Review of Holographic RG - UvA · 2020-06-01 · Review of Holographic RG Hideto Kamei Master’s thesis Supervisor: Prof. Erik Verlinde University of Amsterdam Institute for Theoretical

Review of Holographic RG

Hideto Kamei

Master’s thesis

Supervisor: Prof. Erik Verlinde

University of AmsterdamInstitute for Theoretical Physics

Valckenierstraat 651018 XE Amsterdam

The Netherlands

August 3 2008

Contents

1 Introduction 5

2 Review of Renormalization Group 72.1 The idea of renormalization group . . . . . . . . . . . . . . . . . 72.2 Wilsonian RG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.2.1 Example: balls in 2 dimensional box . . . . . . . . . . . . 82.2.2 Example: Wilsonian RG for scalar field theory . . . . . . 112.2.3 Wilsonian RG for field theory : take perturbation into

account . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.3 Stuckerberg-Petermann RG (Field theoretical RG) . . . . . . . . 162.4 RG-flow equation (Callan-Symanzik equation) . . . . . . . . . . . 19

2.4.1 Example: Balls in the box . . . . . . . . . . . . . . . . . . 202.4.2 Example: partition function of field theory . . . . . . . . 222.4.3 Characteristic equation . . . . . . . . . . . . . . . . . . . 222.4.4 Solving RG flow equation . . . . . . . . . . . . . . . . . . 23

3 AdS-CFT correspondence 253.1 AdS space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.1.1 various parametrization of AdS space . . . . . . . . . . . 253.1.2 boundary of AdS space . . . . . . . . . . . . . . . . . . . 27

3.2 ABC of Conformal field theory . . . . . . . . . . . . . . . . . . . 273.2.1 Why Conformal field theory? . . . . . . . . . . . . . . . . 273.2.2 Conformal invariance = (local) scale invariance . . . . . . 283.2.3 scaling of operators . . . . . . . . . . . . . . . . . . . . . . 293.2.4 conformal anomaly . . . . . . . . . . . . . . . . . . . . . . 30

3.3 AdS/CFT correspondence . . . . . . . . . . . . . . . . . . . . . . 313.4 Natural relation between AdS and CFT . . . . . . . . . . . . . . 32

4 dealing with divergences : Holographic RG 354.1 dealing with divergences . . . . . . . . . . . . . . . . . . . . . . . 354.2 Recipe for computing correlation function . . . . . . . . . . . . . 36

1

2 CONTENTS

5 Wilsonian way of Holographic RG 415.1 first step towards Holographic RG . . . . . . . . . . . . . . . . . 41

5.1.1 Correlation function . . . . . . . . . . . . . . . . . . . . . 415.1.2 Flow of the metric and external fields . . . . . . . . . . . 42

5.2 ADM formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 425.3 Hamilton-Jacobi equation . . . . . . . . . . . . . . . . . . . . . . 47

5.3.1 the method . . . . . . . . . . . . . . . . . . . . . . . . . . 475.3.2 The method tells us about supergravity action . . . . . . 49

5.4 Get RG flow equation out of AdS . . . . . . . . . . . . . . . . . . 495.5 Holographic RG and conformal anomaly . . . . . . . . . . . . . . 525.6 Holographic RG and our universe . . . . . . . . . . . . . . . . . . 535.7 R anomaly from Hamilton-Jacobi method . . . . . . . . . . . . . 54

6 Method of ’new’ Holographic RG 57

7 Holographic c-function 617.1 c-function in CFT . . . . . . . . . . . . . . . . . . . . . . . . . . 617.2 c-function in AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . 637.3 meaning of Holographic c-function . . . . . . . . . . . . . . . . . 647.4 Support for Holographic principle . . . . . . . . . . . . . . . . . . 65

7.4.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . 657.4.2 Similarity of Black hole and light cone . . . . . . . . . . . 657.4.3 Thermodynamics of spacetime . . . . . . . . . . . . . . . 67

7.5 Meaning of holographic c-theorem . . . . . . . . . . . . . . . . . 73

8 future developments 75.1 additional notes on Holographic RG . . . . . . . . . . . . . . . . 77.2 derivation of the curvature formula . . . . . . . . . . . . . . . . . 78

CONTENTS 3

Abstract

There is an idea that gravity theory can be described by one dimensionallower quantum theory. AdS/CFT correspondence is an important example forholography for special class of gravity theory and gauge theory. HolographicRG is one of the most important ideas in AdS/CFT, and claims that each slicesin gravity theory (in AdS space) encodes description of the same gauge theory(CFT) at different scale, therefore the one extra dimension in the gravity theoryhas redundant information.

This thesis introduces various methods to relate the scale of observation ingauge theory, and extra dimension in AdS gravity. After short review of renor-malization group method, which is the tool to analyze the change of observationscale, and review of AdS/CFT correspondence, we will see that we need to reg-ularize the gravity action with respect to the extra coordinate in AdS, just aswe have to regularize field theory with respect to the cutoff scale.

Next, we will see, the equation which governs the change of the scale, willbe derived from ’moving’ extra direction in AdS space.

Additionally, we will also introduce another approach of Holographic RG.Lastly, I will explain the idea of holographic c-theorem. c-theorem is a proof

in 2 dimensional CFT that RG transformation is irreversible, by showing wecan define c-funcion which monotonically decrease as we go to larger scale. It isnatural, since we lose information (and cannot recover) when we derive macro-scopic effective theory by some approximation. In gravity picture, shrinkinglight cone would be corresponding to deriving larger scale physics in CFT. Wecan define holographic c-theorem as expansion rate of the light-cone and showit monotonically decrease when they shrink.

4 CONTENTS

Thanks

I would like to thank all the people who helped me with my thesis. Firstof all, I would like to thank my supervisor, Erik Verlinde. He was nice enoughto allow me to do some project from the first year, and it was really fun. Healso taught me lots of interesting insights into spacetime and gravity. Otherprofessors, Jan de Boer and Kostas Skenderis taught me the idea of holographicRG. I was really lucky because the people in this institute was exactly the peoplewho invented the field. In the institute, Jan Smit and Bernard Nienhuis taughta lot about RG. I also would like to thank people who advised me on my thesis,Albion Lawrence, Tadakatsu Sakai. I would like to thank people who advisedme and checked my thesis, Thomas, Balt, Pierre, and my roommate, Sander,Joorn, taught me about latex program. And, I also thank Kasper, Coco, Milena,Jelena for helping me out a lot. Also thanks to friends in Japan who are enjoyinggood sunshine.

Chapter 1

Introduction

After ’t Hooft [1] and Susskind [2] proposed holographic principle, which claimsthat the degree of freedom in theory of gravity can be considered to live onits surface, many developments were done to know the true degree of freedomof gravity theory. AdS/CFT, which is discovered by Maldacena [3], gave anexcellent experimental field for theoretical physicists to know more about theidea holography. The natural question would be, ’how gravity theory and onedimensional less quantum theory could have exactly same degree of freedom?’.This question could be ansewered by the idea of Holographic RG. If you takearbitrary 4 dimensional slice of AdS space (with gravity), each of these slicesis corresponding to the quantum theory seen in different scales. Therefore, thetrue degree of freedom can be considered as living on the boundary of AdSspace, and the inside of AdS describes larger scale, more rough picture of thesame physics. Therefore, holography can be given a natural interpretation asRG flow, in which you derive larger scale effective physics from the microscopicmodel.

This way of viewing physics is fun; 4 dimensional spacetime coordinate in ourworld and ’scale’ parameter to view the physics is, in dual AdS picture, mixedwith each other, and scale and space-time is on the equal footing. Therefore, itis natural to say that scale could be interpreted as fifth dimension in the world.

The study of Holographic RG is not just phylosophical appeal to claim fivedimension, but more practical use. By comparing the theory of gravity withthe RG picture, we can know a lot about gravity picture. We will see, lookingat so-called c-function, we can know more about how information in gravity isencoded, and how it relates to scale transformation picture. On the other hand,to understand universal structure of low energy quantum physics, we may beable to apply gravity analysis.

Importance of Renormalization group, which is a tool to see larger scale ofthe system, is crucial in modern physics. For example, deriving large scale theoryof QCD from microscopic model of QCD, or obtaining true microscopic theoryfrom ’macroscopic’ description of Standard model, are important problems inphysics, and these behavior of renormalization group may be understood by

5

6 CHAPTER 1. INTRODUCTION

studying gravity theory. The analysis of Holographic RG may teach us morethan just the flow.

Chapter 2

Review of RenormalizationGroup

2.1 The idea of renormalization group

The idea of Renormalization was invented when Quantum Electrodynamics wasdeveloped. At the beginning it was thought of as a mysterious technique to’renormalize’ infinite results into finite sensible physical quantities. However,people found more natural interpretation as ’viewing the physics in differentscales’, and the transformation to study the change of the scale is called ’renor-malization group’. In this thesis, we frequently abbreviate it as RG. The basicidea is to derive macroscopic behavior from microscopic model. Deriv-ing macroscopic behavior is not so trivial as some may think.

What is the use of the method? The first use is in studying phase transition.phase transition is the phenomena where slight change of microscopic param-eter results in dramatic change of macroscopic behavior. The ’group’ propertyof RG enables us to study how microscopic effect is handed into larger and largerscale as we change the observation scale gradually.

RG is also useful in studying near scale invariant point. If the system isalmost scale invariant, we can use perturbation theory to study deviation fromthe system. Actually, the we will see that at the point of phase transition, wealso have scale invariance, and this suggest we can know a lot about physicsnear the point of phase transition.

Last but not least, RG is really crucial in deriving general behavior of macro-scopic scale. Even though you have a bit different microscopic model, it oftenhappens that they behave almost the same in the macroscopic limit. Even ifyou have copper or iron or any metal, these behavior as macro is extremely sim-ilar. This emergence of macroscopic general behavior is called as ’universality’.This universality is the reason why we can explain our world with such simplemodels. The excellent introduction is done by [4].

For what kind of problem can we apply RG? In addition to field theory,

7

8 CHAPTER 2. REVIEW OF RENORMALIZATION GROUP

condensed-matter physics, it can be applied to biology, economics, and every-where there are ’micro’ and ’macro’ scales.

Figure 2.1: Example of scale invariance

There are two different ideas of renormalization group. One is Wilsonian(condensed-matter) RG, and the other is Stuckelberg-Petermann (field theoret-ical) RG.

In condensed matter physics, the appropriate microscopic model is known,and we wish to derive macroscopic physics from it. That is the Wilsonian RG.We see the physics in a bigger and bigger scale, and see which effects becomesbigger and dominant in macroscopic physics.

In high energy physics, on the contrary, usually we don’t know the under-lying microscopic model itself, but we can guess from observation what kind ofmicroscopic model could we have. There are infinitely many models which willmatch low energy observation, but just pick one and assume microscopic modellies at some scale ε. From that model, we will compute other observable quanti-ties. When you do this, it is crucial that observable quantity shouldn’t dependon the scale you assume. That is corresponding to subtraction of infinities infield theory. That process is called Stuckelberg-Petermann RG. 1

2.2 Wilsonian RG

The idea of Wilsonian RG is to change the ’measure’ bigger, and see the samesystem. Renormalization Group is defined as the transformation to ’zoom out’the system. Let’s see it in the simplest example 2 , balls in 2 dimensional box.

2.2.1 Example: balls in 2 dimensional box

We fill in 2 dimensional box with two kinds of balls, one is conducting elec-tricity (copper), and the other is insulating (glass). These balls are randomlydistributed with probability p and (1− p) each. We wish to see if the system is

1Not completely true, but it is just a matter of how you call it.2This example is given by Professor Nienhuis, at University of Amsterdam.

2.2. WILSONIAN RG 9

insulating or conducting at macroscopic scale. We wish to see how the systemlooks like in larger scale, and we proceed in two steps.

Figure 2.2: Two kinds of balls are put into a box.

We wish to know how physics looks like in the bigger scale, so we begin withapproximating three balls with one ball. Let’s say, we take the majority rule forapproximation. If more than two balls are c, the new big ball becomes copper,and vice versa. Therefore, the probability parameter for big copper ball wouldbe p′ = p3 + 3p2(1 − p). The size of the new ball is

√3 times bigger than the

original one, which can be derived from simple trigonometry or from the factthe number of bigger balls is 3 times less than smaller ones and occupies thesame area. This process of approximation is called ’coarse graining’.

Figure 2.3: Three balls are approximated as one ball by majority rule.

In the first step, we have just approximated the system, and nothing fun-damental. In the second step, we perform scale transformation (zooming out)of the system, since we wanted to have bigger view of the physics. If we shrinkthe system by the factor

√3, it exactly looks like the original system with new

parameter p′. Put in a different word, if we use the ’measure’ to see the physicswhich is

√3 times bigger size than the original, the same physics almost exactly

looks like the original one, with new parameter p′. Therefore, combining coarsegraining and scale transformation, we can get the effective description for biggerscale. These two process combined, is called ’renormalization group trans-formation’. First step is just as changing of the resolution of TV monitor,


and second step is stepping away from TV and picture actually gets smaller.Thanks to the first step, the model exactly looks like the same with the changeof parameter from p to p′. As you can see from the figure, the order of coarsegraining and scale transformation doesn’t matter.

Figure 2.4: RG transformation for this system

It is called as renormalization ’group’, since the transformation for zoomingout forms semi-group structure. It is the consequence of the fact that the resultof ’zooming out’ shouldn’t depend on how you do RG, but only on overall ratioof the scale. For example, if we zoom out by the factor of 2, and then by thefactor of 3, the description should be the same as the one we zoom out by thefactor of 6 at once. If we denote the process of Renormalization transformationby the factor of a as Ra, Here R3(R2) = R6 and it forms the semigroup.(Thereis identity, which is ’doing nothing’, but there is not the inverse of the trans-formation, since coarse graining is irreversible.) This group property is crucialwhen we compute critical exponents near the point of phase transition.

Going back to the example, the larger scale behavior of the system is cap-tured by changing the parameter p to p′, and Bottom line is, if p > 0.5, p getsbigger and bigger as we zoom out, and the whole box will be effectively con-ducting, and if p < 0.5, p gets smaller and the whole box will be effectivelyinsulating.

Let me explain few words which is commonly used in RG. The change ofparameter with RG transformation (or, change of the scale) is calles as RGflow. For p = 0, 0.5, 1, even if we perform RG transformation, parameters don’tchange. Such points in parameter space are called fixed point. The point ofp = 0.5 is phase transition, where slight change of microscopic parameterresults in dramatic change of macroscopic physics. At the point p = 0.5, thesystem is scale invariant, and the method of RG is really powerful near thispoint as we will see later.

2.2. WILSONIAN RG 11

Figure 2.5: change of p alongRG transformation

Figure 2.6: Concept ofWilsonian RG

Here, I would like to emphasize the main concept of RG transformation.With coarse graining process, we are able to analyze the same physics using’bigger measure’, and the change of observation scale can just be studied bychanging the parameter of the model. (2.6) That’s what we do in field theory.We study how coupling constant or mass flow along the RG flow, because wecan use the same form of the model for different observation scale.

Here, I would like to summarize the important point in Wilsonian RG. RGtransformation consists of two steps, coarse graining (changing resolution) andscale transformation(zoom out). Scale transformation is the step to view biggerpicture of physics, and coarse graining is crucial that we have exactly samemodel

Summary of Wilsonian RGGolden rule for RG : Coarse graining (changing the resolution)

& scale transformation (zooming out)Idea for RG : change of scale is analyzed by just the change of parameter

2.2.2 Example: Wilsonian RG for scalar field theory

Let’s review field theory and see how universality is derived from simple analysis.Basic knowledge of field theory is needed. This treatment is mostly based on [5].

First question is, how can we introduce scale into field theory? It is doneby introducing momentum cutoff. We assume each component of momentumsatisfies |pi| < 1

ε . In lattice picture, this corresponds to taking lattice spacingas ε, if we take unit where ~ = 1.

We take the action of the scalar field as

S(φ) =∫

d4x(12(∂µφ)2 − m2

2φ2 +

∑n

λnφn)

(2.1)

The φn terms are general perturbation, but we assume that they are muchsmaller than φ2 term. We wish to see how this theory evolves along RG flow.Just as in the previous example, Necessary steps are,


Figure 2.7: momentum cutoff Figure 2.8: lattice

1. Coarse graining

2. Scale transformation

In the first step, we seperate field into high momentum modes and lowmomentum modes as

φ =( ∫ pL d4p

(2π)4φ(p) +

∫ pH d4p

(2π)4φ(p)

)= φL + φH (2.2)

In the coarse graining step, we integrate over high momentum modes, sincethese modes are corresponding to microscopic degree of freedom. In latticepicture, it corresponds to approximating several lattices by one, as we did inthe previous ball example.

Let’s see how it works explicitly. We then seperate the action into ’purelyhigh momentum part’, ’purely low momentum part’, and ’mixture of high andlow momentum part’.

S =∫

d4x(12(∂µφL)2−m2

2φ2

L+∑

n

λnφnL

)+

(12(∂µφH)2−m2

2φ2

H+∑

n

λnφnH

)+

(Smix

)(2.3)

We then would like to see how physics looks like in the low energy fieldtheory, and for that purpose, we see how the parameters for the perturbation(coupling constant m2, λn) changes under RG transformation. We transformthem so that we still have the same partition function, since partition functiondetermines all the quantities we observe.3

Z =∫DφLDφH exp(iS) (2.4)

First, we are assuming parturbation terms (m2, φn terms) are small enough4

compared to the kinetic term, and the contribution of ’mixture’ term only comesfrom kinetic term. However, they are also zero, since they are quadratic in the

3In concept, what we do is nothing different from ball-box example, since the action is likeprobability distribution for each states.

4Since quadratic ’mixed’ terms are zero anyways, we could also assume m2 term is notsmall, which is consistent to the result we will see later. Of course, we can’t ignore perturbationterm for φL, since we take ε ∼ µ, and region for φL is much bigger.


field, and if we integrate over two different modes, it is zero. (∫

φ(k)φ(k′) =0 when k 6= k′) We saw that ’mixed’ terms are zero. We further integrateover high momentum modes, and it just gives constant contribution to thepartition function. Physically this doesn’t change anything, since we alwaystake derivative of log Z whenever we compute physical quantities.

Therefore, after coarse graining step, we can express partition function as,

Z =∫|pi|<1/µ

DφL exp( ∫

d4x(12(∂µφL)2 − m2

2φ2

L +∑

n

λnφnL

))(2.5)

Note that the momentum cutoff for φL is 1/µ.The next step is scale transformation. We take new coordinate x′ = x(ε/µ)

so that again the momentum cutoff will be 1/ε with respect to the new coordi-nate x′.

Z =∫|pi<1/ε|

DφL exp( ∫

d4x′(12(∂′µφL)2

(µ

ε

)2−m2

2φ2

L

(µ

ε

)4 +∑

n

λnφnL

)(µ

ε

)4)

(2.6)We further redefine a new field φ′ = φL(µ/ε) so that the form of the kinetic

term will be the same as the original.5

Finally, the partition function will be

Z =∫Dφ′ exp

( ∫d4x′

(12(∂′µφ′)2 − m2

2φ2

L

(µ

ε

)2 +∑

n

λnφnL

(µ

ε

)4−n))(2.7)

From this expression, you can read off effective parameters for the scale µas,

m2′ = m2(µ

ε

)2 (2.8)

λ′n = λn

(µ

ε

)4−n (2.9)

If we perform this RG transformation again and again, we can know thechange of coupling constant even if macroscopic scale µ is much bigger thanmicroscopic scale ε.6 Note that this behavior can be given by simple dimensionalanalysis.

From this analysis, we can see the change of coupling constant along RG flow.Mass term get bigger and bigger as we go to larger scale, and such terms arecalled renormalizable, or relevant. λn terms get smaller and smaller as we goto larger scale (when n > 4.), and such terms are called nonrenormalizable,or irrelevant. For λ4, we can’t simply conclude how they change from pure

5You may think that this new field φ′ is what we see as a particle at the scale µ, insteadof φL.

6That’s where group property of RG transformation is very useful.


dimensional analysis, and such terms are called marginal. We then have to takeinto account the effect from integrating out perturbation terms. This impliesthat ANY scalar field theory has to be described by φ4 theory (only allowedperturbation is m2 and λ4 terms) at macroscopic scale.7 This emergence ofsimple structure at macroscopic scale is called universality.

Figure 2.9: behavior of coupling constants

I would like to explain ’shortcut’ version of the proof I have given here. Wesaw that the effective field φ′ scaled along the RG flow, but this scaling law isexpected by pure dimensional analysis. Since we are assuming the theory is veryclose to the fixed point, we require the action to be scale invariant. Therefore,for arbitrary rescaling (x → (ε/µ)x, hij fixed), we expect that φ → (ε/µ)φfrom requirement of the scale-invariant property of the action. We require theseperturbation terms has to be scale invariant as well, and we also get the behaviorof other coupling constant λn → (ε/µ)n−4, which is just given by dimensionalanalysis. I would like to stress the property of ’near scale-invariant’ can beseen in our world quite often, since the observation scale is much bigger thanmicroscopic scale of the model, and it approaches the fixed point of RG flow.

The conclusion of the ’shortcut’ method is strong. The scale transforma-tion tells you how parameters will flow along RG flow near scale-invariantpoint. In other words, scale (conformal) transformation near fixed pointis RG transformaion. Later I will see this structure in Holographic RG flow.8

However, the above comment doesn’t take account of the effect from marginalperturbation. We will see that the effect of marginal term will give coupling con-stants slight deviation from naive scaling behavior, which is called ’anomalousdimension’.

7We are assuming we only have even power of fields. This is just the preference of ourtheory to have symmetry φ→ −φ

8In that context, the change of the scale is expressed by changing metric instead of changingthe coordinate, since they are equivalent from diffeomorphism invariance. See the section ofCFT in later chapter.


2.2.3 Wilsonian RG for field theory : take perturbationinto account

In the previous section, we saw that macroscopic effective theory of scalar fieldhave φ2 or φ4 perturbation, but we didn’t derive precise behavior of couplingconstants (especially, we don’t even know λ4 survives at macroscopic scale ornot!) We assumed that all perturbation terms λn are much smaller than kinetic(and mass) term, and the result was self-consistent. All λn (n > 4) termsvanished in macroscopic scale. However, we still don’t know about λ4. Here westill assume this is small and we can treat this term as perturbation, but wewill keep track of this perturbation.9 This analysis is not quite necessary forthe later discussion and it’s a bit detailed, so you can skip this part.

We should take the cutoff to be spherical. Moreover, we take imaginarytime formalism, since it is necessary to evaluate integration of high momentummodes. There are necessary simplification to evaluate integrating high momen-tum modes.10

The change to the imaginary time is done by taking τ ≡ −it as a newcoordinate, and we denote dxdτ as dx′(= idx) and express partition function.The partition function for φ4 theory would be,

Z =∫Dφ exp

(i

∫d4x(

12(∂iφ)2 − m2

2φ2 − λ4

4!φ4)

)=

∫Dφ exp

(−

∫d4x(

12(∂iφ)2 +

m2

2φ2 +

λ4

4!φ4)

) (2.10)

To compute the flow of the couplings λ4 and m precisely, we start from inte-grating out high momentum modes in the ’mixed’ part of the action in equation(2.3). Note that φL is treated as constant since we are integrating over φH

’Mixed’ part is given by

Smix(φ) =∫

d4xλ4

4!(4φ3

HφL + 6φ2Hφ2

L + 4φHφ3L) (2.11)

There is no term quadratic in the field, since integration of two orthogonalmodes are zero as

∫d4xφLφH = 0. These terms will give additional contribution

to coupling constant. Let’s first evalaluate the contribution of perturbation tothe mass up to the first order of λ. The contribution will come from the termwhich has two φL, and it is 6φ2

Hφ2L part. Since propagator of φH is 1/(p2 +m2),

9That is, we take the effect of these perturbations into account when we integrate out highmomentum modes.

10Not completely necessary, but it will make argument much simpler. I heard the renor-malization scheme will not change how the coupling constant flows, but I am not completelysure.


and we with to contract φ(x)φ(x) (same points), it will give 64!

∫d4k

(2π)41

m2+k2 .

δm2 = 2λ4 ·64!

∫1/ε>|k|>1/µ

d4k

(2π)41

m2 + k2

=λ4

2

∫1/ε>|k|>1/µ

2π2

(2π)4

12dL(L−m2)

L

=λ4

32π2(1/ε)2 − (1/µ)2)−m2 log

µ2

ε2

(2.12)

In the second line, we performed the change of the variable from k to L =k2 + m2. We can expand the λ4 term, contract each of them, and make thisδm2 in the exponential again. Therefore, it should be understood as the changeof the mass parameter in coarse graining step.

Let’s then see how λ evolves using the perturbation theory we have used. Thelowest contribution to δλ4 from λ4. First thing we need to is find terms whichhas 4 φLs. you can see above action you need λ2 order expansion to get suchterms. Among these terms, if we expand fields in terms of momentum modes,terms with even number of φH will vanish because of symmetry of integrationregion. Therefore, we need to compute

λ24

16

∫d4xφH(x)2φL(x)

∫d4yφH(y)2φL(y) (2.13)

I will skip process to derive the result, but the main step is first fourier transformall the fields in the above expression, and compute it assuming momentum ofφL is small enough to be ignored compared to momentum of φH and comparethe expression with the fourier transform of

∫d4xφL(x)4. You need integration

λ4 = −4!λ2

4

16

∫1/µ<|k|<1/ε

1(m2 + k2)2

d4k

(2π)4∼ − 3λ4

16π2(log

µ

ε+ m2(ε2 − µ2))

(2.14)We will use this expression when we solve RG flow equation later.

2.3 Stuckerberg-Petermann RG (Field theoret-ical RG)

Then, let’s look at the other approach for Renormalization Group. We saw,in Wilsonian RG case, how to derive macroscopic effective behavior from themicroscopic model, as condensed-matter theorists do. However, when we studyhigh-energy physics, we don’t know what the true microscopic theory looks like,and at what scale it is. The ignorance about the ’true’ model is the differencefrom Wilsonian.

Our task starts from guessing the microscopic model from what we observe,but there are infinitely many possible models which reduces to the same macro-scopic behavior. First, the scale of microscopic model is ambiguous, and second,

2.3. STUCKERBERG-PETERMANN RG (FIELD THEORETICAL RG) 17

we can have arbitrary many irrelevant terms which reduces the same macro-scopic physics. For simplicity, we just assume that the microscopic model isdefined at the scale ε, and it only consists of relevant terms, since they are theonly terms which will survive in the macroscopic theory anyways.

The spirit of Stuckerberg-Petermann RG is cancellation of divergences atmacroscopic scale. From the study of Wilsonian RG, we saw that the smallchange in the relevant (and possibly marginal) terms will grow bigger and biggerat the macroscopic scale. Therefore, the parameter of the ’true’ model has to beextremely small and fine tuned to reproduce exactly what we observe. Therefore,if we miss the value of ’true’ coupling constant by some finite amount, naivelymacroscopic quantity will diverge as the function of the ratio of microscopic scaleε and macroscopic scale µ.This cancellation of divergence gives strict conditiondetermines the ’true’ action, precisely, since if we have wrong value, macroscopicvalues can easily diverge. Once we have ’true’ action, we can easily determinehow things look like at macroscopic scale.

I would like to emphasize the difference of Wilsonian RG and Stuckerberg-Petermann RG. In Wilsonian RG, we fixed microscopic scale and changedmacroscopic scale. In Stuckerberg-Petermann RG, we fix our observation scale,change microscopic scale ε, and we observe divergence. Cancellation of diver-gence at macroscopic scale determines the ’true’ action.

Figure 2.10: Concepts of Wilsonian and Stuckerberg-Petermann.

What kind of observation do we use to determine ’true’ action? It is fixed byso-called ’renormalization condition’. It will fix the value of 4-point function and2-point function for effective field φ′ at some low energy scale µ. Is is nothingdifferent from determining Wilsonian effective action in effect. compute the flowof coupling constant

I would like to explain the concrete procedure for ’field theoretical’ renor-malization and the meaning of counterterms. First, we would like to start outfrom seemingly nonsense question: what if we start out from writing bare ac-tion in terms of (low energy) effective fields and coupling constant? (or, equally,parameters in renormalization condition) Of course, if we plug finite value intobare coupling, you expect the effective parameter would be divergent. It soundsnonsense, but that’s what people tried to do at the beginning of field theory.


Figure 2.11: renormalization condition for φ4 theory.

The expression of the low energy action at the scale µ is,

Sr =∫

d4x(12(∂iφr)2 +

m2r

2φ2

r +λ

4!φ4

r

)(2.15)

We wish to write bare action in the partition function using these renormalized(low energy effective) parameters.

Z =∫|p|<1/ε

Dφb exp( ∫

d4x12(∂iφb)2 +

m2b

2φ2

b +λb

4!φ4

b

)=

∫|p|<1/ε

Dφb exp( ∫

d4x12(∂iφr)2 +

m2r

2φ2

r +λr

4!φ4

r

)+ Sct

(2.16)

where

Sct =∫

d4x(12(Z − 1)(∂iφr)2 +

Zm2b −m2

r

2φ2

r +λbZ

2 − λr

4!φ4

r

)≡

∫d4x

(δZ

2(∂φr)2 +

δm2

2φ2

r +δλ

4!φ4

r

) (2.17)

compute the flow of coupling constant In the second line, we defined Z asφb =

√Zφr. We just seperated bare action into finite action and the difference

of them (This is defined as counterterm). Then we change integration measurefrom φb to φr. Since Z is just constant, (strictly speaking, it is the functionof both scales, but doesn’t depend on any fields) this extra constant doesn’tchange any physics when we compute quantities from partition function.

Therefore,

Z =∫|p|<1/ε

Dφr(−Sr(φr)− Sct(φr)) (2.18)

As we saw in Wilsonian RG, low energy physics should purely be determinedby effective action, and shouldn’t depend on the microscopic scale ε explicitly.Therefore, when we compute n-point function around the energy scale of the

2.4. RG-FLOW EQUATION (CALLAN-SYMANZIK EQUATION) 19

cutoff µ of effective action, ε dependence (divergence) of the effective actioncoming from integrating out high momentum modes and from Sct part shouldcancel out. Let’s see for 2 point function.

Figure 2.12: cancellation of counterterm and loop integral for two point function.

From the evaluation of loop diagram, two point function will be (up to thefirst order of λ)

p2 + m2 +3λ

16π2

( 1ε2−m2 log

1ε2m2

)− δm2 − δZp2 (2.19)

We know the result should not have any ε dependence, so we know that coun-terterm should be

δm2 =3λ

16π2

( 1ε2−m2 log

1ε2m2

)+O(λ2)

δZ = 0 +O(λ2)(2.20)

Therefore, in some context of quantum field theory subtraction of divergencelooks quite arbitrary, but it is the fact of expressing bare action with ’wrong’(effective) coupling constants and fields. In the final result we know that thereis no divergence (ε dependence), so we can just subtract them.

2.4 RG-flow equation (Callan-Symanzik equa-tion)

Next, I would like to explain the idea of RG flow equation (Callan Symanzikequation). This tells you how effective parameters of the theory change along RGflow, and it also gives information on universal behavior between macroscopicquantities.

1. write down the equation. The starting point of the equation is very simple.We use the relation that microscopic ’true’ quantity doesn’t depend on atwhich scale we observe the physics. People believe that the size of the


earth is independent of how you observe, and the equation is as simple asthat. If we denote observation scale as µ, the equation will be

d ’true’ quantitydµ

= 0 (2.21)

2. relate equation to macroscopic quantities. The RG flow equation almostlooks trivial, and you may wonder what you can derive from it. Thekey point is, that we can relate it to observable quantities. Then, theobservable quantites should be described by effective parameters. It willtell you how the effective parameters flow along RG flow. Therefore, theidea of RG flow equation is quite parallel to Wilsonian RG.

3. solve equation. write down and solve differential equation. The techniqueof dimensional analysis helps.

2.4.1 Example: Balls in the box

Let’s see application of RG flow equation for the example we have studied before.We have two kinds of balls, and we consider connected part of one kind of ballsas below. we will derive the size of the cluster using RG flow equation.

Figure 2.13: size of the connected part

We express the typical size of cluster as ξ(p). It has the dimension of length,and we can express it as ξ = εf(p) with some function f of p, since p is theonly parameter we have. This is microscopic ’true’ quantity we consider. Ourgoal is to express this quantity in terms of effective parameters, and derive RGflow equation. Let’s start out from considering what happens to this quantity ifwe perform previous (figure 2.3) RG transformation. With coarse graining, thesize of the cluster doesn’t change as in the figure. Imagine, when you changethe resolution of the TV monitor, the size of the picture doesn’t change. Withscale transformation, the size of the cluster will be ε/µ(= (

√3)−N ) times the

original size.11 The useful observation is, the cluster size of RG transformed11Let’s assume we had N times this RG transformation and the new scale of the theory is

µ instead of ε. Therefore, we have (√

3)N =` µ

ε

´.


picture can be expressed as εf(p′′), where p′′ is probability parameter that wehave RG transformed p by N times. Therefore, we should have

εf(p) · ε/µ = εf(p′) (2.22)

Figure 2.14: relationship of size of the cluster

If p′ and p is very close to 0.5, we can use linear approximation of p′ =p3 +p2(1−p), p′′−0.5 ∼ ( 3

2 )N (p−0.5), where N is number of times we operatedthe RG transformation from three balls to one. and together with µ

ε = (√

3)N

and equation (2.22), we have (p′′ − 0.5) ∼(

32

)log√3(µ/ε). We can express RGflow equation as,

∂ log(εf(p))∂ log µ

= 0

↔∂ log(µf(p′′))∂ log µ

= 0

↔1 +∂ log f(p′′)

∂ log(p′′ − 0.5)∂ log(p′′ − 0.5)

∂ log µ= 0

↔ ∂ log f(p′′)∂ log(p′′ − 0.5)

= − 12(log 3−log 2)

log 3

(2.23)

Solving for f and plugging into ξ, we get the expression for the size of the clusteras,

ξ = ε(p− 0.5)−(log 3)/2(log 3−log 2) (2.24)

Therefore, we could derive universal relation using RG-flow equation. Itis worthy to note that ξ, which is characteristic length of the system divergesat p = 0.5. Actually, we already know that p = 0.5 is the fixed point of RGtransformation, and there should not be any characteristic length, since fixedpoint has scale invariance. At the fixed point, generally, characteristic lengthscale should be 0 or ∞.

Analogous to what we have done for ξ, we could derive n point correlationfunction of field theory using RG flow equation.


2.4.2 Example: partition function of field theory

Let’s remember Wilsonian RG for scalar field theory. We saw that partitionfunction (which is the quantity independent of observation,) can be expressedin terms of bare parameters, and also effective parameters.

We can therefore express partition function Z in two ways as,

Z(ε, gi) = Z(µ, g′i) (2.25)

where g′i is macroscopic effective parameter at scale µ. Callan-Symanzik equa-tion can be written as

∂ log Z

∂ log µ+

∂ log g′i∂ log µ

∂ log Z

∂ log g′i= 0 (2.26)

The coefficient of second term, ∂ log g′i/∂ log µ, tells you how effective pa-rameter g′i flows along the scale, and we call −1 times this quantity as betafunction.12

∂ log g′i/∂ log µ = −βi (2.27)

Let’s derive equation for the flow of the coupling constant. We need to solveso-called characteristic equation.

2.4.3 Characteristic equation

Assume that we have variable x1 · · ·xn, and z, function of x1 · · ·xn, and fi, g,function of x1 · · ·xn and z. We wish to solve equation

n∑i=1

fi∂u

∂xi= g. (2.28)

Let’s consider two dimensional case,

f(x, y, z)∂z

∂x+ g(x, y, z)

∂z

∂y= h(x, y, z) (2.29)

We have three parameters and one equation, and we expect the solution of thisequation to be some 2 dimensional surface. (If we specify a point in the plane,the plane will be determined) We express the surface with z = H(x, y). If weassume that (x, y, z) and (x + dx, y + dy, z + dz) are both on the surface, wehave dx

dydz

·

∂H/∂x∂H/∂y−1

= 0 (2.30)

On the other hand, the equation we wish to solve is

12It is the fact we see the change of coupling constant with respect to length scale. if wesee the change with respect for the energy scale, we don’t need minus signature.


fgh

·

∂H/∂x∂H/∂y−1

= 0 (2.31)

Since we are in 3 dimension, and (dx, dy, dz) consists of two dimensional surface,(∂H/∂x, ∂H/∂y,−1) and (f, g, h) have to be parallel. That gives us a condition,

dx

f=

dy

g=

dz

h(2.32)

Now we have two constraints in three dimension, so if we specify one point inthe parameter space, we get a curve of RG flow in the parameter space. Notethat even when h is always zero, the rest of the constraints still holds.

2.4.4 Solving RG flow equation

Since we have technique at our disposal, let’s solve differential equation. Wesolve RG flow equation for partition function and get the flow of coupling con-stants.

We consider partition function of φ4 theory. We start from the equation’true’(bare) quantity is independent of observation scale,

∂ log Z

∂ log µ= 0 (2.33)

We wish to express the partition function with observable parameters (mr, λr,and momentum pi) and the ratio of the microscopic/ macroscopic scale to obtainuniversal behavior which can be observed in low energy scale. We believe thatthese information is enough to recover the ’true’ theory at the microscopic scale.

If we use dimensional analysis, the partition function should be described as

partition function =[dimensionful part]×[dimentionless part constructed only from dimentionless combination of variables]

(2.34)

Since dimensionally Z is dimensionless,

Z(m2r, λr,xi,

µ

ε) = f(m2

r · µ2, λr,xi

µ,µ

ε) (2.35)

Operate µ ∂∂µ on this should give zero, and we have

∂ log(m2r · µ2)

∂ log µq

∂

∂qf +

∂λr

∂ log µ

∂

∂λrf − xi∇xif +

∂f

∂ log µ= 0 (2.36)

We denoted m2r · µ2 ≡ q. curves which satisfy the above equation should also

satisfies so called Characteristic equation below

dλr

β=

dq

(2 + γ)q= −dx1

x1= · · · = −dxn

xn=

dµ

µ(2.37)


where

γ =∂ log m2

r

∂ log µβ =

∂λr

∂ log µ(2.38)

Note that γ, β both tells you deviation of λr, m2r from naive scaling behavior.

This equation above governs the hypersurface which is along the RG flow.What we are doing is changing the scale of physics slightly, and see the change ofparameters, and repeating the process to change of the scale bigger and bigger.

We can compute the β, γ from what we did in the Wilsonian RG. δm2 , δλshould be identified as m2

r, and λr, and we will have

γ ∼ − λ

8π2

β ∼ − λ2

16π2

(2.39)

Let’s then solve for λr. If we define s = log µ,

∂λr

∂s∼ 3

λ2r

16π2

∆s ∼ 1−3λr/16π2

− 1−3λb/16π2

→ λ =λb

1− λb3

16π2 log µε

(2.40)

Therefore, as I promised, we get universal relationship (flow of couplingconstant) from RG flow equation. The main goal of this thesis is to derive theabove RG flow equation using AdS/CFT correspondence.

Chapter 3

AdS-CFT correspondence

Since Holographic RG is one of the correspondence of AdS-CFT1, we first needto study on the correspondence. It is equivalence of classical supergravitytheory in the (asymptotically) AdS space2 and quantum CFT living on theboundary of the AdS space (with perturbation). We will see, computing su-pergravity action in AdS will give you effective action in CFT, and you cancompute anything from the effective action. Deriving RG-flow equation in CFTusing the correspondence is one of the main goal of this thesis.

Naively we expect classical gravity theory and quantum theory describescompletely different regime of physics, it is expected this will give many in-sights into these physics. Another motivation for studying AdS/CFT comesfrom studying our quantum 4 dimensional world using the CFT side of thecorrespondence. It is expected that QCD, theory of strong force, or even phe-nomena in condensed matter physics, such as superconductivity or quantumhall effect, is expected to have gravity description using AdS-CFT.

Before going into details, let me explain about AdS, and CFT.

3.1 AdS space

3.1.1 various parametrization of AdS space

I will explain various parametrization of AdS space. AdSd+1 space is defined asa manifold with constant negative curvature, which can be embedded ind+2 dimension as below.

ds2 = dy20 + dy2

i

y20 − ηijy

iyj = l2(3.1)

where ηij = diag(−1, 1, ...1), d + 1-dimensional diagonal matrix. In this thesis,I denote roman indices i, j . . . to mean all indices without 0, and denote Greek

1It was originally found by Maldacena [3] as the equivalent description for string theory.2AdS is abbreviation of Anti de Sitter space, here anti implies negative curvature.

25

26 CHAPTER 3. ADS-CFT CORRESPONDENCE

indices µ, ν . . . to take all indices including 0. Later in this paper 0 denotesradial direction of AdS space.

Note here we are considering pure gravity with cosmological constant, andwe will perturb AdS by adding fields to this background. The spacetime will beno longer AdS, but we will assume it will still be Asymptotically AdS.3

We then consider Euclidian version of the metric to make the analysis sim-pler. That is, we replace the time coordinate by imaginary time as t = iτ .Then the weight to evaluate path integral will change eiS to e−S . The metricwill be changed to η = diag(1, 1, ...1). This is mainly done because it will makeboundary compact and the relation of boundary and the bulk will be clearer.

Therefore, the embedded manifold becomes hyperboloid. It would be eas-ier to consider an example in 3 dimension. Here the Euclidian AdS2 space isexpressed as hyperboloid in R3.

y20 − y2

1 − y22 = l2 (3.2)

Figure 3.1: We project hyperboloid to the inside of the circle with radius l.

and you can project this surface from the point (y0, y1, y2) = (−1, 0, 0) tothe y0 = 0 plane. Then the whole AdS2 space will be projected to the circleof radius l on y0 = 0. We can then rescale so that it becomes unit circle. Theinduced metric will be

ds2 =4l2(dy2

1 + dy22)

1− y21 − y2

2

(3.3)

It is further possible to map this into upper half plane by conformal trans-formation (translation and inversion). Note that the inversion of circle withrespect to a point at the boundary of the circle maps it to a straight line. Thenew metric will be,

ds2 =l2

z2(dx2 + dz2) (3.4)

Analogously, for AdSd+1 case, the metric will be

ds2 =l2

z2(dxidxi + dz2) (3.5)

3Asymptotically AdS metric is the same as AdS metric for the leading order near theboundary. See the 3.4

3.2. ABC OF CONFORMAL FIELD THEORY 27

where i takes from 1 to d. Note that ’outside’ of AdS (or, hyperboloid) mapsto small small z, and ’inside’ to big z. The merit of this choice of coordinateis, we will see the z variable in this coordinate is naturally related to the lengthscale of CFT. Therefore, small z corresponds to microscopic scale, and large zcorresponds to macroscopic scale. Here z takes 0 to ∞, and boundary of theAdS is at z = 0.

We can further introduce parameter r to substitute dz so that the coefficientof dr2 in the metric will be 1. That has to satisfy dr = ±l(dz/z), which can besolved to give z = e±

rl (we take minus sign). Here r can take −∞ to +∞, and

boundary is at ∞.4

We get a new metric,ds2 = dr2 + e

2rl dx2 (3.6)

In general d + 1 dimensional case, it is

ds2 = dr2 + e2rl δijdxidxj (3.7)

The parameter r is also interpreted as the ”radial” direction of AdS space.5

It is straightforward to check this metric really describes the space withnegative constant curvature. It gives Ricci scalar , R = −d(d + 1)/l2, and wealso get the value of Λ from Einstein equation, Λ = −d(d− 1)/2l2.

3.1.2 boundary of AdS space

I claimed at the beginning that the CFT lives on the boundary of AdS. However,the ’boundary’ z = 0 corresponds infinitely far away in AdS space, as you cansee from hyperboloid. In order to have a fixed hypersurface for CFT to live,and we need to put boundary by hand. In z coordinate, we put z = εhypersurface and consider CFT is living on this hypersurface.

Next, we would like to see that the boundary of d+1 dimensional AdSd+1

is given by flat space. If you look at the Poincare coordinate(equation (3.5)),metric becomes ds2 = const × dxidxi for z = 0 limit. Therefore, the metric atthe boundary of AdS, where CFT lives, is just flat space.6

3.2 ABC of Conformal field theory

3.2.1 Why Conformal field theory?

Here, I will give the shortest introduction for Conformal field theory (From nowon, we just abbreviate it as CFT), which is relevant to the discussion in themain part of the thesis. I will discuss that CFT can be defined as ’A fieldtheory which is invariant under (local) scale transformation’.

4This is convenient, for example, when we consider light rays going outwards (inwards)AdS space, since radial direction will corresponds to time in that case.

5The main paper of this thesis, [6] takes this choice of coordinate.6We will perturb the metric later, but we assume that boundary metric will be still flat.


CFT is very natural setting to study macroscopic theory. If we see physicsin a bigger and bigger scale, acting RG transformation, we naively expect thetheory to reach the point which is invariant under further RG (scale) transfor-mation.

Actually, in statistical mechanics, lots of models are known whose macro-scopic limit can be described by CFT. Ising models and its variations are suchexamples. We will see, in the main part of the thesis, extra dimension in thesupergravity corresponds to the scale parameter of RG, and it is quite naturalthat CFT, which is at the fixed point of the field theory, lives on the boundaryof the supergravity.

3.2.2 Conformal invariance = (local) scale invariance

We explained conformal field theory as ’a field theory with local scale invariance’,but normally ’conformal symmetry’ can be defined in a bit different way. We willsee that the normal definition reduces to our definition. As its name suggests,conformal symmetry is a symmetry for a transformation which keeps the formof shapes as below;

Figure 3.2: conformal transformation preserves angle.

Formally, you can define conformal transformation as the ’transformationwhich preserves angles between lines’. We consider coordinate transformationx → x′ = x + ε accompanied with transformation of the metric, hij → h′ij(x

′),and look for transformations which preserves angle between vectors.

Therefore, our starting point should be, strictly speaking,

(hij(x)dxidyj)2

(hij(x)dxidxj)(hij(x)dyidyj)=

(h′ij(x′)dx′idy′j)2

(h′ij(x′)dx′idx′j)(h′ij(x′)dy′idy′j)(3.8)

However, we normally starts from easier, equivalent definition of conformaltransformation as below, which indicates that the change of the metric is just

3.2. ABC OF CONFORMAL FIELD THEORY 29

proportional to the original metric.

h′ij(x′) = Ω(x)hij(x) (3.9)

It clearly satisfies the condition that the angle of two vectors will be invariant.7

If we write down the equation for conformal transformation, length ds2 =hijdxidxj should be invariant under the transformation, since the transforma-tion can be thought of as mere a new parametrization.

Therefore, we have

h′ij(x′)d(x′)id(x′)j = hij(x)dxidxj

h′ij(x′) = Ω(x)hij(x)

(3.10)

For normal diffeomorphism without conformal symmetry, we have Ω(x) = 1,and conformal symmetry case can be seen as its generalization.

Above condition (3.10) reduces to

(Ω− 1)hij = −Diεj −Djεi (=2dDkεkhij) (3.11)

Therefore, the definition of conformal transformation (3.9) is equivalent to thisgeneralized form of killing equation. It is called as conformal killing equa-tion. As I promised, let’s move on to show the equivalence of the conformalsymmetry and the local scale symmetry. When we talk about (classical) sym-metry, that means the action is invariant under the transformation. Ifwe define energy-momentum tensor as T ij ≡ 1√

hδS

δhij, local scale symmetry can

be expressed as,

δS = δhijδS

δhij

= (Ω− 1)√

hT ii = 0

(3.12)

From this equation, we wish to show that theory is also conformal invariant.From the above definition of conformal transformation,

δS =√

hδhijTij =

√h

2dDkεkT i

i = 0 (3.13)

We used T ii = 0 in the final step. Therefore, local scale symmetry leads to

conformal symmetry. It is obvious that conformal invariance includes localscale invariance, and they are equivalent.

3.2.3 scaling of operators

Here we will discuss how operators will change by conformal transformation.We parametrize the scale transformation of metric as hij → a2hij .8 When

7It could also be shown that 3.9 is necessary condition for 3.8, but We will not do it here.8Here we are performing scale transformation by changing the metric, but we could use

coordinate transformation h(x)ijdxidxj = h′(x′)ijdx′idx′j and make the metric to be flat


we perform this transformation, what does the operators behave? From groupproperty of RG transformation, we can show they will follow power-law withrespect to the scale parameter a.9

To determine scaling dimension for the source, which couples to the operatorO, let’s look at the perturbation term Spert =

∫d4x

√hOφ in the CFT. If we

demand that Spert is also invariant under scaling, since we have√

h → ad√

h O → a−∆O (3.14)

The source φ should transform as

φ → a∆−dφ (3.15)

∆ in this expression is called as conformal dimension of the operator O. Thisbehavior can also be seen in AdS/CFT correspondence.10

3.2.4 conformal anomaly

We discussed classically conformally invariant theory, and didn’t study quan-tum theory yet. Surprisingly, even if the action itself is invariant under con-formal transformation, the quantum theory is NOT neccessarily symmetric be-cause of the transformation of the integration measure. Symmetry in quan-tum theory is defined as the invariance of the effective action Γ ≡− log[

∫Dφ exp(−S(φ))] (where φ is some fields) since any physical quanti-

ties are computed via this functional. The deviation from conformal sym-metry is usually expressed as taking trace of the energy momentum tensor,⟨T i

i

⟩= anomaly, when classical theory has conformal symmetry and thus van-

ishing trace. This anomaly is called Weyl anomaly, or conformal anomaly,or scale anomaly.

This anomaly is known to be expressed in terms of curvature of the metricor gauge fields. FOr 2d theory coupled to gravity, it is universally⟨

T ii

⟩=

c

24πR (3.16)

and the coefficient c, which is called as central charge, depends on the system.You can see derivation of the conformal anomaly in [7]. In two dimension, itis known that product of two energy-momentum tensor is also related to scaleanomaly as,

〈TzzTzz〉 =c

z4(3.17)

again as well. In that case, conformal transformation can be written as x → ax. Therefore,make hij bigger corresponds to using smaller measure to see the physics. (since we need atimes more change of coordinate observed in new coordinate)

9Since Ra1Ra2 = Ra1a2 , the only allowed transformation is O → O′ = a−∆O. This ∆ iscalled scaling dimension, or conformal dimension of operator O.

10Normally, in the CFT, we fix the metric to be Minkowski and change the coordinate, buthere it is more natural to change the metric fixing the coordinate, since behavior of fields andsources look similar to what we see in AdS/CFT.

3.3. ADS/CFT CORRESPONDENCE 31

where we are taking complex coordinate.In the section of scaling of operators in CFT, we saw that the coupling

constants will change along with metric, and this behavior exactly looks likethe behavior of fields in AdS. Therefore, the value of external fields

3.3 AdS/CFT correspondence

Then we finally explain the correspondence. For some supergravity theory11 inAdSd+1 space12, there is a corresponding CFT living on the boundary of theAdSd+1 space. First we consider perturbing AdS space by putting in some scalarfields φI13 (I denotes different kinds of scalar fields) and gauge fields Aµ, so thatit is still almost AdS space. These scalar fields φI acts as external fields forsome operators OI in the CFT, which we assume to have conformal dimension∆I . Gauge field couples to a current for a global symmetry in CFT.14 Also, themetric of CFT is given by the 4 dimensional part of the metric (induced metric)in AdS space.

Figure 3.3: dynamical fields in AdS becomes external fields at the boundary ofthe CFT.

The statement of the correspondence is15, the effective action 16 for boundaryCFT (assume the boundary is at some small z, z = ε.) with perturbation φI(ε),Ai(ε), metric hij(ε) is exactly same as bulk supergravity ON-SHELL17 action

11If we only consider bosonic part of the supergravity, it is just a gravity theory coupled toscalar fields and gauge fields.

12It is believed that for any sensible gravity theory in AdS there is corresponding CFT, butthe other way is not true.

13If we assume supersymmetry, it has to come along with the metric. In N = 8 supergravitytheory, there are 42 kinds of scalar fields.

14The symmetry is called as R-symmetry.15Precisely speaking, the statement here is low energy approximation of the more general

AdS/CFT correspondence, for which AdS side is described by string theory.16It is just logarithm of partition function, and it is called as ’energy functional’ in the

standard textbook by Peskin [5], but I will be sloppy about the naming and I will just call itas effective action.

17It means theory always satisfies equation of motion. Therefore, if we specify the boundarycondition, we can determine the value of the whole action in AdS.


with boundary condition18 φ(ε) = φI , Ai(ε) = Ai and gij(ε) = hij .19 In formula,this is

Γ = − log〈〈exp(∫

d4x√

h(φIOI + AaJa + δhijT

ij))〉CFT = Ssugra[φI , Aa, hij ]

(3.18)This statement is really powerful. In AdS side, if you specify the boundary

condition, you will get gravity action. Therefore, if you specify the externalfields of CFT, and use this correspondence, we can compute the effective actionof the CFT. Once you get effective action, you can compute everything of CFT.n-point function, anomaly, . . . and RG flow equation.

3.4 Natural relation between AdS and CFT

In this correspondence, we can show that supergravity on-shell action in asymp-totically AdS space automatically has conformal invariance on the boundary.Proof follows from the Paper by Sakai, Matsuura, Fukuma [9]. You could alsoconvince yourself conformal symmetry naturally arises at the boundary of AdSspace and skip this section.

First we define asymptotically AdS space as a metric identical to AdS upto the leading order to AdS metric if we expand in terms of z (since the boundaryis at z ∼ 0 ). In the asymptotically AdS space, the metric is expressed asgij = z−2ηij +O(1), gzi = O(z), gzz = z−2 +O(1)Then we consider diffeomorphism in this space, xµ → xµ+εµ near boundary.(z ∼0) The coordinate transformation can be expanded in terms of z as,

εz(x, z) = ζz(0) + zζz

(1) + z2ζz(2) + ...

εi(x, z) = ξi(0) + zξi

(1) + z2ξi(2) + z3ξi

(3) + ...(3.19)

If we further assume this transformation is diffeomorphism in asymptoticallyAdS space, the change of metric is restricted as,

δgij = ∇iεj +∇jεi = O(1)δgiz = ∇iεz +∇zεi = O(z)δgzz = 2∇zεz = O(1)

(3.20)

If we solve these conditions order by order, we get

εi(x, z) = ξi(0) − z2ηij∂j∂kξk

(0) +O(z3),

εz(x, z) =z

d∂iξ

i(0) +O(z2)

(3.21)

18It is known that the value of fields and supergravity action in Euclidian time AdS isuniquely determined by the boundary condition at z = ε. For real-time AdS, see discussionin [8].

19g denotes five dimensional metric.

3.4. NATURAL RELATION BETWEEN ADS AND CFT 33

And from δgij part, we also get an equation,

∂iξ(0)j + ∂jξ(0)i = 2ζz(1)ηij (3.22)

Here we defined ξi ≡ ηijξj . taking the trace, we can get ζ(1) = ∂iξ

i0/d, therefore

we get

∂iξ(0)j + ∂jξ(0)i =2d∂kξk

(0)ηij (3.23)

This is conformal killing equation, RHS meaning freedom to choose the valueof trace. Since the supergravity action Ssugra(= Γ) is invariant under confor-mal transformation on the boundary, we can conclude that the theory on theboundary is invariant under conformal transformation on the boundary.

This indicates, if the correspondence is correct, that the boundary theoryis automatically conformally invariant, since the on-shell supergravity action issame as generating functional for boundary field Theory.

However, there is a problem in AdS/CFT which makes this statement NOTcompletely correct. When we compute Ssugra, they become infinite expression,and we need to deal with the divergence. When we perform that, it will breakconformal invariance in a mild way. The result will match what we expect formthe result of conformal anomaly in CFT, and the derivation of this anomaly isactually one of the merits in AdS/CFT. The holographic derivation of anomalyis seen by paper by Henningson and Skenderis [10]., which I will discuss in latersection.


Chapter 4

dealing with divergences :Holographic RG

4.1 dealing with divergences

We saw that the AdS/CFT correspondence is a really powerful tool, but thereare two problems. First problem is that the supergravity action in AdS isdivergent. It is natural since we are integrating over infinitely big AdS space.The other problem is, fields hij , φ

I are known to be diverging or going to zeroat the boundary.

First problem comes from the divergence of the action. Generally, super-gravity action looks like

Ssugra = · · ·S(0)ε−1 + S′(0) log ε + S(1)ε = Sct + Γ (4.1)

The action will diverge as we put ε to 0. In order to get finite quantity,according to the prescription of Skenderis et al [10], we will expand the actionfor the power of ε, and we will subtract the divergent part (counterterm Sct) ofthe action.

Second problem is asymptotic behavior of fields in AdS space. Since fieldsdiverge or go to zero when we set ε to 0, as1

φI(ε → 0) ∼ z−∆I+dφI

hij(ε → 0) ∼ z−2hij

Aa ∼ Aa

(4.2)

where hij denotes nonradial d dimensional metric, and φI denotes somescalar field which couples to operator OI with conformal dimension ∆I . Field

1Precisely speaking, this asymptotic behavior is only correct for a particular gauge choiceof metric in which z coordinate is orthogonal to the boundary.

35

36 CHAPTER 4. DEALING WITH DIVERGENCES : HOLOGRAPHIC RG

with hat means z-independent part of the field. We wish to take this φI , hij ascoupling constants in CFT. Note that this expression above is true for a specificchoice of gauge for the metric, as we will discuss.

It may look arbitrary to take care of two divergences, but we will soon seeits correspondence to RG in CFT. This process of handling divergence in theaction is called as holographic RG.

Therefore, the correct version of the AdS/CFT correspondence is,

Γ(φI , Aa, hij) ≡ − log⟨

exp( ∫

d4x(φIOI + AaJa + hijTij

))⟩

= S[φ,Aa, hij ]−div

(4.3)The action on the right hand side is on-shell action with boundary condition atz = ε hypersurface, φI [ε] = φI , gij [ε] = hij , Ai[ε] = Ai.

These treatment for divergence might look a bit arbitrary, but in fact, itis nothing different from RG in field theory. In Stuckelberg-Petermann we ex-pressed ’true’ quantity using effective parameters, and subtracted divergence.Actually you can interpret the action and the fields at the boundary of AdS asthe bare fields and the bare action, as is suggested by [6]. From this perspec-tive, it is much clearer why we regulate action as above, since the regularizationshould be the same process as in QFT.

Even though it sounds almost trivial to subtract divergence here, the actualprocess is technically a bit complicated. I am going to talk about how to proceed.

4.2 Recipe for computing correlation function

Let’s see how to compute correlation function in CFT using this framework,developed by Skenderis et al. [10]. If you would like to know the ’Wilsonian’ wayof doing Holographic RG, you could skip this part since that part is independentfrom this.

The steps are as below;

1. take the coordinate z to be orthogonal to 4d boundary

2. asymptotically expand fields and metric for ε

3. Solve equation of motion

4. Express action as the power of ε and subtract divergent part

5. Express the finite part of the action using 4d covariant fields

I would like to perform these steps for pure gravity AdS. (Although I wouldn’tcompute correlation function)

Setup

We first consider supergravity action in asymptotically AdS spacetime as wedid before. We choose the Gaussian normal coordinate at the boundary of

4.2. RECIPE FOR COMPUTING CORRELATION FUNCTION 37

AdS, in which z direction is orthogonal to the 4d spacetime direction. This isconvenient since in this way it is easier to see the physics in 4d covariant way atthe boundary of AdS. In such coordinate, there is no cross term of radial and 4dimentinal part.

ds2 =l2

4ρ2dρ2 +

hij

ρdxidxj (4.4)

Here ρ is defined as ρ = z2/l2, where z is what we defined in (3.4) and boundaryis at ρ = ε ∼ 0. We may also express the metric with h, where h = ρh, sinceh diverges as h ∼ ρ−1. 2 This choice of metric makes things a lot easier, since4 dimensional spacetime coordinate and scale parameter (or radial direction ofthe bulk AdS) don’t mix with each other.

The action of the supergravity is given by

S =∫

d5x√

g(5R + Λ) +∫

boundary

d4x√

h(Kaa + α) (4.5)

Here gµν is 5 dimensional metric, and hij is 4 dimensional induced metric onthe boundary. extrinsic curvature Kij = hij in this metric. More generaldefinition of Kij and its meaning is given in (5.8). The reason that we have theboundary Ka

a term3 is that we first have to perform integration by parts on5R part so that the bulk action depend at most on the first radial derivative ofthe metric. It is because we would like to solve equation of motion by Euler-Lagrange equation, and this is valid when there is at most first radial derivativeof the fields. If we include this term from the beginning, it will cancel out theterm which comes from integration by parts, and we can ignore that boundarycontribution. Note that the second term of the boundary doesn’t contribute toany dynamics in the theory.

Let’s confirl the above justification for Kaa term. R originally includes 2

derivatives of metric, but from transforming

5R =4 R + KabKab − (Kaa )2 −5 ∇a(na∇cn

c) +∇c(na∇anc) (4.6)

second derivative would not be there anymore. Here nµ is defined as normal(nµnνgµν = 1), orthogonal to 4d boundary. In the above metric4.4, it will benµ = (2ρ/l, 0, 0, 0, 0).

From Stokes’ theorem, the total derivative term can be rewritten as theintegration at the boundary as,∫

boundary

N√

h(nana∇cnc + nc(na∇anc))d4x (4.7)

From the definition of Kab, (5.8), above expression reduces to∫

Kaa

√hd4x.

We should in principle also consider this term in the ’Wilsonian’ methodof Holographic RG, but since we use so-called Hamilton-Jacobi formalism, it

2the above metric is equivalent to the gauge choice N = 1, Ni = 0 in the discussion of thenext chapter.

3it is called as Gibbons-Hawking term.


automatically takes care of well-defined variational principle and you can ignorethese subtleties arising from the boundary counterterm. I will talk about thisin the appendix.

Step1: expand field with respect to ρ

Then, we do asymptotic expansion of fields φ, hij with respect to the parameterρ. For metric, in order to solve supergravity equation of motion order by orderfor ε, and see divergence structure at supergravity action for ε.

h =

h(0) + ρh(2) + ρ2h(4) + · · ·+ ρd/2h(d) + ρd/2 log ρh(d log) + · · · (d = even)h(0) + ρh(2) + ρ2h(4) + · · ·+ ρ(d−1)/2h(d) + · · · (d = odd)

(4.8)Explicitly solving Equation on motion you can see that log term appears inthe asymptotic expansion when d is even, but you can also understand it as aconformal anomaly of one-point function.The detail is in [10].

Step2: solve for fields order by order

The purpose of expanding the fields in terms of ρ was to solve equation ofmotion order by order. Now let’s see how we can solve equation of motion forpure gravity case, where we only have cosmological constant and curvature istherefore constant. We want to solve Einstein equation Rµν − 1

2gµνR = Λgµν .From R = d(d + 1)/l2, and Λ = −d(d− 1)/2l2, it will be

R00 =d

4ρ2

R0i = 0

Rij =hij

ρ

d

l2

(4.9)

This reduces to

ρ(2h′′ − 2h′h−1h′ + Tr(h−1h′)) + l2Ric(h)− Tr(h−1h′)h = 0

(h−1)jk(∇ih′jk −∇kh′ij) = 0

Tr(h−1h′′)− 12Tr(h−1h′h−1h′) = 0

(4.10)

Note that I took matrix notation for metric h. Ric(h) means Ricci tensor madeout of the metric h. We can then solve this equation of motion order by order.

4.2. RECIPE FOR COMPUTING CORRELATION FUNCTION 39

In 4 dimension, it can be solved as

h(2) = −12(Ric(h(0))−

16R(h(0))h(0))

Tr(h−1(0)h(4)) =

116

(RµνRµν −29(R(h(0)))2)

4h(4 log) = 2h(2)h−1(0)h(2) + Ric′(h)|ρ=0 + Tr(2h−1

(0)h(4) − h−1(0)h(2)h

−1(0)h(2))h(0)

(4.11)

Step3: compute counterterm of the action

When we wish to compute physical quantities of boundary CFT, we need tosubtract divergence of the action. The divergent behavior of terms are computedin a straightforward manner. We just plug in the expression of the fields into theaction,performing integral for z coordinate, and see terms with negative powersof ρ as divergent, since we are taking the boundary of AdS ρ = ε to infinitelyfar away by ε → 0.

If we plug the metric we computed into the action (4.5),

S =d

l

∫dρρ−d/2−1

√det h+[ρ−d/2(−2d

l+

4lρ∂ρ

√det g +α

√det g)]ρ=ε (4.12)

The anomaly terms comes from integal term (not from boundary term) and useexpansion of determinant,

δ√

h =

√h

2tr(h−1δh) +

12tr(h−1δhh−1δh) (4.13)

we get

Ssugra =∫

d4x(− 6ε2

+3εR(h(0))− log ε(W) + finite) (4.14)

Here we wish to identify these divergent terms as counterterm and finite partas CFT effective action. Here

W =l3

2(tr[h−1

(0)h(2)]2 − tr[(h−1(0)h(2))2]

)= l3

(− 1

8RijR

ij +124

R2) (4.15)

is conformal anomaly of the action. As I promised you in the section3.4,we have to subtract divergent part of the action Ssugra. Precisely, subtracting−W log ε term will produce counterterm which breaks conformal invariance.Note that this expression of Riemann tensor in the anomaly is constructed outof the metric h(0), and if we wish to compute the real anomaly, we need toexpress it in terms of the field at the boundary, h(ε).4

This subtraction of divergence is analogous to the subtraction of countertermin CFT, identifying going to small ρ as sending ’true’ cutoff to zero (or inmomentum space, to ∞.)

4However, in this case, this replacement doesn’t change the counterterm, and anomaly.


Step4: replace asymptotic value by the value on the boundary

We did asymptotic expansion of fields to solve equation of motion and pluggedthese values into the action, but these expansion coefficients h(i) are NOT thefields which should be used as the source in CFT. When we compute physicalquantity in CFT, actions and fields should be all computed with respect tosome hypersurface ρ = ε. As we have claimed in the AdS/CFT dictionary, ρ

independent part should be extracted as h = ρh.Therefore, we need to invert the series to obtain the expression of h(0) with

respect to h. Let’s compute that up to the second order.

h(0) = h− h(2)

= h− (−12Ric(h0)−

16R(h(0))h(0))

∼ h− (−12Ric(h)− 1

6R(h)h)

(4.16)

This process of inverting the series is a bit laborious, but straightforward.If you wish to compute correlation function for general case, also including

scalar fields in AdS, we can compute the one-point function for OI from theformula for effective action (4.3) as

〈OI〉 =1√h

δΓ

δφI(4.17)

If you want n-point function, vary n times with φI . If you want correlationfunctions for energy momentum tensor or R-current, vary with hij or Aa.

Chapter 5

Wilsonian way ofHolographic RG

In the previous section, we performed how to deal with divergences in AdS/CFT.We took finite, ε independent part of external field, and expressed action withthem, and subtracted infinities. This is quite similar to the process of Stuckelberg-Petermann approach, where we took finite, effective coupling constant, ex-pressed effective action with respect to the effective field, and subtracted thedivergence.

It is natural to think that there may be a way to proceed more in a Wilsonianway. In this picture, it is clearer that the fifth dimension in AdS space, z,is corresponding to the scale parameter in RG flow in CFT. In particular, itshould be possible to derive RG-flow equation, which governs how effectivecoupling constants (or, value of external fields) evolves along the RG flow, fromAdS/CFT correspondence.

5.1 first step towards Holographic RG

Here I will give you several hints that leads us to think radial coordinate zin AdS is corresponding to the scale parameter of CFT. Here I will not givedetailed discussion, but just mention them.

5.1.1 Correlation function

In the context of AdS/CFT, it is known the value of CFT two-point functionis given as 〈OI(x)OJ(y)〉 ∼ exp(−mD(x, y)) where m is the mass of the scalarfield which couples to OI and D(x, y) is the geodesic distance of two pointsmeasured in AdS space. Computed in this way, correlation function deviatesfrom the behavior which is simply expected from conformal field theory, because

41

42 CHAPTER 5. WILSONIAN WAY OF HOLOGRAPHIC RG

Figure 5.1: They move z coor-dinate and see how correlationfunction evolves.

of the regularization at z = ε. The precise expression is

〈OI(x)OJ(y)〉 =ε2ml

(|x− y|+√

ε2 + |x− y|2)2ml. (5.1)

The deviation can be understood as that we have the microscopic scale ε inCFT. This suggests that ε could be understood as fundamental scale in CFT.The detail is in [9].

There is a deeper analysis along this line of research by Kraus and Balasub-ramanian [11]. They move the boundary for CFT to live, z = ε, to bigger z. Thebehavior of correlation function is exactly as expected from that of performingWilsonian RG transformation on CFT, changing the scale of the theory from εto a bigger value. This fact makes it more convincing that z is correspondingto scale of RG flow.

5.1.2 Flow of the metric and external fields

We saw that the metric and external fields changes along the z direction, and theflow of the parameter itself tells us something about CFT. The scaling behaviorof metric and external fields in CFT are exactly the same as the behavior ofmetric and fields in AdS. As you go to bigger z, metric becomes smaller. Ifwe use coordinate transformation to make the metric back to the bigger value,we need to change the coordinate x smaller, which is equivalent to using biggermeasure to see the physics. Therefore, as we go to bigger z, we will see coarse-grained picture of the CFT.

To convince you that z direction is related to the scale, and each slice of AdSspace corresponds to CFT viewed in some scale, I would like to derive RG-flowequation from AdS/CFT correspondence. That is the main goal of the chapter.

5.2 ADM formalism

Our goal is to obtain RG flow equation of CFT from supergravity equation ofmotion in AdS, using AdS/CFT correspondence. Here, I will follow the stepsof [6]. What we have is the boundary condition at z = ε, and we wish to get

5.2. ADM FORMALISM 43

Figure 5.2: er, ei are basis ofcoordinates r, xi and equiva-lent to ∂r, ∂i. e0 is also equiva-lent to ∂e0 .( If you don’t knowit, study differential geometry.Or, convince yourself that ∂and basis vectors have similarproperty.)

supergravity action of whole AdS. Please imagine, if you know the value of thefields at time t, and wish to compute the value of the fields and the action atlater time. You would construct Hamiltonian and solve Hamilton’s equationof motion. Here we proceed analogously. We take z as ’time’ coordinate andconstruct Hamiltonian with respect to z.

How can we make Hamiltonian for gravity? It is nothing different from whatwe do in normal classical mechanics. We treat each components of the metric asfields. If we solve Hamilton’s equation of motion, we can determine the metricinside the AdS, given the metric on the boundary z = ε.

Before constructing Hamiltonian, I would like to explain a convenient choiceof parametrization for a metric. By this choice of parametrization, Hamiltonianlooks extremely simple.

We first take 4 orthonormal basis vectors in r = r0(const) plane. We denotesuch vectors as ∂ea .(a = 1 ∼ 4) and another vector ∂e0 which is orthogonal tor = r0 plane. Then, we define lapse function N , shift function N i as,

∂r = N∂e0 + N i∂i (5.2)

This tells you, when you move along r direction, you will move in 4d space-time direction by N i, and you will move in the direction, which is orthonormalto 4d spacetime, by N .(fig:5.2)

We then transform it into ∂e0 = 1N (∂r − N i∂i), and we compare this with

∂e0 = Eµ0 ∂µ to get Eµ

0 = (1/N,−N i/N) (i = 1...4). E are components oforthonormal vectors with respect to coordinate basis. Then, in 5 dimension, wehave

5Eµa =

1N −N1

N . . . −N4

N0: 4Eµ

a

0

(5.3)

where 4Eµa is component for orthonormal vector in 4 dimension, and 5Eµ

a thatof 5 dimension.


Then, we are prepared to give components of vielbein.1 The components ofvielbein, ea

µ gives transformation of one-form, and they should just be inversematrix of Eµ

a . That’s because of the fact, if we express coordinate with respectto ea by y, and express coordinate with respect to ∂µ as x, ea

µ = ∂ya/∂xµ, andEµ

a = ∂xµ/∂ya. Therefore, the components of vielbein are,

eaµ = (E−1)a

µ =

N N1 . . . N4

0: 4ea

µ

0

(5.4)

the rows are for different µ, and columns are for different a. Calculating E−1

is straightforward, and soon we will notice nonradial part of this (4eaµ) is just

inverse of 4Eµa . Please note that we defined Na ≡ ea

µNµ. Since we know compo-nent of vielbein ea

µ and its inverse Eµa , it is easy to calculate metric from these.

Since

gµν = ea

µηabebν

gµν = Eµa ηabEν

b

, where ηab = ηab = diag(1, 1, . . . 1) here.

gµν =

N2 + NiN

i N1 . . . N4

N1

: hµν

N4

gµν =

1

N2 · · · −Nν

N2 · · ·:

−Nµ

N2 hµν + NµNν

N2

:

(5.5)

Here hµν , hµν is metric for 4 dimensional spacetime.

Next, we wish to express gravity action S =∫ √

5g 5Rd5x (5 means thatthey are constructed from metric in 5 dimension.) by 4d(induced) metric hij ,shift function Ni , and lapse function N .

It is easy to see that√5g = det(5e) = Ndet(4e) = N

√4h. (5.6)

The rest of the work, expression for 5R, is not so easy. In principle, it ispossible to find all radial derivative term of metric in it and find conjugatemomenta for hij , which is πij = ∂L/∂∂rhij , but it helps a lot to introduceextrinsic curvature to do such calculation.

The definition of Extrinsic curvature is as follows;

Kab ≡ −5∇anb (5.7)1vielbein is a set of orthonormal basis vectors in one point of spacetime, whose existence

is assured by equivalence principle, which claims you can take inertial (flat) frame whereveryou are in the spacetime.

5.2. ADM FORMALISM 45

where nµ is covariant vector which is normal (length 1), and orthogonal to 4dspacetime. (5∇ denotes covariant derivative, connection constructed by 5gµν)

This measures how much normal vector is tilted by bringing them to adifferent place. It can measure how much r=const surface is curved.

Let’s write this expression, using N,N i, hij . Since nµ is covariant vector withgµνnµnν = 1, we can take nµ = (N, 0, 0, 0, 0), and then, extrinsic curvature canbe calculated as,

Kab ≡ −5∇anb

=1

2N(4DaNb + 4DbNa − hab)

(5.8)

Here Da denotes covariant derivative in 4 dimension. With extrinsic curva-ture and 4 dimensional curvature scalar, the 5 dimensional curvature scalar canbe expressed as,

5R = 4R + KabKab − (Ka

a )2 (5.9)

I included the steps to derive this in appendix.Since Kab is expressed in terms of N ,Ni,hij we can construct Hamiltonian

with respect to these fields. The rest of the steps to construct Hamiltonian ismore or less straightforward.

Let’s go back to derive Hamiltonian. We start from action,

S =∫ √

5g(5R− 12GIJ(φI)∂µφI∂µφJ − V (φI)− 1

4FµνFµν)d5x (µ = 0, 1, 2, 3, 4)

This is the simplest action which has gravity, scalar fields, and gauge field.Then, we wish to construct Hamiltonian for this action.

You define conjugate momenta of fields φI , hij , Ai as,

pI =∂L

∂∂rφI, πij =

∂L∂∂rhij

P a =∂L

∂∂rAa(5.10)

then we construct Hamiltonian as,

Hamiltonian = pI φI + πij hij + P aAa − L (5.11)

where dot denotes radial derivative. Here, in our example, we have scalar fieldφI , gauge field Aµ, and metric gµν . However, as we see, radial components ofmetric or gauge fields, g00, g0i, A0 don’t have conjugate momenta, since theydon’t have radial derivative. (Example: F00 = 0, so there is no ∂0A0 term!) Wewill see they acts as Lagrange multiplier of the action.

If we plug in the expression for 5R 5.9, we can calculate momenta.

πij =δL

δ∂rhij= −

√h(Kij − hijKa

a )

pI =δL

δ∂rφI= N

√h(−∂rGIJφJ)

P a =δL

δ∂rAa=

δ(− 12F0iF

0i − 14FijF

ij)

δAa

N√

h = −F 0a√

hN.

(5.12)


Then, we wish to replace Kab ,∂rφ and F0a in the expression of Hamiltonian,by conjugate momenta, since they have radial derivatives. The expression is,

Kij =hij

3√

hπa

a −1√h

πij

∂rφI = − N√

hpI + N i∂iφ

F0a = −N2haa′(1√hN

pa′) + N iFia

(5.13)

Using the expression for F0a, we can also express F ab, and Aa without usingradial derivatives. (We use Fij , P

a, ∂aA0 and shift/lapse functions) Using theseexpression, we get Hamiltonian,

Hamiltonian =πij∂rhij + pI∂rφI + paAa − L

=NH0 + N iHi + A0G(5.14)

where

H0 =1√h

πijπkl(hilhjk −13hijhkl)

+√

h[V − 4R +14FijF

ij +12GIJ(φ)∂iφ

I∂iφJ ]− 12√

h[pipjhij + p2

I ]

Hi = −2Diπji + Filp

l + pI∂iφI

G = −∂apa

(5.15)

Note that they are just linear in N , N i, A0. Since there is no radial derivativeof N , N i or A0 in the Lagrangian, their conjugate momenta are identicallyzero, and it results in Hamilton’s equation ∂H

∂N = ∂H∂Ni = ∂H

∂A0= 0, which is

H0 = Hi = G = 0. Since we have Hamiltonian, it is easy to derive equation ofmotion for other fields (without radial components) and momenta as well.

φ =∂H

∂p, p = −∂H

∂φ

hij =∂H

∂πij, πij = − ∂H

∂hij

Aa =∂H

∂pa, P a = − ∂H

∂Aa

(5.16)

If we write radial derivative of fields explicitly, we have

Aa =∂H

∂pa= − pa√

h+ FiaN i + ∂iA0

hij =∂H

∂πij=

2N√h

πij −2N

3√

hπa

ahij + (∇iNj +∇jNi)

φI =∂H

∂pI= − N√

hpI + ∂iN

i

(5.17)

5.3. HAMILTON-JACOBI EQUATION 47

The constraint H0 = 0 is called Hamiltonian constraint, since they roughlycorrespond to the energy in 4 dimension. Since this is related to radial shiftin the bulk theory, and scale transformation in boundary theory, we expect itwill give us RG flow equation, and Weyl anomaly on the boundary theory. Theequation Hi = 0 is used to derive 4d diffeomorphism invariance of CFT, andA0 = 0 is used to derive 4d gauge invariance. We will see how these equationfor symmetries can be derived in the end of the next section.

5.3 Hamilton-Jacobi equation

5.3.1 the method

We have constructed Hamiltonian and written down Hamilton’s equation ofmotion. Equation of motion was given as equation for fields and momenta.Now we wish to solve these equation of motion and compute gravity actiongiven some boundary condition. Hamilton-Jacobi method is a really powerfultool to compute the gravity action, since it relates momenta to the change ofthe whole on-shell action with respect to the change of boundary condition. Informula, it is

δSon-sh

δφbd= pbd (5.18)

Therefore, we can replace momenta by these expressions, and equation of motionbecomes differential equation of action S as functional of boundary condition,which is almost what we want.

example: One particle action L(x, x, t) in 2 dimensionWe fix the starting point of the particle and vary the endpoint of the trajec-

tory, keeping on-shell condition.

Figure 5.3: In case we change x Figure 5.4: In case we change t

When we change x to x + δx, the change of the action is ,

δS =∫ (

δLδx

+δLδx

δx

)dtdx

=∫ (

δLδx− ∂t

δLδx

δx

)dtdx + [

δLδx

δx]endpoint

(5.19)


We operated integration by parts. Since expression inside the integral be-comes 0 by equation of motion, the change of the action just comes from thechange on the boundary, which is

δS =δLδx

δx = pδx (5.20)

Here I just used the definition of p ≡ δL/δx. We then consider changing t tot + δt at the endpoint. The change of the action will be calculated by summingtwo contributions. One contribution comes from comparing two trajectory forthe same time interval, and another from purely the difference of time intervalof two trajectories. First part is evaluated by using the fact it is equivalent tochanging δx = −xδt from the original action.

δS1 =δLδx

δx = −pxδt (5.21)

Then, second part can be evaluated as,

δS2 = Lδt. (5.22)

Therefore, by changing the time at the endpoint, we get action changed by

δS = Lδt− pxδt = −(H)δt (5.23)

where H is the Hamiltonian of the system.Therefore, as I promised, we have shown that variation of the action with respectto the boundary condition is related to momenta (and variation of the positionof the boundary is related to energy).

Extension to fields is straightforward. Changing the value of φ at the bound-ary, r = r0(const), field φ everywhere out of boundary will be also changed asφ → φ + δφ.

δS =∫

rend

p(x)δφd4x

→ δS

δφ(y)=

∫p(x)δ(x− y)d4x = p(y).

(5.24)

When we change the endpoint of r keeping boundary value fixed,

δS =∫

[L − δLδφ

φ]δrd4x

= −∫

Hδrd4x

(5.25)

The discussion for hij and Ai is exactly the same, therefore now we have equa-tions

pI =δS

δφIπij =

δS

δhijP a =

δS

δAa(5.26)

Where these fields are on the boundary of AdS and fields inside AdS is alwayson-shell.

5.4. GET RG FLOW EQUATION OUT OF ADS 49

5.3.2 The method tells us about supergravity action

What is the use of this method? We already constructed Hamiltonian and equa-tion of motion, which gives us constraints for hij , φ, Aa and conjugate momentaof these. If we replace these momenta by variation of action with boundarycondition, we get functional differential equation of the action with respect toboundary value.

Hi = 0 can be rewritten as,

−DjδS

δhij+ F i

l

δS

δAl+ (Diφ)

δS

δφ= 0

⇔∫ √

hd4x(−2εiDjδLδhij

+ (εi(DiAl) + Ai(Dlεi))

δLδAl

+ εi(Diφ)δLδφ

) = 0

⇔(Lεhij)δS

δhij+ (LεAl)

δS

δAl+ (Lεφ)

δS

δφ= 0

(5.27)

Where Lε is Lie derivative. Note here D means covariant derivative in 4dwhich is made out of hij . In the second line, I used integration by parts. There-fore S is invariant under coordinate transformation, x → x + ε.

G = 0 can also be rewritten as,

∂aδS

δAa= 0 ⇔ Λ(x)∂a

δS

δAa= 0 ⇔

∫∂aΛ(x)

δLδAa

d4x = 0 (5.28)

This is, S is invariant under Aa → Aa + ∂aΛ (∂Λ Aa), which means gaugeinvariance of S.

5.4 Get RG flow equation out of AdS

We are almost there. We have Hamilton’s equation of motion, and we canreplace momenta with variation of the action with respect to the boundaryvalue. Our remaining task is

identify divergent part of S and subtract

solve the (functional) differential equation for finite part of Ssugra, Γ asfunctional of φI , Aa, hij

derive RG flow equation

Therefore, philosophy of derivation is not so different from that of Skenderis etal. [10] It is very surprising that RG equation can be derived from 5d gravityequation. It implies 5d covariance should somehow be encoded in RG equationof our world. In the analysis of [6], the derivation of RG equation is done fixingthe gauge N = 1, Ni = 0, but here we take general 5d metric, and 5d covariancein the RG equation may be seen easier.


We saw that the equation of motion with respect to A0 results in gaugeinvariance of the boundary action, and Ni, which was shift in 4d spacetimewhen you move along z direction, results in the diffeomorphism of the action.It is natural to guess that equation of motion with respect to N , which tells youhow much you move in orthogonal direction to spacetime when you move alongz, will include RG-flow equation. The equation of motion for N (5.15), togetherwith Hamilton-Jacobi equation (5.26), is

S, S = L (5.29)

Where A,B is defined to be

1√h

δA

δhij

δB

δhkl(hkihjl −

13hijhkl) +

12√

hGIJ(φ)

δA

δφI

δB

δφJ− hab

2√

h

δA

δAa

δB

δAb(5.30)

And L is, as you can see from Hamiltonian constraint,

L =√

h[R− V − 12GIJ(φ)∂iφ

I∂iφJ − 14FijF

ij ]. (5.31)

Then, how can we identify the divergent part of Ssugra? What we do isnothing special. Since we know the scaling behavior of each fields as equation(4.2), we just count power of ε and see divergence of general terms. Since φI isgoing to zero for ε → 0, (Otherwise φI will dramatically change the metric atthe boundary and it will no longer be AdS space any more,) to know the MOSTdivergent behavior, we just have to concentrace on metric part.

First, we need to notice that we need√

hd4x in the expression of the actionwith respect to boundary values, in order to integrate in 4 dimension covariantly.This

√h is divergent for the power of ε−4 in 4 dimension. The next observation

is, once you have one derivative ∂i or gauge field Ai in the action, you needto introduce hij ∼ ε2 to contract these indices (also to make it diffeomorphisminvariant.) Thus, counting the number of derivatives or gauge fields give youthe worst divergence behavior, since φI just make terms more convergent.

Therefore, the most general divergent term comes from terms with less than4 (0 or 2) derivatives, and we can write down the most general action which is4d diffeomorphism invariant and gauge invariant2 as,

Sct = S(0) + S(2) =∫ √

hd4xU(φI) +∫ √

hd4x(Φ(φI)R +12MIJ(φI)∂iφ

I∂iφJ)

(5.32)S(0), terms with no derivative, is the most divergent part in Sct.

S(0), S(2) are terms with 0 or 2 derivatives, and they are local counterterms.Seperating Hamiltonian constraint (which was the equation of motion for N)into different scaling degrees, I get 3

2They are required by the equation of motion of A0, N i. However, if we include Chern-Simons terms in the AdS, gauge symmetry on the boundary might be broken as we will seelater.

3Strictly speaking, the weight for the gauge field part doesn’t correspond to this expression,since we are dividing S by A, which has the weight of one.

5.4. GET RG FLOW EQUATION OUT OF ADS 51

S(0), S(0) = L(0) (5.33)

2S(0), S(2) = L(2) (5.34)

2S(0),Γ+ S(2), S(2) = L(4) (5.35)

L(0),L(2) are just terms with 0, or 2 derivatives in L. From these equations,we can read off information of Sct and Γ(= Ssugra − Sct). Note that generallyΓ has 4 or more derivatives, since less derivative terms are included in Sct.

From S(0), S(0) = L(0), since δS(0)/δhij =√

hhijU/2, δS(0)/δφ = ∂U√

h,we have

1√hh

4U2 · 4− 1

3(2√

hU)2+1

2√

h(∂U

√h)2 = −

√hV

→ V =13U2 − 1

2GIJ(∂IU)(∂JU)

(5.36)

2S(0), S(2) = L(2) also gives restriction on the parameter Φ(φ), and M(φ).Finally, let’s inspect the third equation. If we believe in the AdS-CFT

dictionary we proposed at the beginning, Γ is the effective action, since wesubtracted divergence from the supergravity action. This equation includs Γ,and it should be the one which includes RG flow equation.

This equation gives you

2hijδΓδhij

+ βIδΓδφI

= anomaly (5.37)

Where βI = −6∂IU/U .It is easy to relate the first term to the change of the scale. First, let’s define

ad ≡√

det h. Then, We will have

2hijδ

δhij= 2hij

δh

δhij

δa

δh

δ

δa= a

δ

δa(5.38)

Now, we have sought-after expression of the RG-flow equation. Note thatwe can locally change the scale, unlike normal RG flow equation. Also notethat there is NO explicit dependence on parameter N , N i. In this sense, 5dcovariance is naturally translated to 4d dynamics. Rather than how much youmove along N or N i, the value of the metric and the fields on the hypersurfacein the AdS is important.

To derive RG flow equation for n-point correlation function, I should varythe expression with respect to

1√h

δ

δφI11

· · · 1√h

δ

δφInn

(5.39)

If this directly acts on effective action Γ, you can easily see that this repro-duces n point function of CFT, and varying with respect to βI will give you


additional terms. When we take variation at different points to get RG flowequation of n-point function, the local expression (anomaly) on the R.H.S. willdisappear. Therefore, we have

(aδ

δa+βI δ

δφI)〈OI1(xI1) · · · OIn

(xIn)〉+

n∑i=1

γJi

Ii〈OI1(xI1) · · · OIi

(xIi) · · · OIn

(xIn)〉 = 0

(5.40)where

γJi

Ii= ∇Ii

βJi − δJi

Ii(5.41)

Where ∇I is covariant derivative with respect to the metric in scalar field space,GIJ .

I should note that there is one more step to make the RG flow equation whatwe usually see, since φI , hij used here is ’bare’ fields and the metric, and weshould convert it to finite, effective ones. Since the fields are defined on infinitelyfar hypersurface, the value of the fields are divergent and we need to extractthe finite part. As we proposed in 4.2, we identify the renormalized fields asdividing fields by r dependent part.

With further identification

βI δ

δφI= βI

R

δ

δφIR

ORI = OJ

δφJ

δφIR

(γR)JI = (γ)K

L

δφL

δφLR

δφJR

δφK

(5.42)

We have

(δ

δa+βI

R

δ

δφIR

)〈ORI1

(xI1) · · · ORIn

(xIn)〉+n∑

i=1

(γR)Ji

Ii〈OR

I1(xI1) · · · OR

Ii(xIi) · · · OR

In(xIn)〉 = 0

(5.43)

5.5 Holographic RG and conformal anomaly

In the previous section, I claimed that R.H.S. of the RG flow equation corre-sponds to anomaly of the theory. This fact can be seen easily. If we just ignorethe scalar field in the theory, the equation will become⟨

T ii

⟩= anomaly (5.44)

let’s compute this in our framework as it is done in [9]. If we assume thatΦ is independent from field φI and just a constant, it becomes Φ = l/2 fromEquation of motion, and anomaly is given as

anomaly = l3(124

R2 − 18RijR

ij) +3

4UFijF

ij (5.45)

When we put all scalar fields to zero. This result is the same as HolographicRG by Skenderis et al(4.15), and also consistent with field theory result.

5.6. HOLOGRAPHIC RG AND OUR UNIVERSE 53

5.6 Holographic RG and our universe

In this section, I would like to show that we can give more natural meaning tocounterterm, as an action for gravity, using the scenario of [12].

In normal AdS/CFT correspondence, we take CFT to be living infinitely farin AdS space, which corresponds to sending cutoff of CFT to 0. However, weexpect the 4d CFT to have some natural cutoff, which is Planck scale. How canwe realize that the theory has the small scale cutoff?

One situation would be put brane-like structure at z = ε by hand, which issimilar to Randall-Sundrum model. [13]4Another merit of putting brane in thefinite radius is, that we can have gravity in 4 dimensional CFT. We saw inthe introduction of AdS/CFT that we consider asymptotically AdS space, andfrom this, 4 dimensional part of the metric should be flat. However, now, weare taking brane at finite distance, and metric on the brane could be differentfrom flat one by the order of ε.

In the scenario of [12], cutoff of the AdS space is introduced by ’the pointAdS × S5 connects to another geometry’, but in effect it will give you similarcutoff of the AdS.

Figure 5.5: The seperation of the action.

The identification of gravity action starts from separating Ssugra into tworegions, putting finite seperation z = µ by hand. (But we assume it is stillsmall enough we can apply AdS/CFT prescription for this cutoff) SUV denotessupergravity action for z < µ, and SIR denotes the action for z > µ. Since SUV

is the diverging part for ε, and SIR will be finite (since it is independent fromε), we can identify

4Randall-Sundrum model is the model to explain ’smallness’ of gravity compared to otherforce in the nature. The scienario is made of two 4 dimensional brane in 5 dimension. (Youcan understand this as D-brane in string theory, but you can just understand this as theobject which has some energy which is extented infinitely for 4 dimension.) On one of thebrane gravity is localized, and we are living on the other brane. If we solve Einstein equation,we get the metric will get exponentially get smaller as you go far from the gravity brane, andthis mechanism of ’decay of gravity in extra dimension’ can be the reason for the weak gravity.


SUV = Sct

SIR = Γ(5.46)

although there is ambiguity for finite part of separation. (This ambiguity cor-responds to moving µ) Since we are always assuming that the gravity action inthe AdS is always on-shell, we have equations

δS

δgµν= 0

δS

δφI= 0

(5.47)

We vary µ independent part of the metric and fields following AdS/CFTprescription, and using the expression for SUV = Sct and plug in to the aboveequation of motion for gij = hij , we get two equations

12hij + Φ(Rij − hij) = 〈Tij〉+ Tφ

ij

φI +∂U

∂φI+

∂Φ∂φI

+ R = 〈O〉(5.48)

Therefore, we can derive 4 dimensional Einstein equation from AdS/CFT cor-respondence.

5.7 R anomaly from Hamilton-Jacobi method

In this part, we consider inclusion of Chern-Simons term into the bulk super-gravity, and see this term will reproduce R-anomaly on the boundary. In asense, Chern-Simons term is required to make the correspondence correct, sincewe do have anomaly for R-symmetry in the CFT.

The steps to derive equations are quite similar to that without Chern-Simonsterm. We wish to express Hamiltonian in terms of gauge field Aa, 4d metrichab, scalar field φI , their momenta P a, πab, pI , and shift/lapse function N,N i.We can also use derivatives of nonradial direction, ∂i in the expression. (i is 1 to4.) Let’s concentrate on the gauge field part, since other part hasn’t changed.Action of gauge field part is,

Lgauge =√

hN

(−1

4FµνFµν + Cεαβγδε 1√

hNAαFβγFδε

)(5.49)

Where

εαβγδε =

1/√

g if αβγδε is even permutation of 01234−1/

√g if αβγδε is odd permutation of 01234

0 else(5.50)

5.7. R ANOMALY FROM HAMILTON-JACOBI METHOD 55

Please remember that εαβγδε behaves like tensor in 5 dimensional spacetime,and

√gεαβγδε is independent of the metric.

First, let’s calculate conjugate momenta for gauge field.

P a =δLgauge

δAa

= −√

hNF 0a + 4C√

hNεαβγ0aAαFβγ (5.51)

We then need to derive expression for F0a, and F ab to express everything interms of gauge fields and derivatives of nonradial components. Combined with

F 0a =hai

N2F0i −

N i

N2hajFij (5.52)

we get

F0i = Nhai1√h

(−P a + 4C√

hεabcdAbFcd) + NkFki

F ab = − N

2√

hhia(P a − 4C

√hεabcdAbFcd)(pi − 4C

√hεijklAjFkl)

+ NaFabPb + P a∂aA0 − 4C

√hεabcdAbFcdFkaNk − C

√hεabcdA0FabFcd

(5.53)

It is straightforward to solve 3 equations of motion (5.15) with these ex-pression, as we did. It can be checked that this indicates gauge anomaly anddiffeomorphism invariance on the boundary.

The Equation of motion for A0 is,

−∂aP a − CεabcdA0FabFcd = 0 (5.54)

Here P a = δS/δAa is identified as R current on the boundary from AdS/CFTcorrespondence, since Aa is regarded as the source to couple to R-current ofthe boundary theory.5 This implies that boundary theory is not R invariantby itself. It should be the same anomaly with the result purely from boundaryanalysis, for example by Fujikawa’s method.

Combined with gravity, scalar part of Hamiltonian and solving equation ofmotion, The equation of motion for Na is,Using −4CεabcdAbFcdFka = CAkεabcdFabFcd,

Hi = −2Djπji + Filp

l + p∂iφ + CAiεabcdFabFcd = 0 (5.55)

If we multiply by εi, infinitesimal shift of coordinate, and perform integration

5Strictly speaking, we also have to claim that counterterm is gauge invariant for abovestatement


by parts, we get

(Diεj + Djεi)δS

δhij+ (εiDiAlp

l + AiDl(plεi)) +δS

δφεi∂iφ + εiCAiε

abcdFabFcd = 0

→LεhijδS

δhij+ LεAl

δS

δAl+ Lεφ

δS

δφ+ Ai(Dlp

l)εi + εiCAiεabcdFabFcd = 0

→LεhijδS

δhij+ LεAl

δS

δAl+ Lεφ

δS

δφ= 0

(5.56)

Therefore, 4d diffeomorphism is maintained as we expected.

Chapter 6

Method of ’new’Holographic RG

There is another method of Holographic RG, developed by Papadimitriou andSkenderis [14]. We will show its equivalence to ’Wilsonian’ Holographic RG byde Boer, Verlinde, Verlinde [6] and the difference in the two method.

General idea for the method

It incorporates two ideas of holographic RG I discussed. The problem of ’old’method by Skenderis et al. [10] was that we had to invert asymptotic expansionof field at infinity into the field on the boundary to make the finaol expression4d covariant.

If we can count the divergence of the action and fields in a more covariantmanner (restricting ourselves to fields on the boundary), and solve equation ofmotion order by order for that degree of divergence obtaining effective actionand fields at the same hypersurface, we don’t have to invert the series.That is done by introducing dilatation (scale transformation) operator in theCFT. positive power of dilatation operator will be divergent as we go to theboundary, and such terms in the action will be regarded as counterterms.

Setup

We consider the same pure gravity action (4.5) and the metric (4.4) as in the’old’ method. First we express action as

S =∫

boundary

d4x√

h(Kaa − λ) (6.1)

First term is the Gibbons-Hawking term, as we have already considered in the’old’ method, and the second term is the on-shell bulk action. This expression isdue to the fact that on-shell bulk action can be written down as the functionalof boundary value of the fields.

57

58 CHAPTER 6. METHOD OF ’NEW’ HOLOGRAPHIC RG

Step1: expand the fields

To see the divergent part of√

h(K − λ) covariantly, we expand K, λ and othermomenta by eigenvalue of dilatation operator, δD.

From the analysis of [6], we already know that, if we take a ≡ (√

h)1/d asthe scale of the CFT and define dilatation operator as δD ≡ d

da , we can expressdilatation operator as

δD =∫

d4x2hijδ

δhij+ (∆− d)φI δ

δφI (6.2)

Please remember it is the operator for CFT to change the scale. (3.2.3) We areassuming that the field φ couples to operator in CFT O with scaling dimension∆. Note that δD ∼ ∂r and it will pick up power of exp((power)r).

Kij =

√h(Ki

j(0) + Ki(2) j · · ·K

ij(d) + Ki

j(0) log e−2r + · · · )

λij =

√h(λi

j(0) + λij(2) · · ·λ

ij(d) + λi

j(0) log e−2r + · · · )

π j =√

h(πij(0) + πi

j(2) · · ·πij(d) + πi

j(0) log e−2r + · · · )

(6.3)

Here, a(scale parameter) dependence of each term is nth power of e−2a, forexample, π(n) ∼ e−2an. Terms with tilde is related to conformal anomaly as wesaw in the ’old’ method.

step 2: solving equation of motion

We use the same equation of motion as the ones in [6]. We wish to solveHamiltonian constraint to solve for the fields, but the difference here is weexpand the field and momenta with the power of dilatation eigenvalue to seethe divergence.

For example, for pure gravity case, from the equation of motion

K2 −KijK

ji = R + d(d− 1)/l2 (6.4)

Since dilatation weight of R is 2,We can solve this order by order and get

K(0) = d

K(2) =R

2(d− 1)

K(2n) =1

2(d− 1)

n−1∑m=1

(Kij (2m)K

ji (2n−2m) −K(2m)K(2n−2m))

(6.5)

step 3: obtaining the action

We wish to express the action in terms of momenta or extrinsic curvature.From Hamilton-Jacobi equation, and the expression of the action above,(6.1)

59

we can express variation of the action in two ways, and we get the equationbelow which can determine λ, bulk action.

πijδhij +∑

πfδf = −δ[√

h(K − λ)] (6.6)

Here f denotes some fields in the supergravity, and φf its momenta. We applythis relation for the change by dilatation to determine λ in terms of momenta.For example, for gravity coupled to scalar, we have

δDS = 2hijδS

δhij+ (∆− d)φI δS

δφI

δDS = d(K − λ)√

h +√

hδD(K − λ)(6.7)

We equate them, and use the expression for momenta (which is explicitly com-puted from the action) πij = − 1

2

√h(Khij − Kij), pI =

√hφI . (dot denotes

derivative with respect to coordinate r)Then we obtain

(1 + δD)K − 2(∆− d)φφ = (d + δD)λ (6.8)

Step4: compute finite part of the action

renormalized action can be identified as

Iren =∫

bd

ddx√

h(K(d) − λ(d)) (6.9)

equivalence of ’new’ method and de Boer, Verlinde, Verlinde

Let’s show that these two approaches are equivalent. First we discuss the equiv-alence of the counterterm. In the original holographic RG, we should subtractthe divergence of the action, but in the ’new’ method, we subtract the diver-gence of the momentum directly. For action, we wish to identify term whichremains finite for large r. Therefore, we want term with δD eigenvalue 0.(If we denote scale of CFT as a, we want scale invariant part as our action.)

ena → subtract

e0a → what we want

e−na → go to zero when we send boundary to infinite

(6.10)

Here n is a positive integer.For example, when we expand πi

j , see πij(d) as momentum, since

πij =

hkj√h

δS

δhik(6.11)

and asymptotic behavior of metric is hij ∼ e2a,√

h ∼ eda, conformal part ofthe action corresponds to π(d). Therefore, we can see πi

j (d) as the regularized

60 CHAPTER 6. METHOD OF ’NEW’ HOLOGRAPHIC RG

one-point function T ij . In a similar manner, for a momentum pI , pI(∆) in the

expansionpI =

√h(

∑d−∆≤s≤∆

pI(s) + pI(∆) log e−2r) (6.12)

is 1- point function of scalar field. We then see the equivalence of n-pointfunction. Let’s see it in the example of 1-point function in such a way that canbe generalized. (Although it is easiest to use momentum itself to derive onepoint function) In [6], they use the definition

T ij ≡ 1√h

δSren

δhij

O ≡ 1√h

δSren

δφ

(6.13)

where hij = hije2r, φ = e(∆−d)rφ when we gauge fix N = 1, Ni = 0. In the new

method, they define

Tij ≡ limr→∞

(e(d−2)r 2√h

δSren

δhij) (6.14)

In ’new’ method, the field itself should be derived from first expressing fieldsin terms of momenta, and plugging in the solution for that. The above expres-sions are obviously equivalent.The basic process to solve equation of motion is exactly the same although againcounting the degree of divergence is a bit in a different manner.

Chapter 7

Holographic c-function

In this section, I would like to review so-called Holographic c-theorem. In con-formal field theory, we can define a ’c-function’, which is extension of centralcharge, and is considered to express degree of freedom of the system. It isknown, using AdS/CFT correspondence, we can define the corresponding quan-tity in AdS, and we can prove that it also has the similar characteristic. Littleis known about c-function in higher dimensional CFT, and this line of researchmight give some new insight into it.

7.1 c-function in CFT

Our setup is general 2 dimensional CFT in flat space. After we perform theWick rotation and have Euclidian time signature, we can further change thecoordinate using complex variable z and z. The metric will be

gij =(

1 00 1

), gαβ =

(0 1/2

1/2 0

)(7.1)

Where i, j take 1 or 2, and underlined indices take z or z.Our assumpsion is as below.

renormalizability of the theory

unitarity of the theory

translational & rotational symmetry of field theory

We wish to confirm that the RG flow in this CFT is irreversible by provingthat there exists a function which monotonically decrease under RGtransformation (want 1). (That is, we will prove that we don’t get thesame CFT as the original one after process of RG flow.) Although it may soundobvious that RG flow is irreversible, this c function might have more implication,since it will be central charge at the fixed point of RG flow (want 2).

61

62 CHAPTER 7. HOLOGRAPHIC C-FUNCTION

Therefore, c function can be understood as the quantity to express degree offreedom in the physical system, since central charge has natural interpretationas the degree of freedom of the conformally-invariant system. For example, for2 dimentional sigma model, where action is S =

∫ √hd2σhijηµν∂iX

µ∂jXν , the

central charge is just given by the number of fields X. (the number of indicesµ can take)

We denote holomorphic part of energy momentum tensor as, T (z) = Tzz/2π

and the trace of the energy momentum tensor as 4Tµµ/2π = 4Tzz/2π ≡ S.

From the conservation of energy-momentum tensor (required from diffeo-morphism invariance of the action), we have

∂iTij = 0 (7.2)

This condition reduces to

∂T +14∂S = 0 (7.3)

where ∂ denotes derivative with respect to z.Then, we define correlation function as,

〈T (z, z), T (0, 0)〉 ≡ F (|z|)/z4

〈T (z, z), S(0, 0)〉 = 〈S(z, z), T (0, 0)〉 ≡ G(|z|)/z3z

〈S(z, z), S(0, 0)〉 ≡ H(|z|)/z2z2

(7.4)

Considering tensor index for the l.h.s, we can conclude F,G,H just dependon the absolute value of z, z. For example, the ’angle’ dependence of 〈T 〉 iscaptured by 1/z4. Maybe you wonder why we define such quantities, but wewill see the F above becomes central charge at the fixed point, and G, H areused to show the decreasing property of F along RG flow.

We then change the scaling of the coordinate and see how these quantitiesflow along the RG flow.1 If we denote the scale of the theory as |z| ≡ R, wehave

Rd

dR= 2|z|2 d

d|z|2= 2z∂ = 2z∂ (7.5)

Using this equation and equation (7.3), we can derive

Rd

dR(F +

14G) =

32G

Rd

dR(G +

14H) = (2G + H)

(7.6)

If we define C function as

C(R) ≡ 2F −G− 38H = 2(F +

14G)− 3

2(G +

14H) (7.7)

1We perform RG flow by changing coordinate instead of changing metric.

7.2. C-FUNCTION IN ADS 63

Then, the change of C under RG flow is,

RdC

dR= 2 · 3

2G− 3

2(2G + H) = −3

2H ≤ 0 (7.8)

We can argue H is positive, using translational invariance of the vacuum,exp(ipx) |0〉 = |0〉, and real property of Tzz, and rewriting H as square of somestate. Therefore want 1 is now proven. This function C is nonincreasingfunction with respect to the scale parameter R, and scale transformation R →(µ/ε)R is irreversible process. This is quite nontrivial, since just the rescaling ofthe coordinate x → ax (accompanied by change of fields) is RG transformationin CFT.

At the fixed point of RG flow, G = H = 0, since S is anomaly for the scaletransformation.

〈T (z, z), T (0, 0)〉f =Cf

2z4(7.9)

f in the above expression denotes fixed point. Now we have also proven want2. C function matches central charge at the fixed point of RG flow. In 2dimensional CFT, it is known 〈T (z, z)T (0)〉 = c/z4 where c is central charge.

7.2 c-function in AdS

In this section, I would like to investigate on the c-function in AdS space. I havealready mentioned that we can understand radial coordinate of the AdS as thescale parameter of RG for CFT. That means, simply looking at the gravity sideof the AdS/CFT correspondence, we also should be able to find the quantitywhich is nonincreasing as we go towards inside of the AdS and reduces to centralcharge at the boundary of AdS. (at the fixed point of RG). Our discussion isbased on [9] and [15].

How should we find the c-function in AdS? The first step is to see the rela-tionship between scale anomaly and central charge. In 2 dimension, for generalCFT which couples to gravity (where metric is fixed as external field), the scaleanomaly is proportional to the curvature of the background metric (or gaugefield), and the coefficient is the central charge.⟨

T ii

⟩∝ cR (7.10)

However, the expression will become more complicated for higher dimen-sional theory. Let me explain in 4 dimentional CFT. The expression for thetrace anomaly can be written as below.⟨

T ii

⟩=

c

16π2W 2

ijkl +a

16π2R2

ijkl + αR+ (7.11)

Where W is Weyl tensor, which is conformally invariant part of anomaly, andW 2

ijkl = R2jikl − 2R2

ij + 13R2 and R2 ≡ ( 1

2εklijRmnkl)2 = Rijkl − 4Rij + R2.

the a in this expression is known to be nonincreasing function in any exampleknown, and considered to be extension for c-function in 4 dimension. Last term


in the anomaly is scheme dependent of regularization, and usually considered asphysically unimportant term. On the other hand, we can also derive conformalanomaly using the method we have analyzed in (5.45), and (4.15) as,⟨

T ii

⟩= l3(

18RijR

ij −R2) (7.12)

Where l is size of the AdS which appears in the metric as

ds2 = dr2 + exp(2r/l)dxidxi (7.13)

Therefore, comparing these two expressions, we can conclude that for CFTwhich has supergravity dual, we have a = c and they are proportional to l3.For two dimensional CFT, we can also show that the anomaly computed fromAdS/CFT is proportional to l. We can see ld−1 to be central charge in ddimensional CFT.

In AdS space, we can define a quantity which is monotonically decreasingand reduces to central charge at the fixed point. If we take gauge-fixed metricfor (asymptotic) AdS space, metric can be expressed as

ds2 = dr2 + exp(A(r))dxidxjηij (7.14)

assuming 4d part of the metric ηij is flat. Let’s define c-function in AdSd+1 as

C ≡(∂A

∂r

)−(d−1) (7.15)

Then, it can be proven that

C is central charge at the fixed point of RG flow

C is nonincreasing for positive z direction (or, negative r direction)

First, let’s show the first condition. Since AdS space metric is (7.13), from thedefinition of c-function, it becomes ∼ ld−1, and therefore it becomes centralcharge at the fixed point. Decreasing property of the c-function will be shownin the last section of this chapter.

7.3 meaning of Holographic c-function

We saw that we can define a quantity

which becomes the central charge of CFT at the fixed point of RG (bound-ary of AdS),

which is nonincreasing along RG flow.

There is a deep meaning in the quantity C = A′−(d−1) as defined above. TheA′ in the expression has the meaning as ’expansion rate of light cone surface’.By several people [16] it is proposed that the entropy in the spacetime enclosed

7.4. SUPPORT FOR HOLOGRAPHIC PRINCIPLE 65

by some light cone is proportional to the area of the light cone. It is calledcovariant entropy conjecture. First I will show you how you can relate A′ toexpansion rate, and next I will consider its relation to entropy.

Let’s imagine light rays going outwards from one point in AdS space. Thenthe expansion rate of the surface is given as,

1Area

∂Area∂r

=∂ log(exp(A))

∂r= A′ (7.16)

Remind the area is expressed as exp(A) for d dimension. In the next section wewill see the meaning of the relation of the ’expansion rate’ and entropy in moredetail.

7.4 Support for Holographic principle

7.4.1 Motivation

There is a strong support to argue that the entropy in the light cone is pro-portional to its surface area, in the paper by Jacobson. [17] He considers ther-modynamics of the space inside the light cone, just as we can consider thermo-dynamics of black holes. By assuming that the entropy of the space inside thelight cone is proportional to the surface area of the light cone, and plugging itinto the thermodynamic equation dS = dQ/T , he derives Einstein equation.

Figure 7.1: Area of the light cone is interpreted as the entropy inside the lightcone.

7.4.2 Similarity of Black hole and light cone

The motivation of the paper [17] clearly comes from Black hole entropy. Beforegoing to indicate how similar they are, I would like to explain what is causalhorizon. You specify a point in the spacetime and consider the domain ofspacetime in which you can send/receive message from the point. In flat space,this is clearly light cone, which is a sphere expanding as you goes future or past.This sphere is boundary of communication. No information can be sent outside


from the point to the outside of the future horizon, and no information can beobtained from outside the past horizon to the point. We wish to consider thethermodynamics of the system inside the light cone, which is horizon in the flatspace.

Figure 7.2: B at some time is outside the lightcone originated from A, and hecan’t know anything about A.

By definition, the horizon of the black hole and the light cone in the flatspace is clearly causal horizon, in the sense they both are the boundary for the’possibility of communication’. As we know that the area of the horizon of theblack hole can be interpreted as the entropy of black hole, it is natural to guessthat the area of the light cone sphere in flat space can also be interpreted as theentropy inside the horizon.

Consider Schwarzschild black hole here.The metric of the Schwarzschild black hole was

ds2 = −(1− 2GM

r)dt2 + (1− 2GM

r)−1dr2 + r2dΩ2 (7.17)

We can take so-called Krusucal coordinate to make coordinate well definedat everywhere, defining coordinates T and R. Then the analogy with flat spacewill be even clearer.

T = (r

2GM− 1)

12 er/4GM sinh(

t

4GM)

R = (r

2GM− 1)

12 er/4GM cosh(

t

4GM)

(7.18)

In this coordinate, the metric becomes

ds2 = −32G3M3

re−r/2GM (dT 2 − dR2) + r2dΩ2 (7.19)

The penrose diagram looks like below. Note that light going in radial direc-tion will move at 45 degree in both diagram.

(Note: intro to Kruskal is necessary before)Here in the left figure, horizon, boundary for communication, is depicted as the


Figure 7.3: r = const observer at the BH horizon and constantly acceleratedobserver escaping from light cone are similar.

straight line which intersects T or R coordinates by 45 degrees, and r = constis depicted as hyperbolic lines outside the horizon. t = const is the straight linewhich go through T = R = 0. Upper part of the diagram corresponds to theblack hole.

It is very clear how it looks like in Minkowskian case, if you look at thediagram diagram 7.3 causal horizon is again depicted as a straight line whichgo through the origin, and you can’t send any information from point p tothe outside of the future horizon. This similarity suggest us, and motivate toconjecture that area of causal horizon, generally, is proportional to the entropy.In this diagram, hyperbolic curve is constantly accelerated observer, instead ofr = const in the Schwarzschild case. The curve is given by

t(τ) = 1α sinh(ατ)

r(τ) = 1α cosh(ατ)

(7.20)

and it is easy to check that the length of acceleration vector

aµ =d2xµ

dτ2(7.21)

is actually α. As you have bigger acceleration, curves are depicted more inwards.I add one comment; Even though the accelerating observer are slower than light,light cone can never catch up the observer when he is constantly accelerating;Achilles can never catch the tortoise when tortoise is constantly accelerating, ifthe velocity of the tortoise becomes infinitely close to the velocity of Achilles.

7.4.3 Thermodynamics of spacetime

Let’s review the thermodynamics of black hole briefly. If we apply thermody-namics, we want the setup to satisfy two assumpsions;First, the system, at least the point of spacetime we consider, has to be in ther-mal equilibrium. For a black hole, we know that at the horizon it is equilibrium


of temperature T = κ2π . This fact can be understood using the concept of Un-

ruh temperature, and I briefly explain it here, since it is crucial in our analysis.Please note we are also implicitly assuming observation is made long enough,otherwise we can’t ignore fluctuation and concept of equilibrium doesn’t makesense. Another assumption is, that observation is made locally, since that isthe place thermodynamics is known to hold. Therefore, when we consider blackhole (inside the BH horizon) as the system, we wish observer to be just outsidethe horizon, and all the quantities such as heat, temperature, are according tohis measurement. In order for him to be really close to the horizon, he hasto be close to light velocity, and he has to be constantly accelerating. Thiscorresponds to curve in inwards of Rindler space.

I explained local observation can be made by infinitely accelerating observernear the horizon. How can we see that the horizon is at the thermal equilibriumthen?

Unruh temperature

The fact is, if you accelerate at a constant rate in vacuum flat space, you will feelthermal radiation even though you are originally in flat space. Please remindthe constantly accelerated observer in Flat space (equation 7.20). For him, τ isthe time he measures. Time translation is clearly ∂τ . What happens if we takeimaginary time? Let’s take θ = −iτ . Then, the curve of the observer is nowgiven as,

Figure 7.4: Wick rotation of the trajectory of observer.

x = 1

α cos αθ

t = 1α sinαθ

(7.22)

Therefore, time translation for the observer is periodic in imaginary time,

θ ∼ θ +2π

α(7.23)

Therefore, we can think of T = α/2π as the temperature the observer feels,since periodicity in imaginary time in field theory is identified as the inverse


of temperature, 1/T . This doesn’t give you any physical picture, but nice forquick derivation.

Black hole again

Then, we go back to thermodynamics of Black hole at the horizon. What tem-perature does the observer at the horizon see? It is natural to set the stateof freely falling observer as vacuum state, since you can’t detect any effect ofgravity at the frame. Then, the observer just outside of the horizon have to ac-celerate to stay outside the horizon. He observes, from Unruh effect, effectively,the thermal radiation coming from Black holes. If we call the acceleration of freefalling observer with respect to static frame as surface gravity, and denote byκ, the temperature observed at the horizon by the static observer is T = κ/2π.Therefore, nearby the horizon of black hole is in equilibrium at temperatureκ/2π.

Figure 7.5: Surface of the black hole is in equilibrium at temperature T = κ/2π.

Now, setup for the thermodynamics is complete, and heat, and temperatureis exactly same as the normal definition for him. We didn’t need to introduceany interpretation or definition. From these quantities, you can compute thechange of entropy, and you can see from Black hole dynamics, that change ofentropy is proportional to the change of area of black hole.( BH eria entropy nokankei mottosetumei!!)

Flat space again

Having reviewed the case of Black hole thermodynamics, the setup for the flatspace is quite obvious. In the flat space, the causal horizon is light cone, andwe wish observer to be always near the horizon. From the Rindler diagram,such observer is obvious; You have to be constantly accelerating to escape fromlight cone. temperature, and heat is that observed by him. Then, our goal is toderive Einstein equation just from the thermodynamic relation and the relationthat entropy increase is proportional to the area increase of the horizon,

δS = ηδA (7.24)


(Here we just assume η is some unknown constant, although its value is known.)Let’s summarize our goal again and what we have in our hands. Our goal

is to show Einstein equation from equation dS = δQ/T , and their gravitationalanalogue are,

T α/2πS area of causal horizon

δQ heat flow into light conemeasured by obserever

It is possible to consider ‘heat‘ as the energy that flows to unobservable de-gree of freedom as Jacobson does, therefore the energy which flows inside thehorizon (note that horizon can’t be seen from escaping observer) can be inter-preted as heat. However, it can also be replaced by normal thermodynamicaldefinition of heat, δQ = ρdV − pdV . Here ρ is energy density, p is pressure, dVis the infinitesimal volume change of the system.

Then, we wish to derive covariant expression for that, since it will be moreconvenient. We introduce proper time τ for the observer, and velocity of light kµ,velocity of the observer χi = dxi/dτ , energy-momentum tensor Tij = (ρ, p, p, p)if we take inertial frame, and ρ, p are the ones observed. A is the area of thehorizon we consider. Lower index for k, χ are time to express 4 velocity.

Figure 7.6: we consider thermodynamics of expanding light cone.

δQ = ρdV − pdV

= Taikiτ (ki

τdτA)

= Taikiτki

τdτA

= Taikiλk

iλdλAdλ

dτ

(7.25)

From the first line to the second, I used that kiτ = (1, 1, 0, 0), and from the

second to third, I transformed vectors from the observer’s frame to static frame.Indices with underlines means static frame. For simplicity, I will omit un-derline from the notation from now on. Here we introduced affine parameter


λ = t+r2 = exp(ατ) to parametrize null geodesics more naturally.

We could write the heat flux in a manifestly covariant way as in [17]. Sincethe dxa/dτ is killing vector, because boost in the Lorentz frame is the symme-try. Then, we can identify the Taidxa/dτ as some current. If we take covariantderivative Di, the one which acts on energy momentum tensor vanishes becauseof conservation of energy momentum tensor, and the one which acts on dxa/dτvanishes because this will give Dadxi/dτ +Didxa/dτ , since the energy momen-tum tensor is symmetric, and it vanishes using the property of killing vector.

Therefore,

δQ ∼ Taikiλk

aλdλAαλ (7.26)

Therefore, we have obtained the tensor expression of heat. By using this equa-tion, we can derive the Einstein equation. Then, let’s go to the tensor expressionof the change of entropy. As we have discussed, entropy inside the light cone isread off from the surface area of the light cone, and the change of the entropyis,

δA = θdλA (7.27)

We define the expansion rate of the area with respect to affine parameter asθ ≡ ∂ logA/∂λ. We wish to know how the infinitesimal area expands along nullgeodesic.Before analyzing the null geodesic case, we first consider the timelike geodesic.Remember, in the whole analysis, we are taking Riemannian normal coordinateat the point we see to make it simple. (That is, gab(x) = ηab, ∂gab(x) = 0 at thepoint x we see) We can take such a coordinate at any point in the spacetime,and we don’t lose generality. Then, we will use the relation, that expansion ofcoordinate is given by the formula

d2rj

dt∼ −Rj

titri (7.28)

Detail is given in [18], but I will briefly explain the idea below. We wish tocompare two test particles on infinitesimally separated geodesics. We assumethey are separated by purely spacelike vector rj . Velocities of these particlesare set to be the same, which means, they are parallel transported from eachother. After time shift of τ , we wish to see how these points accelerates to eachother. That is given by parallel transporting two velocity vectors and compareto each other. The difference of these two are just equal to parallel transportingthe velocity vector (vµ = (1, 0, 0, 0)) along parallelogram made of vµ and rj .Then, expansion rate of rj is given as,

δrj(t) ∼ −ri(0)Rjtitt

2 (7.29)

Therefore, expansion rate of some volume is,


Figure 7.7: shift of coordinate.

δV

V∼ δ log V

∼ δ∑

j

log(rj)

∼∑

j

δrj/rj

∼ −Rjtjtδt

2

(7.30)

Therefore, we can express the change of the volume along timelike geodesicas Rj

tjt. We can, analogously, consider the change of the area along null geodesic.We take, for simplicity, the null vector for the geodesic as ka = (1, 1, 0, 0). (Sincethe change of the area is expressed in a covariant way, the final result doesn’tdepend on the coordinate) We can easily show that the change of the area isgiven by,

δ logA ∼ −12Rabk

akbλ2 (7.31)

Since,

rhs ∼ 12(−R00 −R11)λ2

∼ 12(−R1

010 −R0101 −Ra

0a0 −Ra1a1)λ

2

∼ log(r2) + log(r3)∼ log(A)

(7.32)

Where the index a is sum over 2 and 3. Therefore, along null geodesic, thearea of the surface of the light cone will change by δA, which is expressed as,

δA ∼ ∂logA∂λ

dλA

∼ −λRabkakbdλA

(7.33)

7.5. MEANING OF HOLOGRAPHIC C-THEOREM 73

Now we have necessary ingredients to derive Einstein equation. From equa-tion (7.26), (7.33) and δS = ηδA, and using the thermodynamic relation

δQ = TdS (7.34)

We have a relationTabk

akb =η~2π

Rabkakb (7.35)

This is true for any ka, since we can choose any observer and any light conewhich is chasing the observer. Then, we have

Tab =η~2π

(Rab + f(x)gab) (7.36)

There is ambiguity of adding term which is proportional to gab, where f isarbitrary function of spacetime x. Using Bianchi identity, Tab should be diver-genceless from energy conservation, and since we know the only divergencelesstensor made out of the metric is Einstein tensor Gab = Rab − (1/2)Rgab andmetric itself gab, we know that f has to be of the form f = −R/2 + Λ where Λis constant. Putting all pieces together, now we have Einstein equation;

Tab =η~2π

(Gab + Λgab) (7.37)

7.5 Meaning of holographic c-theorem

In the previous section, we saw that the area of the light-cone is naturallyrelated to entropy. Using these analysis, we can give a meaning to holographicc-function from AdS side.

The definition of the c-function was some power of expansion rate as informula (7.16). Let’s consider setting which is similar to the last section, lightcone expanding in AdS space. The expansion rate can be rewritten as, usingthe expression for entropy inside the light cone S = ηA, as

1Area

∂(Entropy)∂r

(7.38)

Therefore, c-function is related to the increase of the entropy density in the lightcone surface due to the expansion. (You can think the entropy is flowing to thelight cone)

You can further relate the nonincreasing property of the c-funcion to increas-ing property (second law of thermodynamics) of entropy. If we take derivativeof (7.31) two times, since coordinate λ is just proportional to r, it will be

A′′ = −12Rabk

akb (7.39)

If we use Einstein equation, it would be

A′′ ∼ −12Tabk

akb ∝ δS (7.40)


Since, we saw that − 12Tabk

akb becomes δS = ρdV − pdV for the observernear the horizon. Therefore, the decreasing property of the c-function, andthe irreversibility property of RG, is equivalent to the increasing property ofthe entropy.

In order to consider analogy with the RG, let’s imagine shrinking the lightcone instead of expanding. In CFT point of view, this corresponds performingWilsonian RG, and the holographic c-function monotonically decrease towardssmall r. In AdS point of view, this is shrinking light cone, and we could putobserver just inside the light cone, and conclude that entropy outside the lightcone will monotonically increase.

Figure 7.8: shrinking light cone is similar to Wilsonian RG.

Chapter 8

future developments

In this thesis, along the line of AdS/CFT, we saw how the structure of RG flowof CFT is encoded in each slice of AdS. I will briefly introduce several possiblefuture developments in related fields.

AdS/QCDIt is generally expected that application of AdS/CFT to QCD will give a

more conprehensive description of low energy QCD, such as confinement. It isexpected that universal behavior at low energy QCD can be derived using theidea of Holographic RG.

Black hole and condensed matterIn the context of the holographic RG, it is known that black holes in AdS

corresponds to the fixed point of RG. QFT at far outside of AdS correspondsto CFT, and we can think of black hole as another fixed point of RG (CFT)which flows from the boundary CFT.

There is confirmed relation called attractor mechanism, which states thatthe value of the scalar field at the surface of the black hole takes fixed valueswhich is determined only from parameters of black hole, no matter what valuedo they have at the boundary of AdS. This behavior precisely corresponds touniversality in RG flow, in which all the irrelevant parameters go to zero andthe behavior at the large scale fixed point is universal.

There is also a line of research which relates low-energy physics (CFT) withblack hole in AdS. There are applications on quantum hall effect [19], and super-conductivity by [20]. We might be able to understand the universal behavior ofcondensed matter physics using universal behavior of black holes, or vice versa.

Hawking radiation and anomaly [21]

75

76 CHAPTER 8. FUTURE DEVELOPMENTS

There is a line of researches that studies the relationship between the hawk-ing radiation of black hole and anomaly of the field theory at the boundary. Aswe saw in the section of holographic c-theorem, the information in the gravityand the information in RG is related to each other, and it may give a new insightinto hawking radiation and infomation paradox.

.1. ADDITIONAL NOTES ON HOLOGRAPHIC RG 77

.1 additional notes on Holographic RG

There are two ways to derive Ward identities of boundary CFT using AdS/CFTcorrespondence. One way is to consider well-defined variational principle andvary the action by boundary metric. In [22], they derive stress-energy tensor ofpure gravity in this way. The expression is,

T ij = − 18πG

(Kij −Kaahij) (1)

Another way is, as we did in this paper, use Hamiltonian-Jacobi method. If youuse this method, you don’t even have to keep track of well-defined variationalprinciple by adding appropriate boundary term. When we use

δSon-shell

δhijb.c.

=δLδhij

(2)

and expression of the action,

116πG

∫dd+1xN

√h(dR + K2 −KijK

ij − 2Λ) (3)

same expression can be obtained easier.

→ T ij =2√h

δS

δhij=

2√h

δLδhij

=1

8πG(hijK −Kij) (4)

When we consider Chern-Simons term in AdS3, we can do similar analysis.Let’s derive ward identity of CFT considering well-defined variational principleby adding boundary term to the action. We include CS term in AdS3,

Sgauge = − ik

4π

∫d3xAdA + Sboundary

gauge (5)

If we require gauge current on the boundary to be purely holomorphic, thatis J1 + iJ2 = 0, we will need additional boundary term as,

Sboundarygauge = − 1

16π

∫boundary

d2√ggαβAαAβ (6)

Then the current will be Jα = −C(Aα − iεβαAβ), and we can calculate Ward

identity,2DaT ab = J2F

21 + 2CA1F12 (7)

which is identical to the one which is derived by Hamilton-Jacobi formalismwithout knowing anything on the additional boundary term. Therefore, thismethod is really powerful since it automatically takes care of well-defined vari-ational principle by itself.


.2 derivation of the curvature formula

In this part, I will mainly discuss transformation of Einstein-Hilbert action intothe form which only includes first derivative of the metric,1

5R =4 R + KabKab − (Ka

a)2 + total derivative (8)

In the discussion, following equation which connects covariant derivative in 5dimension and 4 dimension will be useful;

DaT bc···ef ··· = ha′

a hbb′h

cc′h

e′

e hf ′

f T b′c′···e′f ′··· (9)

Here T is obviously a tensor in 4 dimension.

We begin with Riemann tensor constructed from 4 dimensional metric andwish to relate them to Riemann tensor constructed from 5 dimensional metric.From

4Rdcabωd = DaDbωc −DbDaωc (10)

Here ω is arbitrary covariant vector in 4 dimension. Then,

DaDbωc = Da(hdbh

ec∇d ωe)

= hfahg

bhkc∇f (hd

ghek∇dωe)

= hfahg

bhkch

dgh

ek∇f∇dωe · · ·α

+ hfahg

bhkc(∇fhe

k)hdg(∇dωe) · · ·β

+ hfahg

bhec(∇fhd

g)(∇dωe) · · · γ

(11)

We can transform β using hek = ge

k + nenk, when nµ is a vector orthogonalto vectors in 4d, and normal.

hfahd

bhkc(∇fhe

k) = hfahd

bhkc∇f (ge

k + nenk)

= hfahd

bhkcn

e∇fnk

= hfahd

bhkcn

eKfk

= hdbn

eKac

(12)

In the second line, I used hkcnk = 0. In the third line, I used the definition of

Kab. (5.8) Remind that Kab didn’t have radial components.For γ, we similarly have

hfahg

bhec(∇fhd

g) = hecn

dKab (13)

Then, we can see, by taking DaDbωc − DbDaωc, this term vanishes from thesymmetry property of Kab.Further, we transform β as,

hdbn

eKac(∇dωe) = hdbKac∇d(neωe)− hd

bKac(∇dne)ωe

= −hdbKacK

edωe

(14)

1The derivation mostly depends on [23].

.2. DERIVATION OF THE CURVATURE FORMULA 79

Note that neωe = 0, since nµ is taken as normal, orthogonal vector to 4d.Therefore, we have

DaDbωc −DbDaωc = hfahg

bhkch

dgh

ek∇f∇dωe − hf

bhgahk

chdgh

ek∇f∇dωe

+ (KacKeb −KbcK

ea)ωe

= KacKebωe −KbcK

eaωe − hf

ahgbh

kch

dgh

ek

5Rg′

gfd(hc′

g′ωe)(15)

Transforming both L.H.S and R.H.S, we have

4Rdcabωd = hf

ahgbh

kch

dl

5Rlkfgωd + (KacK

db −KbcK

da)ωd (16)

Since ω can be any arbitrary covariant vector, we can omit ω from the expression.To get the expression for Ricci scalar, we contract with ha

dhbc, and get

4R = hflh

kg 5Rlkfg + KabK

ab − (Kaa)2

= (gfl + nfnl)(gkg + nkng)5Rl

kfg + KabKab − (Ka

a)2

= 5R + 2 5Rabnanb + KabK

ab − (Kaa ) 2

(17)

Next, we wish to get Ricci tensor, to finally get Ricci scalar. We wish to knowRabn

anb to express 5R in terms of hab, N ,Ni.

Rabnanb = nbRc

acbna

= nb(∇c∇bnc −∇b∇cn

c)

= −(∇cnb)(∇bn

c) + (∇bnb)(∇cn

c) +∇c(nb∇bnc)−∇b(nb∇cn

c)

= −KabKab + (Ka

a)2 +∇c(nb∇bnc)−∇b(nb∇cn

c)(18)

Therefore, we finally have

5R =4 R− 2Rabnanb −KabK

ab + (Kaa)2 +∇c(nb∇bn

c)−∇b(nb∇cnc)

=4 R + KabKab − (Ka

a)2 +∇c(nb∇bnc)−∇b(nb∇cn

c)(19)

Since K only has first derivative of the metric, now our purpose has beenachieved. Please note that there are extra surface terms coming from thisprocess, which is basically why we have to have Gibbons-Hawking boundaryterm.


Keep enjoying!!

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Documents

Review of Holographic RG - UvA · 2020-06-01 · Review of Holographic RG Hideto Kamei Master’s thesis Supervisor: Prof. Erik Verlinde University of Amsterdam Institute for Theoretical