The AdS/CFT correspondence - Foundationsand Applications to Hydrodynamics

Pierre Gratia, August 2008

Abstract

This Master thesis deals with the Anti-de Sitter space/conformal field theoryduality conjecture (AdS/CFT). Firstly put forward in 1997 by the Argen-tinean physicist Juan Maldacena, the AdS/CFT duality conjecture has be-come one of the main research topics in theoretical high-energy physics eversince. In this thesis, I will review the main claims of the correspondence,how it can be used to compute correlation functions in strongly coupled fieldtheories, and how it provides us insights in long-wavelength phenomena infield theories. The latter will be treated in detail, i.e. we will see how onecan compute hydrodynamic quantities such as viscosity or entropy densityby making use of the duality. We will see that the duality, however, is notwell defined in the real-time formulation, but an heuristic approach seems to’work’ nonetheless for two-point functions. It is also shown that a universallimit seems to exist for the viscosity/entropy ratio.

Contents

1 Introduction 4

2 String theory and Conformal field theory 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Dimensional analysis . . . . . . . . . . . . . . . . . . . . . . . 7

2.2.1 String length . . . . . . . . . . . . . . . . . . . . . . . 72.2.2 Slope parameter and tension . . . . . . . . . . . . . . . 72.2.3 Gravitational constant and compactified dimensions . . 8

2.3 Bosonic string theory . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Nambu-Goto action . . . . . . . . . . . . . . . . . . . . 102.3.2 Polyakov action . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Symmetries and equation of motion . . . . . . . . . . . 112.3.4 Solution and boundary conditions . . . . . . . . . . . . 12

2.4 From bosonic to fermionic strings . . . . . . . . . . . . . . . . 152.4.1 Open string . . . . . . . . . . . . . . . . . . . . . . . . 162.4.2 closed string . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 p-Branes and Dp-branes . . . . . . . . . . . . . . . . . . . . . 182.6 T-Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.6.1 T-duality for the closed string . . . . . . . . . . . . . . 192.7 D-branes as solutions of supergravity . . . . . . . . . . . . . . 23

2.7.1 Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.7.2 Electric, magnetic charges . . . . . . . . . . . . . . . . 232.7.3 p-branes as solutions to the low-energy effective action 24

2.8 Conformal Field Theory . . . . . . . . . . . . . . . . . . . . . 262.8.1 Global conformal invariance . . . . . . . . . . . . . . . 262.8.2 conformal invariance in two dimensions . . . . . . . . . 30

2.9 Symmetries and Energy-momentum tensor . . . . . . . . . . . 322.10 Supersymmetry . . . . . . . . . . . . . . . . . . . . . . . . . . 34

2.10.1 Superliealgebras, Bose, Fermi elements . . . . . . . . . 342.10.2 Supersymmetry in String theory . . . . . . . . . . . . . 35

1

3 Anti-de Sitter space 373.1 Lorentzian AdS-space . . . . . . . . . . . . . . . . . . . . . . . 37

4 The AdS/CFT conjecture 424.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.3 Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 444.4 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.5 Correlation functions in AdS/CFT . . . . . . . . . . . . . . . 46

4.5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . 464.5.2 Correlators from AdS/CFT . . . . . . . . . . . . . . . 474.5.3 Two-point function . . . . . . . . . . . . . . . . . . . . 484.5.4 Currents in the gauge theory . . . . . . . . . . . . . . . 514.5.5 Massive fields in AdS . . . . . . . . . . . . . . . . . . . 51

4.6 Minkowski prescription for two-point functions . . . . . . . . . 534.6.1 Review of thermal Green’s functions . . . . . . . . . . 544.6.2 Problem with the Minkowski formulation . . . . . . . . 554.6.3 ’Solution’ to the problem . . . . . . . . . . . . . . . . . 57

5 Hydrodynamics and Linear Response theory 585.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 585.2 Review of classical hydrodynamics . . . . . . . . . . . . . . . . 59

5.2.1 General expression for the surface forces. . . . . . . . . 605.3 Relativistic hydrodynamics . . . . . . . . . . . . . . . . . . . . 61

5.3.1 Expression for σµν . . . . . . . . . . . . . . . . . . . . . 635.4 Linear Response Theory . . . . . . . . . . . . . . . . . . . . . 65

5.4.1 Response function . . . . . . . . . . . . . . . . . . . . . 65

6 Finite temperature AdS/CFT 696.1 Hawking temperature . . . . . . . . . . . . . . . . . . . . . . . 696.2 Entropy density . . . . . . . . . . . . . . . . . . . . . . . . . . 726.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.3.1 Viscosity/Entropy ratio . . . . . . . . . . . . . . . . . 776.4 M5-brane setup . . . . . . . . . . . . . . . . . . . . . . . . . . 77

6.4.1 entropy density . . . . . . . . . . . . . . . . . . . . . . 786.5 M2-branes setup . . . . . . . . . . . . . . . . . . . . . . . . . 796.6 CFT on Sn−1 × S1 . . . . . . . . . . . . . . . . . . . . . . . . 806.7 Confinement and AdS/CFT . . . . . . . . . . . . . . . . . . . 826.8 Correlators of the R-charge current and the diffusion constant 846.9 The membrane paradigm . . . . . . . . . . . . . . . . . . . . . 87

2

7 Conclusion 93

3

Chapter 1

Introduction

In 1997, Juan Maldacena came up with an idea that has truly revolution-ized the way physicists think of string theories. He conjectured that a stringtheory defined on an Anti-de Sitter (AdS) space is equivalent (dual) to aconformal field theory living on the AdS boundary. With over 5000 cita-tions, Maldacena’s paper ranks among the top five cited high-energy theorypapers ever. Schools and Workshops are being centered on the AdS/CFTcorrespondence and its applications, and a general lowering in interest is notin sight. On the contrary: AdS/CFT gives us a first glance at what couldbe a non-perturbative formulation of string theory. It states that there isa deep connection between string theories and quantum field theories. Theformer naturally containing gravity and the latter being a quantum field the-ory, one also speaks of the gauge-gravity correspondence. More precisely, astring theory defined on n+1 dimensional Anti de Sitter space (AdS) shouldbe completely equivalent to (i.e., contain the same degrees of freedom than)an n-dimensional conformal field theory living on the boundary of the AdSspace. The thesis is structured as follows. I will start with providing thenecessary background in string theory and conformal field theory in Chap-ter 1. Chapter 2 is dedicated to a detailed description of AdS spacetime,in different coordinate systems. The choice of a coordinate system playsan important role, since some of them only cover a patch of the full AdSspacetime, others contain closed timelike curves violating causality. This hasconsequences for the AdS/CFT formulation. The chapter also includes avery basic introduction to supersymmetry, since it is a supersymmetric for-mulation of gravity which is present in the bulk spacetime. In chapter 3,I will motivate and state the duality conjecture. One usually considers thelow-energy limit in the bulk string theory, which is supergravity. As we willsee, the supergravity approximation is the framework in which the correla-tion functions are computed, so strictly speaking, nothing can be said about

4

the full string theory. In other words, we do not know if AdS/CFT is stilltrue for high energies where the supergravity approximation breaks down,but the duality is believed to hold in all regimes. An important point is topoint out how the parameters on both sides of the conjecture are related.We will see that the string coupling constant equals the square of the Yang-Mills coupling constant in the field theory (up to a constant factor). Also,we will define another important parameter, the ’t Hooft coupling constantλ. It plays an important role in the AdS/CFT formulation. Still in chapter3, a receipe to calculate correlation functions in the AdS/CFT framework isgiven, and an explicit calculation of a two-point function is done. Also, I willmotivate the use of AdS/CFT to extract information of the CFT in the hy-drodynamic (i.e. long-wavelength) limit. Indeed, AdS/CFT is very successfulin deriving hydrodynamic quantities. For this, one needs some backgroundin hydrodynamics, which is summarized in chapter 4. Furthermore, I willpoint out the differences between a Euclidean and Minkowski formulation ofAdS/CFT, and discuss a heuristic approach to compute two-point functionsin the Minkowski prescription, which does not directly follow from the mainAdS/CFT statement, but which still makes use of the conjecture that a grav-ity theory in the bulk can be used to compute correlators in the quantumfield theory on the boundary. Finally we will be able to compute explicitlyentropy density and the viscosity of the so-called quark-gluon plasma, whichis done in chapter 5.

5

Chapter 2

String theory and Conformalfield theory

2.1 Introduction

This chapter is entirely devoted to introduce the necessary concepts involvedin the AdS/CFT correspondence. AdS/CFT relates a string theory to aconformal field theory, so these are the two theories that are going to betreated here. Only the concepts directly related to the duality conjectureare presented. Also, this chapter is insufficient in both content and depthfor a good introduction to string theory and conformal field theory, even forthe subjects treated here, and a knowledge of string theory of the level of []is recommended. However, for a first exposure to AdS/CFT, I believe thisthesis is self-contained enough to grasp the main points, without getting lostin technical details.

String theory is the most promising theory of unification to date. It nat-urally contains all the four interactions observed in Nature, and has providedvaluable insight in many areas in theoretical physics and mathematics. Thebasic claim of string theory is that the most elementary building blocks ofNature are not zero-dimensional point particles, but one-dimensional strings.If one starts with this assumption, one can go on and treat the string rela-tivistically, quantize it, etc. The result is a theory defined in ten spacetimedimensions containing one parameter only: the string length ls, or, equiva-lently, the string tension. Indeed, as we will see, the string length is inverselyproportional to the square of the string tension. Furthermore, string theorydoes not have any dimensionless parameter that needs to be ’adjusted’. Wesay that string theory has no adjustable parameters.

6

2.2 Dimensional analysis

This section introduces the main parameters used in string theory and dis-cusses their relation between each other. We start with the string length.

2.2.1 String length

The string length is the only dimensionful parameter in string theory. It isclosely related to the slope parameter α′, which is defined below. Since stringtheory is supposed to describe the physics at the most elementary, smallestpossible scale, the string scale defined by string length ls is thought to bemuch smaller than the nuclear scale. This is why it is impossible to ’observe’strings.

2.2.2 Slope parameter and tension

As we will see, the action is proportional to the proper worldsheet-area sweptout by the string as it propagates in spacetime. One can now do the followingreasoning. Since an action has units of energy times time, i.e. [ML2

T], and the

worldsheet area has units of area, we must include in the action a quantityof dimension [M

T] = force

velocity, which does not spoil the Lorentz symmetry of

the action.1 The velocity is taken to be the (Lorentz invariant) speed oflight, and the force is going to be the string tension, as we will soon see.We can introduce one more parameter, the slope parameter α′. It is definedas the proportionality constant between the angular momentum of a rigidlyrotating open string in units of ~, and its energy squared:

J

~= α′E2 (2.1)

The units of α′ are thus seen to be [E−2]2. It can be shown that J is in-deed proportional to E2 by evaluating the angular momentum of an rigidlyrotating open string in a plane. The result is

J =E2

2πT0c(2.2)

where T0 is now the tension of the rotating string. Thus we have the followingrelation between the string tension and the slope parameter α′:

α′ =1

2πT0

or T0 =1

2πα′(2.3)

1As for most physicists, Lorentz invariance is considered to be an exact symmetry ofNature, and in fact all of string theory builds on this more than reasonable assumption

2J has units of ~.

7

With this result in hand, we can construct a characteristic length in termsof ~, c and α′, which is indeed the string length ls:

ls = ~c√α′ (2.4)

2.2.3 Gravitational constant and compactified dimen-sions

In Nature, the three fundamental constants are Newton’s constant G, thespeed of light c, and Planck’s constant ~. They are dimensionful, and bycombining them in an appropriate way, it is possible to get quantities withdimension length, mass, and time. Here they are:

Planck length lP =

√G~c3

(2.5)

Planck mass MP =

√c~G

(2.6)

Planck time tP =

√G~c5

(2.7)

Also, it is important to recall that the units of the gravitational constant Gdepend on the number of spacetime dimensions. This can be seen as follows.The gravity field ~g can be expressed in terms of a potential:

~g = −∇V (2.8)

Furthermore, ~g has units of force per unit mass (from ~F = m~g). Conse-

quently, V has units of energy per unit mass: [V ] = [Energy]M

. Finally, Vsatisfies a Poisson equation

∇2V = 4πGρ (2.9)

where ρ is a mass density. The quantity on the left hand side always hasthe same units, independent of the number of dimensions. Since on the righthand side, the units of ρ do depend on d by [ρ] = M

Ld , we conclude that Galso depends on d. We will now find the exact relation for GD when D − 4dimensions are compactified, i.e. we will find an expression for G in D di-mensions, given that D − 4 of them are compactified. We will see that inthat case, G(D) is expressed in terms of the four-dimensional gravitationalconstant, Newton’s constant G(4)3. Firstly, we consider the case D = 5: Con-sider a five-dimensional spacetime where the fifth dimension is curled up into

3with none compactified of course

8

a circle of radius R. We also place a ring of mass M extending in the fifthdimension, going around the circle, at (x1, x2, x3) = (0, 0, 0). This configura-tion makes that V (5), the potential in five dimension, is independent of thecompactified x4 direction and can be seen as an effective four-dimensionalpotential. Poisson’s equation in five dimensions reads

∇2V = 4πG(5)ρ(5) (2.10)

If we can express ρ5 in terms of ρ4, that is to say, if we find the proportionalityconstant C between the two, we can see this equation as an effectively four-dimensional one by identifying CG(5) = G(4), since then all the quantitiesin the equation are effectively four dimensional4. So let’s look at the massdensities. In five dimensions, the mass density is, in our example,

ρ(5) = mδ(x1)δ(x2)δ(x3) (2.11)

where m is the mass per unit length, m = M/2πR. The right hand side ofthis equation can be integrated over the whole space5 to give 2πRm, as itshould, so this is indeed the correct expression for ρ(5). For an observer infour dimensions, however, the ring looks like a point particle of mass M , andthus

ρ(4) = Mδ(x1)δ(x2)δ(x3) (2.12)

Comparing 2.11 and 2.12, we see that ρ(5) = 12πR

ρ(4). Plugging this back into2.10, we have

∇2V = 4πG(5) 1

2πRρ(4) ≡ 4πG(4)ρ(4). (2.13)

where G(4) = G(5)

2πR. We have thus found a relation between the gravitational

constant in five dimensions and the four-dimensional one:

G(5) = 2πRG(4) (2.14)

Note that 2πR is the length of the compactified dimension. Generally, in Ddimensions with D − 4 of them compactified, one has

G(D) = (2πR)D−4G(4) (2.15)

4∇V is the same in any dimension, so the left hand side trivially satisfies the conditionthat the equation is effectively four-dimensional

5including∫ 2πR

0dx4

9

2.3 Bosonic string theory

2.3.1 Nambu-Goto action

The main quantity describing the dynamics of a theory is the action. Byextremizing the action, one gets the equation of motion. The action shouldbe the same for any Lorentz observer. As a trivial example, take a relativisticpoint-particle propagating in spacetime. By doing so, it swaps out a world-line. We want the action to be such that for any given trajectory of theparticle, any Lorentz observer measures the same value of the action. In thisspecific case of a point-particle in flat spacetime, the quantity which is thesame for any Lorentz observer is the proper time τ = ds/c. In order to getthe dimensions right, one multiplies ds/c with mc, another Lorentz invariantquantity (m is the rest mass of the particle). The action thus reads:

S = −mc∫ds (2.16)

In string theory, we replace point-particles with strings. Consequently, thepath traced out by a string is not a world-line but a worldsheet. It turns outthat the action for the relativistic string is proportional to the area of theworldsheet. In other words, the action we are going to find is an exampleof an area functional. Let’s write down the area functional of a surface inorder to be able to compare it to the string action defined afterwards. Anarea can be parameterized by two variables ξ1, ξ2. Strictly speaking, ξ1 andξ2 are variables in a parameter space, and the functions ~x(ξ1, ξ2) describe amapping from parameter space to the target surface embedded in a three- orhigher dimensional space. The area functional then reads

A =

∫dξ1dξ2

√( ∂~x∂ξ1

· ∂~x∂ξ1

)( ∂~x∂ξ2

· ∂~x∂ξ2

)−( ∂~x∂ξ1

· ∂~x∂ξ2

)2

(2.17)

The minimization of this action gives the physical area of an object. Now let’sturn to the string worldsheet. Being an area, we need two parameters, andwe call them τ , σ. The mappings from the parameter space to the physicalspacetime are the string coordinates Xµ(τ, σ), and the functional thus reads

S = −Tc

∫ τf

τi

dτ

∫ σ1

0

dσ

√(∂X∂τ

· ∂X∂τ

)(∂X∂σ

· ∂X∂σ

)−(∂X∂τ

· ∂X∂σ

)2

(2.18)

This is the Nambu-Goto action, and it is reparameterization invariant. It ispossible to express this action in term of the induced metric on the world-sheet. We have

ds2 = ηµνdXµdXν = ηµν

∂Xµ

∂ξα

∂Xν

∂ξβdξαdξβ (2.19)

10

where α and β sum over τ and σ. If we define

hαβ ≡ ηµν∂Xµ

∂ξα

∂Xν

∂ξβ=∂X

∂ξα· ∂X∂ξβ

(2.20)

we can write the action in the form

S = −Tc

∫dτdσ

√−h, where h ≡ det(hαβ) (2.21)

2.3.2 Polyakov action

The square root in the Nambu-Goto action makes it difficult to deal with.There exists an equivalent action, the Polyakov action, which gives thesame equations of motion but which does not contain the square root. ThePolyakov action contains an additional, so-called auxiliary field on the world-sheet, which does not affect the dynamics resulting from the action. Thisauxiliary field is the worldsheet metric γαβ (not to be confused with themetric induced on the worldsheet hαβ).

S = −T2

∫dτdσ

√−γγαβ∂αX

µ∂βXνηµν (2.22)

To check that the two actions are indeed equivalent, take the equation ofmotion for γαβ to eliminate it from the Polyakov action, which gives backthe Nambu-Goto action.

2.3.3 Symmetries and equation of motion

The Polyakov action contains several symmetries: it is Poincare invariant,diffeomorphism invariant, and Weyl invariant in two dimensions. Recall thatPoincare invariance reads

Xµ → ΛµνX

ν + aµ (2.23)

where Λ is such that ΛηΛT = η, and aµ is a translation four-vector. ThePoincare symmetry is a global symmetry. Diffeomorphism invariance meansthat

X ′µ(τ ′, σ′) = Xµ(τ, σ) (2.24)

∂ξ′γ

∂ξα

∂ξ′δ

∂ξβγ′γδ(τ

′, σ′) = γαβ(τ, σ) (2.25)

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Finally, Weyl invariance in two dimensions is expressed as

X ′µ(τ ′, σ′) = Xµ(τ, σ) (2.26)

γ′αβ(τ, σ) = e2ω(τ,σ)γαβ(τ.σ) (2.27)

where ω(τ, σ) can be any function. A Weyl transformation is defined as alocal rescaling of the metric. Local as opposed to global, because the scalingfunction in the exponent depends on the coordinate, i.e. the transformationdepends on the coordinates. Note that in our case, it is the worldsheetmetric which is Weyl invariant, which was the auxiliary field that had beenintroduced. Hence, the Nambu-Goto action, lacking this auxiliary field, doesnot possess such a symmetry. We can use the diffeomorphism invarianceof the Polyakov action to simplify it. Generally, in two dimensions, anycoordinate reparameterization allows an arbitrary metric hαβ on the surfaceto have the form

hαβ = ρ2(ξ)ηαβ (2.28)

hαβ is called a conformally flat metric. On the worldsheet, we can thus goto the conformal gauge, which consists of the coordinates that result in aconformally flat metric. In the conformal gauge, γαβ can take the form

γαβ = e2ω(τ,σ)ηαβ (2.29)

This can be plugged into the action, and noting that√−γγαβ = ηαβ, we find

S =T

2

∫dτ

∫ L

0

dσ(∂τX · ∂τX − ∂σX · ∂σX) (2.30)

The equation of motion is thus

Xµ ≡ (−∂2τ + ∂2

σ)Xµ = 0 (2.31)

with the condition that the boundary term in the action vanishes: [Xµ∂σXµ]L0 =

0. This is the equation of motion for the string coordinates Xµ.

2.3.4 Solution and boundary conditions

We can distinguish two cases:

• If the string is closed, there are no boundary conditions, and thus theboundary term must vanish. This translates as

Xµ(τ, σ + L) = Xµ(τ, σ) (2.32)

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The most general solution to the equation of motion for the closedstring is

Xµ(τ, σ) = qµ+1

TLpµτ+

i√4πT

∑n6=0

1

n

[αµ

ne−2πin(τ+σ)/L+αµ

ne−2πin(τ−σ)/L

](2.33)

The first two terms describe the center of mass motion, and the lastterm describes the oscillations of the string. If they were to vanish, theequation describes the motion of a point particle.

• If the string is open, the vanishing of the boundary term introducesadditional constraints. Indeed we must have that

∂σXµδXµ|σ=0,σ=L = 0 (2.34)

These two conditions can be satisfied in two different ways:

∂σXµ|σ=0,σ=L = 0 Neumann boundary condition (2.35)

δXµ|σ=0,σ=L = 0 Dirichlet boundary condition (2.36)

Dirichlet boundary conditions only apply to space coordinates. Indeed,we always have δX0 6= 0, because the string endpoints have fixed σ = 0or L, and the only parameter τ has to vary as time flows. Thus wehave to exclude the value µ = 0. The most general solutions for theopen string equations of motion satisfying a given set of open stringboundary conditions are:

Xµ(τ, σ) = q + 2α′µpµτ + i√

2α′∑

n6=0,n∈Z

1

nαµ

n cos(nσ)e−inτ (2.37)

when both endpoints have Neumann boundary conditions (NN), ∂σXµ|σ=0,π =

0.

Xµ(τ, σ) = qµi +

1

π(qf − qi)

µσ +√

2α′∑

n6=0,n∈Z

1

nαµ

n sin(nσ)e−inτ (2.38)

when both endpoints have Dirichlet boundary conditions (DD),Xµ|σ=0 =qµi , Xµ|σ=π = qµ

f , µ 6= 0. Finally,

Xµ(τ, σ) = qµf + i

√2α′

∑r∈Z+1/2

(2.39)

when one endpoint satisfies Neumann boundary conditions and theother Dirichlet boundary conditions (ND), ∂σX|σ=0 = 0, X|σ=π = qf .

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As one can suspect now, the coordinates qi and qf correspond to theposition of the D-branes on which the endpoints lie. Note that only NNboundary conditions allow the open string’s center of mass to move; itis fixed in space for the ND and DD boundary conditions.

Neumann boundary condition on an open string’s endpoint coordinateXµ(τ, σ =0, L) restrict the string’s endpoint to be stationary with respect to σ, whereasa Dirichlet boundary condition on an open string’s endpoint coordinate re-strict that coordinate to be stationary with respect to time. It turns out thatthe total momentum of an open string is conserved for Neumann boundaryconditions, but not for Dirichlet boundary conditions. Since a string endpointwith Dirichlet boundary conditions is fixed in one or more space directions(depending on the number of Dirichlet boundary conditions), translation in-variance is broken in these directions. Since translation invariance impliesmomentum conservation by Noether’s theorem, we should not expect themomentum to be conserved in the given directions. The apparent violationof momentum conservation is resolved by letting the open string endpointsend on a Dirichlet brane, or shortly D-brane, which is an extended object inspacetime. The momentum then ’flows’ from the open string to the brane,and the total momentum of the string-brane system is conserved. More aboutbranes will be said later.

So far we have only considered the relativistic, classical (as opposed toquantum) string. We need to quantize the relativistic string, and this can bedone by usual canonical quantization. The procedure to follow is not beingdeveloped here, but it can be read off from any string theory textbook. Herewe just state the commutation relations for the operators:

[qi, pj] = iδij (2.40)

[αin, α

jm] = nδn+m,0δ

ij (2.41)

[αin, α

jm] = nδn+m,0δ

ij (2.42)

These commutation relations apply for both the open and the closed string.Furthermore, it is possible to show that Lorentz invariance requires the quan-tized strings (both open and closed) to live in 26 spacetime dimensions. Thefact that the theory is consistent in a given number of dimensions only isquite remarkable; ordinary quantum field theories, for example, do not ex-hibit this constraint and can be defined in different spacetime dimensions.The closed string spectrum consists of three massless states plus a tower ofmassive excitations. The massless states are the graviton Gij, an antisym-metric tensor particle Bij, and the dilaton Φ. It turns out that the dilaton Φis related to the closed string coupling constant gs via gs = eΦ0 , where Φ0 is

14

the vacuum expectation value of the dilaton. The open string spectrum con-tains a massless vector particle as its first excitation, the ground state beinga tachyonic (i.e. negative mass squared) particle that needs to be removedfrom the theory.

Finally, the Polyakov action can be generalized in such a way that all themassless string modes are explicitly written down:

S = − 1

4πα′

∫worldsheet

dτdσ√−γ[γαβ∂αX

µ∂βXνGµν(X)+εαβ∂αX

µ∂βXνBµν(X)+α′R(γ)Φ(X)

](2.43)

We recognize the graviton, the antisymmetric tensor particle Bµν , and thedilaton. The massless gauge bosons from the open string first excited stateare not written down. They live on the boundary of the worldsheet. Ofcourse, the graviton Gµν corresponds to the spacetime metric in which thestring propagates. R(γ) is the Ricci scalar for the worldsheet metric γµν

2.4 From bosonic to fermionic strings

So far, only bosonic string theory has been introduced. Since our worldcontains fermions, we should be able to formulate an action incorporatingfermionic excitations. The action we need is

S = −1

2

∫dτdσ

[T∂αX

i∂αX i − i

πΨ

iρα∂αΨi

](2.44)

This action contains supersymmetric particles. It can be derived from a gen-eralized Polyakov action, and the two dimensional field theory on the world-sheet now contains bosons and fermions. All of them couple to supergravity,as opposed to the gravity coupling of the particles of the bosonic action. Theψi are the ’leftovers’ of the supersymmetry partner of the Xµ, the Ψµ, aftermaking use of symmetries and gauge choice. This action is formulated in thelight-cone gauge. To be able to use this gauge, one firstly has to switch tolight-cone coordinates:

X+ =1√2(X0 +X1) (2.45)

X− =1√2(X0 −X1) (2.46)

It is then possible to show that X+(τ, σ) = βα′p+τ . The light-cone gaugegives X+ a quite simple form, and this allows for an explicit solution to theequations of motion. Let’s come back to the action. Both the Ψi and the X i

15

form a vector of SO(D-2), and the ρα are two-dimensional Dirac matrices,i.e. they satisfy ρα, ρβ = −2ηαβ. The equations of motion are:

X i = 0 (2.47)

ρα∂αΨi = 0 (2.48)

The first equation is just the previous wave equation for the bosonic string,and the second equation translates as

ρα∂αΨi = 0 ⇔

∂−ψ

i = 0

∂+ψi= 0

(2.49)

where we chose the following basis: ρ0 =(

0 −ii 0

), ρ1 =

(0 ii 0

).

2.4.1 Open string

Again, there are boundary conditions to be satisfied (from the surface termin the action), and for the open string they read

ΨTρ0ρ1δΨ|σ=0,π = (ψδψ − ψδψ)|σ=0,π = 0 (2.50)

where ΨT = (ψ, ψ). There are different choices of sign that solve theseconditions. Usually one chooses the same sign at σ = 0 and considers twodifferent choices at σ = π:

ψi(τ, 0) = ψi(τ, 0) (2.51)

ψi(τ, π) = ±ψi(τ, π) (2.52)

The plus sign corresponds to the Neveu-Schwarz sector (NS) of the theory,whereas the minus sign corresponds to the Ramond sector (RR) of the theory.Now one can write down the solutions to the equations of motion:

ψi =∑

r∈Z(∈Z+1/2)

ψire−ir(τ+σ) (2.53)

ψi=

∑r∈Z(∈Z+1/2)

ψire−ir(τ−σ) (2.54)

for the R (NS) sector, respectively. As usual for fermions, we introduceanticommutation relations when we quantize the theory:

ψir, ψ

js = δr+s,0δ

ij (2.55)

16

Just like in the bosonic case the critical dimension was found to be d = 26, onefinds that if fermions are incorporated into the theory, the critical dimensionhas to be d = 10. This is again found by looking at the first excited statein the NS-sector, which has to be a SO(D-2) vector (i.e. a massless state)for Lorentz invariance to be preserved. Since the first excited state hasm2 ∼ 10−D, we must have D = 10. The line of reasoning is the same thanin the bosonic case.

2.4.2 closed string

The ’boundary conditions’ for the closed string are the following periodicityconditions, leaving the action invariant:

ψi(τ, σ + π) = ±ψi(τ, σ) (2.56)

ψi(τ, σ + π) = ±ψi(τ, σ) (2.57)

(2.58)

Hence we have four possible sectors for the closed string:

• (R,R) both signs plus

• (NS,NS) both signs minus

• (R.NS) first plus, second minus

• (NS,R) first minus, second plus

The (R,R) and (NS,NS) sectors contain spacetime bosons, whereas the (R,NS)and (NS,R) sectors contain spacetime fermions. To end up, we just state thatfor reasons of consistency one must have invariance of the partition functionunder a specific kind of diffeomorphisms, also called modular invariance ofthe partition function. To achieve modular invariance of the fermionic string,one has to project out the tachyonic states which violate causality. This isdone by the Gliozzi-Scherk-Olive projection, or GSO-projection. The closedfermionic string permits two possible GSO-projections. The spectrum in the(NS,NS) sector is the same for both of them: it contains a graviton Gij, anantisymmetric tensor particle Bij, and the dilaton φ. The (R,R) sector isdifferent, inasmuch as one gets achiral states for one projection and chiralstates for the other. One refers to type IIA and type IIB superstring the-ory, respectively. Recall from above that the (NS,NS) and the (R,R) sectorscontain spacetime bosons only. Spacetime fermions appear in the (NS,R)

17

and (R,NS) sectors. In this thesis, we are mainly interested in the type IIBtheory, so for the record, here are the massless bosonic excitations of it:

(NS,NS) sector : Gµν , Bµν ,Φ (2.59)

(R,R) sector : scalar χ, 2-form B′µν , 4-form Aµνρσ with self-dual field strength F = dA

(2.60)

We now continue to describe the extended objects on which open strings havetheir endpoints, D-branes.

2.5 p-Branes and Dp-branes

A brane is an extended object in spacetime that can have any dimensionsmaller than or equal to the dimension of space itself. To specify the dimen-sion, one adds a number p which stands for the number of spatial dimensionsof the brane; for example, a 0-Brane is a particle, a 1-brane a string. Gen-erally, p-branes are solutions of supergravity, the low-energy limit of stringtheory (ls → 0). p-branes can carry charge, but the value of the charge isbounded by the mass of the branes, |Q| 6 M . If Q = M , we say that thep-brane is extremal. However, not every p-brane is a Dp-brane. Dp-branesare objects on which open strings have their endpoints. The ’D’ in Dp-branestands for Dirichlet and specifies a particular boundary condition, the Dirich-let boundary condition for open strings, which states that the open stringendpoints are fixed throughout the motion. As an example, take a flat D2-brane in three-dimensional space (x1, x2, x3), at x3 = 0, say. Open stringsattached to that brane are free to move in the x1 and x2 direction, but theirx3 coordinate stays fixed at x3 = 0. A Dp-brane has a certain energy densityand thus curves spacetime around it. It has been proven by Joseph Polchinskiin 1995 that Dp-branes and extremal p-branes are the same objects; conse-quently, open strings end on objects that are solutions of supergravity. Aswe shall see, the theory of open strings living on a D3-brane is N = 4 Su-per Yang-Mills, which has a well-known gravity dual. Thus, D3-branes areof major importance in this document. The D3-branes have a tension thatscales as

TD3 ∼1

gsl4s(2.61)

This relation follows partly from dimensional analysis: The tension of thethree-dimensional D3-brane has units of energy (or mass) per unit volume.In natural units, mass has units of inverse length, and thus the presence of the1/l4s factor. The dependence on the inverse string coupling constant follows

18

from T-duality, which is developed below. It follows that the gravitationalradius must scale as gsNc. The exact relation is

R4

l4s= 4πgsNc (2.62)

2.6 T-Duality

One might wonder about why introducing at all branes. One could sim-ply impose Neumann boundary conditions for all of the open string coordi-nates (recall a Dirichlet boundary condition in direction xi forces the stringendpoint to be fixed at in the xi direction). There are two main reasonswhy one cannot disregard the concept of branes. The first argument comesfrom T-duality, a symmetry transforming Neumann boundary conditions intoDirichlet boundary conditions. Clearly, one cannot then ’get rid’ manuallyby avoiding Dirichlet boundary conditions to begin with, because there au-tomatically exists a same theory involving Dirichlet boundary conditions.More often than not, T-duality is the first of a large web of dualities in stringtheory one encounters when studying it. The reason is that T-duality can beunderstood with a very elementary background in string theory (basically,all one needs to know is the solutions to the equations of motion and theoperator formalism when quantizing the theory in the way we mentionedabove). So it is very useful to have a closer look at T-duality, and how thisduality begs for the introduction of D-branes.

We consider bosonic string theory in 26 dimensions; T-duality also appliesfor superstring theory involving fermions, but the restriction to the bosoniccase is more straightforward.

2.6.1 T-duality for the closed string

If one compactifies one of the space dimensions, the X25 say, into a circle ofradius R, one gets:

X25(σ + π, τ) = X25(σ, τ) + 2πRW, W ∈ Z (2.63)

for a closed bosonic string. W is called the winding number and countsthe number of times the string winds around the compactified dimension.Clearly, if the string is open, there is no such thing as a winding number, butfor now let’s move on with the closed string, as the results will simplify thediscussion for the open string. The compactification of the 25th dimensionsimplies that the component of the momentum in that direction is quantized.

19

Indeed, the quantum mechanical wave function is proportional to eip25x25,

and it should return to its initial value after one turn around of the circle:

eip25(x25+2πR) = eip25x25 ⇒ p25 =K

R, whereK ∈ Z (2.64)

K is called the Kaluza-Klein excitation number. Let’s introduce anotherparameter that will be useful later, the windings w:

w ≡ RW

α′(2.65)

w naturally enters in the expressions of the zero modes, on the same footagethan the momentum p (w has indeed dimensions of momentum in naturalunits), and as a consequence, it also enters the expression for the so-called

Virasoro generators L⊥0 and L⊥0 , which have an expansion in terms of the

modes. This in turn implies that the difference between the two numberoperators is not zero, as it would without the compactification, but equals

N⊥ −N⊥

= α′pw (2.66)

The right hand side can be modified by making use of the expression forquantized momentum and equation ():

N⊥ −N⊥

= α′K

R

RK ′

α′= KK ′ (2.67)

where K,K ′ ∈ Z. Now some comments on the mode expansion. For boththe closed and open string, one can divide the mode expansion (that is, thesolution to the equation of motion) into two terms: a left-moving wave anda right moving wave:

Xµ(τ.σ) = XµL(τ + σ) +Xµ

R(τ − σ) (2.68)

For the closed string, these terms have to satisfy periodicity conditions, andfor the open string, they satisfy boundary conditions. The consequence ofadding the periodicity of one space direction into the mode expansion of theclosed string is that the mass squared operator becomes

M2 =K

R2+K ′R2

α′+

2

α′(N⊥ +N

⊥ − 2) (2.69)

Taking into account (64) and (65), an observation can be made: equation(65) remains unchanged if we replace the radius of compactification R byR ≡ α′

R. This is what we call T-duality of the closed string. There is another

20

way to express T-duality for the closed string. For this, we write down theexplicit form of the left- and right movers,

XL(τ + σ) =1

2(x0 + q0) +

α′

2(p+ w)(τ + σ) + i

√α′

2

∑n6=0

αn

ne−in(τ+σ)

(2.70)

XR(τ − σ) =1

2(x0 − q0) +

α′

2(p− w)(τ − σ) + i

√α′

2

∑n6=0

αn

ne−in(τ−σ)

(2.71)

The full mode expansion is the sum of the two:

X(τ, σ) = x0 + α′pτ + α′wσ + i

√α′

2

∑n6=0

e−inτ

n(αne

−inσ + αneinσ) (2.72)

q0 is a coordinate added by hand and which vanishes when summing the twoterms. However, if we now look what happens when we define a ’dual modeexpansion’ X(τ, σ),

X(τ, σ) ≡ XL(τ, σ)−XR(τ, σ) (2.73)

we see that it is x0 that vanishes instead of q0:

X(τ, σ) = q0 + α′wτ + α′pσ + i

√α′

2

∑n6=0

e−inτ

n(αne

−inσ + αneinσ) (2.74)

Comparing X with X, we see that the roles of p and w have been inter-changed, and q0 replaces x0 in X. ¿From this we conclude that for q0, theassociated momentum is w and the associated winding number is p. It canbe shown that the Hamiltonian is the same for both choices of the modeexpansion. Furthermore, recall that the mode expansion X(τ, σ) is the modeexpansion for the compactified dimension, thus x0 lives on the circle of radiusR, and its conjugate momentum p is quantified, p = K

R, K ∈ Z. Since we

have w = RK′

α′, K ′ ∈ Z, we conclude from the form of X that q0 lives on the

circle of radius R ≡ α′

R. Thus for closed string theory, a compactification of

radius R or R = α′

Ris indistinguishable. The two theories are effectively the

same.

T-duality for the open string

We can now move on to the open string case. As already said above, foropen strings, there is no winding number. But let’s go on and write down

21

the expressions for the left- and right movers of the open string satisfyingNeumann boundary conditions in all directions, in particular the one we areinterested in, the compactified one:

XL(τ + σ) =1

2(x0 + q0) +

1

2p(τ + σ) + i

√α′

2

∑n6=0

1

nαne

−in(τ+σ) (2.75)

XR(τ − σ) =1

2(x0 − q0) +

1

2p(τ − σ) + i

√α′

2

∑n6=0

1

nαne

−in(τ−σ) (2.76)

Again we introduce X ≡ XL − XR, and we can compare X with X in thecompactified direction:

X(τ, σ) = x0 + pτ + i√

2α′∑n6=0

1

nαne

−inτ cos(nσ) (2.77)

X(τ, σ) = q0 + pσ +√

2α′∑n6=0

1

nαne

−inτ sin(nσ) (2.78)

Again, note the interchange x0 ↔ q0. Furthermore, X does not possess mo-mentum in the compactified dimension (the term linear in τ has vanishedand been replaced by a term linear in σ). Now take σ = 0, π: we see thatfor these values of σ, the position of the open string is fixed in the compact-ified direction, the string now satisfies a Dirichlet boundary condition! Inparticular, we have:

X(τ, 0) = q0 (2.79)

X(τ, π) = q0 +πK

R= q0 +

RπK

α′= q0 + 2π

K

2α′R (2.80)

For open strings, T-duality transforms Neumann boundary conditions intoDirichlet boundary conditions in the compactified directions. The open stringhas momentum but no winding number at the beginning, and after a T-duality transformation it has winding number but no momentum in the com-pactified directions. The interpretation is that the dual open string, ends ona fixed hypersurface, a Dp-brane. Thus, after one compactification, the dualopen string ends on a D24 brane. The open string’s endpoints are free tomove in the 24 directions on the brane (corresponding to Neumann boundaryconditions), but they are fixed in the direction that has been compactified.

What we can conclude from the discussion of T-duality in this section isthat D-branes naturally appear in theories dual (i.e.equivalent) to theorieswith no branes. This is one of the main arguments why one wishes to studythem. We now turn to the second argument, which involves the low-energyeffective action of the IIB superstring theories.

22

2.7 D-branes as solutions of supergravity

We switch to a different viewpoint now. The massless fields of the IIB theoryoutlined above are still present in the low-energy limit which we will considernow. Their dynamics is encoded in the low-energy effective action. If at thesame time, we take the limit of the string coupling constant to be small,gs → 0, and the string length to zero, i.e. α → 0, one gets IIA and IIBsupergravity theories. We will see that branes are solutions to the low-energy effective action in the above limits. In the low-energy effective action,a variety of massless antisymmetric tensor fields are present; therefore werecall some of their basic properties here.

2.7.1 Forms

Recall the action of an exterior derivative on an n-form:

dA =∂Aµ1µ2...µn

∂xµαdxµα ∧ dxµ1 ∧ ... ∧ dxµn (2.81)

If one defines F = dA, then dF = 0, since d(dA) = 0 for any form. The Hodgedual of a p-form is an (d−p)-form, where d is the number of dimensions. Forexample, if ω = Adx+Bdy+Cdz, then ∗ω = Ady∧dz+Bdx∧dz+Cdx∧dy,in three dimensions. In index notation, we have for a k-form Mµ1µ2...µk

:

(∗M)µk+1µk+2...µd =1

2√−g

εµk+1µk+2...µdµ1µ2...µkMµ1µ2...µk(2.82)

where ε... is the totally antisymmetric Levi-Civita tensor in d dimensions.

2.7.2 Electric, magnetic charges

The coupling of an electrically charged particle in four dimensions is

qe

∫CA = qe

∫Cdτdxµ

dτ(2.83)

where C is the world-line of the particle. The electric charge can be expressedin terms of ∗F , if we define F = dA the field strength:

qe =

∫S2

∗F (2.84)

Equation 2.83 can be generalized to higher dimensions. Generally, a p-dimensional charged object coupled to a (p+ 1)-form gauge field as

qe

∫Cp+1

Ap+1 (2.85)

23

where qe is now p-dimensional, and Cp+1 is the world-volume of the chargedobject. These objects are called p-branes if they couple to (p+1)-dimensionalgauge fields that arise in the (R,R) sector of the theory. We can write downthe electric charge of a p-brane:

qe =

∫SD−p−2

(∗F )D−p−2 (2.86)

A p-brane6 looks like a point in the transverse (D−p−1)-dimensional space,and it can be surrounded by a (D − p − 2)-sphere. The trivial example isa point-particle (p = 0) surrounded by a two-sphere in three-dimensional(D = 4) space. One can also define the corresponding dual magnetic chargeof a dual brane, given by

qm =

∫SD−(D−p−2)

F (D−(D−p−2)) =

∫Sp+2

F (p+2) (2.87)

What kind of a brane does a p + 2-sphere surround? The brane must lookpointlike in p + 2 + 1 = p + 3 directions, thus it is a (D − (p + 3) − 1) =(D − p − 4)-brane. This shows that an electrically charged p-brane is dualto a magnetically charged (D − p− 4)-brane. 7

2.7.3 p-branes as solutions to the low-energy effectiveaction

What one can say from the above analysis is that given a p-brane solutionto the equations of motion originating from the low-energy effective action,there must be, in the (R,R) sector, a gauge field A(p+1) to which the p-branecouples8. Also, the p-brane solution must contain the spacetime metric whichcouples to the energy-momentum tensor of the potential A(p+1). The low-energy effective action that gives this p-brane solution is

S =1

2κ2

∫d10x

√−g(R− 1

2gµν∂µφ∂νφ−

1

2(p+ 2)!e

3−p2 F 2

(p+2)

)(2.88)

where F 2(p+2) = Fµ0µ1...µp+1F

µ0µ1...µp+1 and κ2 = 12(2π)7α′4g2

s . This action im-plies the supergravity equations of motion. We are interested in a particular

6p denotes the number of space dimensions, whereas D stands for the number of di-mensions of the whole spacetime in which the p-brane is embedded.

7Note that for D = 10 and p = 3, the 3-brane is self-dual; the same goes for D = 4,p = 0, where the 0-brane, or point-particle is self-dual. The latter means that in fourspacetime dimensions, one can have particles that are both electrically and magneticallycharged, the dyons.

8recall that by definition a p-brane couples to a (p + 1)-form from the (R,R) sector

24

solution, the brane solution. Therefore one can make an ansatz what formthe solutions ought to take, and plug them back into the equations of motion.For example, the ansatz for the dilaton is

eφ = C(r) (2.89)

The dependence on r only stems from the fact that we want the sphericalsymmetry in the space transverse to the brane. By plugging this back into

the equation of motion ∇2φ = (3−p)4(p+2)!

e(3−p)

2φF 2

(p+2), one finds

C(r) = f(r)3−p4 , where f(r) = 1 +

kpN

r7−p(2.90)

for (9− p)-dimensional transverse space (we assume D = 10), and

kp =2κ2Tp

(7− p)Ω(8−p)

, Ω8−p = V ol(S(8−p)) (2.91)

This looks quite complicated, but in the case p = 3, one sees that the dila-ton is constant. It was Polchinski who showed in 1997 that these solutionsdescribe Dp-branes. Details of how he made the identification can be foundin his paper. It is possible to generalize the solution for one brane to a stackof N parallel branes. In that case, the function f(r) becomes:

f(r) = 1 +N∑

i=1

kp

|~y − ~yi|7−p(2.92)

We focus on the case p = 3, and we consider N coincident p-branes, in whichcase

f(r) = 1 +kpN

r4= 1 +

R4

r4, R4 ≡ 4πgsα

′2N (2.93)

The metric is

ds2 = f(r)−1/2(−dt2 + d~x2) + f(r)1/2(dr2 + r2dΩ25) (2.94)

By looking at this metric, one easily sees that for r → ∞, it reduces to theten-dimensional Minkowski metric. It is less obvious that for r → 0, this isthe metric of AdS5 × S59.

9r → 0 ⇒ f(r) → R4

r4 , and a change of variables ρ ≡ R2

r makes the AdS5 × S5 metricmanifest.

25

2.8 Conformal Field Theory

This section constitutes an elementary introduction to conformal field the-ory. In order to understand the AdS/CFT correspondence, one needs tounderstand what it means for a field theory to be conformal and what theconsequences of conformal invariance are.

2.8.1 Global conformal invariance

A conformal transformation is an invertible mapping of the spacetime coor-dinates which leaves the metric invariant up to a scale:

x −→ x′ g′µν(x′) = Λ(x)gµν(x) (2.95)

¿From this requirement it is possible to derive all the allowed coordinatetransformation in a straightforward way. This thesis is supposed to be asself-consistent as possible, therefore we are going to derive the main trans-formations explicitly. Consider an arbitrary infinitesimal coordinate trans-formation

xµ → x′µ = xµ + εµ(x) (2.96)

Under this transformation, the metric tensor changes as

gµν → gµν − (∂µεν + ∂νεµ) (2.97)

We want our transformation to be conformal. From the definition, we requirethat

∂µεν + ∂νεµ = f(x)gµν (2.98)

Our aim is now to find what form the function f is allowed to take. Takinganother partial derivative on this equation gives10

∂ρ∂µεν + ∂ρ∂νεµ = ∂ρf(x)gµν (2.99)

Interchanging the indices ν ↔ ρ and µ↔ ρ,

∂ν∂µερ + ∂ν∂ρεµ = ∂νf(x)ηµρ

∂µ∂ρεν + ∂µ∂νερ = ∂µf(x)ηρν

Adding the two previous equations gives

2∂µ∂νερ = ηµρ∂νf + ηρν∂µf − ∂ν∂ρεµ − ∂µ∂ρεν

10From now on, we switch to ηµν .

26

and from 2.99, the last two terms on the right hand side are −ηµν∂ρf . Thuswe arrive at

2∂µ∂νερ = ηµρ∂νf + ηρν∂µf − ηµν∂ρf (2.100)

Now contract with ηµν , which gives

2∂2ερ = δνρ∂νf + δµ

ρ∂µf − d∂ρf

= (2− d)∂ρf

Thus,(2− d)∂ρf = 2∂2ερ (2.101)

Now apply ∂ν on 2.98 and 2.101:

(2− d)∂ν∂µf = 2∂ν∂2εµ (2.102)

∂ν∂µεν + ∂2εµ = ∂νfηµν (2.103)

Plugging 2.103 into 2.102 gives

(2− d)∂ν∂µf = 2∂ν [∂νfηµν − ∂ν∂µεν ] (2.104)

= 2∂2fηµν − 2∂2∂µεν (2.105)

The last term is −(2− d)∂µ∂νf (from 2.101), and we get

(2− d)∂µ∂νf = ∂2fηµν (2.106)

Contracting with ηµν gives

(2− d)∂2f = d∂2f ⇔ (d− 1)∂2f = 0 (2.107)

¿From eqs. (E3) and (E4), we can find the form of f and thus of the conformaltransformation. Let us firstly consider the case d > 3. Equation (E3) gives∂2f > 0 and thus, from (E3):

∂µ∂νf = 0 (2.108)

We see that f is a linear function of the spacetime coordinates:

f(x) = A+Bµxµ (2.109)

¿From equation(), we see that the linearity of f implies that ∂µ∂νερ is con-stant. Hence,

εµ = aµ + bµνxν + cµνρx

νxρ (2.110)

with cµνρ = cµρν . εµ(x) is at most quadratic in the coordinates. The con-straints on f hold for all x, so we can treat each power of the coordinate

27

separately. The constant term aµ is seen to be free of constraints. It standsfor an infinitesimal translation. Let’s turn to the linear term. Substitute itinto (C1):

∂µ(bνρxρ) + ∂ν(bµρx

ρ) = f(x)ηµν =2

d∂ρ(b

ρλxλ)ηµν

⇔ bνρδρµ + bµρδ

ρν =

2

dbρλδρληµν

⇔ bνµ + bµν =2

dbλληµν

implying that bµν = αηµν +mµν , where mµν is antisymmetric: mµν = −mνµ.The trace part αηµν represents an infinitesimal scale transformation, andthe antisymmetric part is an infinitesimal rotation. We come now to thequadratic term. Thus, quadratic term in f implies an (infinitesimal) trans-formation of the form

x′µ = xµ + 2(x · b)xµ − bµx2 (2.111)

This is the special conformal transformation (SCT). To summarize, the in-finitesimal conformal transformations are:

translation: aµ (2.112)

scale transformation: αηµν (2.113)

rigid rotation: mµν (2.114)

SCT: ηµρbν + ηµνbρ − ηνρbµ (2.115)

The finite transformations are deduced from the infinitesimal ones:

translations: x′µ = xµ + aµ (2.116)

dilatations: x′µ = αηµνxν = α (2.117)

xµrotations: x′µ = Mµν x

ν (2.118)

SCT: x′µ =xµ−bµx2

1− 2bx+ b2x2(2.119)

We are now in the position to derive the generators of these transformations.Before, let us recall the definition of a generator. If we have an action infield theory, S =

∫L(φ(x), ∂µφ(x))dmx, then its infinitesimal transformation

involves infinitesimal transformations on both the coordinates and the fields:

x′µ = xµ + ωaδxµ

δωa

φ′(x) = φ(x) + ωaδF

δωa

(x)

28

where φ′(x′) = F (φ(x)) is a function of φ(x). We work at first order in ωa.The generator Ga of a symmetry transformation is defined as follows:

δωφ(x) = φ′(x)− φ(x) = −iωaGaφ(x) (2.120)

But to first order in ωa, we also have:

φ′(x′) = φ(x) + ωaδF

δωa

(x) = φ(x′)− ωaδxµ

δωa

∂µφ(x′) + ωaδF

δωa

(x′) (2.121)

Comparing with (3), we see that

iGaφ(x) =δxµ

δωa

∂µφ(x)− δF

δωa

(x) (2.122)

As a trivial example, consider an infinitesimal translation ωµ. Then δxµ

δων = δµν ,

and δFδων = 0. Thus iGνφ = ∂νφ, and the translation generator is Pν = Gν =

−i∂ν . Similarly, one finds the other generators of conformal transformations.They are:

translation: Pµ = −i∂µ (2.123)

dilatations: D = −ixµ∂µ (2.124)

rotations: Lµν = i(xµ∂ν − xν∂µ) (2.125)

SCT: Kµ = −i(2xµxν∂ν − x2∂µ) (2.126)

The commutations relations are zero except the following:It is worth noting that all these transformations have been deduced

from the simple initial condition of scale invariance of the metric: g′µν(x) =Λ(x)gµν(x). All what followed was based on this assumption only, and yetwe found that the allowed transformations are constrained to four differenttypes only. Let’s count the number of components for the parameters ofa conformal transformation in d dimensions: d for translations, 1 for scaletransformations, 1

2d(d − 1) for rotations (antisymmetric matrix), and d for

the SCTs, which gives a total of (d+1)(d+2)2

components. There is a way toregroup the conformal generators into an antisymmetric (d + 2) × (d + 2)matrix, where

Jµ,d+1 =Kµ − Pµ

2Jµ,d+2 =

Kµ + Pµ

2Jd+1,d+2 = DJµν = Lµν (2.127)

The algebra defined by the original generators Kµ, PµD,Lµν is isomorphic tothe algebra of SO(2,d) and can be put in the standard form of the SO(2,d)algebra. In other words, conformal invariance in flat (1,d-1) dimensions (d¿2)corresponds to the symmetry group SO(2,d). The generators of SO(2,d) areprecisely the elements of the matrix JMN . But SO(2,d) is also the symmetrygroup of d+1 dimensional AdS space! This is a first hint that d-dimensionalconformal field theory might be related to a gravity theory on AdSd+1.

29

2.8.2 conformal invariance in two dimensions

Let us examine the conformal invariance in two-dimensional field theory morein detail. Consider the complex plane with (z0, z1) coordinates. For anarbitrary coordinate transformation in two dimensions, the metric transformsas

zµ → wµ(x)gµν → (δwµ

δzα)(∂wν

∂zβ)gαβ (2.128)

This transformation is conformal provided the following conditions are sat-isfied:

(∂w0

∂z0)2 + (

∂w0

∂z1)2 = (

∂w1

∂z0)2 + (

∂w1

∂z1)2 (2.129)

(∂w0

∂z0)(∂w1

∂z0) + (

∂w0

∂z1)(∂w1

∂z1) = 0 (2.130)

There are two solutions:

∂w1

∂z0=∂w0

∂z1

∂w0

∂z0= −∂w

1

∂z1(2.131)

or∂w1

∂z0= −∂w

0

∂z1

∂w0

∂z0=∂w1

∂z1(2.132)

These sets of equations are known as the Cauchy-Riemann equations forholomorphic and anti-holomorphic functions, respectively. We can switch tocomplex coordinates in the usual way:

z = z0 + iz1

z = z0 − iz1

∂z =1

2(∂0 − i∂1)

∂z =1

2(∂0 + i∂1)

The Cauchy-Riemann equations become:

∂zw(z, z) = 0 (2.133)

The solution to this equation is any holomorphic mapping w(z) that has noz dependence. If we want the conformal transformation to be global, it mustbe both defined on the whole plane and be invertible. It is not difficult toshow that this reduces the set of possible coordinate transformations to thefollowing mappings:

z → f(z) =az + b

cz + d(2.134)

30

with ad−bc = 1. These transformations clearly belong to the group SL(2,C),the group of invertible complex 2 × 2 matrices with determinant 1. If wedo not restrict ourselves to global conformal transformations, we can havelocal transformations which are holomorphic but not necessarily invertible.Infinitesimally, we can express a holomorphic transformation in terms of aLaurent expansion:

z′ = z + ε(z), where ε(z) =+∞∑

n=−∞

cnz−n+1 (2.135)

The same holds for the antiholomorphic transformations, with the replace-ment z → z and ε(z) → ¯ε(¯)z. If we act with this mapping on a spinless scalarfield φ(z, z) on the plane, we get:

φ′(z′, z′) = φ(z, z) = φ(z′, z′)− ε(z′)∂φ(z′, z′)

∂z′− ε(z′)

∂φ(z′, z′)

∂z′(2.136)

Thus, δφ = −ε(z)∂φ∂z− ε(z)∂φ

∂z. Making use of the expansion of ε(z) and ε(z),

and defining the generators

Ln = −zn+1 ∂

∂z(2.137)

Ln = −zn+1 ∂

∂z(2.138)

we find:δφ =

∑n

[cnLnφ(z, z) + cnLnφ(z, z)] (2.139)

It is useful to write down the commutation relations of the generators:

[Ln, Lm] = (n−m)Ln+m

[Ln, Lm] = (n−m)Ln+m

[Ln, Ln] = 0

The algebra contains an infinite number of generators. This is a specialfeature of conformal invariance in two dimensions. However, if we consideronly the three generators L−1. L0 and L1, we see that they define a finitesubalgebra. This subalgebra is precisely the algebra associated with theglobal conformal group, where the most general transformation was of theform f(z) = az+b

cz+d, ad− bc = 1. Indeed, we have:

L−1 = −∂z translations

L0 = −z∂z scale transformations

L1 = −z2∂z SCTs

31

Recall that quasi-primary fields with zero spin transform under a conformaltransformation as

φ(x) →(∂x′∂x

)−∆dφ(x) (2.140)

In the particular case of two dimensions, a field φ(z, z) (with or without spin)is quasi-primary if it transforms under z → w(z), z → w(z) like

φ′(w, w) =(dwdz

)−h(dwdz

)−h

φ(z, z) (2.141)

where h = 12(∆ + s), h = 1

2(∆ − s). Note the total derivatives in the trans-

formation: w(z)(w(z)) depends on z(z) only. h(h) is defined to be the holo-morphic (antiholomorphic) conformal dimension of the field φ(z, z). When afield φ transforms like the above under any local conformal transformation,it is called a primary field. All primary field are also quasi-primary, but thereverse is not true: a field may transform like the above under global confor-mal transformations (i.e. elements if SL(2,C) but not under local conformaltransformations. If that is the case, the field is called secondary.

2.9 Symmetries and Energy-momentum ten-

sor

An infinitesimal transformation on a field φ reads

φ(x) → φ′(x) = φ(x) + α∆φ(x) (2.142)

where α is an infinitesimal parameter, and ∆φ(x) the field deformation. Ifthe action stays invariant or at most changes by a surface term under thistransformation, we say that the transformation is a symmetry of the action.A change by a surface term is allowed because this does not affect the equa-tions of motion. In other words, such a symmetry leaves the Lagrangianinvariant up to a four-divergence:

L→ L+ α∂µJµ(x) = L+ α∆L (2.143)

Let’s vary the fields involved:

α∆L = α∂L

∂φ∆φ+ α

∂L

∂(∂µφ)∆(∂µφ) (2.144)

= α∂µ[∂L

∂(∂µφ)∆φ] + α[

∂ :

∂φ− ∂µ(

∂L

∂(∂µφ))]∆φ (2.145)

32

But we also haveα∆L = α∂µJ

µ (2.146)

Thus we get

∂µJµ = ∂µ[

∂L

∂(∂φ)∆φ] (2.147)

or

∂µjµ = 0 where jµ =

∂L

∂(∂µφ)∆φ− Jµ (2.148)

Now consider an infinitesimal translation

xµ → x′µ = xµ + εµ (2.149)

A scalar field transforms under this like

φ(x) → φ(x+ a) + aµ∂µφ(x) (2.150)

The Lagrangian, being a scalar, must transform the same way:

L→ L+ aµ∂µL = L+ aν∂µ(δµνL) (2.151)

But above we saw that L transforms up to a four-divergence if the transfor-mation is to be a symmetry of the action. Here the infinitesimal transforma-tion is not a number, but a Lorentz vector εµ; consequently, the conservedcurrent will a two-component tensor Jµν instead of Jµ. The change in theLagrangian becomes

L→ L+ aν∂µJµν (2.152)

Comparing this with (), we find that Jµν = δµ

νL, and we can write dpwn theconserved currents by plugging this value for Jµ

ν into equation () defining theconserved current. There are indeed a set of four conserved currents, one foreach ν. These conserved currents are regrouped in a tensor which is calledthe energy-momentum tensor T µ

ν :

T µν =

∂L

∂(∂µφ)∂νφ− δµ

νL (2.153)

By definition, ∂µTµν = 0. We have shown that an arbitrary coordinate trans-

formation implies the conservation the energy-momentum tensor Tµν . Butwe are interested in conformal transformations, so let’s see what happens ifwe impose this constraint. Under a general coordinate transformation, theaction changes as

δS = −1

2

∫ddxT µν(∂µεν + ∂νεµ) (2.154)

33

Conformal invariance implies that ∂µεν + ∂νεµ = 2d∂ρε

ρgµν , thus

δS =1

d

∫ddxT µ

µ ∂ρερ (2.155)

The action is invariant under conformal transformations if the energy momen-tum tensor is traceless: T µ

µ =0. The converse is not true, however, because∂ρε

ρ is not an arbitrary function.

2.10 Supersymmetry

This section just gives a couple of notions related to supersymmetry.

2.10.1 Superliealgebras, Bose, Fermi elements

Supersymmetry enlarges the Poincare algebra which consists of translationsand Lorentz transformations. The Poincare algebra has translation generatorPµ and Lorentz transformation generator Lµν . The Poincare symmetry con-sists of these generators, and it is an external one: it changes the spacetimecoordinates. There are also internal symmetries: the local U(1) symmetryof electromagnetism is an example. The corresponding generators obey acommutation rule

[Tr, Ts] = f trsTt (2.156)

This bracket operation is bilinear and anticommutative. The U(1) generatorsalso form a vector space, and they obey the Jacobi identity:

[A, [B,C]] + [C, [A,B]] + [B, [C,A]] = 0 (2.157)

These three requirements being fulfilled, we have a Lie algebra. Now wemight ask the following question: Can we combine generators of internalsymmetries with generators of external symmetries with a non-trivial result?In other words, can we have [Ts, Pa] 6= 0or[Ts, Jab] 6= 0? The answer tothis question is no, and it involves the famous Coleman-Mandula theorem.In a nutshell, the theorem states that if internal and external symmetrieswould be combined, the S matrices for all processes would vanish. However,there is a loophole: the theorem is only valid when the final algebra is a Liealgebra. Physicists came up with a new proposal that allows for a non-trivialcombining of internal and external symmetries, provided the final algebra isa graded Lie algebra. Surprisingly, mathematicians had overlooked this richstructure with all its amazing consequences, and started to investigate it onlyafter physicists came up with the idea. What is a graded Lie algebra? It is a

34

vector space which has two types of elements; we call them Bose and Fermi.With the null element being the only exception, each vector in this spaceis either Bose or Fermi, never both of them or none of them. This meansthat each vector is graded (modulo 2), and we call Bose the even, grade 0elements, and fermi the odd, grade 1 elements. We can now define properlya Lie superalgebra s:

• s is a mod 2 graded vector space over C

• s is endowed with a binary operation, the bracket, which is bilinear,superanticommutative and mod 2 grade additive.

• the bracket operation obeys the super-Jacobi identity, (−1)ac[A, [B,C]]+(−1)ba[B, [C,A]] + (−1)cb[C, [A,B]] = 0

An explanation of the second point is in order. The bracket operation is saidto be superanticommutative when

[A,B] = −[B,A]when A,B are either both Bose or one Bose, one Fermi(2.158)

[A,B] = [B,A]when A,B are both Fermi (2.159)

In the second case, the bracket is an anticommutator, which is usually de-noted as A,B. However, knowing the grade of the vectors, there is no ambi-guity in writing [A,B] from now on. If we denote by a, b, c, ... the grades ofthe vectors A,B,C, ... mod 2 grade additive means

[A,B] = C ⇐⇒ a+ b = c (mod 2) (2.160)

Recall that the grades can only take the values 0 (Bose) or 1 (Fermi). Thus,the bracket of two Bose elements or two Fermi elements is Bose, whereasthe bracket of one Bose element and one Fermi element is Fermi. A Liesuperalgebra is simple if for any subsuperalgebra l of s, L ∈ l and A ∈ s, wehave [L,A] is trivial, i.e. l = 0 or l = s.

2.10.2 Supersymmetry in String theory

When we discussed string theory containing fermions, we did not elaborateon the two sectors (R,NS) and (NS,R). Contrary to the (NS,NS) and (R,R)sectors which contain only spacetime bosons, the (R,NS) and (NS,R) sectorscontain spacetime fermions, and it is possible to show that the spectrum ofthese sectors has an N = 2 space-time supersymmetry. This means thatthere are N = 2 spinors. Indeed, the supersymmetry charges transform as

35

doublets under the Lorentz group.11 The supersymmetry generators obeythe graded Lie algebra defined in the previous paragraph.

It can be shown that the space-time supersymmetry of the type II stringtheories arises from the world-sheet supersymmetry of the Polyakov action forthe fermionic string. Now comes an important point: to first order in α′, thelow-energy effective actions for the open string are super Yang-Mills theorieson the world-volumes of the branes. This is the regime we are working in,so we naturally get these particular gauge theories on the branes. The gaugegroup then depends on the brane configuration; N parallel D3-branes give forexample four-dimensional N = 4 super Yang-Mills theory, with gauge groupU(N).

11since they change the spin of a particle, they carry spin themselves, namely spin 1/2.

36

Chapter 3

Anti-de Sitter space

Anti-de Sitter space is a maximally symmetric spacetime with constant nega-tive curvature. Spacetime is maximally symmetric if it is both homogeneousand isotropic. Roughly speaking, homogeneity means invariance under trans-lations and isotropy invariance under rotations. Clearly, our Universe is notAnti-de Sitter, since it looks different at different times. In the AdS-CFTduality we are going to use, however, supergravity is defined on AdS5XS

5,hence the importance of AdS space. This chapter is entirely devoted to anexposure of Anti-de Sitter space. There are many different parameterizationsof the metric, and it is important to understand how these metrics are relatedto each other. Furthermore, we will show which symmetries are inherent ofAdS space. It turns out that the symmetry group of d+1 dimensional AdSspace is identical to that of d-dimensional N=4 super Yang-Mills theory onthe AdS boundary. Anti-de Sitter space is a solution to the Einstein equationin the vacuum with a negative cosmological constant:

3.1 Lorentzian AdS-space

We will consider d+1 dimensional Anti-de Sitter space. It can be representedas a hyperboloid of radius R

X20 +X2

d+1 −d∑

i=1

X2i = R2 (3.1)

which is embedded in d+2 dimensional flat space with metric

ds2 = −dX20 − dX2

d+1 +d∑

i=1

dX2i (3.2)

37

These coordinates are called embedding coordinates. We want to find ametric on the hyperboloid. This is done by parameterizing the embeddingcoordinates Xi, i = 0, ..., d+ 1 in the following way:

X0 = R sec ρ cos τ (3.3)

Xi = R tan ρ Ωi, i = 1, ..., d (3.4)

Xd+1 = R sec ρ sin τ (3.5)

where sec ρ is the secant of ρ, defined as the multiplicative inverse of cos ρ:sec ρ = 1

cos ρ. Here 0 < ρ < π

2, −π < τ < π, and −1 < Ωi < 1. The

coordinates ρ, τ and Ωi represent the whole hyperboloid, and they are calledglobal coordinates. It is easy to see that we must have

∑di=1 Ωi = 1. Indeed,

we have:

X20 +X2

d+1 −d∑

i=1

X2i = R2

(cos2 τ

cos2ρ+

sin2 τ

cos2 ρ− tan2 ρ

d∑i=1

Ω2i

)(3.6)

=R2

cos2 ρ(1− sin2 ρ

d∑i=1

Ω2i ) (3.7)

This equals R2 only if∑d

i=1 Ωi = 1. We can write the metric in terms ofthese new coordinates:

dX0 = Rtan ρ

cos ρcos τdρ−R

1

cos ρsin τdτ (3.8)

dX20 = R2 tan2 ρ

cos2 ρcos2 τdρ2 +R2 1

cos2 ρsin2 τdτ 2 − 2R2 tan ρ

cos2 ρcos τ sin τdρdτ

(3.9)

dXi =R

cos2 ρΩidρ+R tan ρdΩi (3.10)

dX2i = R2 1

cos4 ρΩ2

i dρ2 +R2 tan2 ρdΩ2

i + 2R2 tan ρ

cos2 ρΩidρdΩi (3.11)

dXd+1 = R2 tan2 ρ

cos2 ρsin2 τdρ2 +R

1

cos ρcos τdτ (3.12)

dX2d+1 = R2 tan2 ρ

cos2 ρsin2 τdρ2 +R2 1

cos2 ρcos2 τdτ 2 + 2R2 tan ρ

cos2 ρcos τ sin τdρdτ

(3.13)

38

Putting everything together, we get

ds2 =R2

cos2 ρ(− tan2 ρ cos2 τ)dρ2 − sin2 τdτ 2 − tan2 ρ sin2 τdρ2 − cos2 τdτ 2

(3.14)

+d∑

i=1

(1

cos2 ρΩ2

i dρ2 + sin2 ρdΩ2

i + 2 tan ρΩidρdΩi) (3.15)

=R2

cos2 ρ(− tan2 ρdρ2 − dτ 2 +

dρ2

cos2 ρ+ sin2 ρ

d∑i=1

dΩ2i + 2 tan ρdρ

d∑i=1

ΩidΩi)

(3.16)

=R2

cos2 ρ(−dτ 2 + dρ2 + sin2 ρ

d∑i=1

dΩ2i ) (3.17)

This metric covers all of the d+1 dimensional AdS space; the AdS-boundaryis at ρ = π

2. It is easily seen that ρ = π

2corresponds to spatial infinity: the

Xi, i = 0, ..., d + 1 coordinates all diverge at this point. By allowing ρ totake this value, we compactify AdS space, i.e. the whole space is covered bycoordinates ranging over a finite interval only, in this case 0 < ρ < π

2.

Poincare coordinates Another very useful coordinate system describingAdS space are the so-called Poincare coordinates. Define light-cone coordi-nates in the following way:

u =X0 −Xd

R2andv =

X0 +Xd

R2(3.18)

and defined furthermore

xi =Xi

Ruand t =

Xn+1

Ru(3.19)

Then X0 = R2

2(u + v), Xd = R2

2(v − u), Xi = Ruxi, and Xd+1 = Rut. By

plugging into the equation for the hyperboloid (), we get

R4uv +R2u2(t2 −

d−1∑i=1

(xi)2)

= R2 (3.20)

Solving for v, we find

v =R2 −R2u2(t2 −

∑d−1i=1 (xi)2)

R4u(3.21)

39

Substitute this expression for v into X0 = R2

2(u + v) and Xd = R2

2(v − u),

and we find

X0 =1

2u

(1 + u2(R2 − t2 +

d−1∑i=1

(xi)2))

(3.22)

Xd =1

2u

(1 + u2(−R2 − t2 +

d−1∑i=1

(xi)2)), together with (3.23)

Xi = Ruxi, i = 1, ...d− 1 (3.24)

Xd+1 = Rut (3.25)

We have now all the Xi, i = 0, ..., d+ 1 expressed in terms of the variableu. Now we make a last change of coordinates:

z =1

u(3.26)

which gives

X0 =1

2z

(z2 +R2 − t2 +

d−1∑i=1

(xi)2)

(3.27)

Xd =1

2z

(z2 −R2 − t2 +

d−1∑i=1

(xi)2))

(3.28)

Xi =Rxi

z, i = 1, ..., d− 1 (3.29)

Xd+1 =Rt

z(3.30)

We have coordinates (z, t, x1, ...1xd−1), and the metric in terms of these co-ordinates reads

ds2 =R2

z2

(−dt2 + dz2 +

d−1∑i=1

(xi)2)

(3.31)

z is a radial coordinate in this metric. It divides the AdS space into twodistinct regions, one where z > 0, and one where z < 0. Since 1

z= u = X0−Xd

R2 ,this corresponds to X0 > Xd and X0 < Xd, respectively. The coordinatesXi, i = 1, ..., d − 1 are kept fixed. These two regions are called Poincarecharts, and the Poincare AdS space denotes one of these charts, usually theone where z > 0. The hyperplane dividing the space into these two regions isat X0 = Xd, i.e. at z = ∞. The points on the hyperplane satisfying X0 = Xd

40

or z = ∞, equation () becomes

X2d+1 −

d−1∑i=1

(xi)2 = R2 (3.32)

or, in Poincare coordinates,

t2 +d−1∑i=1

(xi)2 = R2z2 (3.33)

Thus we see that on the hyperplane, we have t→ ±∞.

mapping between global and Poincare coordinates As we have seen,in Poincare coordinates, z = 0 corresponds to the AdS boundary separatingthe region where z > 0 from the region where z < 0. Now we seek to obtaina matching between the Poincare coordinates (t, z, xi, i = 1, ..., d−1) and theglobal coordinates (τ, ρ,Ωi, i = 1, ..., d). This is easily done (though tediousto work out) by equating the expressions for Xi, i = 0, 1, ..., d in both globalcoordinates ((3.3),(3.4),(3.5)) and Poincare coordinates ((3.27)-3.30)). Forexample, by identifying

(3.3)2 + (3.5)2 = (3.27)2 + (3.30)2, (3.34)

we find

sec ρ =1

2R|z|

√√√√(z2 +R2 − t2 +d−1∑i=1

)2

+ (2Rt)2 (3.35)

In this way, every global coordinate has an expression in terms of the Poincarecoordinates.

41

Chapter 4

The AdS/CFT conjecture

4.1 Introduction

We now proceed towards a motivation of the conjecture. This is a first, notyet rigorous approach motivating why two theories should be dual to eachother. Evidently, the coupling constants in both theories should be related ina way or another. Additionally, the symmetries should be the same. Thesetwo conditions by themselves do not guarantee at all any equivalence betweenthe theories, but they provide a first hint that there might be more behind. Infact, as we will see by the end of this chapter, the AdS/CFT correspondencepostulates that the partition functions of both theories agree. This is a highlynon-trivial statement. The duality is not (yet) proven, but we assume it tohold in both weak and strong coupling of the conformal field theory.

4.2 Motivation

Consider a collection of N coincident D3 branes in ten dimensional flatMinkowski spacetime. Since the D3 branes carry energy, they curve space-time around them. But how much do they curve it? This question canbe answered by comparing the gravitational radius of the branes with thenatural length scale in string theory, the string length ls. We work in thelow-energy limit, which means that we consider energies smaller than thestring length energy scale:

E 1

ls(4.1)

This condition is equivalent to keeping all energies bounded and letting lstend to zero. In this limit, after quantization of the open strings on the D-branes, the massive states are not accessible, and we are left with massless

42

U(N) Yang-Mills fields. Furthermore, the interactions between closed stringsthat propagate over the whole of spacetime involve Newton’s constant G,and in ten dimensions, G(10) ∼ g2l8s . At low energies, i.e. ls → 0, G(10) → 0,and the closed strings become free (no interactions). The same holds for theinteractions between the U(N) fields on the branes and the spacetime fields.Thus, at low energies we have

• decoupled closed strings on 10D Minkowski spacetime

• Supersymmetric U(N) Yang-Mills theory on the D3 branes

The U(N) theory reduces further to SU(N), because one gauge field decouples.Let’s now look at the situation from another point of view. The D3

branes, carrying energy and Ramond-Ramond charge, are a solution of thefield equations for the massless fields in type IIB string theory. The solutiondescribing these D3 branes includes a horizon that lies at the ’end’ of aninfinite throat. The throat contains a point-mass at an infinite distanceto the bottom. We see that asymptotically, the throat becomes an infinitecylinder, and the radius of a circle surrounding the point-mass reaches anasymptotic limit equal to the radius of the infinite cylinder. The circle atthe infinite ’end’ of the throat is called the horizon, and this horizon cannotbe reached, as well as the point-mass surrounded by the former. This is incontrast to the Schwarzschild black hole solution for instance, where boththe horizon and the singularity can be reached. The D3-branes stretch alongthe x1, x2, x3 directions, and appear point-like along x4, ..., x9 (recall thatwe are in 10D Minkowski spacetime). In the transverse six dimensions, thebranes are surrounded by five-spheres, analogous to the point-mass beingsurrounded by a circle. Just like the circle approached a constant radiuswhen going down the throat, the 5-spheres reach some constant radius. Thisis the radius of the horizon.

Having described the geometry of spacetime, let’s analyze how an observerfar away from the throat (i.e. living in 10D Minkowski), perceives finiteenergy excitations. Clearly, a finite energy excitation near the horizon isred-shifted, such that the observer sees a vanishingly small excitation. It isimpossible for her to distinguish a finite energy excitation emanating fromthe nearby horizon from an infinitesimal energy excitation far away fromthe horizon. These two kinds of excitations decouple; this means that theexcitations far away from the branes are almost never captured by them, andthose near the branes cannot escape to infinity. Thus, we have two decoupledsystems:

• closed strings on 10D Minkowski spacetime

43

• a near horizon system

What is the near horizon system? It is AdS5 × S5! The horizon radius isnothing but the radius of the 5-sphere, and the AdS5 space arises from thefour-dimensional D3 branes and the radial direction on the transverse space.

Since we described the same configuration (D3 branes in 10D Minkowski),the systems we found should agree. Consequently, we should have that closedstrings in AdS5 × S5 are physically equivalent to (describe the same physicsthan) SU(N) Super Yang-Mills theory.

This is the conjecture:N=4 SU(Nc) Super Yang-Mills is equivalent to IIB String theory on

AdS5 × S5.This is the strongest version of the AdS/CFT duality conjecture. Since

computations outside the limits described above are extremely difficult, amore conservative statement would be that the duality only holds in theselimits, i.e.

N=4 SU(Nc) Super Yang-Mills is equivalent to supergravity on AdS5×S5.

4.3 Parameters

On the gauge theory side, we have the coupling constant gY M and the con-stant Nc. Combined, they give λ = g2

Y MNc, the ’t Hooft coupling constant.On the string theory side, we have the string coupling constant gs, and RAdS5

and RS5 , the size of AdS5 and S5, respectively. Since these spaces are maxi-mally symmetric, this is their only scale. Furthermore, the spaces have equalradii, which we call R. The theories are equivalent if

gs = g2Y M (4.2)

andR2

l2s=R2

α′∼√gsNc ∼

√λ (4.3)

The first relation suggests to examine both theories at weak couplingand compare the results. However, the second relation then states that thesize of S5 is very small, and we do not know how to deal with IIB Stringtheory in this regime. Thus what would need is both weak coupling andlarge R2

α′= λ. However, we cannot deal with Super Yang-Mills at large

λ. Despite of these incompatibilities, there are quantities that can be cal-culated, due to supersymmetry. Indeed, supersymmetry implies that thereexist λ-independent quantities which we can compute at zero-coupling on the

44

gauge theory side and compare with the predictions on the gravity side. Fur-thermore, in the large λ limit, the massive string modes decouple and stringtheory reduces to supergravity. This implies that large-λ effects on the gaugeside which are extremely difficult to compute can readily be calculated in theten-dimensional supergravity theory, and this is what is being done in thisthesis. Another relation which will be useful when computing the entropydensity using AdS/CFT is

16πG = (2π)7g2s l

8s (4.4)

expressing the ten-dimensional gravitation constant in terms of the stringlength and string coupling constant.

4.4 Symmetries

If IIB string theory on AdS5 × S5 is to be dual to N = 4 super Yang-Millstheory, a necessary condition is that the symmetries on both sides are thesame. Let’s start with the symmetries on the string theory side. AdS5 has anisometry SO(4,2) in the Minkowski formulation1. The five-sphere S5 has therotation symmetry SO(6). The respective covering groups are SU(2,2) andSU(4). This means we have an SU(2, 2) × SU(4) symmetry for AdS5 × S5.These are the global symmetries on the string theory side. Let’s now turnto the global symmetries of N = 4 super Yang-Mills theory. This is a con-formally invariant theory; in particular, it does not contain a scale. It isinvariant under global super.conformal transformations, and the correspond-ing supergroup is SU(2, 2) × SU(4)R. We recognize the conformal group infour dimensions SU(2, 2). SU(4)R is the R-symmetry group. By definition,an R-symmetry is a symmetry that does not commute with the supersym-metries. It can be seen as the rotation group of the space transverse to thebranes. One sees that the global symmetries are the same in both theories,supporting the claim that they are equivalent. One can make another obser-vation. IIB String theory lives in ten-dimensional spacetime whose metric is

ds2 =r2

R2(−dt2 + dx2

1 + dx22 + dx2

3) +R2

r2dr2 +R2dΩ2

5 (4.5)

where R2dΩ25 is the metric on the 5-sphere. r = 0 (r = ∞) corresponds

to the horizon (boundary) of AdS5. The coordinate r can be interpreted asthe renormalization group scale in the gauge theory. This can be seen as

1SO(5,1) in the Euclidean formulation

45

follows: N=4 SYM is a conformal field theory, and as such it is invariantunder dilations:

D : xµ → Λxµ (4.6)

where Λ is a constant. The metric 4.5 is invariant under such a transformationprovided r is rescaled like r → r

Λ. We can thus make the following statement:

short-distance physics (Λ 1)in the gauge theory is associated to physicsnear the AdS boundary and long-distance physics in the gauge theory(Λ 1)is associated to physics near the AdS horizon. It is now evident that we canidentify r with the renormalization group scale in the gauge theory.

4.5 Correlation functions in AdS/CFT

After the groundbreaking paper by Maldacena in 1997, Edward Witten wasthe first to propose a more precise correspondence between the correlationfunctions on both sides of the theory, that is, a matching between correlatorsof supergravity and those of the conformal field theory. However, his pre-scription relates the correlators of euclidean AdS5 to those of euclidean R4.To get a Minkowski formulation, one has to analytically continue euclideanspace to Minkowski space, and this continuation does not come without sub-tleties. Firstly, however, we shall deal with the euclidean formulation.

4.5.1 Motivation

As we now know, the boundary of AdS5 × S5 is identified with the spacewhere N = 4 super Yang-Mills is defined. On that boundary, the field φ hasa value φ0 (we will become more precise on this in the next paragraph), andthis value should somehow have some impact to the boundary field theory(since it is defined on the boundary). A natural guess would be to say thatφ0 is a source for some operator O. In that case, we can construct a partitionfunction ZO[φ0] which is the generating functional for correlation functionsof O. Note however that O should be gauge-invariant, for otherwise we couldnot define correlators involving it. We have

ZO[φ0] =

∫D[SYMfields]e−SSY M+

Rd4xO(x)φ0(x) (4.7)

An arbitrary correlator is then

< O(x1)...O(xn) >=1

Z0

δn

δφ0(x1)....δφ0(xn)ZO[φ0]|φ0=0 (4.8)

46

where Z0 is the generating functional with no source (φ0 = 0). This isthe usual Feynman path-integral method to calculate correlation functions:adding a term

∫Jφ to the action where J is a source and φ the operator,

then taking the n-th functional derivative with respect to J for an n-pointfunction, and putting J to zero at the end. To avoid any confusion, let’srecapitulate: φ0 plays the role of J and O plays the role of φ:

J ↔ φ0 source φ↔ O operator (4.9)

4.5.2 Correlators from AdS/CFT

If our claim in the previous paragraph is correct (the boundary value of thebulk field is a source for an operator on the boundary field theory), we areable to compute correlation functions of this operator in the field theory.AdS/CFT now claims that we do not have to compute this correlator on thefield theory side! The main statement of AdS/CFT is the following mostimportant relation, which defines the duality:

ZCFT = Zstring (4.10)

or, in Poincare coordinates,

< eR

d4xO(~x)φ0~x >CFT = Zstring

[φ(~x, z)|z=0 = φ0(~x)

](4.11)

The partition functions on both sides are identical. Another way of say-ing this is that the degrees of freedom on both sides are the same. In thisequation φ0 can be any function. The right hand side contains a boundarycondition for the field φ, but apart from this, φ is arbitrary. We have thus aone-to-one correspondence between fields in the string theory and operatorsin the field theory. This is also known under the name field-operator corre-spondence. The fact that we can compute quantities defined on one side byusing the dual theory is extremely interesting. For instance, if we are in thestrong coupling regime of the field theory, we have no clue how to computecorrelators there. As we have seen previously, strong coupling (large λ,Nc)in the field theory corresponds to weak coupling in the theory in the bulk,which is supergravity. We know how to compute things in supergravity! Onthe other hand, we also know how to compute things in weakly coupled fieldtheory, and get information about the full string theory in the bulk. SoAdS/CFT allows us to switch to the weakly coupled theory when we wantto calculate a correlator in the strongly coupled theory. In this thesis, wewill do the calculations on the supergravity side to get information aboutthe strongly coupled boundary field theory. It is evident that one could go

47

the other way round and calculate things in the weakly coupled field theory(making use of perturbation theory as usual) and get information about thestring theory in the bulk. However, the problem is that we do not know towhat perturbatively computed quantities in the field theory correspond inthe string theory! In fact, string theory itself is defined perturbatively, andthe quantities we would get from the weakly coupled field theory should havean interpretation as a quantity in a non-perturbatively formulation of stringtheory, on which our knowledge is very poor. This is why one usually makesuse of AdS/CFT in one way only, namely extract information about stronglycoupled field theories by considering supergravity in the bulk.

As outlined in the beginning, we work in the regime where both thenumber of colors N and the ’t Hooft coupling constant λ = g2

Y MN are large.On the AdS side, this corresponds to taking the supergravity limit, that is,letting α′ → 0. By doing so, we ignore so-called stringy corrections (in α′)and loop corrections (in κ = gsα

′2). Equation () then becomes:

e−W = ZCFT = Zstring = e−SSUGRA (4.12)

where W = −log < eR

d4xO(x)φ0(x) >.

4.5.3 Two-point function

Let us start with a first computation making use of the AdS/CFT framework:the two-point function of a massless scalar field. The procedure to follow isas follows. Firstly we try to find an expression of the bulk action in terms ofthe boundary value of the field, φ0. This action, by definition of AdS/CFT,can then be used to find the two-point function of the corresponding operatorliving on the boundary (for which φ0 is a source). The two-point function isobtained by taking the second functional derivative of this action with respectto φ0 and letting φ0 = 0 in the end. This is why we want to express S in

terms of φ0 in the first place, and AdS/CFT tells us that δ2S[φ0]

δφ20

corresponds

indeed to the two-point function of the corresponding operator. We startwith the simplest possible action, that of a massless scalar field. Recall weare in AdS-space, and we assume it has d+1 dimensions. The action is

S[φ] =1

2

∫√g∂µφ∂µφdd+1x (4.13)

The equation of motion is determined by the Euler-Lagrange equation andreads

φ = 0 (4.14)

48

which is just the massless Laplace equation. We work in euclidean AdSd+1,and we need to define a metric for that space. If we consider (d+1) dimen-sional euclidean space Rd+1(x0, ..., xd), we can define an open unit ball Bd+1

consisting of all points satisfying∑d

i=0 x2i < 1. Then we can identify AdSd+1

with Bd+1 with the metric

ds2 =4∑d

i=0 dx2i

(1− x2)2(4.15)

or

ds2 =1

x20

d∑i=0

(dx2i ) (4.16)

where x0 > 0. In this d+1 dimensional metric, the boundary is Rd locatedat x0 = 0, together with a single point P at x0 = ∞. We are going to do thecalculation in this metric. Recall the definition of the box operator:

1√g∂µ(√ggµν∂ν) (4.17)

Here√g =

√det g = x

−(d+1)0 . The metric only depends on the x0 coordinate,

so we get

= xd+10

∂

∂x0

(x−d−10 x2

0

∂

∂x0

)// = xd+10

∂

∂x0

(x−d+10

∂

∂x0

) (4.18)

Hence, the equation to solve is

φ = 0 ⇐⇒ ∂

∂x0

(x−d+10

∂

∂x0

)K(x0) = 0 (4.19)

We want to find a solution K of that equation which has as boundary valuea delta function at some point P on the boundary. We choose P to be thepoint at x0 = ∞. This means we want K to vanish on the boundary exceptat P ≡ ∞. The solution which satisfies this condition is

K(x0) = cxd0 (4.20)

where c is a constant. We see that K diverges for x0 = ∞. That this cor-responds to a delta function can be seen when one makes an SO(1,d+1)transformation of P mapping it to a finite point. The following transforma-tion maps P to the origin:

xi →xi

x20 +

∑dj=1 x

2j

(4.21)

49

where i = 0, ..., d. Indeed, the point x0 = ∞ is mapped to the point x0 = 0.The function K(x0) is mapped to

K(x) = cxd

0

(x20 + (

∑dj=1 x

2j))

d(4.22)

We see that K(x) is a delta function: when x0 → 0, it vanishes everywhereexcept at x1 = x2 = ... = xd = 0. Now we can write down φ(x) in terms ofφ0:

φ(x0, xi) = c

∫dx1dx2...dxd

xdo

(x20 +

∑dj=1(xj − x′j)

2)dφ0(x

′i) (4.23)

As said in the beginning of this paragraph, we want to express the actionS in terms of φ0. If we integrate the action by part, one sees indeed thatthis should be possible, since the nonzero term is a surface term, and φ0 isdefined on the surface:

S[φ] =1

2

∫√g∂µφ∂µφdd+1x (4.24)

=1

2

∫dd+1x

√gDµ(φDµφ)− 1

2

∫dd+1xφDµD

µφ (4.25)

The second term disappears because the equation of motion is satisfied (inother words, the action is on-shell). The first term can be expressed as aboundary term by Stoke’s theorem, so that we get

S[φ] = limε→0

1

2

∫Tε

ddx√hφ(~n · ~∇)φ (4.26)

Tε is the surface x0 = ε, n is a unit normal vector to Tε. We need tocompute the quantity ~n · ~∇)φ = x0

∂φ∂x0

(we also write dx1dx2...dxd ≡ d~x and∑dj=1(xj − x′j)

2 ≡ |~x− ~x′|2):

∂φ

∂x0

= cdxd−10

∫d~x

φ0(x′i)

(x20 + |~x− ~x′|2)d

− 2cdxd+10

∫d~x

φ0(x′i)

(x20 + |~x− ~x′|2)d+1

(4.27)For x0 → 0, the second term is negligible with respect to the first, and in4.26, φ → φ0. Furthermore, in 4.26, h is the induced metric on the slicex0 = ε, thus

√h = x−d

0 which simplifies with the xd−10 from the the first term

in 4.27. Putting everything together, we find at x0 → 0:

S[φ0] =cd

2

∫d~xd~x′

φ0(~x)φ0(~x′)

|~x− ~x′|2d(4.28)

50

The two-point function is obtained by taking two functional derivatives withrespect to φ0 (since it is a source in the field theory):

〈O(x)O(x′)〉 =δ2S[φ0]

δφ0(x)δφ0(x′)∼

1

|~x− ~x′|2d(4.29)

This is exactly what we found when we computed the two-point functionof two operators of the same conformal dimension, equation (). Since theoperators are the same here, the conformal dimensions are trivially the same.

4.5.4 Currents in the gauge theory

It is natural to try to extend the duality by relating other bulk quantities toquantities on the boundary. For example, in the bulk AdS space, we couldhave gauge fields Aa, belonging to some gauge group of dimension s. In thatcase, a = 1, 2, .., s. In electromagnetism, the group is the one-dimensionalU(1), and there is just one gauge field A. We consider here the generalcase where Aa belongs to an s-dimensional gauge group G. From symmetryarguments, G is also a global symmetry group in the boundary conformalfield theory. To each gauge field Aa corresponds a boundary current Ja, andwe can compute correlators of these currents by making use of AdS/CFTand computing them in the bulk supergravity approximation. Again, thepartition functions are conjectured to be the same on both sides:

〈eR

Sd JaAa0〉CFT = Zstring(A0) = e−SSUGRA(A) (4.30)

Here A0 is the value A approaches at the boundary. The integration rangeover the d-sphere Sd is just the boundary of the bulk Bd+1. The procedureto follow for computing correlators of currents J is the same than before foroperators: express the supergravity action in terms of the boundary valuesAa

0 of the massless gauge fields Aa, and take this action as the generatingfunctional for correlation functions of operators Ja in the boundary fieldtheory whose sources are the Aa

0.

4.5.5 Massive fields in AdS

In this paragraph, we seek to find the two-point function of a massive scalarfield. The only difference in the action compared to the previous one is thusthe presence of a mass term. We will see that one obtains an equation relatingthe mass of the bulk field to the conformal dimension of the boundary field.We will use again the metric

ds2 =1

x20

(dx20 +

d∑i=1

dx2i ), x0 ≥ 0. (4.31)

51

Just like in the massless case, we start with the supergravity action, and weconsider a scalar field with mass m:

S[φ] =1

2

∫dd+1x

√gDµφD

µφ+m2φ2 (4.32)

The equation of motion is the massive Laplace equation and reads

φ−m2φ = 0 (4.33)

Again we want to find a solution K to this equation that vanishes everywhereon the boundary except at one point. And again we take this point to bex0 = ∞. The metric being the same than before, the box operator is alsothe same, and the equation to solve is(

xd+10

d

dx0

x−d+10

d

dx0

−m2)K(x0) = 0 (4.34)

Before solving this equation, a sidestep is in order. We are going to expressthe massive Laplace equation in a different metric describing AdSd+1:

ds2 = dx2 + sinh2 xdΩ2, 0 ≤ x <∞ (4.35)

In this metric, the boundary is at x = ∞, and dΩ2 is the metric on the unitsphere. In this metric, the equation of motion is found to be(

− 1

(sinhx)d

d

dx(sinhx)d d

dx+

L2

sinh2 x+m2

)φ = 0 (4.36)

where L is the angular momentum, the angular part of the Laplacian. Sincewe are interested in the near-boundary behavior, i.e. large x, we can neglectthe term involving L. It turns out that there are two linearly independentsolutions to this equation, which behave as e±λx for large x, and λ± are thetwo roots to the equation

λ(λ+ d) = m2 (4.37)

Now we can come back to our previous equation 4.34 and write down asolution:

K(x0) = xd+λ+

0 (4.38)

As it should, K(x0) vanishes at x0 = 0. If we perform an inversion

xi →xi

(x20 + |~x|2)

(4.39)

52

we get

K(x0) =x

d+λ+

0

(x20 + |~x|2)d+λ+

(4.40)

Just as before, K vanishes when x0 → 0 unless ~x = 0. Now that we have anexpression for the Green’s function K(x0), we can use it to write down thesolution φ in terms of φ0:

φ(x) = c

∫d~x′

xd−λ+

0

(x20 + |~x− ~x′|2)d+λ+

φ0(~x′) (4.41)

This solution goes like x−λ+

0 φ0(x) for x0 → 0. With this expression in hand,we can plug it back into the on-shell action, which (after integration by parts)only contains a surface term, and find

S[φ0] =c(d+ λ+)

2

∫d~xd~x′

φ0(~x)φ0(~x′)

|~x− ~x′|2(λ++d)(4.42)

From this it follows directly that

〈O(x)O(x′)〉 ∼ |~x− ~x′|−2(d+λ+) (4.43)

in accordance with the CFT result for the two-point function of a conformalfield of dimension ∆ = λ+ + d. Now, recall equation 4.37, where the mass isrelated to λ. If we replace λ by ∆− d, 4.37 becomes:

∆(∆− d) = m2 (4.44)

As promised, we have found a relation between the mass of the bulk scalarfield φ and the conformal dimension of the field O for which φ0 is a source.The equation can be solved to give

∆ =1

2(d+

√d2 + 4m2) (4.45)

4.6 Minkowski prescription for two-point func-

tions

Let us now turn to the AdS/CFT formulation in the Minkowski formulation.In the Minkowski signature, subtleties arise when trying to formulate theAdS/CFT correspondence and computing correlators, and we will look atthis issue in more detail in this section. The problem arises when one tries

53

to formulate the AdS/CFT conjecture (with the supergravity approximationin the bulk):

〈eiR

δM φ0O〉 = eiSSUGRA[φ] (4.46)

The main problem is that the wave equation in the bulk admits several nor-malizable solutions, which have the same boundary value all of which areregular at the horizon. This means that given a boundary value φ0 and thecorresponding operator to which φ0 couples, there can be more than onebulk field: the boundary field φ0 (with the requirement of regularity) doesnot uniquely specify the bulk field, there is no one-to-one correspondence be-tween the bulk fields and the gauge-invariant operators on the boundary fieldtheory. This is in contrast to the Euclidean formulation, where this corre-spondence is unique. One can then try, in the Minkowski formulation, to putan additional boundary constraint on the field: the incoming wave boundarycondition. This boundary condition only allows waves to travel towards theregion behind the horizon, and forbids them to be emitted from the horizon.Of course, after noticing all these problem with the Minkowski formulationthat do not arise in the Euclidean case, one could ask the question: whywould we like to have a Minkowski formulation of AdS/CFT? The reason isthat many quantities in the boundary gauge theory can only be computedfrom real-time Green’s functions. Green’s functions involve correlators andcorrelators are directly involved in the AdS/CFT formulation. This is whywe seek for a Minkowski signature (i.e. real-time) formulation of AdS/CFT.In particular, hydrodynamic quantities such as the viscosity or diffusion con-stant can only be found in the real-time prescription of AdS/CFT. One couldthink of analytically continuing the result from euclidean signature to theMinkowski version, by making use of analytical properties of the Green’sfunctions. However, this is only possible if the Euclidean correlators areknown for all Matsubara frequencies. In this section, a particular method isproposed to compute two-point functions directly in the Minkowski-signatureformulation of AdS/CFT.

4.6.1 Review of thermal Green’s functions

Recall from chapter 3 that the retarded propagator, or retarded Green’sfunction for the operator O is defined as

GR(k) = −i∫d4xe−ikxθ(t)〈[O(x),O(0)]〉 (4.47)

Note that this is the Minkowski space retarded Green’s function. There areseveral Green’s functions in the Minkowski formulation, but only one in the

54

Euclidean formulation. It is called the Matsubara propagator, and it reads

GE(kE) =

∫d4xEe

−ikE ·xE〈TEO(xE)O(0)〉 (4.48)

where TE stands for time-ordering. This propagator is defined only for dis-crete values of ω ≡ k0

E.

4.6.2 Problem with the Minkowski formulation

Consider a scalar field in a metric which has the following form:

ds2 = gzzdz2 + gµν(z)dx

µdxν(+Sn) (4.49)

For example, this could be an AdS metric in Poincare coordinates. Theaction of the scalar is

S = K

∫d4x

∫ zH

zB

dz√−g[gzz(∂zφ)2 + gµν∂µφ∂νφ+m2φ2] (4.50)

The coefficient of proportionality K is not important for the rest of thecalculation; suffice it to say that it is a normalization constant depending onthe scalar field one is considering. The region of integration goes from theboundary zB to the horizon zH . In Poincare coordinates, the AdS boundarywould be at zB = 0. The equation of motion follows straightforwardly:

1√−g

∂z(√−ggzz∂zφ) + gµν∂µ∂νφ−m2φ = 0 (4.51)

The first two terms come from the usual definition of the box operator, = 1√

−g∂µ(√−ggµν∂νφ), and because

√−g depends on z only, it drops out

of the derivative in the second term and cancels with its inverse. As usual, werequire a well-defined boundary value φ0(x) ≡ φ(zB, x) for the scalar. Thesolution to the equation of motion can be expressed in terms of a Fouriertransform, after a separation of variables:

φ(z, x) =

∫d4k

(2π)4eik·xfk(z)φ0(k) (4.52)

where fk(z) satisfies the equation

1√−g

∂z(√−ggzz∂zfk)− (gµνkµkνφ+m2)fk = 0 (4.53)

55

φ0(k) of course is a boundary value and needs to be such that

φ0(x) ≡ φ(zB, x) =

∫d4k

(2π)4eik·xφ0(k) (4.54)

So we have φ(z, k) expressed in terms of its boundary value φ0(k) ≡ φ(0, k)).Comparing 4.52 and 4.54, we see that we must have fk(zB) = 1. Furthermore,fk(z) has to satisfy the incoming wave boundary condition. Now we proceedin the same way than we did in the euclidean case. The action, being on-shell, only has a surface term after integration by part of the first term. Also,we go to momentum space:

∫d4x→

∫d4k

(2π)4. So finally we have the following

expression for the action:

S =

∫d4k

(2π)4φ0(−k)F(k, z)φ0(k)|z=zB

z=zH(4.55)

whereF(k, z) = K

√−ggzzf−k(z)∂zfk(z) (4.56)

Now that we have the action expressed in terms of the boundary value of thefield, φ0, we can take its second functional derivative a with respect to φ0 toget the two-point correlator. Consequently, we should also get the retardedGreen’s function by equation (), and the quantity we get is

G(k) = −F(k, z)|zHzB−F(−k, z)|zH

zB(4.57)

Here is where the problem arises: G(k) cannot be the retarded Green’s func-tion, and the explanation is the following. The quantity F(k, z) has a realand an imaginary part, and let us find an expression for Im(F). Generallyone has Im(x) = 1

2i(x− x∗). In our case, this gives

Im(F) =K

2i

√−ggzz[f−k(z)∂zfk(z)− f ∗−k(z)∂zf

∗k (z)] (4.58)

Since f ∗k (z) is also a solution to equation (), we have f ∗k (z) = f−k(z), and wecan write

Im(F) =K

2i

√−ggzz[f ∗k (z)∂zfk(z)− fk(z)∂zf

∗k (z)] (4.59)

¿From this we conclude that ∂z

(ImF

)= 0. Since we can write equation ()

in the form

G(k) =

∫ zH

zB

−∂zF(k, z)− ∂zF(−k, z) (4.60)

we see that the imaginary part of F(k, z) and F(−k, z) vanishes: G(k) is real.As we know, however, a retarded Green’s function is complex in general. soG(k) cannot be the retarded Green’s function we are looking for.

56

4.6.3 ’Solution’ to the problem

Son and Starinets conjectured that, in the Minkowski formulation of AdS/CFT,the retarded Green’s function is

GR(k) = −2F(k, z)|zB(4.61)

This means that the contribution from the horizon should completely bedisregarded. The conjecture that G(k) should take this form is nothingmore than an ’educated guess’, and it works only for two-point functions,which is why it cannot account as a complete solution to the problem. Inparticular, it is not a consequence of the Minkowski AdS/CFT identity 4.46.However, by taking 4.61 as the retarded Green’s function, one gets correctresults for the two-point correlators, as we will see in the last chapter, wherewe compute viscosity and diffusion constant in this framework. It is thus acomplete ad-hoc prescription to circumvent the fact that the quantity GR, ifcomputed ’correctly’ in the Minkowski formulation, is real and thus cannotbe the correct result for the retarded Green’s function.

57

Chapter 5

Hydrodynamics and LinearResponse theory

5.1 Introduction

Since quite a while now, it has been known that one can associate thermody-namic quantities to black holes. Indeed, they have a well defined entropy thatscales with the area of the horizon1 In other words, the number of degreesof freedom beyond the horizon scales like the horizon area. Associated withentropy comes a temperature: one can indeed compute the Hawking temper-ature of a black hole, and this will be done in the next chapter. This showsthat a black hole can be seen as a thermodynamical system. In the pastten years, it has been realized that black holes also exhibit hydrodynamicproperties. Hydrodynamics deals with systems that are out of equilibriumby a small amount only; black holes are indeed subject to fluctuations thatmove it out of an equilibrium state. Generally, the non-equilibrium behav-ior of many-particle systems is extremely complex. However, things simplifydrastically if one only considers small variations in time and space, that is,the low-frequency, long distance behaviour. The most developed theory thatstarts with these assumptions is ordinary fluid dynamics, or hydrodynam-ics.2 In the hydrodynamic limit, the system is characterized by five partialdifferential equations (PDEs). A natural question to ask is then, why doessuch a simplification occur? The reason is that at any time, we can con-sider the system in thermodynamic equilibrium, such that a temperature,

1generally, one speaks of the horizon volume, not area, since black holes exist in higherdimensions too. These higher-dimensional black holes are also called black branes.

2It should be emphasized that there is no proof that any interacting theory at finitetemperature can be described by hydrodynamics in the limits outlined above; however,this should not concern us in this thesis.

58

pressure and density can be defined. These quantities are local values ofthermodynamic variables, and they are interrelated by the system of PDEs.However, hydrodynamics is an effective theory: it contains parameters whosevalue is not given by the theory, they cannot be computed in the frameworkof hydrodynamics. A more fundamental theory is needed. These parame-ters are the transport coefficients such as the viscosity, diffusion constant,conductivity. One way to compute the transport coefficients is by means oftime-dependent correlation functions. The correlation functions completelydescribe the system if its departure from equilibrium is small. AdS/CFTcan be used to compute quantities such as viscosity or diffusion of a fluid, sothe aim of this chapter is to properly introduce the concepts and relations inhydrodynamics and linear response theory. The first part of this chapter isa brief review of classical and relativistic hydrodynamics, with an emphasisonly on topics needed in the next chapter. The second part is an introductionto linear response theory. We will see that one can derive a precise expressionfor transport coefficients in the hydrodynamic limit.

5.2 Review of classical hydrodynamics

A general velocity gradient in three space dimensions reads, to first order:

dvi =3∑

j=1

( ∂vi

∂xj

)dxj ≡

3∑j=1

Gijdxj. (5.1)

Gij can be decomposed into a symmetric and an antisymmetric piece:

Gij =1

2

( ∂vi

∂xj

+∂vj

∂xi

)+

1

2

( ∂vi

∂xj

− ∂vj

∂xi

)≡ eij + ωij (5.2)

It is easy to show that the diagonal terms of eij represent the velocity ofelongation in the corresponding direction, while the cross terms represent thevelocities of local angular deformation. Thus, eij stands for any deformationof the fluid element. It is useful to decompose eij in a diagonal term and aterm of zero trace:

eij =1

3δijell +

[eij −

1

3δijell

]≡ tij + dij (5.3)

where tij stands for dilatation and dij for constant volume deformations ofthe fluid element. Also, ell ≡ e11 + e22 + e33.

59

5.2.1 General expression for the surface forces.

Consider a surface element of area dS in a fluid. Firstly we want to define aconstraint tensor σ, whose components σij represent the force per unit surfacein the i-direction on a surface whose normal vector is in the j-direction. Tan-gential constraints, σij, i 6= j, appear for moving fluids. They arise because offriction between different layers of the fluid. Friction is a consequence of vis-cosity, hence the viscosity tensor is closely related to the constraint tensor3.Recall that viscosity is a transport coefficient that quantifies the momentumtransport from zones of high velocities towards zones of low velocities. It iseasy to determine the constraint σ~n on an arbitrary surface ~dS = dS~n4. Theway to go is to write down Newton’s second law along each direction Ox,Oy, Oz, and let the volume element dV tend to zero. The result is: σx~n

σy~n

σz~n

=

σxx σxy σxz

σyx σyy σyz

σzx σzy σzz

nx

ny

nz

(5.4)

or, equivalently,σi~n = σijnj (5.5)

σx~nis the ~x-component of the constraint on the unit surface dS~n. It is

to be noted that for a fluid at rest, the constraint is perpendicular to thesurface and one number, the pressure, determines its magnitude. In thatcase, the only non-zero components are on the diagonal (pressure forces actperpendicular to surfaces); furthermore, they are isotropic, i.e. the threecomponents have the same value. We then write

σij = σ′ij − pδij, (5.6)

where p is the pressure. The minus sign accounts for the fact that a fluidat rest is usually under compression, i.e. the pressure forces act in the op-posite direction to the surface vector. The first term on the right hand sideis what we define as the viscosity tensor ; it is the part of σij that accountsfor the deformation of the fluid elements. It can be shown that σ′ij is sym-metric: σ′ij = σ′ji, by analyzing the torques on a small cubic volume elementdV = dxdydz. We now want to find a relation between the viscosity tensorand the fluid deformation. If the fluid element moves without deformation,the pressure is the only constraint. In particular, the elements of σ′ij are

3We will see that the viscosity tensor does not contain the perpendicular forces, i.e.the pressure will be removed from the constraint tensor σ to get the viscosity tensor σ′.

4The constraint σ~n is the force on a unit surface element whose normal vector is ~n

60

independent of translations and rotations of the element. Therefore, the vis-cosity tensor cannot have any dependence on components of ωij, since thistensor accounts for rotations. This agrees with fact that ωij is antisymmetric,while σ′ij is symmetric. Hence, the components of σ′ij can only depend onthe components of eij defined above. We suppose the fluid newtonian; thismeans that the σ′ij’s depend linearly on the deformation values. If we furtherassume isotropy, σ′ij can be written as follows:

σ′ij = 2Aeij +Bδijemm (5.7)

The derivation is done by using the fact that σ′ij is symmetric. Recall thatwe decomposed eij into a diagonal term and a traceless term; we can do thesame for σ′:

σ′ij = η(2eij −

2

3δijemm

)+ ζ(δijemm) (5.8)

The first term on the right hand side corresponds to a deformation at constantvolume; η is the viscosity. The second corresponds to an isotropic dilatation.It vanishes for an incompressible fluid. ζ is called the second viscosity. Notethat the first term is zero for i = j.

5.3 Relativistic hydrodynamics

Hydrodynamics is an effective theory that describes dynamics at large dis-tances and large scales. It deals with a huge number of particles, and insteadof associating a four-momentum to each of the particles, it considers the sys-tem as a fluid, to which we can associate an overall four-momentum field. Todescribe the fluid completely, we need to introduce the energy-momentumtensor T µν . This symmetric (2,0) tensor can be thought of the flux of four-momentum pµ across a surface of constant xν . The energy-momentum tensorof a perfect fluid contains two parameters: the rest-frame energy density ρand the rest frame pressure p. Assuming isotropy, that is, no flux of a mo-mentum component in an orthogonal direction, it follows that T µν is diagonalin its rest frame, and that T 11 = T 22 = T 33. In its rest frame, T µν reads thus

T µν =

ρ 0 0 00 p 0 00 0 p 00 0 0 p

(5.9)

In the rest frame, the four-velocity has components u0 = −1, ui = 0.The expression in a general frame giving 5.9 in the rest frame with the given

61

values of ui is5

T µν = (ρ+ p)uµuν + pgµν (5.10)

Since it reduces to 5.9 in the rest frame, this is the right expression for theenergy-momentum tensor of a perfect fluid. However, since we are interestedin dissipative processes such as viscosity and thermal conduction, we mustgo to the next order. For example, entropy is conserved for a perfect fluid.This can be seen as follows, using the conservation of T µν :

∂µTµν = 0 ⇔ ∂µ

[(ε+ p)uµuν + pgµν

]= 0 (5.11)

⇔ ∂µ

[(ε+ p)uµ

]uν + (ε+ p)uµ∂µu

ν + ∂νp = 0 (5.12)

(5.13)

Multiplying by uν :

∂µ

[(ε+ p)uµ

]uνuν + (ε+ p)uµuν∂µu

ν + uν∂νp = 0 (5.14)

The second term is zero, since

uν∂µuν = ∂µ(uνu

ν)− uν(∂µuν) = −uν(∂µuλ)gλν = −uλ∂µu

λ ≡ −uν∂µuν

(5.15)Thus uν∂µu

ν = 0, and 5.14 reduces to:

− ∂µ

[(ε+ p)uµ

]+ uν∂

νp = 0 (5.16)

⇔ −∂µ

[nuµ

(ε+ p

n

)]+ uν∂

νp = 0 (5.17)

⇔ ∂µ

(nuµ

)ε+ p

n+ nuµ

[∂µ

(ε+ p

n

)− 1

n∂µp]

= 0 (5.18)

By going from the first line to the second, we have multiplied and divided bythe number density n of the particles of the fluid at rest. This implies thatthe first term in the third line vanishes, since it is just the four-derivative ofthe particle flux vector nuµ which is conserved by the equation of continuity6.Also, the third term comes from the fact that uµuν∂

ν = ∂µp. The secondterm can be modified by noting that from dε = Tds and dp = sdT , we get

d(ε+ p

n

)= Td

( sn

)+

1

ndp, (5.19)

5We use the convention (−+ ++)6−∂tn +∇~u = 0

62

which implies that eq.5.18 becomes

uµ[T∂µ

( sn

)]= 0. (5.20)

It follows that∂µ(suµ)− s

n∂µ(nuµ) = 0, (5.21)

and using the equation of continuity, ∂µ(nuµ) = 0, we finally have shownthat the entropy flux is conserved:

∂µ(suµ) = 0. (5.22)

In order to have entropy production, we need to add a dissipative termto the energy-momentum tensor:

T µν = (ε+ P )uµuν + pgµν − σµν (5.23)

The particle flux density vector now reads:

nµ = nuµ + νµ (5.24)

In relativistic mechanics, an energy flux involves mass flux. Thus, itmakes no sense anymore to define the velocity (of a heat flux, for instance)in terms of a mass flux density. We define the four-velocity in such a waythat in the rest frame of a fluid element, the element’s momentum be zero,and its energy be expressed in terms of the other thermodynamics quantities,by the same formulae as when dissipative processes are absent. This meansthat in the proper frame of the fluid element,

σ00 = σ0i = 0. (5.25)

Since we also have ui = 0 in this frame, it follows that

σµνuν = 0, (5.26)

Being a tensor equation, this equation is valid in any frame.

5.3.1 Expression for σµν.

Let us find a general expression for σµν in flat spacetime. The generalizationto curved spacetime is straightforward. We will make use of the continuityequation, which reads in the presence of dissipation:

∂µ(nuµ − νµ) = 0. (5.27)

63

Energy conservation yields (recall 5.14)

−∂µ(nuµ)ε+ P

n− nuµ∂µ

(ε+ P

n

)+ uν∂

νP −(∂µσ

µν)uν = 0 (5.28)

where the dissipative term is now present. The first term can be replacedusing the continuity equation 5.27. Applying the same procedure than before,we get

− ∂µνµ ε+ p

n− T∂µ(suµ) +

Ts

n∂µ(nuµ)− (∂µσ

µν)uν = 0 (5.29)

⇔ −T∂µ(suµ)−(ε+ p− Ts

n

)∂µν

µ − uν∂µσµν = 0 (5.30)

⇔ T∂µ(suµ) + µ∂µνµ + uν∂µσ

µν = 0 (5.31)

When going from 5.30 to 5.31, we used the following thermodynamical iden-tity:

d(ε+ p− Ts

n

)=dp

n− s

ndT +

dε

n− T

nds (5.32)

The last two terms cancel, and the remaining is just dµ. Using σµνuν = 0,we find

∂µ

(suµ − µ

Tνµ)

= −νµ∂µ

(µT

)+σµν

T∂µuν (5.33)

The left hand side must be the four-divergence of the entropy flux, and onthe right we have the increase in entropy due to dissipative processes. Wesee that the entropy flux density four-vector reads

σµ ≡ suµ − µ

Tνµ (5.34)

In order to make the r.h.s. positive, σµνand νµ must be linear functions ofthe gradients of velocity and thermodynamic quantities. This together withthe relations σµνuµ = 0 and νµu

µ = 0 yields

σµν = P µαP νβ[η(∂αuβ + ∂βuα −

2

3gαβ∂λu

λ)

+ ζgαβ∂λuλ], (5.35)

where P µν = gµν + uµuν is the projection operator onto the directions per-pendicular to uµ. Note the similarity to eq.5.8.

The generalization to curved spacetime reads

σµν = P µαP νβ[η(∇αuβ +∇βuα) +

(ζ − 2

3η)gαβ∇ · u

](5.36)

64

Now, let hij(t) be a small time-dependant perturbation of gµν :

gij(t, x) = δij + hij(t), hij 1 (5.37)

g00(t, x) = −1, g0i(t, x) = 0. (5.38)

Substituting this metric together with uµ = (1, 0, 0, 0) into (80), only thespatial components are nonzero. For example,

σxy = η∂0hxy. (5.39)

One can compute the same in linear response theory, and compare bothexpressions. In the end, we will find:

GRxy,xy(ω,~0) =

∫dtd~x eiωtθ(t) <

[Txy(t, ~x), Txy(0,~0)

]>= −iηω +O(ω2)

(5.40)As we will see, this is the zero spatial momentum, low-frequency limit of theretarded Green’s function of T xy.

5.4 Linear Response Theory

This section describes how a system initially in equilibrium ’responds’ af-ter being slightly perturbated. Kubo’s formula then gives an expressionfor the response formula in terms of the applied perturbation. As alreadymentioned, linear response theory deals with small perturbations from equi-librium; consequently, considering the response of the system to be linear isa valid approximation.

5.4.1 Response function

A response function exists for any physical observable. It relates an ensemble-averaged physical observable to an applied external force. Let f(t) be thesmall external perturbation applied to our system initially in equilibrium.The Hamiltonian of the unperturbed system being H0, the total Hamiltonianreads:

H = H0 +H1 (5.41)

where H1 = −Af(t). A is a constant operator associated with the perturba-tion. For example, f(t) could be an external magnetic field that is turned onas a small perturbation for a system of uncharged spin 1/2 particles. In thatcase, A is the magnetization of the system, and H1 = −Af(t) represents theinteraction of the external field with the system. For an unperturbed system

65

in equilibrium, the ensemble average of any observable is constant and reads

< B >= Tr(ρ0B), (5.42)

where ρ0 is the (constant) density operator of the system, ρ0 = 1Ze−βH0 . Once

the perturbation is present, ρ becomes time-dependent (through H1(t)), andso does the ensemble-average of our operator:

< B(t) >= Tr(ρ(t)B) (5.43)

The linear relation between < B(t) > and the external perturbation is ex-pressed as follows:

δ < B(t) >=< B(t) > − < B >=

∫ t

−∞φBA(t− t′)f(t′)dt′ (5.44)

This equation states that the differential change of < B(t) > is proportionalto the external disturbance f(t′) and the duration of the perturbation δt′.Note that the response function φBA depends on t − t′; this comes fromthe condition that perturbations at different times act independently of eachother. Furthermore, causality requires that φBA(t−t′) be independent of anyperturbations after t. This implies

φBA(t− t′) = 0 for t′ > t. (5.45)

As a consequence, 5.44 can be integrated from−∞ to +∞. We now introducethe Laplace transform of φBA(t):

χBA(s) =

∫ ∞

0

φBA(t)e−stdt→ χBA(z) =

∫ ∞

0

φBA(t)eiztdt, (5.46)

where s has been replaced by −iz, z = z1 + iz2. We want the perturbationto be turned on in an adiabatic way. Therefore we replace in 5.44 f(t′) byf(t′)e−εt′ , ε > 0. Taking the limit ε→ 0 and the Fourier transform of (4), wefind

< B(ω) >= χBA(ω)f(ω) (5.47)

χBA(ω) is called the generalized susceptibility.We want to find an expression for the response function φBA(t) in terms of

the operators B, A and the unperturbed Hamiltonian H0. For this purpose,we are going to express δ < B > in a different form using the density matrixformalism. Generally, we have

< B(t) >= Tr(ρS(t)B) = Tr(ρHB(t)) (5.48)

66

where ρS and ρH are the density matrices in the Schroedinger and Heisenbergpicture, respectively. In the interaction picture, the density operator becomes

eiH0t

h ρ(t)e−iH0t

h = ρI(t) (5.49)

Taking the time-derivative gives the equation of motion:

ρI(t) =i

he

iH0th [H1, ρ0 + ρ1(t)]e

−iH0th (5.50)

Since we go to first order in the perturbation f(t), we may neglect the secondterm containing ρ1 in the commutator. Thus, we arrive at

ρI(t) = − i

h[e

iH0th H1e

−iH0th , ρ0] (5.51)

= − i

h[A0(t), ρ0]f(t) (5.52)

(5.53)

whereA0(t) = e

iH0th Ae

−iH0th (5.54)

¿From 5.49, we find

ρ(t) = e−iH0t

h ρIeiH0t

h (5.55)

= e−iH0t

h

[∫ t

−∞

d

dt′ρI(t

′)dt′ + ρ0

]e

iH0th (5.56)

= ρ0 +i

h

∫ t

−∞[A0(t

′ − t), ρ0]f(t′)dt′ (5.57)

to first order in the perturbation. Finally,

δ< B(t) > = Tr((ρ(t)− ρ0)B) (5.58)

=i

hTr[∫ t

−∞[A0(t− t′), ρ0]Bf(t′)dt′

](5.59)

(5.60)

which can be rewritten as

δ< B(t) > =i

h

∫ t

−∞Tr[ρ0[B, A0(t

′ − t)]]f(t′)dt′ (5.61)

=i

h

∫ t

−∞< [B0(t), A0(t

′)] >0 f(t′)dt′ (5.62)

67

Comparing this with the expression found in 5.44, we conclude that

φBA(t− t′) =i

hθ(t− t′) < [B(t), A(t′)] > (5.63)

This is Kubo’s formula. It gives the response φBA(t − t′) of an observableB(t) after a small perturbation has been applied.

68

Chapter 6

Finite temperature AdS/CFT

There is one main reason for studying the correspondence at finite temper-atures: both supersymmetry and conformal invariance are broken at finitetemperature in N = 4 Super Yang-Mills. Furthermore, QCD is expectedto have a deconfining temperature Tdec above which there is a quark-gluon-plasma phase (QGP). There is reasonable hope that some properties of finitetemperature N = 4 Super Yang-Mills theory are shared by the QCD plasma.In the following two paragraphs, we shall describe some properties of Anti-deSitter space and show that at finite temperature, Anti-de Sitter space con-tains black holes. Also, the Hawking temperature in the bulk is identified,through AdS/CFT, with the temperature of the field theory.

6.1 Hawking temperature

Definition The easiest way to determine the Hawking temperature is toexpress the metric in Euclidean time τ = it, and then examine the period-icity of the coordinate. The period is found by requiring that the metricbe regular at the horizon, and this period can be identified with the inversetemperature. The reasoning goes as follows. In ordinary quantum mechan-ics, time evolution is given by the operator U ≡ e−iH(tf−ti), where H is theHamiltonian of the system and tf and ti the final and initial times, respec-tively. The transition amplitude between the two states qi, ti and qf , tf isthen:

< qf , tf |qi, ti >=< qf |e−iH(tf−ti)|qi > (6.1)

This amplitude can also be written as a path integral:

< qf , tf |qi, ti >=

∫q(ti)=qi,q(tf )=qf

Dq(t)eiR tf

tidtL[q(t),q(t)] (6.2)

69

Now let’s make the following replacement: tf − ti → −iβ. At this point,this is just a change of notation (but also, we are going to imaginary time asthe factor of i is there). In particular, β is not (yet) to be interpreted as aninverse temperature. At this stage, it is just a parameter. When we equalizethe right hand sides of 6.1 and 6.2, after this replacement, they become

< qf |e−βH |qi >=

∫q(0)=qi,q(β)=qf

Dq(t)e−R β0 dtELE [q(tE),q(tE)] (6.3)

This is the first result. Now, consider the partition function of a finite tem-perature field theory:

Z[β] = Tr(e−βH) =

∫dq〈q|e−βH |q〉 (6.4)

Here, β is the inverse temperature β = 1T

of the field theory. This is thesecond result. An important observation can now be made. If we take, in6.3, the initial and final states identical, we have

< q|e−βH |q >=

∫q(0)=q(β)

Dq(tE)e−R β0 dtELE [q(tE),q(tE)] (6.5)

But the left hand side is just the partition function of a field theory attemperature T = 1

β! So we can express the partition function as

Z[β] =

∫q(0)=q(β)

Dq(tE)e−R β0 dtELE [q(tE),q(tE)] (6.6)

Z[β] now has a path integral representation over closed euclidean paths wherethe path reaches it starting point after euclidean time β = 1

T. We come to

the conclusion that one can identify a periodicity of imaginary time with theinverse Hawking temperature. This means that if we want to compute theHawking temperature of a black hole, we look at the metric in the imaginary-time formulation and find a periodicity condition on the imaginary time co-ordinate (which usually contains a coefficient) by regarding it as an angularcoordinate. Then, through the argumentation above, we identify the period-icity of tE we just found with the inverse Hawking temperature.

Modification of the metric At finite temperature, the AdS5 part of themetric gets a modification, so that we have:

ds2 =r2

R2(−fdt2 + dx2

1 + dx22 + dx2

3) +R2

r2fdr2 +R2dΩ2

5 (6.7)

70

where

f(r) = 1− r40

r4(6.8)

r0 is a constant with dimensions of length related to the temperature. Forlarge r, the metric is the same than before. Since large r corresponds tothe UV region in the gauge theory, we see that UV physics is unaffectedby the temperature. However, IR physics is very different from the zerotemperature case. Indeed, there is a finite-area horizon at r = r0, whoseHawking temperature is identified with the temperature of the dual CFT.

In our case, the Euclidean metric reads (only the radial and time partsare considered here):

ds2E =

r2

R2

[(1− r4

0

r4

)dt2E

]+R2

r2

(1− r4

0

r4

)−1

dr2 (6.9)

We have euclidean signature, and as a consequence, this metric should notmake sense inside the horizon where r < r0, for the signature will not be eu-clidean anymore. This means that, although there should not be a singularityat the horizon, the metric should ’end up’ there.

Define ρ such thatr = r0(1 + ρ2) (6.10)

Substituting for r in the metric and taking the near-horizon limit ρ→ 0, weget

ds2E = R2

(4r20ρ

2

R4dt2E + dρ2

)(6.11)

This metric must be periodic for it to make sense, and we know determine theperiodicity. We see that it corresponds to a flat plane in polar coordinates,ds2 = dρ2 + ρ2dφ2, where φ = 2r0

R2 tE. However, a flat metric of the form

ds2 = dρ2 + ρ2dφ2 (6.12)

can have a singularity at ρ = 0, depending on the periodicity of the angularvariable φ. Generally, on has

φ ∈ [0, 2π −∆], (6.13)

and for any ∆ 6= 0, ρ = 0 is a so-called conical singularity. Thus ∆ mustbe zero in order to avoid the singularity. This translates in our case thatφ = 2r0

R2 tE is periodic with periodicity 2π, and thus the periodicity of euclidean

time tE is β = πR2

r0. We have found the Hawking temperature of the AdS5

black hole:

TH =1

β=

r0πR2

(6.14)

71

6.2 Entropy density

In this paragraph, the entropy density in large Nc N=4 SYM at strong cou-pling is computed. This quantity can easily be calculated on the String theoryside, using the AdS/CFT correspondence. In the Nc, λ → ∞ limit, we canuse the supergravity approximation. The entropy is then the Bekenstein-Hawking entropy SBH = A

4G. In the metric 6.9, the horizon is at r = r0 and

constant tE, and the area reads

A =

∫d3xd5Ω

√g. (6.15)

For constant r = r0 and constant t,√g =

r30

R3 ×√gS5 . The volume of the

unit 5-sphere being π3, we get:

A =r30

R3× π3R5 × V3 ≡ aV3. (6.16)

where V3 =∫d3x is the infinite volume in the 123-directions. The entropy

density sBH = SBH

V3then can be expressed in terms of reads, using 4.3 and

4.4 to eliminate G and ls:

sBH =a

4G=

1

2πR−6r3

0N2c (6.17)

Since TH = r0

πR2 , this yields

sBH =1

2π2N2

c T3H , (6.18)

independent of the string coupling constant. Remarkably, the entropy densityis finite at infinite coupling λ → ∞. Another interesting fact is that thisexpression differs from the entropy density of a free gas of N=4 particles bya factor of 3/4:

sBH =3

4sfree (6.19)

One could, on the basis of the two results for the entropy density (free andstrongly coupled), extrapolate and try to find a function f(λ) such that

s = f(λ)π2

2N2T 3 (6.20)

with f(λ) satisfying f(∞) = 1, f(0) = 4/3. It is believed that there is asmooth transition of f(λ) from one regime to the other, but the exact form

72

of the function is still unknown. Suffice it to say that corrections have beencalculated to the first order in the two regimes:

f(λ) =4

3− 2λ

π2+ ... λ 1 (6.21)

f(λ) = 1 +15ζ(3)

8λ3/2+ ... λ 1 (6.22)

6.3 Viscosity

In this section, the shear viscosity of theN = 4 plasma is derived. As we haveseen in the previous chapter, the viscosity has an expression in terms of aGreen’s function, and the relation is an example of a Kubo formula. The taskis thus to find this Green’s function. We will see that the retarded Green’sfunction involves the two-point correlator of the energy-momentum tensor ofthe field theory, and, through AdS/CFT, it couples to a source which is themetric in the bulk. Following the AdS/CFT prescription, we should be ableto find this correlator by computing the second functional derivative of thebulk action with respect to the boundary value of the metric tensor. It turnsout that this is not an easy task, and there are several subtleties arising on theway. The main difficulty is that one has to work in the real-time formalism inorder to get the viscosity (real-time retarded Green’s functions), and we willuse the prescription for two-point functions proposed by Son and Starinets.This prescription is unsatisfactory in the sense that it does not follow froman identification of the partition functions on both sides, as it is the casein the imaginary-time, euclidean formulation of AdS/CFT. The reason whyit is used nonetheless is that ’it works’ for two-point functions. As we haveseen in the previous section, the retarded Green’s function determines theresponse of the system when an external source is present which couples tothe current1. We can write the retarded Green’s function as follows:

GRµν(ω, ~q) = −i

∫d4xe−iq·xθ(t)〈[jµ(x), ν(0)]〉 (6.23)

Again, q = (ω, ~q) and x = (t, ~x). Here, the perturbation is at x = 0, and GR

gives the response of the system at x. For the hydrodynamics approximationto apply, we need ω and ~q to be small. The Green’s function can have poles,and we provide here an easy example. Consider Fick’s law,

∂0j0 = D∇2j0 (6.24)

1The existence of a current implies that the theory has a conserved global charge

73

where j0 is the spatial density of the fluid. Then one can consider a solution∼ e−i(−ωt+~q·~x), which gives the dispersion relation

ω = −iDq2 (6.25)

Consequently, the retarded Green’s functions involving j0 have a pole at thisvalue of ω. Recall from chapter 4 that the prescription for calculating two-point functions in the Minkowski formalism, when solving the linearized fieldequations, one has to impose an additional boundary condition, namely theincoming wave boundary condition. One gets a solution of the form φ(z, q) =fq(z)φ0(q)

2, which can be plugged into the retarded Green’s function

GR(q) = A(z)f−q(z)∂zfq(z)|z→0 (6.26)

We start with the non-extremal (finite temperature) black three-brane metricwhich is a solution of the IIB supergravity equations of motion,

ds2 = H−1/2(r)(−f(r)dt2 + d~x2) +H1/2(r)( dr2

f(r) + r2dΩ25

)(6.27)

where H(r) = 1 + R4

r4 and f(r) = 1− r40

r4 . In the zero temperature case, thereis no horizon, r0 = 0, and f(r) = 1. We consider the near horizon limitr R, and we make a change of variables z = r2

0/r2, which amounts the

horizon to be at z = 1 and the boundary at z = 0. This is the metric 6.7,where the coordinate r plays the role of z:

ds2 =r20

R2(−f(z)dt2 + d~x2) +

R2

4z2f(z)dz2 +R2dΩ2

5 (6.28)

For what follows, we leave out the part of the metric describing the five-sphere, since it does not enter the calculations. Since we want to computethe viscosity, we need to find the two-point function of the stress-energytensor Tµν , which enters the Kubo formula for the viscosity:

η = limω→0

1

2ω

∫dtd~xeiωt〈[Txy(x), Txy(0)]〉 (6.29)

What we first need to do is to add a small perturbation to the metric 6.28.

gµν = g(0)µν + hµν (6.30)

2we work in momentum space and use Poincare coordinates

74

Then 6.28 describes g(0)µν . It can then be shown that the Ricci tensor Rµν

can be expanded in a sum where the first term is the Ricci tensor for theunperturbed metric3:

Rµν = R(0)µν +R(1)

µν (h) +R(2)µν (h) + ... (6.31)

Then, the Einstein equations read, to first order:

R(1)µν = − 4

R2hµν (6.32)

The Einstein-Hilbert action gets modified in the presence of a perturbation:

S =π3R5

2κ210

(∫ 1

0

dz

∫d4x√−g(R− 2Λ) + 2

∫d4x√−hK

)(6.33)

where κ10 = 2π5/2R4/N is the ten-dimensional gravitational constant, and itis related to the number N of coincident branes and the gravitational radiusR. The second term contains a negative cosmological constant Λ, and thelast term is the Gibbons-Hawking-York boundary term. It has to be includedin the action defined on a manifold which has a boundary for otherwise thetechnique of varying the action is not well-defined.

Let’s look at the case where hxy 6= 0 and all other components of theperturbation vanish. Starting from the definition of the Riemann tensor,

Rαβµν = ∂µΓα

βν − ∂νΓαβµ + Γα

σµΓσβν − Γα

σνΓσβµ (6.34)

and working to first order in hµν , it is straightforward, but tedious to workout, the field equation for hxy following from 6.32:

h′′xy +1− 3z2

zf(z)h′xy +

R4

4r20zf

2(z)

(f(z)

∂2hxy

∂z2− ∂2hxy

∂t2

)− 1 + z2

z2f(z)hxy = 0 (6.35)

It should be kept in mind that hxy = hxy(t, z, ~x). By defining a new function

φ(t, z, ~x) = zhxy(t,z,~x)

r20

, and then expressing it in terms of a Fourier decompo-

sition,

φ(z) =

∫d4x

(2π)4e−iωt+i~q~xφ(t, z, ~x) (6.36)

eq.6.35 becomes

φ′′k −1 + z2

zf(z)φ′k +

ω2dl − ~q2

dlf(z)

zf 2(z)φk = 0 (6.37)

3This is shown by computing the difference of the covariant derivatives in gµν and g(0)µν .

See the textbook Gravitation by Misner, Thorne, Wheeler, page 966

75

Here we have defined a dimensionless energy ωdl ≡ ω2πT

and dimensionlessmomentum qdl ≡ q

2πT, where T = r0

πR2 is the Hawking temperature. Thisequation does not look easy to solve, and in fact the procedure to find asolution is not going to be outlined here. Rather, we proceed with writingdown the solution and look which form the action takes. The solution ofeq.6.37 which satisfies the incoming-wave boundary condition is

φk(z) = (1− z)−iωdl/2Fk(z) (6.38)

where

Fk(z) = 1−( iωdl

2+ q2

dl

)ln

1 + z

2+ higher order terms (6.39)

We have seen in Chapter 4 that the function Fk(z) has to be regular at thehorizon z = 1. Also, we recall here the ad-hoc definition of the retardedGreen’s function in the Minkowski formulation:

GR(k) = −2F(k, z)|zB(6.40)

whereF(k, z) = K

√−ggzzF−k(z)∂zFk(z) (6.41)

With the expression 6.39 for Fk(z) in hand, we find, in the metric (6.28 andthe limit z → 0:

F(k, z) =−π2N2T 4

8

[ iωdl

2− q2

dl

(1− 1

z

)]+ higher order terms (6.42)

Finally, making use of 6.40 the retarded Green’s function is

Gxy,xy(ω, ~q) = −N2T 2

16(2iπTω + q2) (6.43)

For the viscosity, we need to take ~q = 0:

GR(ω,~0) = −iπωN2T 2

8(6.44)

Plugging this into eq.6.29, we find that the viscosity is

η =π

8N2T 3 (6.45)

76

6.3.1 Viscosity/Entropy ratio

An important observation can now be made. If one takes the ratio of theviscosity to the entropy density, one gets4

η

s=

πN2T 3/8

π2N2T 3/2=

1

4π(6.46)

This ratio seems to be universal: any large N , large λ (i.e. strongly coupled)finite temperature field theory which has a dual gravity description satisfiesthis ratio. Furthermore, the ratio is very small: water, for example, hasη/s ' 380/4π, liquid helium η/s ' 9/4π. There have been found correctionsbeyond the leading order, increasing the ratio. Also, calculations have shownthat it becomes very large in the weak coupling regime. This suggests thepostulation of a lower bound:

η

s≥ 1

4π(6.47)

6.4 M5-brane setup

In this and the following section, we investigate the spacetime induced bya stack of M-branes in 11 dimensions, at temperature T . We also computeentropy density and Hawking temperature. We will see that, similarly tothe D3-brane setup, the spacetime close to the M-branes separates into anasymptotically AdS space and a sphere. Recall that a spacetime is asymptot-ically AdS if it can be conformally compactified into a region whose boundarystructure is the same than that of one half of the static Einstein Universe.Chapter 1 contains much more information about that. At zero temperature,the eleven-dimensional metric is (we are in the euclidean time formulation):

ds2 = H(r)−1/3[dt2 + d~x2] +H(r)2/3[dr2 + r2dΩ24] (6.48)

where H(r) = 1+ R3

r3 . Between the first pair of brackets is the six-dimensionalLorentz metric (in euclidean time formulation), and between the second thefive-dimensional Euclidean metric. The horizon is at r = 0. Near the horizon,where r 1, the metric becomes

ds2 =r

R[dt2 + d~x2] +

(Rr

)2

dr2 +R2dΩ24 (6.49)

4In natural units; restauring the constants gives η/s = ~/(4πkB) = 6.08× 10−13K · s

77

By doing the change of variables r = 4R3

z2 , we see that this metric describesan AdS7 × S4 spacetime:

ds2 =1

z2

[(2R)2(dt2 + d~x2 + dz2)

]+R2dΩ2

4 (6.50)

In contrast with the D3-branes setup, where the radius of the AdS5 was thesame than that of the five-sphere S5, the radius of the AdS7 is twice theradius of the S4. Let’s move on to the more interesting finite temperaturecase. The metric is modified by a function f(r) and becomes

ds2 = H(r)−1/3[f(r)dt2 + d~x2] +H(r)2/3[ dr2

f(r)+ r2dΩ2

4

](6.51)

where f(r) = 1− r30

r3 . We will now compute the Hawking temperature of thisblack hole. The horizon is at r0 = r and the boundary at r = ∞. Near thehorizon, the metric reduces to (again, we consider only the relevant parts ofthe metric (radial and time direction):

ds2 =r

R

(1− r3

0

r3

)dt2 +

R2

r2

(1− r3

0

r3

)−1

dr2 (6.52)

By doing the change of variables r = r0(1 + ρ2), and after that taking thelimit ρ→ 0, the metric has the form of a conical:

ds2 =4

3R2[dρ2 +

9r04R3

ρ2dt2]

(6.53)

Since the metric should be regular at the horizon, we need the angular

variable φ ≡ 3r1/20

2R3/2 t to be 2π-periodic. This amounts for t to have period

β = 4πR3/2

3r1/20

. As we have seen, β can be identified with the inverse Hawking

temperature, thus

TH =1

β=

3

4π

r1/20

R3/2(6.54)

Note that the calculation was exactly the same than in the D3-brane case.

6.4.1 entropy density

We now turn to the calculation of the entropy density. As we have seen fromthe D3-branes case, the entropy density can be expressed in terms of theHawking temperature. Recall that we need to find the area of the horizon,

78

for S = A4G

. We consider the near-horizon metric 6.52, but we include allvariables:

ds2 =r

R

(1− r3

0

r3dt2 + d~x2

)+R2

r2

((1− r3

0

r3

−1

)dr2 + r2dΩ24

)(6.55)

The horizon is at r = r0 and t = const.. This means that the horizon area is

A =

∫d5xd4Ω

√g (6.56)

where√g =

(r0

R

)5/2√gS4 . The volume of the four-sphere is

∫d4Ω = V ol(S4) =

83π2, and

√gS4 =

√R8 = R4. By dividing out the infinite volume directions

d5x, we get

a ≡ A∫d5x

=8

3π2 × (

r0R

)5/2R4 =

8π2r5/20 R3/2

3(6.57)

We can express R in terms of the string length, the string coupling constantand the number of colors of the gauge group N :

R4

l4s= 4πgsN (6.58)

Similarly, G(11), Newton’s constant in eleven dimensions, is expressed interms of the string length and coupling constant. Expressing r0 in termsof TH from 6.54, we find that the entropy density is independent of thestring length and coupling constant, and the

sBH =a

4G(11)

=27π3

36N3T 5 (6.59)

6.5 M2-branes setup

The procedure to follow is exactly the same than before. The metric reads

ds2 = H(r)−2/3[dt2 + dx2 + dy2] +H(r)1/3[dr2 + r2dΩ27] (6.60)

where H(r) = 1 + R6

r6 . At finite temperature, the metric becomes

ds2 = H(r)−2/3[f(r)dt2 + dx2 + dy2] +H(r)1/3[dr2

f(r)+ r2dΩ2

7] (6.61)

79

where f(r) = 1− r60

r6 . The Hawking temperature is

TH =3

2π

r20

R3(6.62)

The entropy density is

sBH =8√

2π2

27N3/2T 2 (6.63)

6.6 CFT on Sn−1 × S1

There is a reason why we consider conformal field theories on Sn−1 × S1, orRn−1 × S1. It has been shown by Hawking and Page that this manifold isthe boundary of two possible n+ 1-dimensional manifolds: A quotient spaceisomorphic to Z, and a Schwarzschild black hole in AdSn+1. The conformalfield theory on the boundary can then be studied by taking into account bothbulk spacetimes. However, it turns out that at high temperatures, only theAdS black hole contributes and the boundary becomes Rn−1 × S1. Since weare interested in the high temperature regime, we will focus on that space.The AdS black hole metric reads

ds2 =(r2

b2+ 1− wnM

rn−2

)dt2 +

1(r2

b2+ 1wnM

rn−2

)dr2 + r2dΩ2, (6.64)

where

wn =16πGN

(n− 1)V ol(Sn−1)(6.65)

with GN the N-dimensional gravitational constant and V ol(Sn−1) the volumeof the unit n− 1-sphere. b is the radius of curvature of Anti-de Sitter space.The factor wn is there in order for M to be the mass of the black hole. Tosee this, let’s do some dimensional analysis. As we saw earlier, the units of Gdepend on the dimension. Since the quantity wnM

rn−2 needs to be dimensionless,the inclusion of wn effectively gives M the dimension of mass. Hawking andPage also found that the above spacetime is restricted to the region r > r+,where r+ is the largest solution to the equation

r2

b2+ 1

wnM

rn−2= 0 (6.66)

The period of t needs to be

β0 =4πb2r+

nr2+ + (n− 2)b2

(6.67)

80

For large M , this reduces to

β0 =4πb2

n(wnb2)1nM

1n

(6.68)

Now we want to change the metric in such a way that we have asymp-totically (i.e. at the boundary) Rn−1 × S1 instead of Sn−1 × S1. To do so,we make the following change of variables:

r =(wnM

bn−2

) 1nρ, t =

(wnM

bn−2

)− 1nτ (6.69)

The prefactor in the metric becomes, for large M:

r2

b2+ 1− wnM

rn−2−→

(wnM

bn−2

)n2(ρ2

b2− bn−2

ρn−2

)(6.70)

Furthermore, the period for τ is computed easily:

τ =

(wnM

bn−2

) 1n

t =(wnM

bn−2

) 1n

(t+ β0) (6.71)

= τ +

(wnM

bn−2

) 1n

β0 (6.72)

Thus, the period is β1 =

(wnMbn−2

) 1n

β0. For large M, we use expression () for

β0 to find

β1 =4πb2

n(6.73)

In the new variables ρ, τ , the metric reads

ds2 =(ρ2

b2− bn−2

ρn−2

)dτ 2 +

(ρ2

b2− bn−2

ρn−2

)−1

dρ2 +(wnM

bn−2

) 2nρ2dΩ2 (6.74)

In the last term, we see that the radius of the Sn−1 goes like (ρM1n )2 instead

of the usual ρ2. Consequently, as M → ∞, the space looks locally flat. Wecan introduce coordinates yi on a patch of Sn−1, such that at some pointon this patch, we have dΩ2 =

∑i dy

2i . Finally, making the variable change

yi =(

wnMbn−2

)− 1n, we get for the metric

ds2 =(ρ2

b2− bn−2

ρn−2

)dτ 2 +

(ρ2

b2− bn−2

ρn−2

)−1

dρ2 + ρ2

n−1∑i=1

dx2i (6.75)

81

This space goes to Rn−1 × S1 as ρ→∞, which is what we were looking for:an AdS space which has an Rn−1 × S1 boundary. To summarize: we wouldlike to have an n+1 dimensional AdS-space with a boundary on which wecan define a conformal field theory. Hawking and Page provided us with asuitable AdS boundary Sn−1×S1. There are two different bulk spaces havingthis boundary, one of which is the Schwarzschild AdS black hole. At largetemperatures, only the black hole solution contributes.

6.7 Confinement and AdS/CFT

In this section, I will briefly discuss a very interesting issue in several gaugetheories and how AdS/CFT comes into play. The discussion will remain qual-itative, for a detailed review would be out of the scope of this thesis. Untilnow, we have focused on the simplest example of the AdS/CFT correspon-dence: IIB string theory on AdS5 × S5 dual to N = 4 Super Yang-Mills onthe AdS5 boundary. N = 4 Super Yang-Mills is very different from quantumchromodynamics, the theory of the strong interactions, the main differencebeing that it does not exhibit confinement (an explanation of this will fol-low). In order to make contact with the ’real world’, it would be nice to havea gauge theory with a gravity dual that shares properties with QCD. This iswhat we are discussing in this section.

Confinement, deconfinement Here we will only mention a couple of no-tions related to gauge theories exhibiting a deconfining phase. For a rigorousapproach to such theories in general or the theory of the strong interactionsin particular, the reader is invited to consult one of the many field theorybooks on the market. Ref.[] is a very good starting point.

Quantum chromodynamics (QCD) is a gauge theory whose gauge fieldsare the gluons and whose degrees of freedom are the quarks. The gaugegroup is SU(3). Quarks are charged under the SU(3); their charge is referredto as color. The hadrons we observe in Nature are ’color-neutral’: they arecomposed of quarks in such away that the color charge of the constitutingquarks sums up to zero. Since there are no neutral-colored quarks, one neverobserves a single quark in Nature. This fact is known as color confinement. Itshould be noted that there exists no rigorous proof that QCD should exhibitconfinement; a price of one million dollars is to be handed out to the personproducing a proof. At extremely high temperatures or densities, one suspectsa phase transition to occur from the confining phase to a deconfining phasein which the quarks and gluons are ’free’. This phase is known as the Quark-gluon plasma and is highly sought after in current particle experiments. The

82

theoretical indication that a phase transition should happen at about 175MeV comes from lattice simulations or lattice gauge theory. In fact, the colorcharge is too large to allow the consistent use of perturbation theory such asfor QED. In lattice gauge theory, spacetime is discretized on a lattice andone is then able to carry out simulations on the computer and get meaningfulresults. So a phase transition is expected to happen, and here we would liketo investigate a gauge theory having both this property and a gravity dual.By doing so, we will be able to see to what the phase transition in the gaugetheory corresponds in the dual gravity theory in the bulk. Our starting pointwill be a stack of near-extremal D4 branes, as opposed to the D3-branes weconsidered previously. On these D4-branes lives a SU(N) Super Yang-Millstheory in five dimensions. Let’s now compactify one of the space dimensions,so that we have a four-dimensional theory. The compactification is done bygiving one of the four space coordinate (anti-)periodicity conditions, i.e. wecompactify it on a circle S1 of given length L. After imposing antiperiodicboundary conditions, supersymmetry is broken, and furthermore fermionsacquire mass of order 1

L. Moreover, the scalar particles also acquire mass

through quantum effects. The zero modes of the gauge fields are the onlydegrees of freedom that remain massless. At energies E M , the massivedegrees of freedom are not present, so we have a Yang-Mills theory containingonly massless particles. This comes close to QCD, where at the deconfiningenergy scale, the quarks can be considered massless, so that in that regime,QCD effectively has no massive degrees of freedom either. In addition, QCDis not supersymmetric, just like the effective four-dimensional theory we justconstructed. What is the bulk supergravity theory that has near-extremalD4-branes as a boundary? The correct metric is

ds2 =( rR

)3/2(−f(r)dt2+dx2

1+dx22+dx

23+dy

2)+(Rr

)3/2 dr2

f(r)+R3/2r1/2ds2(S4)

(6.76)

with f(r) = 1− r30

r3 . Let us compute the Hawking temperature for this blackbrane solution. As usual, we are interested in the radial and time directiononly, and we switch to euclidean signature:

ds2 =( rR

)3/2

f(r)dt2 +(Rr

)3/2 1

f(r)dr2 (6.77)

After a change of variable r = r0(1 + ρ2) and taking the near-horizon limitρ→ 0, the metric is

ds2 = 4R3/2r1/20

(dρ2 +

3r04R3

ρ2dt2)

(6.78)

83

giving a Hawking temperature of

TH =1

β=

√3r0

4πR3/2(6.79)

We come to an important point now: if in the metric 6.77 we interchangethe variables t and y, the resulting metric,

ds2 =( rR

)3/2(dt2 + d~x2 + f(r)dy2

)+(Rr

)3/2 dr2

f(r)(6.80)

is a solution of the supergravity equations, too! The Lorentzian version ofthis metric is of course obtained by placing a minus sign in front of the timecoordinate. But this metric, as similar as it may look compared to 6.77,differs strongly from the latter: it has no horizon, whereas in 6.77 there isone at r = r0. This no-horizon metric is known under the name AdS-soliton.This means we have now two different metrics describing the same brane-configuration, and consequently the same gauge theory living on the brane.In the AdS/CFT language, we say that the low-energy string partition func-tion (i.e. the supergravity partition function) in the bulk possesses two saddlepoints. Their respective contributions to the thermal partition function ise−βFiV , i = 1, 2, and Fi is the free energy density of the particular solution.We can take the ratio to compare the contributions:

ebetaF1V

e−βF2V= e−β∆FV (6.81)

The volume V in which the gauge theories are defined is infinite, so for anyfinite free energy difference ∆F , one contribution is completely suppressedwith respect to the other. This means that there is a critical point at F1 =F2, and any deviation from this point completely suppresses one of the twogeometries. A phase transition occurs at the critical point, and it is knownat the Hawking-Page phase transition. As one can suspect, this corresponds,on the gauge theory side, to the confinement/deconfinement transition. Theconfinement phase corresponds to the AdS-soliton, and the deconfining phaseto the black hole.

6.8 Correlators of the R-charge current and

the diffusion constant

The R-charge is related to the R-symmetry, on which we will say nothingmore than that it is, by definition, a symmetry which does not commute

84

with the supersymmetries. The R-charges have a dual field in the bulk: theMaxwell gauge fields F a

µν . We use the ten-dimensional , finite temperaturemetric

ds2 =r20

R2

(−f(z)

zdt2 + d~x2

)+

R2

4z2f(z)dz2 +R2dΩ2

5 (6.82)

The action we need is

S ∼∫d5x√−gF a

µνFµνa (6.83)

We work in Fourier space,

Ai =

∫d4q

(2π)4e−iωt+i~q·~xAi(q, z) (6.84)

What we need to do is to solve Maxwell’s equations for the bulk gauge field,DµF

µν = 0, with the boundary condition limz→0Aµ = A0µ. For the action

above, we get

1√−g

∂ν

(√−ggµρgνσ(∂ρAσ − ∂σAρ)

)= 0 (6.85)

For a more convenient notation, from now on we replace ω → R2

2r0ω and

q → R2

2r0q. If we consider the spatial momentum boing aligned along the

x3-axis, q = (ω, 0, 0, q), the equations of motion are:

ωA′t + qfA′3 = 0 (6.86)

A′′t −1

zf(q2At + ωqA3) = 0 (6.87)

A′′3 +f ′

fA′3 +

1

zf 2(ω2A3 + ωqAt) = 0 (6.88)

A′′α +f ′

fA′α +

1

zf

(ω2

f− q2

)Aα = 0 (6.89)

The last equation applies for the x and y components, α = x, y. From 6.88,we have

A3 = −zf2

w2

[A′′3 +

f ′

fA′3 +

wq

zf 2At

](6.90)

We can replace A′3 and A′′3 in this equation by using 6.86 which states

A′3 = − ω

qfA′t (6.91)

85

and, taking the derivative of 6.91,

A′′3 =wf ′

qf 2A′t −

w

qfA′′0 (6.92)

Now we can plug these expressions for A′3 and A′′3 back into 6.90, so that wehave an expression for A3 in terms of At and A′′t :

A3 = −zqωAt +

zf

qωA′′t (6.93)

Taking the derivative of 6.87,

A′′′t +(uf)′

(uf)2(q2At + wqA3)−

1

uf(q2A′t + wqA′3) = 0 (6.94)

and replacing A3 and A′3 by their expressions in terms of At and A′t givenabove, we finally find the following third-order equation for At:

A′′′t +(zf)′

zfA′′t +

ω2 − q2f

zf 2A′t = 0. (6.95)

To solve this equation, we firstly solve for A′t. The point z = 1 is singular(since f(1) = 0), and the rule for solving the equation goes that first onehas to find the behavior near the singular point. If we substitute A′t =(1 − z)νF (z) into 6.95, with F (z) regular, we find that F (z) satisfies thefollowing complicated equation:

F ′′ +(1− 3z2

zf+

iω

1− z

)F ′ +

iω(1 + 2z)

2zfF +

ω2(4− z(1 + z)2)

4zf 2F − q2

zfF = 0

(6.96)In addition, there are two allowed values for ν: ν± = ±iω/2. This is wherethe incoming wave boundary condition becomes necessary, which tells usthat we need to choose ν−. Let’s go on and solve eq. 6.97. We consider thelow-energy, long-wavelength limit, i.e. we take ω and q2 small. Then we canexpand the solution in terms of these two parameters:

F (z) = F0 + ωF1 + q2G1 + ω2F2 + ωq2H1 + q4G2 + ... (6.97)

We need the first three term only for the two-point function to be correct tofirst order:

F0 = C (6.98)

F1 =iC

2ln

2z2

1 + z(6.99)

G1 = C ln1 + z

2z(6.100)

86

Now we substitute the expression for A′t back into eq.(129) and take thelimit z → 0. We shall just state the result here, since the technical detailsshould not overlap the main line of reasoning. From substituting A′t = (1−z)−iω/2F (z) into 6.95 with F (z) given in 6.97 by the first three terms, we getthe value of the constant C in terms of the boundary values A0

t , A035:

C =q2A0

t + ωqA03

Q(ω, q)(6.101)

where Q(ω, q) = iω − q2 + higher order terms. The constant C being de-termined, we have found an exact expression for the solution A′t(z). Thisimplies that we also know A′3(z). Finally, near the horizon, z = ε 1, allthe components of A′ are expressed in terms of boundary values A0, whichis what we wanted. Now we apply the prescription for the retarded Green’sfunction,

GR = −2F(k, z)|zB(6.102)

with F(k, z) = K√−ggzzF−k(z)∂zFk(z). This means that we have the re-

tarded Green’s function expressed in terms of the components of A′, whichin turn are expressed in terms of ω and q. For example, to first order, wehave

Gabtt =

N2Tq2δab

16π(iω −Dq2)+ ... (6.103)

As expected, there is a pole at ω = −iDq2, confirming the result gettingfrom pure hydrodynamics.

6.9 The membrane paradigm

Let us now compute the diffusion constant D with a different method. Themembrane paradigm states that the event horizon of a spherically symmetricblack hole can be viewed as a conducting spherical surface. To get an intu-itive feeling for this, let us turn to an example of classical electrodynamics.Imagine that one approaches a positive charge towards a conducting, un-charged metallic sphere. By doing so, the free negatively charged electronsin the sphere accumulate at a surface region closest to the positive charge:the sphere is polarized. The analogy with black hole physics is as follows: Ifone computes the field line configuration of a positive charge in the vicinityof a black hole, it turns out that the field topology resembles a lot to the

5limz→0 At(z) = A0t , limz→A3(z)=A0

3

87

aforementioned metallic sphere. Indeed, one can view the (spherically sym-metric) event horizon as a thin spherical shell or stretched horizon. This shellhas a zero net charge, but if a charge approaches it, it becomes polarized.

The fact that there exist hydrodynamic modes in thermal field theorycomes from the existence of poles of the retarded correlators on the gravityside. Now, if gravity counterparts of the hydrodynamic normal modes exist,there must exist linear gravitational perturbations of the metric that havethe dispersion relation identical to that of the shear hydrodynamic mode,ω ≡ −iq2, and of the sound mode, ω = csq − iγq2. Let us construct thegravitational counterpart of the shear mode.

We use the following metric:

ds2 = gttdt2 + grrdr

2 + gxxd~x2 (6.104)

The metric we take is seen to be diagonal; furthermore, it should have ahorizon at r = r0 near which

gtt = −γ0(r − r0) (6.105)

andgrr =

γr

r − r0(6.106)

where γ0 and γr are some positive constants. This metric is the metric of adimensionally reduced background on which gravitational/vector fields prop-agate. It can be shown that this is equivalent to the problem of analyzingy-independent tensor/vector perturbations in the non-reduced background,where y represents coordinates that parametrize some d-dimensional compactspace. The existence of an event horizon is a consequence of the dimensionalreduction. Let’s compute the Hawking temperature for this metric. In eu-clidian time, we have

ds2 = γ0(r − r0)dt2E +

γr

r − r0dr2 (6.107)

where we ignore the non radial part which is not relevant for the computation.Again, we define a variable ρ such that

r = r0(1 + ρ2) (6.108)

Substituting this into the metric, we get (for any ρ):

ds2 = 4r20γr

( γ0

4r20γr

ρ2dt2E + dρ2)

(6.109)

88

which has the formds2 = dρ2 + ρ2dφ2 (6.110)

where φ =√

γ0

4r20γrtE. Giving φ a 2π-periodicity, we get a periodicity for tE

which we identify with β, the inverse Hawking temperature. This impliesthat

TH =1

4πr0

√γ0

γr

(6.111)

This is also the Hawking temperature for the full D-dimensional black-branebackground metric, which is

ds2 = g00(r)dt2 + grr(r)dr

2 + gxx(r)

p∑i=1

(dx)2 + Z(r)Kmn(y)dymdyn (6.112)

where p is the number of non-compact spatial dimensions, and Z(r) is aso-called ”warping factor”.

The dynamics of the vector perturbations is governed by Maxwell’s action

Sgauge =

∫dp+2x

√−g( 1

g2eff

F µνFµν

)(6.113)

where geff is an effective gauge coupling that is a function of the radialcoordinate r if Z(r) is not constant, i.e. if the compact space does notfactorize.

Maxwell’s equations follow directly from the action and read

∂µ

( 1

g2eff

√−gF µν

)= 0 (6.114)

We take geff constant so that Maxwell’s equations become ∂µ

(√−gF µν

)=

0. The results for the r-dependent geff can be restored by replacing√−g by√

−g

g2eff

in the final answers.

We want to find the dispersion law ω = ω(q). Therefore we considerfluctuations of the gauge field that behave as Aµ ≡ e−iωt+i~q~x; we also needappropriate boundary conditions. If for q → 0, ω(q) = −iDq2 with Dconstant, there is diffusion and D is the diffusion constant. Instead of solvingMaxwell’s equations like we did before, we proceed differently this time.Indeed, we want to find a conserved current jµ, because this current canbe related to Fick’s law ji = −D∂ij

0 from which we deduce the diffusionconstant D. Our task is thus to prove the validity of Fick’s law.

89

Before we can define the current, we need to introduce the stretchedhorizon. This is, in our case, a flat spacelike surface located at a constantr = rh slightly larger than r0, the location of the event horizon:

rh = r0 + ε, ε r0 (6.115)

It can be shown that the current which is associated with the stretchedhorizon is

jµ = nνFµν |rh

(6.116)

where nν is a normal vector along the r-direction, i.e. perpendicularly to thestretched horizon. The antisymmetry of F µν implies nµj

µ = 0, the currentis parallel to the horizon. Furthermore, Maxwell’s equations together withthe fact that all metric components gµν depend on r only imply that jµ isconserved for any rh: ∂µj

µ = 0. The components of the current near thehorizon read

j0 = F 0rnr = − F0r

γ0γ1/2r (rh − r0)1/2

(6.117)

ji = F irnr =(rh − r0)

1/2

gxxγ1/2r

Fir (6.118)

We will show now that if Aµ varies slowly in space and time, then F0i isproportional to ∂iF0r for appropriate choices of rh, and thus that relations(90) and (91) imply Fick’s law. Before doing that, we need discuss theboundary conditions. Let us choose the wave vector along the x ≡ x1 axis:

Aµ = Aµ(t, r)eiqx (6.119)

We assume q TH ; generally q is arbitrarily small if we consider the hy-drodynamic limit. We now impose incoming-wave boundary conditions onthe horizon: waves can be absorbed by the horizon but not emitted. Thisaccounts for the violation of time reversal of Fick’s law. The situation is ana-logue to the relation ~B = −~n× ~E for plane waves on a nonreflecting surfacein classical electrodynamics. The relevant Maxwell equations and Bianchiidentities are

g00∂tF0r − gxx∂xFrx = 0 (6.120)√−gg00gxx∂tF0x + ∂r(

√−ggrrgxxFrx) = 0 (6.121)

∂r(√−ggrrg00F0r) +

√−ggxxgoo∂xF0x = 0 (6.122)

∂tFrx + ∂xFor − ∂rF0x = 0 (6.123)

90

Taking the time-derivative of 6.120, substituting ∂tFrx from 6.123, and finallyusing ??, we get the following equation:

∂2t F0r − iqg00g

xx∂rF0x − q2g00gxxF0r = 0 (6.124)

Similarly, taking the time-derivative of 6.121 and substituting ∂tFrx from6.123, we obtain:

∂2t F0x + g00∂r(g

rr∂rF0x)− g00∂r(grr∂xF0r) = 0 (6.125)

Assume that the fields vary over a typical time-scale Γ−1, so that ∂2t ≡ Γ2.

In the near-horizon region, r − r0 r0, and equation 6.124 simplifies to

F0r ≡r − r0r0

q

Γ2∂rF0x (6.126)

If we haver − r0r0

Γ2

q2, (6.127)

we can neglect the third term on the left hand side in 6.125, which reducesto a ”wave equation”:

∂2t F0x −

γ0

γr

(r − r0)∂r[(r − r0)∂rF0x] = 0 (6.128)

The general solution of this equation is

F0x(t, r) = f1

[t+

√γr

γ0

ln(r − r0)]

+ f2

[t−√γr

γ0

ln(r − r0)]

(6.129)

f1 and f2 are arbitrary functions. The incoming-wave boundary conditionsimply

∂tF0x =

√γ0

γr

(r − r0)∂rF0x (6.130)

Finally, from 6.123 and 6.126, we find that

Frx −√γr

γ0

F0x

r − r0(6.131)

is independent of t. Since we expect the solution to decay as t → ∞, thisexpression is zero, and we have found that

Frx =

√γr

γ0

F0x

r − r0(6.132)

91

We are now in the position to derive Fick’s law. We work in the radialgauge Ar = 0. For q = 0, A0 satisfies the Poisson equation (see 6.122)

∂r(√−ggrrg00∂rA0) = 0 (6.133)

The solution with the boundary condition A0(r) = 0 for r →∞ is

A0(r) = C0

∫ ∞

r

dr′g00(r

′)grr(r′)√

−g(r′)(6.134)

Consequently, the ratio A0

F0rapproaches a constant near the horizon:

A0

F0r

|r=r0=

√−g(r0)

g00(r0)grr(r0)

∫ ∞

r0

drg00(r)grr(r)√

−g(r)(6.135)

This yields Fick’s law:

jx = − F0x

gxx

√γo(r − r0)

(6.136)

=∂xA0

gxx

√γ0(r − r0)

(6.137)

=(A0

F0r

) ∂xF0r

gxx

√γo(r − r0)

(6.138)

= −D∂xj0 (6.139)

where the diffusion constant D reads

D =

√−g(r0)

gxx(r0)g2eff (r0)

√−g00(r0)grr(r0)

∫ ∞

r0

dr−g00(r)grr(r)g

2eff (r)√

−g(r)(6.140)

for a gauge field with r-dependent coupling. As stated above, for a constantcoupling, just remove the g2

eff ’s. Taking the black three-brane metric, whichwe recall is

ds2 =r2

R2(−fdt2 + d~x2) +

R2

r2fdr2 +R2dΩ2

5 (6.141)

with f = 1− r40

r4 , and taking geff =constant (R-charge current), the diffusionconstant reads

D =1

2πT(6.142)

which is in agreement with the AdS/CFT computation.

92

Chapter 7

Conclusion

In this thesis, we have seen how the AdS/CFT correspondence permits usto get information on strongly coupled gauge theories. This is a remarkablefact: for the first time, it is possible to get analytical results in the strongcoupling regime of a gauge theory. To achieve this goal, we switched to thebulk supergravity theory, computed the quantities there (subject to boundaryconditions), and found the results for the gauge theory through AdS/CFT.The importance of the AdS/CFT duality cannot be overemphasized, andeven though a proof is still missing, AdS/CFT is believed to hold in allregimes, that is, for all values of the ’t Hooft coupling λ. In this thesis, onlya part of ’checks’ have been done, and many more are available, all of themsupporting the validity of AdS/CFT. The main difficulty in finding a proofof the conjecture is that there is no known way of defining string theory otherthan in a perturbative manner. Nonperturbative properties are known, buta precise definition in a non-perturbative way is still lacking. So a proofof AdS/CFT would involve non-perturbative properties of the string theory,which makes the task very difficult. In this thesis, we have only dealt withAnti-de Sitter spacetimes. It would be interesting to see inasmuch one canextend the duality to include de Sitter spaces. Part of the reason why oneshould try to find a dS/CFT, or even a dS/nonCFT correspondence comesfrom the observation that we live in a de Sitter space with a close to zero,but positive cosmological constant.

93

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