansion (128), the scalar function P (ε) can be identified as the thermodynamic pressure (in equilibrium, when all the gradients vanish), while η(ε) and ζ(ε) are the shear and bulk viscosities. The expansion coefficients P , η and ζ are determined by the physics of the fast (non-hydrodynamic, microscopic) modes that have been integrated out. 4.1.3. Conformal invariance in first-order hydrodynamics It is straightforward to check that if T μν transforms as in Eq. (124) and Tμμ = 0, then its covariant divergence transforms homogeneously: ∇μ T μν → e(d+2)ω ∇μ T μν , hence the hydrodynamic equation ∇μ T μν = 0 is Weyl invariant [52]. Let us now see what restrictions conformal invariance imposes on the first-order constitutive equations (128). First, the tracelessness condition Tμμ = 0 forces ε = (d − 1)P and ζ = 0. Since in a conformal theory ε = const·T d , we shall trade ε variable for T in what follows. Since gμν uμ uν = −1 the conformal weight of uμ is 1.

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By definition (127) and by (124) ε has conformal weight d and therefore T → eω T,

uμ → eω uμ

(132)

in accordance with the simple rule (126). By direct computation we find that σ μν → e3ω σ μν ,

(133)

i.e. σ μν transforms homogeneously with conformal weight 3 independent of d (in agreement with (126)). For conformal fluids η = const · T d−1 , and therefore T μν transforms homogeneously under Weyl transformation as in Eq. (124). 4.2. Second-order hydrodynamics of a conformal fluid In this Section we shall continue the derivative expansion (128). We shall write down all possible second-order terms in the stress-energy tensor allowed by Weyl invariance. Then we shall compute the coefficients in front of these terms in the N = 4 SYM plasma by matching hydrodynamic correlation functions with gravity calculations in Section 4.4. 4.3. Second-order terms Rewriting Eq. (127) we introduce the dissipative part of the stress-energy tensor, Πμν : T μν = εuμ uν + P Δμν + Πμν ,

(134)

which contains only the derivatives and vanishes in a homogeneous equilibrium state. The tensor Πμν is symmetric and transverse, uμ Πμν = 0. For conformal fluids it must be also traceless gμν Πμν = 0. To first order Πμν = −ησ μν + (2nd order terms),

(135)

where σ μν is defined in Eq. (130). We will also use the notation for the vorticity Ωμν =

1 μα νβ Δ Δ (∇α uβ − ∇β uα ) . 2

(136)

We note that in writing down second-order terms in Πμν , one can always rewrite the derivatives along the d-velocity direction D ≡ u μ ∇μ

(137)

(temporal derivative in the local rest frame) in terms of transverse (spatial in the local rest frame) derivatives through the zeroth-order equations of motion: 1 D ln T = − (∇⊥ · u), d−1 (138) Duμ = −∇μ⊥ ln T, ∇μ⊥ ≡ Δμα ∇α . Notice also that ∇⊥ · u = ∇ · u. With the restriction of transversality and tracelessness, there are eight possible contributions to the stress-energy tensor: ∇μ ln T ∇ν ln T, σ μν (∇·u), Ω

μ

λΩ

νλ

,

∇μ ∇ν ln T,

σ μ λ σ νλ , uα R

αμνβ

σ μ λ Ωνλ , uβ ,

R

μν

(139) .

By direct computations we find that there are only five combinations that transform homogeneously under Weyl tranformations. They are  O1μν = Rμν − (d − 2) ∇μ ∇ν ln T  − ∇μ ln T ∇ν ln T , (140) O2μν = Rμν − (d − 2)uα Rαμνβ uβ , O3μν

=σ

μ

, O4μν = σ μ λ Ωνλ , λσ O5μν = Ωμ λ Ωνλ .

(141)

νλ

(142)

In the linearized hydrodynamics in flat space only the term O1μν contributes. For convenience and to facilitate the comparision with the IsraelStewart theory we shall use instead of (140) the term 1  σ μν (∇·u) (143) Dσ μν  + d−1 which, with (138), reduces to the linear combination: O1μν −O2μν −(1/2)O3μν −2O5μν . It is straightforward to check directly that (143) transforms homogeneously under Weyl transformations. Thus, our final expression for the dissipative part of the stress-energy tensor, up to second order in derivatives, is   σ μν (∇·u) Πμν = −ησ μν + ητπ  Dσ μν  + d−1   + κ Rμν − (d − 2)uα Rαμνβ uβ + λ1 σ μ λ σ νλ + λ2 σ μ λ Ωνλ + λ3 Ωμ λ Ωνλ .

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(144) The five new constants are τπ , κ, λ1,2,3 . Note that using lowest order relations Πμν = −ησ μν , Eqs.(138) and Dη = −η ∇·u, Eq. (144) may be rewritten in the form 

d Πμν (∇·u) Πμν = −ησ μν − τπ  DΠμν  + d−1   + κ Rμν − (d − 2)uα Rαμνβ uβ



λ2 λ1 + 2 Πμ λ Πνλ − Πμ λ Ωνλ + λ3 Ωμ λ Ωνλ . η η (145) This equation is, in form, similar to an equation of the Israel-Stewart theory. In the linear regime it actually coincides with the Israel-Stewart theory. We emphasize, however, that one cannot claim that Eq. (145) captures all orders in the momentum expansion. Further remarks are in order. First, the κ term vanishes in flat space. If one is interested in solving the hydrodynamic equation in flat space, then κ is not needed. Nevertheless, κ contributes to the two-point Green’s function of the stressenergy tensor. We emphasize that the term proportional to κ is not a contact term, since it contains uμ . The λ1,2,3 terms are nonlinear in velocity, so are not needed if one is looking at small perturbations (like sound waves). For irrotational flows λ2,3 are not needed. The parameter τπ has dimension of time and can be thought of as the relaxation time. This interpretation of τπ can be most clearly seen from Eq. (145). 4.3.1. Kubo’s formulas To relate the new kinetic coefficients with thermal correlators, first let us consider the response of the fluid to small and smooth metric perturbations. We shall moreover restrict ourselves to a particular type of perturbations which is simplest to treat using AdS/CFT correspondence. Namely, for dimensions d ≥ 4 we take hxy = hxy (t, z). For d = 3, there are only two spatial coordinates, so we take hxy = hxy (t). Since it is a tensor perturbation the fluid remains at rest: T = const, uμ = (1, 0). Inserting this into

Eq. (144) we find, for d ≥ 4, ¨ xy T xy = −P hxy − η h˙ xy + ητπ h κ ¨ xy + ∂ 2 hxy ] . − [(d − 3)h z 2

(146)

The linear response theory implies that the retarded Green’s function in the tensor channel is (ω, k) = P − iηω + ητπ ω 2 Gxy,xy R κ − [(d − 3)ω 2 + k 2 ] . 2

(147)

For d = 3 there is no momentum k, and the formula becomes (ω) = P − iηω + ητπ ω 2 , Gxy,xy R

d = 3 . (148)

Thus the two kinetic coefficients τπ and κ can be found from the coefficients of the ω 2 and k 2 terms in the low-momentum expansion of (ω, k) in the case of d ≥ 4, and just from Gxy,xy R the ω 2 term in the case of d = 3. 4.3.2. Sound Pole We now turn to another way to determine τπ , which is based on the position of the sound pole. The fact that we have two independent methods to determine τπ allows us to check the selfconsistency of the calculations. To obtain the dispersion relation, we consider a (conformal) hydrodynamic system in stationary equilibrium, that is, with fluid velocity uμ = (1, 0), homogeneous energy density ε = const · T d and Πμν = 0. The speed of sound is defined by c2s = dP (ε)/dε. In conformal theory it is a constant: c2s = 1/(d − 1). Now let us slightly perturb the system and denote the departure from equilibrium energy density, velocity, and stress as δε, ui , and Πij . For small perturbations, one can neglect the nonlinear terms in Eq. (145) and the hydrodynamic equations are identical to those of the Israel-Stewart theory. For completeness, we rederive here the sound dispersion in this theory. To linear approximation in the perturbations, we have δT 00 = δε,

δT 0i = (ε + P )ui ,

δT ij = c2s δε δ ij + Πij .

(149)

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For sound waves travelling in x direction we take ux and Πxx as the only nonzero components of ui and Πij , and dependent only on x and t. Energy-momentum conservation implies ∂0 (δε) + (ε + P )∂x ux = 0 , x

(ε + P )∂0 u +

c2s ∂x (δε)

+ ∂x Π

xx

(150)

= 0.

(151)

Eq. (145) has the form τπ ∂0 Πxx + Πxx = −

2(d − 2) η∂x ux . d−1

(152)

For a plane wave, equations (150), (151) and (152) give the dispersion relation 3

2

− ω τπ − iω + + ωk

− 2) η + ik 2 c2s = 0. d−1 ε+P

2 2(d

ω1,2



c2s τπ

Γ − 2

(153)

k

3

+ O(k ) ,

d−2 η . d−1 ε+P

=

0.

(158)

From Eq. (145) we find τπ ∂0 Πxy + Πxy = −η∂x uy .

−ω 2 τπ − iω + k 2 (154)

(159)

η =0 ε+P

(160)

so the shear mode dispersion relation in the limit k → 0 becomes

where (155)

The third solution is given by ω3 = −iτπ−1 + O(k 2 ) .

(157)

The dispersion relation is determined by



4

Γ=

Πxy (t, x) ,

(ε + P )∂0 uy + ∂x Πxy

At small k, the two solutions of this equation corresponding to the sound wave are Γ = ±cs k − iΓk ± cs

uy (t, x),

such that we get from ∂μ δT μν = 0

ωk 2 c2s τπ

2

can be fully determined only in third-order hydrodynamics. To illustrate that, we shall compute this correction here, taking the second-order theory literally and pretending the third-order terms are not contributing. We shall than find the expected mismatch between this (incorrect) result and the AdS/CFT computation in the strongly coupled N = 4 SYM theory. The perturbation corresponding to the fluid flowing in the y direction with velocity gradient along the x direction (shear flow) involves the variables

(156)

Since ω3 does not vanish as k → 0, but remains on the order of a macroscopic scale, this third solution lies beyond the regime of validity of hydrodynamics. 4.3.3. Shear pole In hydrodynamics, there exists an overdamped mode describing fluid flow in a direction perpendicular to the velocity gradient, e.g., with uy ∼ e−iωt+ikx . First-order hydrodynamics gives the leading-order dispersion relation, ω = −iηk2 /(ε+ P ). The next correction to this dispersion relation is proportional to k 4 and thus is beyond the reach of the second-order theory. This correction

ω = −ihk2 − ih2 τπ k 4 + O(k 6 ), η h= . ε+P

(161)

The second solution, ω = −iτπ−1 + O(k 2 ), is obviously beyond the regime of validity of the hydrodynamic equation. It is easy to see that expression (161) unjustifiably exceeds the precision of the second-order theory: the kept correction is O(k 2 ) relative to the leading-order term, instead of being O(k). We can trace this to Eq. (160), in which we keep terms to second order in ω and k. For shear modes, however, ω ∼ k 2 , and the term ω 2 that we keep in Eq. (160) is of the same order of magnitude as terms O(k4 ) omitted in Eq. (160). The latter term can appear if the equation (159) for Πxy contains a term ∂x3 uy that may appear in third-order hydrodynamics. This is beyond the scope of this paper.

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4.3.4. Bjorken Flow So far, we have studied only quantities involved in the linear response of the fluid, for which linearized hydrodynamics suffices. In order to determine the coefficients λ1,2,3 , one must consider nonlinear solutions to the hydrodynamic equations. One such solution is the Bjorken boostinvariant flow [53], relevant to relativistic heavyion collisions. Since hydrodynamic equations are boostinvariant, a solution with boost-invariant initial conditions will remain boost invariant. The motion in the Bjorken flow is a one-dimensional expansion, along an axis which we choose to be z, with local velocity equal to z/t. The most convenient are the comoving coordinates: proper time √ for each local element τ = t2 − z 2 and rapidity ξ = arctanh (z/t). In these coordinates each element is at rest: (uτ , uξ , u⊥ ) = (1, 0, 0). The motion is irrotational, and thus we can only determine the coefficient λ1 , but not λ2 or λ3 . Since velocity uμ is constant in the coordinates we chose, the only nontrivial equation is the equation for the energy density: Dε + (ε + P )∇ · u + Πμν ∇μ uν = 0.

(162)

Boost invariance means that ε(τ ) is a function of τ only. The metric is given by ds2 = −dτ 2 + τ 2 dξ 2 + dx2⊥ and it is easy to see that the only nonzero component of ∇μ uν is ∇ξ uξ = τ . Using P = ε/(d − 1) we can write: d ε = −τ Πξξ . (163) d−1 τ For large τ , the viscous contribution on the r.h.s. in (163) becomes negligible and the asymptotics of the solution is thus given by ∂τ ε +

ε(τ ) = C τ −2+ν + (viscous corrections), (164) d−2 , ν≡ d−1 and C is the integration constant. As we shall see, the expansion parameter in (164) is τ −ν . Calculating the r.h.s. of Eq. (163) using Eq. (144) we find   d−3 2νη 2ν 2 . (165) −τ Πξξ = 2 + 3 ητπ − 2λ1 τ τ d−2

Integrating equation (163), one should take into account the fact that kinetic coefficients η, τπ and λ1 in Eq. (165) are functions of ε, which in a conformal theory are given by the following power laws: η = Cη0 λ1 =

 ε (d−1)/d

Cλ01

, τπ = τπ0

C  ε (d−2)/d C

 ε −1/d C

, (166)

,

where, for convenience, we defined constants η0 , τπ0 and λ01 , and we chose the constant C to be the same as in Eq. (164). Integrating Eq. (163) we thus find  2(d−1) 2 ε(τ ) −2+ν −2 =τ η0 − 2η0 τ + C d   d−3 d−2 η0 τπ0 − 2λ01 τ −2−ν + . . . . (167) − d−1 d−2 In Section 4.4.4 we shall match the Bjorken flow solution in the strongly-coupled N = 4 SYM theory found in [54] (see also [55]) using AdS/CFT correspondence and determine λ1 in this theory. In order to compare our results to the ones obtained in Ref. [54], we shall write here the equations of second-order hydrodynamics using also the alternative representation (145) for Πξξ in (163). We obtain the following system of equations for the energy density and the component of the viscous flow, which we define as Φ ≡ −Πξξ (see [56]; c.f. [57] for λ1 = 0): Φ d ε + , d−1 τ τ d τπ 2(d−2) η −Φ− Φ τπ ∂τ Φ = d−1 τ d−1 τ d−3 λ1 2 − Φ . d−2 η 2 ∂τ ε = −

(168)

(169)

As should be expected, the asymptotics of the solution of this system coincides with Eq. (167). Equation (169) is different from the one used in [54] by the last two terms proportional to τπ and λ1 .

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4.4. Second-order hydrodynamics for strongly coupled N = 4 supersymmetric Yang-Mills plasma In this Section, we compute the parameters τπ , κ, and λ1 of the second-order hydrodynamics for a theory whose gravity dual is well-known: N = 4 SU(Nc ) supersymmetric Yang-Mills theory in the limit Nc → ∞, g 2 Nc → ∞ [2–4]. According to the gauge/gravity duality conjecture, in this limit the theory at finite temperature T has an effective description in terms of the AdS-Schwarzschild gravitational background with metric ds25 =

 π 2 T 2 L2

−f (u)dt2 + dx2 + dy 2 + dz 2 u L2 du2 , (170) + 4f (u)u2

where f (u) = 1 − u2 , and L is the AdS curvature scale [21]. The duality allows one to compute the retarded correlation functions of the gaugeinvariant operators at finite temperature. The result of such a computation would in principle be exact in the full microscopic theory (in the limit Nc → ∞, g 2 Nc → ∞). As we are interested in the hydrodynamic limit of the theory, here we compute the correlators in the form of lowfrequency, long-wavelength expansions. In momentum space, the dimensionless expansion parameters are k ω  1, q≡  1. (171) w= 2πT 2πT Comparing these expansions to the predictions of the second-order hydrodynamics obtained in Sections 4.3.1, 4.3.2 and 4.3.4 for d = 4, we can read off the coefficients τπ , κ, λ1 . One must be aware that the N = 4 SU(Nc ) supersymmetric Yang-Mills theory posesses conserved R-charges, corresponding to SO(6) global symmetry. Therefore, complete hydrodynamics of this theory must involve additional hydrodynamic degrees of freedom – R-charge densities. Our discussion of generic conformal hydrodynamics without conserved charges can be, of course, generalized to this case. This is beyond the scope of this paper. Here we only need to observe that since the R-charge densities are not singlets under the SO(6) they cannot contribute at linear

order to the equations for T μν . These contributions are therefore irrelevant for the linearized hydrodynamics we consider in Sections 4.3.1, 4.3.2 and 4.3.3. For the discussion of the Bjorken flow in Section 4.3.4 they are also irrelevant, since (and as long as) we consider solutions with zero Rcharge density. 4.4.1. Scalar channel We start by computing the low-momentum expansion of the correlator GR xy,xy (ω, k). To leading order in momentum, this correlation function has been previously computed from gravity in [6,8]. Following [8], here we obtain the next to leading order term in the expansion. The relevant fluctuation of the background metric (170) is the component φ ≡ hyx of the graviton. The retarded correlator in momentum space is determined by the on-shell boundary action  grav Stot [H0 , k] = lim Sboundary [H0 , , k]+

→0  Sc.t. [H0 , , k] , (172) following the prescription formulated in [6]. Here H0 (k) = H(, k) is the boundary value (more precisely, the value at the cutoff u =  → 0) of the solution to the graviton’s equation of motion (Eq. (6.6) in [8]) H(u, k) = H0 (k)

φk (u) . φk ()

(173)

A perturbative solution φk (u) to order w2 , q2 is given by Eq. (6.8) in [8]. The gravitational action (Eq. (6.4) in [8]) reduces to the sum of two terms, the horizon contribution and the boundary contribution. The horizon contribution should be discarded, as explained in [6] and later justified in [7]. The remaining boundary term, grav [H0 , , k], is divergent in the limit  → 0, Sboundary and should be supplemented by the counterterm action Sc.t. [H0 , , k] following a procedure known as the holographic renormalization.2 In the case 2 The holographic renormalization [58] corresponds to the usual renormalization of the composite operators in the dual CFT.

D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

of gravitational fluctuations, the counterterm action is [59] 3N 2 Sct = − 2 c 4 4π L

 √ L2 P− d x −γ 1 + 2 u=   L4 kl P Pkl − P 2 log  , 12 

4

(174)

where γij is the metric (170) restricted to u = , and   1 1 ij P = γ Pij , Rij − Rγij . (175) Pij = 2 6 Evaluating (174), we find the total boundary action3 Stot

 π 2 Nc2 T 4 H()H  () H 2 () V4 − + =− 8  2  2 2 2 (q − w )H () + O(w3 , wq2 , ). (176) − 

The boundary action (176) is finite in the limit 0  → 0. Its fluctuation-independent part is Stot = 2 2 4 −P V4 , where P = π Nc T /8 is the pressure in N = 4 SYM, V4 is the four-volume. The part quadratic in fluctuations gives the two-point function. Substituting the solution (173) into Eq. (176) and using the recipe of [6], we find GR xy,xy = −

  π 2 Nc2 T 4 1 iw − w2 + q2 + w2 ln 2 − 4 2 + O(w3 , wq2 ) .

(177)

Comparing Eq. (177) to the hydrodynamic result (147) we obtain the pressure [60], the viscosity [5] and the two parameters of the second-order hydrodynamics for N = 4 SYM: π2 2 4 π Nc T , η = Nc2 T 3 , 8 8 η 2 − ln 2 , κ= . τπ = 2πT πT

P =

(178)

3 Terms quadratic in H in Eq. (176) should be understood as products H(−ω, −k)H(ω, k), and an integration over ω and q is implied.

241

4.4.2. Shear channel The dispersion relation (161) manifests itself as a pole in the retarded Green’s functions GR ty,ty , R R Gty,xy , Gxy,xy in the hydrodynamic approximation. To quadratic order in k this dispersion relation was computed from dual gravity in Section 6.2 of Ref. [8]. Here we extend that calculation to quartic order in k. This amounts to solving the differential equation for the gravitational fluctuation G(u) [8] G −



  2u iw iw q2 1 − G + 2+ − f 1−u f 2 u  2 2 w [4 − u(1 + u) ] G = 0 (179) + 4uf

perturbatively in w and q assuming w ∼ q2 . The solution G(u) is supposed to be regular at u = 1 [8]. Such a solution is readily found by writing G(u) = G0 (u) + wG1 (u) + q2 G2 (u) + w2 G3 (u) + wq2 G4 (u) + q4 G5 (u) + · · · (180) and computing the functions Gi (u) perturbatively4 . The functions Gi (u) are given explicitly in Appendix of Ref. [12]. To obtain the dispersion relation, one has to substitute the solution G(u) into the equation (6.13b) of [8] and take the limit u → 0. The resulting equation for w, q4 + 2q2 − 4iw − iwq2 ln 2 + 2w2 ln 2 = 0 , (181) has two solutions one of which is incompatible with the assumption w  1. The second solution is w=−

i(1 − ln 2)q4 iq2 − + O(q6 ) . 2 4

(182)

If we naively compare Eqs. (161), (182), we would get τπ = (1 − ln 2)/(2πT ), which is inconsistent with the value obtained from the Kubo’s formula, Eq. (178). As explained in Section 4.3.3, this happens because the O(k 4 ) term in the shear dispersion relation is fully captured only in third-order hydrodynamics. In other words, we confirm that Eq. (161) has an error at order O(k 4 ). 4 Note that, for u real, G∗ (u, −w) = G(u, w). This implies Im G0,2,3,5 = 0, Re G1,4 = 0.

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4.4.3. Sound channel The sound wave dispersion relations (154) appear as poles in the correlators of the diagonal components of the stress-energy tensor in the hydrodynamic approximation. These correlators and the dispersion relation to quadratic order in spatial momentum were first computed from gravity in [9]. A convenient method of studying the sound channel correlators was introduced in [32]. In this approach, the hydrodynamic dispersion relation emerges as the lowest quasinormal frequency of a gauge-invariant gravitational perturbation of the background (170). According to [32], the sound wave pole is determined by solving the differential equation 3w2 (1 + u2 ) + q2 (2u2 − 3u4 − 3)  Z uf (3w2 + q2 (u2 − 3)) 1 [3w4 + q4 (3 − 4u2 + u4 ) + 2 2 uf (3w + q2 (u2 − 3))

Z  −

+ q2 (4u5 − 4u3 + 4u2 w2 − 6w2 )]Z = 0

(183)

with the incoming wave boundary condition at the horizon (u = 1) and Dirichlet boundary condition Z(0) = 0 at the boundary u = 0, and taking the lowest frequency in the resulting quasinormal spectrum. The exponents of the equation (183) at u = 1 are ±iw/2. The incoming wave boundary condition is implemented by choosing the exponent −iw/2 and writing Z(u) = f −iw/2 X(u) ,

(184)

where X(u) is regular at u = 1. Thus we obtain the following differential equation for X(u)   4q2 u 2u iw 1 + u2 − − X X  + f uf 3w2 + q2 (u2 − 3)  q2 (1 + u + u2 )w2 − + u(1 + u)f uf  2 3 4q u (1 + iw) X = 0 . (185) − uf (3w2 + q2 (u2 − 3)) This equation can be solved perturbatively in w  1, q  1 assuming w ∼ q (the expected scaling in the sound wave dispersion relation). Rescaling w → λw, q → λq, where λ  1, we

look for a solution in the form X(u) = X0 (u) + λX1 (u) + λ2 X2 (u) + · · · . (186) The functions Xi (u) are written explicitly in Appendix of Ref. [12]. The Dirichlet condition X(0) = 0 leads to the equation for w(q): − iwq2 + −

 q2 3w2 w4 2 − + π − 12 ln2 2 + 24 ln 2 2 2 16

q4 (2 ln 2 − 8) 12  w2 q2 2 − π − 12 ln2 2 + 48 ln 2 = 0 . 48

(187)

To order q3 , the solution is given by q iq2 (3 − 2 log 2)q3 √ ± w = ±√ − + O(q4 ) . (188) 3 3 6 3 This is the dispersion relation for the sound waves to order q3 . The complete dispersion relation can be obtained by solving the equation (183) numerically [32]. The sound dispersion curve is shown in Fig. 3. Comparing Eq. (188) to Eq. (154) we find the relaxation time τπ for the strongly coupled N = 4 SYM plasma: 2 − ln 2 . (189) 2πT The result (189) coincides with the one obtained in Section 4.4.1, which is a nontrivial check of our approach. τπ =

4.4.4. Bjorken flow In order to determine λ1 , we match Eq. (167) with the solution found by Heller and Janik [54] given by5  Nc2 −4/3 ε(τ ) = − 2η0 τ −2 τ 2π 2   10 2 6 ln 2 − 17 −8/3 √ η + +τ , 3 0 36 3 1 . (190) η0 = √ 2 33/4 5 The quantities in Eq. (190) can be thought of as dimensionless combinations of quantities in Eq. (167) with an appropriate power of an arbitrary scale parameter τ0 : τ /τ0 , ετ0d , η0 τ0ν , Cτ0dν etc. Due to conformal invariance, a rescaled solution is also a solution, and the scale τ0 can be used instead of the integration constant C, to parameterize the solutions in Eq. (167).

D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

cs 1.5

1.0

0.5

0.0 0.0

 0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 3. Sound dispersion cs = cs (q) in N = 4 SYM plasma. The dark (blue) curve shows the sound speed dependence√on wavevector, cs (q) = Re w/q, with cs (0) = 1/ 3 (this plot is based on numerical data first obtained in [32]). The light (red) curve corresponds to analytic approximation derived from Eq. (188) and valid for sufficiently small q.

Matching by using C = Nc2 /(2π 2 ), and τπ = (2 − ln 2)/(2πT ) from Eq. (189), together with ε = 3π 2 Nc2 T 4 /8 and Eq. (166) gives λ1 =

η . 2πT

(191)

Note that Heller and Janik [54] found a different value for τπ since they matched to the IsraelStewart equations for hydrodynamics, and not the more general (nonlinear) equation (145). 5. NONRELATIVISTIC HOLOGRAPHY There exist, in nonrelativistic physics, another prototype of strong coupling: fermions at unitarity [61–63]. This is the system of fermions interacting through a short-ranged potential which is fine-tuned to support a zero-energy bound state. The system is scale invariant in the limit of zerorange potential. Since its experimental realizations using trapped cold atoms at the Feshbach resonance [64–69], this system has attracted enormous interest. One may wonder if there exists a gravity dual of

243

fermions at unitarity. If such a gravity dual exists, it would extend the notion of holography to nonrelativistic physics, and could potentially bring new intuition to this important strongly coupled system. Similarities between the N = 4 super– Yang-Mills theory and unitarity fermions indeed exist, the most important of which is scale invariance. The have been some speculations on the possible relevance of the universal AdS/CFT value of the viscosity/entropy density ratio [34] for unitarity fermions [70–72]. Despite these discussions, no serious attempt to construct a gravity dual of unitarity fermions has been made to date. In this paper, we do not claim to have found the gravity dual of the unitary Fermi gas. However, we take the possible first step toward such a duality. We will construct a geometry whose symmetry coincides with the Schr¨odinger symmetry [73,74], which is the symmetry group of fermions at unitarity [75]. In doing so, we keep in mind that one of the main evidences for gauge/gravity duality is the coincidence between the conformal symmetry of the N = 4 field theory and the symmetry of the AdS5 space. On the basis of this geometric realization of the Schr¨ odinger symmetry, we will be able to discuss a nonrelativistic version of the AdS/CFT dictionary— the operator-state correspondence, the relation between dimensions of operators and masses of fields, etc. In this lecture d always refers to the number of spatial dimensions in the nonrelativistic theory, so d = 3 corresponds to the real world. 5.1. Review of fermions at unitarity and Schr¨ odinger symmetry In this section we collect various known facts about fermions at unitarity and the Schr¨ odinger symmetry. The goal is not to present an exhaustive treatment, but only to have a minimal amount of materials needed for later discussions. Further details can be found in [76]. We are mostly interested in vacuum correlation functions (zero temperature and zero chemical potential), but not in the thermodynamics of the system at nonzero chemical potential. The reasons are twofold: i) the chemical potential breaks the

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Schr¨ odinger symmetry and ii) even at zero chemical potential there are nontrivial questions, such as the spectrum of primary operators (see below). One way to arrive at the theory of unitarity fermions is to start from noninteracting fermions, L = iψ † ∂t ψ −

|∇ψ|2 , 2m

(192)

ψ(x1 , x2 , . . . ; y1 , y2 , . . .) (where xi are coordinates of spin-up particles and yj are those of spin-down particles) which satisfy the following boundary conditions when a spin-up and a spin-down particle approach each other, ψ(x1 , x2 , . . . ; y1 , y2 , . . .) →

add a source φ coupled to the “dimer” field ψ↓ ψ↑ [77], L = iψ † ∂t ψ −

|∇ψ|2 + φ∗ ψ↓ ψ↑ + φψ↑† ψ↓† , 2m

(193)

and then promote the source φ to a dynamic field. There is no kinetic term for φ in the bare Lagrangian, but it will be generated by a fermion loop. Depending on the regularization scheme, one may need to add to (193) a counterterm ∗ c−1 0 φ φ to cancel the UV divergence in the oneloop φ selfenergy (such a term is needed in momentum cutoff regularization but not in dimensional regularization.) The theory defined by the Lagrangian (193) is UV complete in spatial dimension 2 < d < 4, including the physically most relevant case of d = 3. This system is called “fermions at unitarity,” which refers to the fact that the s-wave scattering cross section between two fermions saturates the unitarity bound. Another description of fermions at unitarity is in terms of the Lagrangian L = iψ † ∂t ψ −

|∇ψ|2 − c0 ψ↓† ψ↑† ψ↑ ψ↓ . 2m

(194)

where c0 is an interaction constant. The interaction is irrelevant in spatial dimensions d > 2, and is marginal at d = 2. At d = 2 +  there is a nontrivial fixed point at a finite and negative value of c0 of order  [78]. The situation is similar to the nonlinear sigma model in 2 +  dimensions. In the quantum-mechanical language, unitarity fermions are defined as a system with the free Hamiltonian H=

 p2 i , 2m i

(195)

but with a nontrivial Hilbert space, defined to contain those wavefunctions

C |xi − yj |

+ O(|xi − yj |). (196) where C depends only on coordinates other than xi and yj . This boundary condition can be achieved by letting the fermions interact through some pairwise potential (say, a square-well potential) that has one bound state at threshold. In the limit of zero range of the potential r0 → 0, keeping the zero-energy bound state, the twobody wave function satisfies the boundary condition (196) and the physics is universal. Both free fermions and fermions at unitarity have the Schr¨ odinger symmetry—the symmetry group of the Schr¨ odinger equation in free space, which is the nonrelativistic version of conformal symmetry [75]. The generators of the Schr¨ odinger algebra include temporal translation H, spatial translations P i , rotations M ij , Galilean boosts K i , dilatation D (where time and space dilate with different factors: t → e2λ t, x → eλ x), one special conformal transformation C [which takes t → t/(1 + λt), x → x/(1 + λt)], and the mass operator M . The nonzero commutators are [M ij , M kl ] = i(δ ik M jl + permutations) [M ij , P k ] = i(δ ik P j − δ jk P i ), [M ij , K k ] = i(δ ik K j − δ jk K i ), [D, P i ] = −iP i ,

[D, K i ] = iK i ,

(197)

[P i , K j ] = −iδ ij M, [D, H] = −2iH, [D, C] = 2iC, [H, C] = iD. The theory of unitarity fermions is also symmetric under an SU(2) group of spin rotations. The theory of unitarity fermions is an example of nonrelativistic conformal field theories (NRCFTs). Many concepts of relativistic CFT, such as scaling dimensions and primary operators, have counterparts in nonrelativistic CFTs.

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A local operator O is said to have scaling dimension Δ if [D, O(0)] = −iΔO(0). Primary operators satisfy [K i , O(0)] = [C, O(0)] = 0. To solve the theory of unitarity fermions at zero temperature and chemical potential is, in particular, to find the spectrum of all primary operators. In the theory of unitarity fermions, there is a quantum-mechanical interpretation of the dimensions of primary operators [79,80,76]. A primary operator with dimension Δ and charges N↑ and N↓ with respect to the spin-up and spin-down particle numbers (the total particle numbers is N = N↑ + N↓ ) corresponds to a solution of the zero-energy Schr¨ odinger equation: ⎛ ⎞  ∂2  ∂2 ⎝ ⎠ 2 + 2 ∂x ∂y i j i j ψ(x1 , x2 , . . . , xN↑ ; y1 , y2 , . . . , yN↓ ) = 0,

(198)

which satisfies the boundary condition (196) and with a scaling behavior ψ(x1 , x2 , . . . , y1 , y2 , . . .) = Rν ψ(Ωk ),

(199)

where R is an overall scale of the relative distances between xi , yj , and Ωk are dimensionless variables that are defined through the ratios of the relative distances. Equations (7) and (8) define, for given N↑ and N↓ , a discrete set of possible values for ν. For example, in three spatial dimensions, for N↑ = N↓ = 1, there are two possible values for ν: 0 and −1. For N↑ = 2, N↓ = 1, the lowest value for ν is ≈ −0.22728. Each value of ν corresponds to an operator with dimension Δ, which is related to ν by Δ=ν+

dN . 2

(200)

It has also been established that each primary operator corresponds to a eigenstate of the Hamiltonian of unitarity fermion in an isotropic harmonic potential of frequency ω [79,80,76]. The scaling dimension of the operator simply coincides with the energy of the state: E = Δ¯hω.

(201)

The first nontrivial operator is the dimer ψ↓ ψ↑ . It has dimension Δ = d in the free theory, and

Δ = 2 in the theory of fermions at unitarity. This corresponds to the fact that the lowest energy state of two fermions with opposite spins in a harmonic potential is d¯ hω in the case of free fermions and 2¯ hω for unitarity fermions. 5.2. Embedding the Schr¨ odinger group into a conformal group To realize geometrically the Schr¨ odinger symmetry, we first embed the Schr¨ odinger group in d spatial dimensions Sch(d) (d = 3 for the most interesting case of the unitarity Fermi gas) into the relativistic conformal algebra in d + 2 spacetime dimensions O(d+2, 2). The next step will be to realize the Schr¨ odinger group as a symmetry of a d + 3 dimensional spacetime background. That the Schr¨ odinger algebra can be embedded into the relativistic conformal algebra can be seen from the following. Consider the massless Klein-Gordon equation in ((d+1) + 1)dimensional Minkowski spacetime, 2φ ≡

−∂t2 φ

+

d+1 

∂i2 φ = 0.

(202)

i=1

This equation is conformally invariant. Defining the light-cone coordinates, x± =

x0 ± xd+1 √ , 2

(203)

the Klein-Gordon equation becomes 

 ∂ ∂ ∂i2 −2 − + + ∂x ∂x i=1 d

 φ = 0.

(204)

If we make an identification ∂/∂x− = −im, then the equation has the form of the Schr¨ odinger equation in free space, with the light-cone coordinate x+ playing the role of time,   ∂ (205) 2im + + ∂i ∂i φ = 0. ∂x This equation has the Schr¨ odinger symmetry Sch(d). Since the original Klein-Gordon equation has conformal symmetry, this means that Sch(d) is a subgroup of O(d+2, 2).

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Let us now discuss the embedding explicitly. The conformal algebra is ˜ αβ ] = i(η μα M ˜ νβ + perm.), ˜ μν , M [M ˜ μν , P˜ α ] = i(η μα P˜ ν − η να P˜ μ ), [M ˜ K ˜ μ ] = iK ˜ μ, ˜ P˜ μ ] = −iP˜ μ , [D, [D,

(206)

˜ ν ] = −2i(η μν D ˜ +M ˜ μν ), [P˜ μ , K where Greek indices run 0, . . . , d+1, and all other commutators are equal to 0. The tilde signs denote relativistic operators; we reserve untilded symbols for the nonrelativistic generators. We identify√the light-cone momentum P˜ + = (P˜ 0 + P˜ d+1 )/ 2 with the mass operator M in the nonrelativistic theory. We now select all operators in the conformal algebra that commute with P˜ + . Clearly these operators form a closed algebra, and it is easy to check that it is the Schr¨ odinger algebra in d spatial dimensions. The identification is as follows: M = P˜ + , H = P˜ − , P i = P˜ i , ˜ ij , K i = M ˜ i+ , D = D ˜ +M ˜ +− , M ij = M (207) ˜+ K C= . 2 From Eqs. (206) and (207) one finds the commutators between the untilded operators to be exactly the Schr¨ odinger algebra, Eqs. (197). 5.3. Geometric realization of the Schr¨ odinger symmetry To realize the Schr¨odinger symmetry geometrically, we will take the AdS metric, which is is invariant under the whole conformal group, and then deform it to reduce the symmetry down to the Schr¨ odinger group. The AdS space, in Poincar´e coordinates, is ημν dxμ dxν + dz 2 . (208) ds2 = z2 The generators of the conformal group correspond to the following infinitesimal coordinates transformations that leave the metric unchanged, P μ : xμ → xμ + aμ , D : xμ → (1 − a)xμ , 2

z → (1 − a)z,

K : x → x + a (z + x · x) − 2xμ (a · x) μ

μ

μ

μ

(209)

(here x · x ≡ ημν xμ xν ). We will now deform the metric so to reduce the symmetry to the Schr¨ odinger group. In particular, we want the metric to be invariant under ˜ +M ˜ +− , which is a linear combination of D=D ˜ +− and the a boost along the xd+1 direction M ˜ but not separately under scale transformation D, ˜ The following metric satisfies this ˜ +− or D. M condition: ds2 = −

2(dx+ )2 z4 −2dx+ dx− + dxi dxi + dz 2 + . z2

(210)

It is straightforward to verify that the metric (210) exhibits a full Schr¨ odinger symmetry. From Eqs. (207) and (209) one finds that the generators of the Schr¨ odinger algebra correspond to the following isometries of the metric: P i : xi → xi + ai , H : x+ → x+ + a, M : x− → x− + a, K i : xi → xi − ai x+ , x− → x− − ai xi , D : xi → (1 − a)xi , z → (1 − a)z, x+ → (1 − a)2 x+ , x− → x− ,

(211)

C : z → (1 − ax+ )z, xi → (1 − ax+ )xi , x+ → (1 − ax+ )x+ , a x− → x− − (xi xi + z 2 ). 2 We thus hypothesize that the gravity dual of the unitarity Fermi gas is a theory living on the background metric (210). Currently we have very little idea of what this theory is. We shall now discuss several issues related to this proposal. i) The mass M in the Schr¨ odinger algebra is mapped onto the light-cone momentum P + ∼ ∂/∂x− . In nonrelativistic theories the mass spectrum is normally discrete: for example, in the case of fermions at unitarity the mass of any operator is a multiple of the mass of the elementary fermion. It is possible that the light-cone coordinate x− is compactified, which would naturally give rise to the discreteness of the mass spectrum. ii) In AdS/CFT correspondence the number of color Nc of the field theory controls the mag-

D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

nitude of quantum effects in the string theory side: in the large Nc limit the string theory side becomes a classical theory. The usual unitarity Fermi gas does not have this large parameter N , hence the dual theory probably has unsuppressed quantum effects. However, there exists an extension of the unitarity Fermi gas with Sp(2N ) symmetry [78,81]. The gravity dual of this theory may be a classical theory in the limit of large N , although with an infinite number of fields, similar to the conjectured dual of the critical O(N ) vector model in 2+1 dimensions [82]. iii) We can write down a toy model in which the metric (210) is a solution to field equations. Consider the theory of gravity coupled to a massive vector field with a negative cosmological constant,  S=

d

d+2

x dz

√

 −g

1 1 R − Λ − Hμν H μν 2 4  m2 μ Cμ C , (212) − 2

where Hμν = ∂μ Cν − ∂ν Cμ . One can check that Eq. (210), together with C − = 1,

(213)

is a solution to the coupled Einstein and Proca equations for the following choice of Λ and m2 : 1 Λ = − (d + 1)(d + 2), 2

m2 = 2(d + 2). (214)

iv) Although the g++ metric component has z singularity at z = 0, the metric has a planewave form and all scalar curvatures are finite. For example, the most singular component of the Ricci tensor, R++ , has a z −4 singularity, as the C+i+i and C+z+z components of the Weyl tensor. However, since g ++ = 0, any scalar constructed from the curvature tensor is regular. −4

v) In terms of a dual field theory, the field Aμ with mass in Eq. (214) corresponds to a vector operator Oμ with dimension Δ, which can be found from the general formula (Δ − 1)[Δ + 1 − (d + 2)] = 2(d + 2),

(215)

247

from which Δ = d + 3. We thus can think about the quantum field theory as an irrelevant deformation of the original CFT, with the action  S = SCFT + J dd+2 x O+ . (216) 5.4. Operator-field correspondence Let us now discuss the relationship between the dimension of operators and masses of fields in this putative nonrelativistic AdS/CFT correspondence. Consider an operator O dual to a massive scalar field φ with mass m0 . We shall assume that it couples minimally to gravity,  √ (217) S = − dd+3 x −g(∂φ∗ · ∂φ + m20 φ∗ φ). Assuming the light-cone coordinate x− is periodic, let us concentrate only on the Kaluza-Klein mode with P + = M . The action now becomes  d+2 d x dz S= (2iM z 2 φ∗ ∂t φ − z 2 ∂i φ∗ ∂i φ z d+3 − m2 φ∗ φ), (218) where the “nonrelativistic bulk mass” m2 is related to the original mass m20 by m2 = m20 +2M 2 . Contributions to m2 can arise from interaction terms between Cμ and φ, for example |C μ ∂μ φ|2 , |C μ C ν ∂μ ∂ν φ|2 , etc. We therefore will treat m2 as an independent parameter. The field equation for φ is   d+1 m2 2 2  ∂z φ+ 2M ω − k − 2 φ = 0. (219) ∂z φ− z z The two independent solutions at small z are

(d+2)2 d/2+1±ν . (220) , ν = m2 + φ± = z 4 As in usual AdS/CFT correspondence, one choice of φ± corresponds to turning a source for O in the boundary theory, and another choice corresponds to a condensate of O. One can distinguish two cases: 1. When ν ≥ 1, φ+ is non-normalizable and φ− is renormalizable. Therefore φ+ corresponds to the source and φ− to the condensate. The correlation function of O is OO ∼ (k 2 − 2M ω)2ν ,

(221)

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which translate into the scaling dimension Δ=

d+2 + ν. 2

(222)

2. When 0 < ν < 1 both asymptotics are normalizable, and there is an ambiguity in the choice of the source and condensate boundary conditions. These two choices should correspond to two different nonrelativistic CFTs. In one choice the operator O has dimension Δ = (d + 2)/2 + ν, and in the other choice Δ = (d + 2)/2 − ν. It is similar to the situation discussed in [83]. The smallest dimension of an operator one can get is Δ = (d + 2)/2 − ν when ν → 1. Therefore, there is a lower bound on operator dimensions, d . (223) 2 This bound is very natural if one remember that operator dimensions correspond to eigenvalues of the Hamiltonian in an external harmonic potential. For a system of particles in a harmonic potential, one can separate the center-of-mass motion from the relative motion. Equation (223) means that the total energy should be larger than the zero-point energy of the center-of-mass motion. The fact that there are pairs of nonrelativistic conformal field theories with two different values of the dimensions of O is a welcome feature of the construction. In fact, free fermions and fermions at unitarity can be considered as such a pair. In the theory with free fermions the operator ψ↓ ψ↑ has dimension d, and for unitarity fermions, this operator has dimension 2. The two numbers are symmetric with respect to (d + 2)/2:

Δ>

d+2 d−2 d+2 d−2 + , 2= − . (224) 2 2 2 2 Therefore, free fermions and fermions at unitarity should correspond to the same theory, but with different interpretations for the asymptotics of the field dual to the operator ψ↓ ψ↑ . A similar situation exists in the case of Fermi gas at unitarity with two different masses for spinup and spin-down fermions [84]. In a certain interval of the mass ratios (between approximately d=

8.6 and 13.6), there exist two different scaleinvariant theories which differ from each other, in our language, by the dimension of a three-body p-wave operator. At the upper end of the interval (mass ratio 13.6) the dimension of this operator tends to 5/2 in both theories; at the lower end it has dimension 3/2 in the theory with three-body resonance and 7/2 in the theory without threebody resonance. 5.5. Turning on sources Let us now try to turn on sources coupled to conserved currents in the boundary theory. That would correspond to turning on non-normalizable modes. For the fields that enter the model action (212), the general behavior of the nonnormalizable part of the metric and the field Cμ near z = 0 is 2e−2Φ (dx+ −Bi dxi )2 z4 2e−Φ − 2 (dx+ −Bi dxi )(dx− −A0 dx+ −Ai dxi ) z gij dxi dxj + dz 2 + O(z 0 ), + z2 C − = 1. (225) ds2 = −

= 0. We have chosen the gauge gμz The non-normalizable metric fluctuations are parametrized by the functions A0 , Ai , Φ, and Bi of x+ ≡ t and xi . These functions are interpreted as background fields, on which the boundary theory exists. Following the general philosophy of AdS/CFT correspondence, we assume that the partition function of the high-dimensional theory with the boundary condition (225) is equal to the partition function of an NRCFT in the background fields, Z = Z[A0 , Ai , Φ, Bi , gij ].

(226)

This partition function should be invariant with respect to a group of gauge transformations acting on the background fields, which we will derive. The gauge condition gμz = 0 does not completely fix the metric: there is a residual gauge symmetry parametrized by arbitrary functions of

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t and xi (but not of z): t → t = t + ξ t (t, x), x− → x− = x− + ξ − (t, x),

(227)

x → x = x + ξ (t, x), i

i

i

i

and another set of infinitesimal transformations characterized by a function ω(t, x), z → z  = z − ω(t, x)z, 1 xμ → xμ = xμ + g μν ∂ν ω. 2

(228)

Consider first (227). Under these residual gauge transformations, the fields entering the metric (225) change in the following way: δA0 =ξ˙− − A0 ξ˙t − Ai ξ˙i − ξ μ ∂μ A0 , δAi =∂i ξ − − A0 ∂i ξ t − eΦ gij ξ˙j − ξ μ ∂μ Ai − Aj ∂i ξ , δΦ =ξ˙t − Bi ξ˙i − ξ μ ∂μ Φ, δBi =∂i ξ + Bi (ξ˙t − Bj ξ˙j ) − ξ μ ∂μ Bi

(229)

− gkj ∂i ξ k − gik ∂j ξ k , where ξ ∂μ ≡ ξ ∂t + ξ ∂i . The residual gauge symmetry implies that the partition function of the boundary theory should be invariant under such transformations, δZ = 0.

i

(230)

Can one formulate NRCFTs on background fields with this symmetry? In fact, it can be done explicitly in the theory of free nonrelativistic particles. One introduces the interaction to the background fields in the following manner:  i √ dt dx g e−Φ eΦ (ψ † Dt ψ − Dt ψ † ψ) 2 Bi g ij Di ψ † Dj ψ − (Dt ψ † Di ψ + Di ψ † Dt ψ) − 2m 2m  B2 † − Dt ψ Dt ψ , (231) 2m 

S=

In fact, this invariance is an extension of the general coordinate invariance previously discussed in [85]. The invariance found in [85] corresponds to restricting Φ = Bi = ξ t = 0 in all formulas. To linear order in external field, the action is   S = S[0] + dt dx A0 ρ + Ai j i + Φ + Bi j i  1 + hij Πij , (233) 2

• hij is coupled to the stress tensor Πij ,

• (Φ, Bi ) are coupled to the energy current (, j ).

δgij = − (Bi gjk + Bj gik )ξ˙k − ξ μ ∂μ gij

t

(232)

• Aμ is coupled to the mass current (ρ, j),

− Bj ∂i ξ j ,

μ

δψ = imξ − ψ − ξ μ ∂μ ψ.

and from Eq. (231) one reads out the physical meaning of the operators coupled to the external sources:

j

t

where g ij is the inverse matrix of gij , g ≡ det |gij |, B i ≡ g ij Bi , B 2 ≡ B i Bi , and Dμ ψ ≡ ∂μ ψ − imAμ ψ. One can verify directly that the action (231) is invariant under the transformations (229), if ψ transforms as

The invariance of the partition function with respect to the gauge transformations (229) leads to an infinite set of Takahashi-Ward identities for the correlation functions. The simplest ones are for the one-point functions. The fact that the group of invariance includes gauge transformation of Aμ : δAμ = ∂μ ξ − guarantees the conservation of mass. The fact that the linear parts in the transformation laws for Φ and Bi look like a gauge transformation, δΦ = ξ˙t +· · · and δBi = ∂i ξ t +· · · leads to energy conservation in the absence of external fields:     ∂ ln Z ∂ ln Z + ∂i = 0. ∂t ∂Φ ∂Bi Aμ =Φ=Bi =hij =0 (234) Energy is not conserved in a general background (which is natural, since the background fields exert external forces on the system). Similarly, momentum conservation ∂t j i + ∂j Πij = 0 (and the

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fact that momentum density coincides with mass current) is related to terms linear in ξ i in δAi and δgij : δAi = −ξ˙i + · · · , δgij = −∂i ξ j − ∂j ξ i + · · · . Let us now turn to the transformations (228), under which δΦ = 2ω,

δgij = −2ωgij .

(235)

The invariance of the partition function with respect to this transformation implies 2 = Πii ,

(236)

which is the familiar relationship between energy and pressure, E=

d P V, 2

(237)

valid for free gas as well as for Fermi gas at unitarity. The action (231) is not invariant under (235), but it can be made so by replacing the “minimal coupling” by a “conformal coupling” to external fields. Therefore, the proposed holography is consistent with conservation laws and the universal thermodynamic relation between energy and pressure. 6. NONRELATIVISTIC HOLOGRAPHIC HYDRODYNAMICS Using a solution-generating transformation, the authors of [86–88] constructed a twoparameter family of black hole spacetimes with Schr5 asymptotics. These black holes can be shown to solve the five-dimensional effective equations of motion obtained by a Kaluza-Klein reduction of the 10-dimensional type IIB theory on the X5 . The simplest five dimensional effective action is composed of gravity coupled to a massive vector field and a single scalar field [86], which is a truncation of a more general five-dimensional Lagrangian involving three scalar fields. The latter, remarkably, is a consistent truncation of type IIB supergravity on X5 [87]. This black hole solution was used to study the equilibrium thermodynamic properties of the field theory and was shown to be dual to the grand canonical ensemble for the dual field theory. The thermodynamics is, not surprisingly, consistent

with non-relativistic scale invariance in two spatial dimensions. In particular, it was found that ε = P , where ε is the energy density and P is the pressure, as required in non-relativistic CFTs. Furthermore, non-equilibrium transport properties of the non-relativistic plasma were also explored in [86,88]. In particular, the shear viscosity η of the non-relativistic fluid was calculated and found to take the universal value η/s = 1/4π typical of strongly interacting field theories with gravity duals. Nonrelativistic fluids, in addition to viscosity, also possess another transport coefficient— the thermal conductivity. To compute this coefficient, we describe the general reduction of a relativistic stress tensor on the light cone to obtain a non-relativistic stress tensor complex. The structure of the relativistic conformal stress tensor implies that for any non-relativistic conformal theory obtained in this way, the thermal conductivity κ of the non-relativistic fluid is κ = 2η

ε+P ρT

(238)

where ρ is the mass density, or, even more succinctly, Pr = 1

(239)

where Pr is the Prandtl number. In Section 6.1 we wll discuss how the relativistic fluid equations are reduced on the light-cone to the non-relativistic Navier-Stokes equations. This will allow us to explore the general properties of the non-relativistic stress-tensor complex. 6.1. Light-cone reduction of relativistic fluids Consider a relativistic fluid in Minkowski space in d + 2 spacetime dimensions; we will use lightcone coordinates {x+ , x− , x} and take the metric to be ds2flat = ημν dxμ dxν = −2 dx+ dx− + dx2 . (240) Suppose we view this fluid in the light-cone frame and evolve it in light-cone time x+ . Then, for fixed light-cone momentum P− , we obtain a system in d + 1 dimensions with non-relativistic in-

D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

variance. This is of course familiar from the discrete light-cone quantization (DLCQ) of quantum field theories. In fact, one of the models for studying non-relativistic conformal field theories holographically, suggested in refs. [89,90], was that one could consider pure AdS, with the relativistic conformal symmetry broken to Galilean symmetry simply by compactification of the x− coordinate, which singles out a preferred lightcone direction. Note that in this case we are not only compactifying the light-cone direction in the boundary where gravity is non-dynamical (and the metric flat, (240)), we are also required to compactify the coordinate in the bulk AdS spacetime. This involves introducing closed null curves in the geometry and the validity of supergravity becomes questionable [87]. We will return to a different gravitational background where x− does not need to be compactified to achieve Galilean symmetry. Note however that it is still useful to take x− compact, so that the momentum P− is integer quantized, since P− is interpreted as particle number in the dual theory. Relativistic hydrodynamics in d + 2 dimensions is formulated in terms of pressure (or, equivalently, the temperature) and the four velocity uμ , subject to the condition that ημν uμ uν = −1. This gives d + 2 degrees of freedom. At the same time, non-relativistic hydrodynamics in d spatial, one temporal dimensions can be formulated in terms of the mass density ρ, the pressure P , and the spatial velocities v i , also giving d + 2 degrees of freedom. We would like to find a mapping between the degrees of freedom of the (d + 2)-dimensional theory to the degrees of freedom of the d + 1 dimensional theory such that the relativistic hydrodynamic equations imply the non-relativistic hydrodynamic equations. We would also like to find how the thermodynamic quantities of the two formulations are related. Finally, we plan to use the map to find out the thermal conductivity of the non-relativistic theory. We will first begin with ideal hydrodynamics and then discuss dissipative terms.

251

6.1.1. Ideal fluids The relativistic hydrodynamics equations are just the conservation of energy and momentum ∇μ T μν = 0.

(241)

An ideal relativistic fluid has a stress tensor given by6 T μν = (rel + Prel ) uμ uν + Prel η μν ,

(242)

where the energy density rel is related to the pressure Prel by a thermodynamic equation of state. Equations (241) and (242) define a system of d + 2 equations for the d + 2 unknowns. Non-relativistic ideal hydrodynamics is described by the continuity equation,

 (243) ∂t ρ + ∂i ρ v i = 0, together with the equation of momentum conservation, (here i = 1, . . . , d) ∂t (ρ v i ) + ∂j Πij = 0, Πij = ρ v i v j + δ ij P , and the equation of energy conservation,   1 2 ∂t ε + ρ v + ∂i jεi = 0, 2 1 jεi = (ε + P ) v 2 v i . 2

(244)

(245)

where v 2 = v i v i . Consider the relativistic equations (241) on the light-cone. We will consider only solutions to the relativistic equations that do not depend on x− ; that is, all derivatives ∂− vanish. The coordinate x+ corresponds to the non-relativistic time t. The equations of energy-momentum conservation are, ∂+ T ++ + ∂i T +i = 0 , ∂+ T +i + ∂j T ij = 0 , ∂+ T

+−

+ ∂i T

−i

(246)

= 0,

which reduce to the non-relativistic equations under the following identification: identify T ++ 6 We use the subscript “rel” for quantities in relativistic hydrodynamics and indicate quantities in non-relativistic hydrodynamics without subscripts.

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D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

with the mass density, T +i with the mass flux (which is equal to the momentum density), T ij with the stress tensor, T +− with the energy density, and T −i with the energy flux, T ++ = ρ, T

+−

T +i = ρ v i , T ij = Πij , 1 = ε + ρ v2 , T −i = jεi . 2

The component of the relativistic velocity u− can be determined using the normalization condition uμ uμ = −1 to be   1 1 + 2 + u v . (249) u− = 2 u+ While the analysis has been for a general relativistic fluid with an equation of state rel (Prel ), we will soon focus on conformal fluids. Conformal invariance requires that the stress tensor for the relativistic theory be traceless, Tμμ = 0, which gives us the equation of state rel = (d+1) Prel . In the non-relativistic theory we can once again use the conformal invariance to learn that 2 ε = d P . 6.1.2. Viscous fluids We now wish to extend our mapping of relativistic hydrodynamics into non-relativistic hydrodynamics to first order in derivatives on both sides. The ideal stress energy tensor (242) can be supplemented with dissipative terms, which can be expanded systematically in terms of derivatives of the velocity field and pressure. Specifically, we have (250)

where π μν incorporates all the dissipative contributions. For first order viscous hydrodynamics we have π μν = −2 ηrel τ μν ,

τ

μν

 1 μα νβ = P P ∇α uβ + ∇β uα 2  2 γ ηαβ ∇γ u − d+1

(252)

(247)

It is now easy to convince oneself based on (247) that the precise mapping between relativistic and non-relativistic hydrodynamic variables is

1 ρ + , ui = u+ v i , = u 2 ε+P Prel = P , rel = 2 ε + P. (248)

T μν = (rel + Prel ) uμ uν + η μν Prel + π μν ,

where

(251)

is the shear tensor and we have introduced the spatial projector P μν = η μν + uμ uν . We will use the zeroth-order equations of motion to simplify the viscosity term. By using zeroth-order equations, we make an error of second order in derivatives, which can be neglected. Namely, we use the ideal hydrodynamic equations in the following form, uμ ∇μ rel + (rel + Prel ) ∇μ uμ = 0, u ν ∇ν u μ +

∇μ⊥ Prel = 0, rel + Prel

(253)

to rewrite the stress-energy tensor as T μν = (rel + Prel )uμ uν + Prel η μν  2 P μν ∇α uα − ηrel ∇μ uν + ∇ν uμ − d+1  (uμ ∇ν⊥ + uν ∇μ⊥ )Prel − . rel + Prel

(254)

On the non-relativistic side, we use the ideal hydrodynamic equations in the form ∂t ρ + ∂i (ρ v i ) = 0, 1 ∂t v i + v j ∂j v i + ∂i P = 0, ρ i ∂t ε + ∂i (ε v ) + P ∂j v j = 0.

(255)

The first-order contributions to the (spatial) stress tensor and the energy flux are Πij = ρv i v j + P δ ij − η σ ij , 2 σ ij = ∂i vj + ∂j vi − δ ij ∂k v k , d  ρv 2 i jε = ε + P + v i + ησ ij v j − κ∂i T, 2

(256)

where κ is the thermal conductivity and T is the temperature.

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D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

By using (254) and (256), we now establish the mapping between relativistic and non-relativistic viscous hydrodynamics. First, we find τ ++ = 0, and therefore T

++

+ 2

= (rel + Prel ) (u ) .

(257)

Regarding the other components of the stress tensor, after some calculations involving many cancellations, one discovers that T +− is T +− =

1 1 (rel − Prel ) + ρ v 2 , 2 2

(264)

The identification T ++ = ρ implies then that

ρ u+ = , (258) rel + Prel

which means that the relationship between relativistic and non-relativistic energy densities remain unchanged,

unchanged from the ideal hydrodynamic level (248). Next, we find   u+ ∂i Prel . (259) τ i+ = −ηrel ∂i u+ − 2 (rel + Prel )

Finally, for T −i we find

On the other hand, we still want to map T +i = ρ v i . This means that there is now a correction to the relation between ui and v i :    ηrel u+ ∂i Prel i + i + u =u v + ∂i u − . ρ 2(rel + Prel ) (260) For T ij , after some algebra, we find  T ij = ρ v i v j + Prel δ ij − ηrel u+ ∂i vj + ∂j vi  2 − δ ij ∂k v k , (261) d which implies that the pressures on the two sides still coincide, Prel = P,

(262)

rel = 2 ε + P.

T

−i

ηrel

(263)

Note that our identifications automatically give a first-order correction σij in the non-relativistic theory with the correct tensor structure. That is, the trace-free relativistic shear tensor gives a trace free spatial stress tensor in the nonrelativistic theory.

 1 2 i ε + P + ρv v − ηrel u+ σ ij vj 2 ∂j u+ u+ ij δ ∂j P. + ηrel δ ij + 2 − ηrel (u ) ρ

 =

(266)

Thus we have to require that ηrel

∂i u+ u+ ∂i P = −κ∂i T. − η rel (u+ )2 ρ

(267)

In order to see that the left hand side is indeed proportional to ∂i T , we need to use the mapping (258) and the equation of state for a nonrelativistic theory. Focusing specifically now on conformally invariant fluids, using (258) and ε = d2 P , we find ηrel

∂i u+ u+ ∂i P − ηrel + 2 (u ) ρ   (d+4)/(d+2)  P ε+P ∂i ln . = −ηrel 2ρ ρ

(268)

Recalling that the equation of state of the holographic non-relativistic liquid is [91] 

and the relationship between the viscosities is η = +. u

(265)

P =α

T2 μ

(d+2)/2 ,

(269)

the argument of the logarithm in (268) is T 2 up to a constant. Therefore, the left hand side of (267) is indeed proportional to ∂i T , and one reads out the value for the thermal conductivity: κ = 2η

ε+P . ρT

(270)

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D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

Let us now compute the Prandtl number. The Prandtl number is defined as the ratio of the kinematic viscosity ν and the thermal diffusivity χ, Pr =

ν , χ

where η ν= , ρ

(271)

χ=

κ , ρ cp

(272)

where cp is the specific heat at constant pressure. We note the definition of the heat capacity at constant volume:   ∂H , (273) Cv = ∂T P,N where H = E + P V is the enthalpy. Write H = w V = w N/n. We then find   ∂ w N w ∂n =− 2 (274) Cp = N ∂T n P n ∂T P where we have used the fact that w = (d/2 + 1) P and is fixed at fixed P . At fixed P , μ ∼ T 2 , and n = ∂P/∂T ∼ 1/T 2 , and ∂n/dT = −2n/T . Therefore Cp =

2wN Tn

(275)

and cp = Cp /M (M being the total mass) is equal to 2w/ρ T . Thus we find: Pr =

2wη = 1. ρT κ

(276)

Note that this result is valid for any nonrelativistic conformal fluid obtained from the DLCQ of a relativistic conformal fluid. 7. CONCLUSION In these lectures, we covered only a small part of the applications of AdS/CFT correspondence to finite-temperature quantum field theory. Here we briefly mention further developments and refer the reader to the original literature for more details. In addition to N = 4 SYM theory, there exists a large number of other theories whose hydrodynamic behavior has been studied using the

AdS/CFT correspondence, including the worldvolume theories on M2- and M5-branes [10], theories on Dp branes [29], and little string theory [94]. In all examples the ratio η/s is equal to 1/(4π), which is not surprising because the general proofs of Sec. 3.6 apply in these cases. We have concentrated on the shear hydrodynamic mode, which has a diffusive pole (ω ∼ −ik 2 ). One can also compute correlators which have a sound-wave pole from the AdS/CFT prescription [9]. One such correlator is between the energy density T 00 at two different points in space-time. The result confirms the existence of such a pole, with both the real part and imaginary part having exactly the values predicted by hydrodynamics (recall that in conformal field theories the bulk viscosity is zero and the sound attenuation rate is determined completely by the shear viscosity). Some of the theories listed above are conformal field theories, but many are not (e.g., the Dpbrane worldvolume theories with p = 3). The fact that η/s = 1/(4π) also in those theories implies that the constancy of this ratio is not a consequence of conformal symmetry. Theories with less than maximal number of supersymmetries have been found to have the universal value of η/s, for example, the N = 2∗ theory [92], theories described by Klebanov-Tseytlin, and MaldacenaNunez backgrounds [36]. A common feature of these theories is that they all have a gravitational dual description. The bulk viscosity has been computed for some of these theories [93,94]. Besides viscosity, one can also compute diffusion constants of conserved charges by using the AdS/CFT correspondence. Above we presented the computation of the R-charge diffusion constant in N = 4 SYM theory; for similar calculations in some other theories see Ref. [10,29]. Recently, the AdS/CFT correspondence was used to compute the energy loss rate of a quark in the fundamental representation moving in a finite-temperature plasma [95–98]. This quantity is of importance to the phenomenon of “jet quenching” in heavy-ion collisions. So far, the only quantity that shows a universal behavior at the quantitative level, across all theories with gravitational duals, is the ratio of

D.T. Son / Nuclear Physics B (Proc. Suppl.) 195 (2009) 217–257

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