d-branes and ads/cft...d-branes and ads/cft by junaid saif khan department of physics lahore...

D-Branes and AdS/CFT

By

JUNAID SAIF KHAN

Department of PhysicsLAHORE UNIVERSITY OF MANAGEMENT SCIENCES (LUMS)

A thesis submitted to the Lahore University of Man-agement Sciences (LUMS) in accordance with the re-quirements of the degree of MASTER OF SCIENCE in theFaculty of Physics.

MAY 2018

ABSTRACT

Despite the successes of string theory in describing perturbative gravity, a non-perturbative description of quantum gravity is still elusive and not understood.At the same time, non-perturbative, strong coupling dynamics of gauge theories

has not been amenable to direct analytical methods. AdS/CFT duality connects gravityin AdS space to gauge theories living on the boundary of this space via a strong to weakcoupling duality thus giving the strong coupling definition of either theory in terms ofthe week coupling perturbative processes in the other. This idea has got more generalapplicability as a concept called holograph connecting gravitational theories to quantumtheories in one lower dimension.

In this thesis, we have studied this correspondence by studying the dynamics ofstrings and branes. D-Branes are multidimensional objects on which open strings canend. AdS/CFT correspondence has been derived by studying the low energy action ofopen strings on a stack of N D3-branes, which is N = 4, Super Yang-Mills action andlow energy limit of open strings, which is supergravity. The D-brane acts as a source insupergravity and AdS5 ×S5 metric is derived in a particular limit. At low energies andsmall string coupling, the gravity theory in the bulk and the gauge theory on the branedecouple and describe the same physics giving rise to AdS/CFT. After this derivation,we verify that the correlation functions of gravity and gauge theories are the same onboth sides of the duality and using this, we study holographic principle and verify thatAdS/CFT correspondence is a realization of the holographic principle.

i

AUTHOR’S DECLARATION

LAHORE UNIVERSITY OF MANAGEMENT SCIENCES

Department of Physics

CERTIFICATE

I hereby recommend that the thesis prepared under my supervision by Junaid SaifKhan on D-Branes and AdS/CFT be accepted in partial fulfillment of the requirementsfor the MS degree.

Supervisor: Co-supervisor:Dr. Babar Ahmed Qureshi Dr. Adam Zaman Chaudhry

__________________________ __________________________

Recommendation of Thesis Defense Committee :

Dr. Ata ul Haq ————————————————————Name Signature Date

Dr. Ammar Ahmed Khan ——————————————-Name Signature Date

ii

DEDICATION

Dedicated to Sher Muhammad and my parents for their endless love, support andencouragement.

iii

ACKNOWLEDGEMENTS

F irst and foremost, I offer my sincerest gratitude to my supervisor Asst/Prof BabarAhmed Qureshi, who has supported me throughout my MS candidature with hispatience and knowledge whilst allowing me the room to work in my own way.

He has taught me, both consciously and unconsciously, how good theoretical physicsis done. The joy and enthusiasm he has was contagious and motivational for me. I amalso thankful for the excellent example he has provided as a successful physicist andprofessor.

I would also like to say thanks to Dr. Faryad, Chairman of MS, and Dr. Ata ul Haqwho always have opened their doors to facilitate me and used to spend a lot of time fordiscussing my ideas. My thanks also to my fellow group members - Arslan, Hassan, Asif,Waqar, Noman, Subhan, Bilal, Hassan Wasalat, Usman, Ayesha, Yasir, Bilal Khalid, andAsad - with whom I had various illuminating discussions and providing an environmentconducive in 9-123 room to research. In particular, I would like to thank Mr. ArshadMaral and Abu Bakr Mehmood, they were very supportive and helped me out in everydifficult situation.

I should also express my gratitude towards Sofia, Awais Ahmed, Ghulam Mehmood,Waris, Qaiser, M. Usman, Wakeel, and Omer for being my excellent support systemduring the entire time. All of these my friends have been very encouraging and accom-modating for me.

Finally, I would like to thank my entire family for supporting me. They believed inme even when I doubted myself.

iv

FOREWORD

TTo understand the behavior of strongly coupled gauge theories has been a longstanding problem. Since the relevant coupling constants are large, a perturbationexpansion is not workable for these theories. An alternate approach must therefore

be formulated. Fortunately, the AdS/CFT correspondence conjecture offers a way out.This correspondence is the main subject of this work.

The analysis of gauge theories in the dual gravity picture relies heavily on theconcept of D-branes which are the subject of first two chapters. In an attempt to makethis thesis as self-contained as possible, a detailed discussion of conformal field theoriesand anti de sitter spaces, the two main halves of the correspondence, has been includedin the chapter 3. Chapter 4 provides motivation for how the AdS/CFT correspondencecomes about and chapter 5 demonstrates how the correspondence may be applied tostudy correlation functions of gauge and gravity theories through which we studied theholographic principle.

v

TABLE OF CONTENTS

Page

List of Figures viii

1 Basics of string theory 1

1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 The Point Particle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 Action for a bosonic point particle . . . . . . . . . . . . . . . . . . . . 3

1.2.2 The BRST quantization of the point particle . . . . . . . . . . . . . . 6

1.3 Relativistic Classical Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.1 The Nambu-Goto action . . . . . . . . . . . . . . . . . . . . . . . . . . 9

1.3.2 The Polyakov action . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

1.3.3 Light-Cone relativistic classical strings . . . . . . . . . . . . . . . . . 19

1.4 Relativistic Quantum Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.5 Relativistic Superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1.5.1 Quantization of superstrings . . . . . . . . . . . . . . . . . . . . . . . 33

2 D-branes and T-dualities 37

2.1 Gauge fields on D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2.2 Parallel Dp-branes and open strings . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 String and D-brane charges . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

2.4 T-dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

2.4.1 Closed strings and T-duality . . . . . . . . . . . . . . . . . . . . . . . 44

2.4.2 Open strings and T-duality . . . . . . . . . . . . . . . . . . . . . . . . 47

vi

TABLE OF CONTENTS

3 AdS Space and Conformal Field Theory 50

3.1 Anti de Siter Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.1.2 Geometry of AdSn+1 space . . . . . . . . . . . . . . . . . . . . . . . . 51

3.1.3 Boundary of AdSn+1 space . . . . . . . . . . . . . . . . . . . . . . . . 53

3.1.4 Example: AdS2+1 space . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2 Conformal Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.2 The Conformal Group . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.2.3 The Conformal Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.4 Conformal Group in d> 3 . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.2.5 Conformal Group in d= 2 . . . . . . . . . . . . . . . . . . . . . . . . . 57

4 D-Branes and AdS/CFT Correspondence 59

4.1 The Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.2 From Symmetry Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.3 The ’t Hooft Large N Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.4 The Large λ Limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5 AdS/CFT Correlation Functions and Holography 66

5.1 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

5.1.1 Correlation Functions in CFT . . . . . . . . . . . . . . . . . . . . . . 67

5.1.2 Correlators Functions in Gravity Theory . . . . . . . . . . . . . . . . 69

5.1.3 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Holographic Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

5.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

Bibliography 72

vii

LIST OF FIGURES

FIGURE Page

1.1 Motion of open string in spacetime. . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.2 Complete parametrization of worldsheet. . . . . . . . . . . . . . . . . . . . . . . 14

2.1 Parallel D2-branes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.2 Interaction of D-branes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.1 Interaction between chain of pair of gluons. . . . . . . . . . . . . . . . . . . . . . 62

4.2 Closed string interaction from two open string splitting interactions. . . . . . 63

viii

CH

AP

TE

R

1BASICS OF STRING THEORY

This chapter relies heavily on the introduction of string theory then we will

overlook some basic topics of string theory that will help us further. First we

will see the point particle. This isn’t excessively astounding as the point particle

can be obtained in the limit as the string falls to a point. Albeit one may surmise that

the relativistic free point particle is a somewhat trivial framework, it is a framework

with constraints and must be quantised with comparing care. In this section we give

the traditional depiction of the point particle and afterward quantise it utilizing Becchi,

Rouet, Stora, Tyutin (BRST) strategy. Afterwards we will see classical and quantum

relativistic strings, its quantization and finally see the superstrings and its spectrum.

1.1 Introduction

The quest in physics has been truly overwhelmed by unwinding the simplicity of the

physical laws, pushing increasingly toward the rudimentary. Even though this isn’t

ensured to succeed uncertainly, it has been vindicated up until this point. Recent people

study string theory is essentially falls into two categories:

One may be a unified quantum theory of all interactions that incorporate gravity

1

CHAPTER 1. BASICS OF STRING THEORY

and it’s really the reason why string theory became popular in the eighties, suddenly

we seem to have a theory that could do everything and of course that very difficult but

very interesting as well if it’s supposed to do everything. It would do for you the particle

physics, the kind of standard model ideas, cosmology because string theory in particular

contains gravity.

Another reason people do string theory is because of its applications and the applica-

tions run into very many areas and probably they are all very interesting:

• One that perhaps is not too relevant for us but fascinating is its applications to

mathematics. If you look the various areas of string theory, you cover pretty much

all of mathematics depending what you specialize on. You can cover Riemann

surfaces with two-dimensional surfaces, cover Riemannian manifolds which is the

higher dimensional theory of Riemann, cover number theory, differential geometry

or group theory etc. It’s unfathomable that string theory has suggested generalized

ideas, for example branch of mathematics called generalized geometry which is

very popular these days. There’s enormous amount of stuff going on here and its

quite fascinating.

• The certainly applications to gravitation and black holes.

• Applications to gauge theory.

Three among the four fundamental forces known were described by gauge theories, the

other one known as gravity which is also based on a local invariance, albeit of a different

type stands apart so far. It’s sort of funny how it works out because it turned out that

string theory was initially most exciting because it could do gravity, quantum gravity,

something you could not do before and do it in a nice simple way. You used to get infinites,

now you don’t get them anymore. It was just marvelous but suddenly what has happened

is that in some conditions and situations the string theories that are supposed to be

describing gravity can also describe gauge theory through gravity so you can have a

description of gauge theory as a gauge theory and you have a description of gauge theory

2


as a gravity or as a string theory which is the key idea of AdS/CFT correspondence.

Therefore, it has many applications to gauge theory:

• Heavy ion physics (quark-gluon plasma)

• Applications to condensed matter physics.

Application of gauge theory, heavy ion physics and condensed matter physics fall under

the rubric of AdS/CFT correspondence. If you are engineered to do AdS/CFT efficiently

then you don’t have to learn everything about string theory. You can just zoom into some

direction that are going to be most useful. AdS/CFT dominates, there’s a lot of work going

on nowadays. Probably 60% of the string theory communities nowadays are working on

AdS/CFT regime, small number are on the applications of mathematics and certainly a

good number, maybe 20% are on unified quantum theory.

1.2 The Point Particle

1.2.1 Action for a bosonic point particle

In order to describe the dynamics of a physical system one can write either the equations

of motion or, alternatively, an action. For the free point particle of rest mass m > 0, it is

easy to write down the equations of motion but we would formulate the action because

action principle gives a great method to get predictions, writing the interactions of

objects as a Lagrangian on some manifold, which basically parameterizes the dynamical

variables of the system. It is worth noticing that action has the dimensions of [energy] .

[time] or [momentum] . [length] and its SI unit is J-s.

We begin our analysis with non-relativistic point particle. The action for a free

nonrelativistic point particle is given by the time integral of the kinetic energy:

S =∫

Ldt =∫

12

mv2dt. (1.1)

3


According to Hamilton’s principle, S would only be stationary if velocity remains constant

and we get the equation of motion.ddt~v = 0. (1.2)

This action ‘S’ allows the particle to move with any constant velocity, even one that

exceeds the velocity of light. So S cannot be the relativistic action.

Now we construct a relativistic action S′ for the free point particle. As the par-

ticle moves through a Minkowski space-time of dimension D with coordinates xµ,

µ = 0,1, ...,D − 1, it traces out a path in space-time called the world line which we

choose to parameterise by τ. The action must be lorentz invariant because if the action is

lorentz invariant then the equations of motion would also be lorentz invariant. Here rises

the question that how do we find the lorentz scaler? A particle is moving relativistically

in space-time so there are many possible worldlines between stationary and ending

points, only elapsed proper time is the quantity on which all lorentz observers agree on

so lets take the action to be related proper time associated to it.

S′ =−mc2∫

dsc

, (1.3)

where mc2 is the lorentz scaler and ds is the square root of the interval between

stationary and ending points.

As we recall that the interval between two events is defined as −ds2 = −ηµνdxµdxν,

where ηµν is the Minkowski metric, which in our conventions is given by ηµν = diag

(−1,+1,+1, ...,+1). Now the action (Eq. 1.3) becomes

S′ =−mc∫ τ f

τi

√−ηµν dxµ

dτdxν

dτdτ, (1.4)

where τi and τ f represent the values of the parameter at the initial and final points of

the world-line. As a result, we conclude that the point particle moves so as to extremise

its proper time. Eq. 1.4 is the explicit form of the action when the path has been specified

by a parametrization xµ(τ) with parameter τ. The choice of parameterisation of the

worldline is of no physical significance and indeed the action of equation (Eq. 1.4) is

invariant and we get the same action, by chain rule, if we change our parameter from

4


τ→ τ′. So this action is reparametrization invariant.

As noted above our action becomes lorentz as well as reparametrization invariant, we

can get the equations of motion from it by applying Hamilton’s principle.

δS′ =−mc∫δ(ds). (1.5)

After some simplification we get,

δS′ = mc∫ τ f

τi

d(δxµ)dτ

dxµds

dτ. (1.6)

δS′ =∫ τ f

τi

d(δxµ)dτ

pµdτ, (1.7)

where pµ is the momentum of the point particle.

δS′ =∫ τ f

τi

d(δxµpµ)dτ

dτ−∫ τ f

τi

δxµ(τ)dpµdτ

dτ. (1.8)

At the boundaries of the worldline, the coordinates are fixed so the first term vanishes

out and we get,

δS′ =−∫ τ f

τi

δxµ(τ)dpµdτ

dτ. (1.9)

As δxµ(τ) is arbitrary, so in order to fulfill the condition of Hamilton’s principle we obtain

the equation of motionddτ

pµ = 0. (1.10)

This is the required equation of motion for the relativistic free point particle with any

parametrization. This equation implies that the momentum of the point particle is

constant along its worldline. If we parametrize the worldline with the proper time, we

get the following equation of motion:

d2

ds2 xµ = 0. (1.11)

This is the equivalent version of the equation of the motion. Eq. 1.10 is more general and

Eq. 1.11 is just its special case when the parameter would become proper time.

Now we write the action (Eq. 1.4) in an alternative but classically equivalent way.

S′ = 12

∫dτ

e−1 .xµ .xνηµν−m2e

, (1.12)

5


where xµ and e are independent bosonic and einbein fields. Suppose we have a metric of a

worldline, it changes from point to point. we can make our metric to minkowski metric by

a change of transformation or by considering that there is a field called einbein through

which we can connect our metric to the minkowski’s hence e is an induced metric and its

role is to make action invariant under reparametrization. This form of action has couple

of advantages over Eq. 1.4:

1. It works even for massless particles.

2. Easier to calculate because there is no annoying square root in this action.

Momentum form of Eq. 1.12:

S′ =∫

dτ

.xµpνηµν− e2

(pµpνηµν+m2)

. (1.13)

Apply Hamilton’s principle on Eq. 1.12 and repeat the same steps as we did above, we

get the following equations of motion:

ddt

(e−1 .xµ)= 0, e2m2 + .xµ .xνηµν = 0. (1.14)

1.2.2 The BRST quantization of the point particle

BRST quantization signifies an exhaustive scientific way to deal with quantizing a field

theory with a gauge symmetry. The central thought of the BRST construction is to find

out the solutions of the constraints with the cohomology classes of a specific nilpotent

operator, the BRST operator Ω, under which the action remains invariant. In order to

develop this kind of operator, we require another class of variables, the ghost variables.

The ghosts are anticommuting variables (odd elements of a Grassmann algebra) for those

theories which don’t include fermionic fields foremostly. [1]. Later, it was found that the

introduction of these ghosts occurred naturally in a path integral formulation by Dewitt

and independently by Faddeev and Popov. Finally, it was realized by Becchi, Rouet, Stora

and by Tyupin that the gauge fixed action including the ghost terms possessed a rigid

symmetry with a Grassmann odd parameter.

6


Once the gauge is fixed, a global fermionic symmetry is present there. Due to this,

there would be so many states in the large fock space (sum of a set of Hilbert spaces rep-

resenting zero particle states, one particle states so on and so forth). BRST quantization

helps us to identify the physical states out of it.

We defined the Lagrangian action ‘S’ earlier in terms of momentum and velocity by the

integral:

S =∫

dτ

.xµpνηµν− e2

(pµpνηµν+m2)

. (1.15)

To produce the analogue of the Lagrangian action, one performs the Legendre transfor-

mation to produce the Hamiltonian by:

H =∂S∂

.xµ.xµ+ ∂S

∂.e

.e−L. (1.16)

After simplification, we get

H = 12

e(pµpµ+m2). (1.17)

From Eq. 1.17, one can recognize the part of the einbein it is a Lagrange multiplier,

implementing the constraining condition:

dGdτ

= 0, (1.18)

where G = 12

(pµpµ+m2).

Also, there is a group of reparametrizations of the phase space which follow up on

this system, identified by the einbein. As the action is invariant under the changes of

this field (gauge field) so we can fix the gauge by set the e = 1. By doing this, another

field B is introduced whose parametrization is controlled by δ functional. Once the gauge

is fixed then ghost-antighosts term appears in the action. The purpose of introducing

these new fields is to produce the BRST charge Ω which encodes the constraint equation

and against which the action is invariant. The charge is then defined as:

Ω= bα(Gα+Mα), (1.19)

where bα are grassmann numbers generated by fermionic fields Mα is some quantity

and specifically, this is chosen so that the action of Ω is nilpotent i.e., Ω,Ω= 0. So that

7


for the sake of easiness we may take M = 0. Hence Ω= bG.

As we know that the action in terms of momentum and velocity is:

S =∫

dτ

.xµpµ− e2

(pµpµ+m2)

. (1.20)

Now, Faddeev-Popove procedures train us to broaden the classical action by relating the

ghost and anti-ghost fields to find BRST action:

SBRST =∫

dτ

.xµpµ+ ιbb− 12

(pµpµ+m2). (1.21)

Here the integration is over the B field which fixes the gauge.

Now we would see that the action is invariant under Becchi-Rouet-Stora-Tyutin

(BRST) transformation. Firstly, we would compute the transformations (δΩ) on the

individual fields by using Poisson bracket but as we introduced a single ghost-antighost

pair b and b, so the poisson bracket would be extended by the general formula:

Q,R= ∂Q∂pµ

∂R∂xµ

− ∂Q∂xµ

∂R∂pµ

+ ι(−1)|Q|(∂Q∂bα

∂R

∂bα+ ∂Q∂cα

∂R∂cα

). (1.22)

Hence, the transformations on the individual fields are:

δΩ(xµ)= Ω, xµ= bpµ

δΩ(pµ)= Ω, pµ= 0

δΩ(b)= Ω,b= 0

δΩ(b)= Ω,b= ι

2(pµpµ+m2).

(1.23)

Now,

δΩSBRST =∫

dτddτ

(pµbpµ+ ιbι1

2(pµpµ+m2)+0

)=

∫dτ

ddτ

[12

b(p2 −m2)]. (1.24)

This shows that SBRST is invariant under the action of BRST operator Ω. The BRST

operator is a Grassmannian operator – a fermionic one, so it’s eigenvalues are not real.

This is very much like some supersymmetry generators that may be nilpotent. The

nilpotency physically implies that such an Ω might be envisioned as switching you

amongst fermions and their bosonic superpartners. Ω may map electron to a selectron,

8


but when you apply Ω again, you get zero. So the nilpotency implies that the second step

is zero because there are finite number of particles whose properties are connected to

each other by supersymmetry. So whenever we are dealing with a nilpotent operator

Ω2 = 0, (1.25)

it is very easy to find some states∣∣Ψ⟩

which are BRST invariant satisfies Ω∣∣Ψ⟩= 0 (the

definition of physical states due to nilpotence condition) and so does∣∣Ψ⟩+Ω ∣∣χ⟩

for any

arbitrary state∣∣χ⟩

in the extended state space because there are more states invariant

under the BRST charge than there are physical states, so now our goal is to produce the

physical states out of the system and this is done by the cohomology of Ω group which is

defined as:

HBRST = K erΩImΩ

, (1.26)

where HBRST is the BRST cohomology while K erΩ gives the states which annihilate

under Ω and ImΩ gives the states which are obtained by acting Ω on any arbitrary state∣∣χ⟩.

1.3 Relativistic Classical Strings

1.3.1 The Nambu-Goto action

From the concepts of general relativity, we know that particles trace out paths that

minimize proper length, they trace out world lines. The natural generalization of this to

a string is obviously a world sheet wherewith proper area is minimized. We are going to

find the invariant surface in spacetime. The string in space is just a line as it evolves

in time, it traces the surface, a surface in spacetime that is the surface we are trying

to compute it’s invariance. We map a rectangle from parameter space to a target space

on worldsheet whose sides are d#»

b1 = ∂ #»x∂ξ1 dξ1 and d

#»

b2 = ∂ #»x∂ξ2 dξ2, where the woldsheet is

parametrized by ξ1 and ξ2.

9


From the formula for the area of a parallelogram, the infinitesimal area has the

following expression:

dA =∣∣∣d #»

b1

∣∣∣∣∣∣d #»

b2

∣∣∣∣∣sinθ∣∣=√∣∣∣d #»

b1

∣∣∣2∣∣∣d #»

b2

∣∣∣2 −∣∣∣d #»

b1

∣∣∣2∣∣∣d #»

b2

∣∣∣2 cos2θ . (1.27)

Writing in terms of dot product gives:

dA =√

(d#»

b1.d#»

b1)(d#»

b2.d#»

b2)− (d#»

b1.d#»

b2)2 . (1.28)

Using the expressions for d#»

b1 and d#»

b2, the proper area is subsequently given by:

A =∫

dξ1dξ2

√(∂#»x∂ξ1 .

∂#»x∂ξ1

)(∂#»x∂ξ2 .

∂#»x∂ξ2

)−

(∂#»x∂ξ1 .

∂#»x∂ξ2

)2. (1.29)

To give worldsheet a more physical outlook, let us replace τ and σ with our calling

parameters ξ1 and ξ2. These are related to the spacelike and timelike directions on the

worldsheet. The detail is skipped here in favor of brevity.

So, what is left to this point is to give this expression the dimensions of action. The

proper area is therefore multiplied by −T0/c. Switching to the convention of denoting

worldsheet coordinates by Xµ in string theory and denoting:

.Xµ = ∂Xµ

∂τ, (1.30)

and

Xµ′ = ∂Xµ

∂σ. (1.31)

SNG =−T0

c

∫ τ f

τi

dτ∫ σ1

0dσ

√( .X .X ′)2 − .

X2X ′2 . (1.32)

There are two things that need to mentioned here. Firstly, in Eq. 1.32 the order of the

terms is switched inside the radical. It can be shown that the term under the radical is

always positive. Secondly, the minus sign is put in there to ensure that the lagrangian

has the correct form in the non-relativistic limit. Moreover, it is an easy exercise to show

that the term in the radical in Eq. 1.32 is equal to −Γ where Γ is the determinant of the

induced metric on the worldsheet of the string. We can therefore write our action in the

following nice compact form:

SNG =−T0

c

∫dτdσ

p−Γ , Γ= det(Γαβ). (1.33)

10


Here T0 is a constant of proportionality. We will see shortly that it is the tension of

the string, meaning the mass per unit length. The lagrangian is just a scalar and the

measure is manifestly lorentz invariant. The action therefore is lorentz invariant. This

is called the Nambu-Goto action for a relativistic classical string.

Our next step is to vary the action in the usual way to derive the Euler-Lagrange

equations. Using the following notation:

Πτµ =

∂L

∂.

Xµ, (1.34)

and

Πσµ =

∂L∂Xµ′ . (1.35)

The variation of the action, after getting rid of terms which contribute total derivatives,

is given by:

δSNG =∫ τ f

τi

dτ[δXµΠσ

µ

]σ10 −

∫ τ f

τi

dτ∫ σ1

0dσδXµ

(∂Πτ

µ

∂τ+∂Πσ

µ

∂σ

). (1.36)

Due to Hamilton’s principle, the second term on R.H.S of the above equation must vanish

for all variations, so we get the following constraint:

∂Πτµ

∂τ+∂Πσ

µ

∂σ= 0. (1.37)

In Eq. 1.36 the first term deals with the string end points. There are two ways in which

this term can be made to vanish. Either δXµ(τ,σ∗)= 0 at the string end points where σ∗is σ1 or 0. This is what is called the Dirichlet boundary condition. Dirichlet boundary

conditions are only possible for space directions. Otherwise, we can put Πσµ(τ,σ∗) = 0.

This is the free endpoint or the Neumann boundary condition. In string theory, we

generically have open strings as well as closed strings. Obviously only open strings

require boundary conditions. Also, as is clear from Eq. 1.36, we must fix a boundary

condition for terms corresponding to each coordinate. In general, a mixture of Neumann

and Dirichlet boundary conditions is chosen. For Dirichlet boundary conditions, the

string end points are fixed in time and can therefore be imagined to be attached to some

sort of a surface. This surface is what is called a D-brane. A Dp-brane is a brane with

11


p spatial dimensions. If a Dirichlet boundary condition is imposed corresponding to a

particular coordinate, the string is free to move in the directions tranverse to it and

so we have a situation wherewith a string endpoints are free to move in the directions

tangent to the D brane.

Once we parametrize the worldsheet by imposing static gauge condition (X0(τ,σ)≡cτ), at that point the components of X (τ,σ) are the string spatial coordinates, its deriva-

tive with respect to time is by all accounts the nearest thing we have to a velocity but

this velocity, be that as it may, relies on the decision of σ. Its direction, for instance,

comes along the lines of constant σ (Fig. 1.1).

(a) (b)

Figure 1.1: (a) Parameter space for open strings. (b) Open string worlsheet in targetspace.

Since σ can be picked arbitrarily, keeping σ constant in taking the derivative is

obviously not physically noteworthy! Settling physically the σ parameterization of a

string is unobtrusive because the string is an object with no substructure. When look-

ing at a string at two close-by times, it isn’t conceivable to state that a point moved

from one location to the next. To talk about the points on the string we require a σ

parameterization, and reparameterization invariance influences it to clear to us that

this parameterization isn’t remarkable. This recommends longitudinal movement on the

string isn’t physically significant. There is a reparameterization invariant velocity that

can be characterized on the string. This is, be that as it may, a transverse velocity.

12


The transverse velocity v⊥ at any point on the string is a vector orthogonal to

the string and tangent to the string spatial surface. Keeping in mind the end goal to

characterize the transverse speed v⊥, it is valuable to have a unit vector which is tangent

to the string. To this end, we now define a parameter l which is more physical than

our nearly arbitrary σ: l measures length along the string. Give us a chance to work

with the string at a fixed time, and characterize l(σ) to be the length of the string in the

interval [0,σ]. Along these lines, for instance, l(0)= 0, and l(σ1) is the length of a whole

open string. Since dl is the length of the infinitesimal vector d#»X which emerges from an

interval dσ along the string, we have:

dl =∣∣∣d #»

X∣∣∣=∣∣∣∣∣∂

#»X∂σ

∣∣∣∣∣∣∣dσ∣∣ . (1.38)

By chain rule, we get:

dl =∣∣∣∣∣∂

#»X∂l

∣∣∣∣∣∣∣∣∣ ∂l∂σ

∣∣∣∣∣∣dσ∣∣=∣∣∣∣∣∂#»X∂l

∣∣∣∣∣∣∣dl∣∣ . (1.39)

This implies that ∂#»X /∂l is a unit vector which is tangent to the string. Suppose any

vector #»a ; its a⊥ component can be calculated as #»a − (#»a .#»n )#»n , where #»n is the unit vector.

Therefore, using our unit vector ∂#»X /dl along the string, we have

#»v ⊥ = ∂#»X∂t

−(∂

#»X∂t

.∂

#»X∂l

)∂

#»X∂l

. (1.40)

Using the static gauge and after some simplifications, we get

√( .X .X ′)2 − .

X2X ′2 = cdldσ

√1− v2

⊥c2 . (1.41)

The Nambu-Goto action in terms of transverse velocity is given by:

SNG =−T0

∫dτ

∫ σ1

0dσ

( dldσ

)√1− v2

⊥c2 . (1.42)

Eq. 1.42 is valid for open and closed strings. Albeit it still prompts complicated equations

of motion except the most symmetrical situations. Keeping in mind the end goal to get

simple equation of motion, we should be shrewd in our decision of σ. For open strings,

likewise, we should see how the endpoints move. We have seen that the endpoints move

with the speed of light and move transversely to the string.

13


FIGURE 1.2. Complete parametrization of worldsheet.

Till now, by imposing static gauge condition, we have partially fixed the parametriza-

tion of the worldhseet. Now we are going to find a useful σ parametrization of the string

surface. In this way, we would completely parametrized the worldsheet. The σ parame-

terization of a given string can be used to construct lines of constant σ that are always

perpendicular to the lines of constant t. In this parameterization of the string surface,

the tangent ∂#»X /∂σ to the strings (lines of constant t) and the tangent ∂

#»X /∂t to the lines

of constant σ are perpendicular to each other at any point (Fig. 1.2):

∂#»X∂σ

.∂

#»X∂t

= 0

=⇒ ∂#»X∂l

.∂

#»X∂t

= 0.

(1.43)

Therefore, Eq. 1.40 becomes#»v ⊥ = ∂

#»X∂t

. (1.44)

This velocity is for all points and not only at the endpoints. This is the advantage of

choosing this σ parametrization.

In conclusion, we have following advantages for the complete parametrization of the

worldsheet:

14


• When we have only τ parametrization, then the velocity of the string is given by

Eq. 1.40 everywhere except at the endpoints and at the endpoints is given by Eq.

1.44 but when we have both τ as well as σ parametrization, then the velocity of

the string is given by Eq. 1.44 everywhere.

• Eq. 1.34 and Eq. 1.35 get more simple by this parametrization which is another

advantage of this parametrization.

Πτµ = T0

c2

∂l∂σ√

1− v2⊥

c2

∂Xµ

∂t. (1.45)

Similarly,

Πσµ =−T0

√1− v2

⊥c2

∂Xµ

∂l. (1.46)

We simplified these two above equations because these form the equation of motion (Eq.

1.37). Now first we will solve the Eq. 1.37 for the time component (µ= 0) and then for the

space component. After solving these, we get following results:

wave equation:∂2 #»

X∂σ2 − 1

c2∂2 #»

X∂t

= 0, (1.47)

parametrization condition:∂

#»X∂t

.∂

#»X∂σ

= 0, (1.48)

parametrization condition:(∂

#»X∂σ

)2+ 1

c2

(∂

#»X∂t

)2= 1, (1.49)

boundary condition:∂

#»X∂σ

∣∣∣∣σ=0

= ∂#»X∂σ

∣∣∣∣σ=σ1

= 0. (1.50)

We obtained another result σ1 = E/T0 if the string has total energy E. By combining Eq.

1.48 and Eq. 1.49 we get following equations called Virasoro constraints:(∂

#»X∂σ

± 1c∂

#»X∂t

)2= 1. (1.51)

After solving the wave equation for the open strings and apply boundary conditions

and Virasoro constraints, we obtain following results:

#»X (t,σ)= σ1

πcos

πσ

σ1

(cos

πctσ1

,sinπctσ1

). (1.52)

15


and the energy density of the string is given by:

ε(d)= T0√1− 4d2

l2

. (1.53)

At the center (d = 0), the string does not move and gives ε(0)= T0 and at the endpoints

(d =±l/2) the energy density diverges but the total energy would be finite.

Now we are going to solve the wave equation for closed strings, there would be no

boundary conditions for closed strings but rather a periodic condition which is

#»X (t,σ+σ1)= #»

X (t,σ). (1.54)

As σ increases by σ1 we are back to the same point on the closed string from where we

started. When we solve the wave equation for this case we would see that at σ=σ0, we

have a cusp: a point on the string where the two outgoing string directions form zero

angle. In conclusion, as time goes by cusps will appear and disappear periodically at

different points on the string.

1.3.1.1 Conserved quantities on world-sheet

If the string is moving freely, there would be a conserved current on the worldsheet and a

conserved current gives rise to a conserved charge which is related to conserved momen-

tum. Since every component of pµ ought to be independently conserved, this is where we

have a set of conserved charges. We have seen in the Nambu-Goto action (Eq. 1.32) that

the lagrangian density (L ) is integrated over τ and σ which are worldsheet coordinates

not spacetime coordinates. From this, we conclude that the conserved currents live on

the worldsheet. Conserved currents arise only when the lagrangian density has some

symmetries. Conserved currents are defined such a way,

∂α jαi = 0, (1.55)

and the quantities jαi defined by

$i jαi ≡ ∂L

∂(∂αϕα)∂ϕα, (1.56)

16


where $ are some constants and ϕ are the fields.

To find the conserved currents for our case, we require a field variation ∂Xµ such that

the lagrangian density does not change and the one is given by

∂Xµ(τ,σ)=$µ, (1.57)

where $ is a constant that does not depend on τ or σ which indicates that each point on

the worldsheet is displaced by this same vector which is a spacetime translation. Put Eq.

1.57 in Eq. 1.63, we get following expressions for the currents:

jαµ =∂L

∂(∂αXµ)−→ ( j0

µ, j1µ)=

(∂L

∂.

Xµ,∂L

∂Xµ′

). (1.58)

After comparing with Eq. 1.34 and Eq. 1.35, one can conclude that

jαµ = ( j0µ, j1

µ)= (Πτµ,Πσ

µ). (1.59)

Hence, now the equation for current conservation becomes

∂αΠαµ =

∂Πτµ

∂τ+∂Πσ

µ

∂σ= 0. (1.60)

and one can see that this is simply the equation of motion for the relativistic string (Eq.

1.37).

To get charges, integrate the zeroth components of the currents over space means

over σ:

Qµ −→ pµ(τ)=∫ σ1

0Πτµ(τ,σ)dσ. (1.61)

Here we called conserved charges pµ because they give the spacetime momentum carried

by the string as we have seen that these charges arises from spacetime translational

invariance. Hence Πτµ is the spacetime momentum density which is carried by the string.

To check the conservation, differentiate Eq. 1.61 w.r.t τ and use the Eq. 1.60, we get:

dpµdτ

= 0. (1.62)

Aside from charge conservation, there is another thing indicated by this equation that

the derivative is not with respect to t but with respect to worldsheet coordinates (τ and

17


σ) so one can say that the pµ is conserved in worldsheet time only but this is true for

Minkowski time too if we choose the static gauge t = τ. The momentum carried by the

string along the space direction may not be conserved for open strings with Dirichlet

boundary conditions which implies that the above equation does not guarantee to be zero.

In open string theory, Dirichlet boundary conditions show up when we have D-branes

that are not space filling so the momentum of the string would not able to be conserved,

but the total momentum of the string and the D-brane is conserved.

Our next task is to generalize the Eq. 1.61 not only for constant lines of τ but for any

lines of τ and t which is given by:

pµ =∫

(Πτµdσ−Πσ

µdτ). (1.63)

This equation is valid for both open and closed strings.

1.3.2 The Polyakov action

Due to occurrence of the square root, the Nambu-Goto action is difficult to deal with

so we are going to find an action which has no square root, no primary constraint and

equivalent to the Nambu-Goto action in the same as einbein trick for the relativistic

point particle. There’s introduced another field gαβ at the expense of the elimination of

square root.

SP =−T0

2

∫d2σ

p−g gαβ∂αXµ∂βXνGµν. (1.64)

Here g ≡ detg and this action is called the Polyakov action. gαβ is a worldsheet metric

while Gµν is a spacetime metric. gαβ is the two dimensional analogon to the einbein

function e. This action has no primary constraint. In Eq. 1.64, we take Gµν = ηµν which

implies that we work with flat spacetime metric. In order to do so, Polyakov action has

following symmetries:

1. Global spacetime Poincare symmetry:

Xµ(τ,σ)→ Xµ(τ,σ)+ cµ (Translational) (1.65)

Xµ(τ,σ)→ΛµνXν(τ,σ) (Lorentzian) (1.66)

18


2. Local gauge symmetries:

a) Reparametrization symmetry of the worldsheet:

Xµ(τ,σ)→ Xµ′(τ′,σ′)= Xµ(τ,σ). (1.67)

gαβ(τ,σ)→ g′αβ(τ

′,σ′)= ∂σ

′α

∂σc∂σ

′β

∂σd gcd(τ,σ). (1.68)

b) Local Weyl symmetry:

gαβ(τ,σ)→Λ2(τ,σ)Gαβ(τ,σ), (1.69)

where Gαβ is the spacetime metric and Λ2(τ,σ) = e−2ω(τ,σ) is the pre-factor

under which this symmetry holds and Polyakov action remains invariant.

To find the equations of motion, varying Eq. 1.64,

δSP =−T∫

d2σ

(p−g Tαβδgαβ+2∂α(p−g gαβ∂βXµ

)δXµ

)+bnd term, (1.70)

where Tαβ is the energy momentum tensor of the woldsheet, the factors appearing

when varying SP with respect to gαβ,

Tαβ =−4π∂SPp−g ∂gαβ

. (1.71)

The classical equations of motions are:

Tαβ = ∂αXµ∂βXµ− 12

gαβglm∂l Xµ∂mXν = 0, and ∂α

(p−g gαβ∂βXµ

)= 0. (1.72)

At the ends of the string, one can impose either Neumann boundary condition (gασ∂αXµ =0) or Dirichlet condition (δXµ = 0).

1.3.3 Light-Cone relativistic classical strings

Each oscillation of a string produces a particle in spacetime. Massless open strings

propose particles of spin 1 known as photon, gluon etc. while massless closed strings

produce particles whose spin is 2 known as gravitons. The gauge symmetries (2a and 2b)

19


demonstrate the repetition in degrees of freedom but we would fixed them when we quan-

tize the strings. We would use canonical quantization because it leads our expressions to

simpler ones and do it in a simpler and faster way than other quantizations.

In canonical quantization we should have classical equation of motion (which we have

already) then we would fix the gauge symmetries and find out the complete classical

solution. Once it’s done then we would move to quantum relativistic strings and quantize

them in order to read the spectrum of the strings and discover some remarkable results.

In light-cone quantization, at the classical level one first solves the Virasoro con-

straints explicitly, find out the independent degrees of freedom, and then quantize only

those. By reparametrization symmetry of the worldsheet, one can work with flat space-

time metric Gαβ = ηαβ and Weyl scaling can be used to get gαβ = ηαβ just for the sake

of convenience. Even with these assumptions, there’s still gauge redundancy, called the

conformal symmetry. With gαβ = ηαβ, second equation of motion of Eq. 1.72 becomes free

scaler field and can be written as:

∂2Xµ

∂τ2 − ∂2Xµ

∂σ2 = 0. (1.73)

and the first equation of motion of Eq. 1.72 can be further simplified:

T00 = Tττ = Tσσ = 12

(∂Xµ

∂τ

∂Xµ

∂τ+ ∂Xµ

∂σ

∂Xµ

∂σ

)= 0. (1.74)

Tτσ = Tστ = ∂Xµ

∂τ

∂Xµ

∂σ= 0. (1.75)

For open string, Neumann boundary condition becomes:

∂σXµ(τ,σ= 0,π)= 0. (1.76)

As we have fixed the gauge now we will compute the solution of the equation of motion.

As Eq. 1.73 is a linear wave equation so it’s solution is

Xµ(τ,σ)= xµ+vµτ+ Xµ

R(τ−σ)+ Xµ

L(τ+σ), (1.77)

where Xµ

R(τ−σ) is a right-moving wave and Xµ

L(τ+σ) is left-moving wave and xµ and vµ

are arbitrary constants. This is a complete solution of the equation of motion for both

20


open and closed strings. Now insert Eq. 1.77 in Eq. 1.76 (Neumann condition), we get

XR(τ)= XL(τ) (at σ= 0) (1.78)

XR(τ−π)= XL(τ+π) (at σ=π) (1.79)

The above equation tells us that XL = XR is periodic in 2π.

Now we still have to solve Eq. 1.74 and Eq. 1.75. So what these equations do is to

impose some nontrivial constraints between XL and XR and xµ and vµ etc. When we

plug Eq. 1.77 into Eqs. 1.74 and 1.75, we obtain bit complicated equations which are very

hard to solve so we find some ingenious way to solve those constraint equations and then

quantize only those, which is key idea of light-cone quantization.

After fixing gαβ = ηαβ, there are actually still some residue gauge freedom (conformal

symmetry). By residue gauge degrees of freedom means we can still find the combination

of coordinate transformation and the Weyl scaling so that after those operations, we

still going back to the minkowski metric (ηαβ). So, to see this we have to introduce the

light-cone coordinate on the worldsheet:

σ± = τ±σp2

(1.80)

and the worldsheet metric can be written as:

ds2 =−dτ2 +dσ2 =−2dσ+dσ−. (1.81)

This metric is preserved by the following coordinate transformation:

σ+ → σ+ = j(σ+), σ− → σ− = k(σ−), (1.82)

where j and k are some arbitrary functions. When we plug above equation in Eq. 3.7,

under this coordinate transformation we get

ds2 → ds2 =−2dσ+dσ−

=−2 j′(σ+)k

′(σ−)dσ+dσ−. (1.83)

21


Under this coordinate transformation, our metric is only transform by an overall pre-

factor ( j′(σ+)k

′(σ−)) and we can get rid off it by Weyl scaling. So under these new

coordinates, we define τ as:

τ= 1p2

(σ++ σ−)

= 1p2

(j(τ+σ)+k(τ−σ)

)⇒ τ, (1.84)

which has the same form as the classical solution of Xµ (Eq. 1.77), then by choosing

appropriate j and k one can fixed the gauge completely. So τ is given by any combination

of solutions of Xµ because those solutions have precisely this form. Therefore, Eq. 1.84

can be written in any combination of those Xµ, and the smart choice is what we call the

light-cone gauge is given below:

τ= X+

c+⇒ X+ = τv+, (1.85)

X± ≡ 12

(X0 ± X1). (1.86)

where v+ is some constant. This infers that the worldsheet time is settled by the space-

time light-cone coordinate.

The Virasoro constraints in light-cone gauge with Xµ = (X+, X−, X i) become:

2v+∂X−

∂τ=

(∂X i

∂τ

)2+

(∂X i

∂σ

)2, (1.87)

v+∂X−

∂σ= ∂X i

∂τ

∂X i

∂σ, (1.88)

therefore, X− is completely expressed in terms of X i(τ,σ). So we conclude that the

independent degrees of freedom are only X i. We already know X+ by fixing the gauge (Eq.

1.85) and now X− which is in terms of X i. This will make our life easier. Because X i are

just free scalar fields. So now we can quantize them. Before doing the quantization, let’s

expand the Xµ

R and Xµ

L (in Eq. 1.77) in terms of Fourier modes so that the quantization

task become easy.

22


For closed strings, Xµ

R and Xµ

L are independent periodic functions with period 2π,

therefore Eq. 1.77 can be written in terms of Fourier expansion as:

Xµ(τ,σ)= xµ+vµτ+ ι√α

′

2

∑n 6=0

1n

(αµne−ιn(τ+σ) + αµne−ιn(τ−σ)

). (1.89)

For open strings, Xµ

R and Xµ

L are equal and leads to the following expression:

Xµ(τ,σ)= xµ+vµτ+ ι√

2α′ ∑n 6=0

1nαµne−ιnτ cosnσ. (1.90)

The coefficients αµn and αµn tell us about the modes of oscillation of the strings. Open

strings can be portrayed as standing waves in light of the fact that the right-moving

and the left-moving contributions are the same while for the close strings they are

independent.

Zeroth mode of oscillation (non-oscillating): The first two terms in Eq. 1.89 and Eq. 1.90

give us the track of the string in terms of proper time for both closed and open strings.

These terms also represent the center of mass motion of the strings.

For closed strings, plug Eq. 1.89 in Eqs. (1.87 and 1.88) and then equating the coefficients

of different Fourier modes we get following set of equations:

2v+v− = v2i +α

′ ∑n 6=0

(αi−nα

in + αi

−nαin

). (1.91)

∑n 6=0

αi−nα

in = ∑

n 6=0αi−nα

in. (1.92)

Eq. 1.92 is recognized as the level of matching condition. It means it is the consequence

of no special point along a string.

Similarly, for open strings after putting Eq. 1.90 in Virasoro constraints, we obtain:

2v+v− = v2i +2α

′ ∑n 6=0

αi−nα

in. (1.93)

and Eq. 1.88 trivially satisfies for open strings.

23


Now move forwards to the worldsheet’s conserved currents which is the string mo-

mentum in spacetime and by Eq. 1.63 we get:

pµ = l2π

vµα

′ (1.94)

⇒ vµ = 2πα′

l. (1.95)

Here the momentum density Πτµ along the string which is appeared in Eq. 1.63 is given

by Noether procedure (here I am putting just result):

Πτµ =

12πα′ ∂τXµ. (1.96)

For open string, Eq. 1.93 in terms of pµ, we get:

2p+p−− p2i =

12α′

∑n 6=0

αi−nα

in. (1.97)

and we know that

2p+p−− p2i =−pµpµ = M2. (1.98)

Here the mass of the string is represented by M. Hence, Eq. 1.97 becomes:

M2 = 12α′

∑n 6=0

αi−nα

in. (1.99)

For closed strings, Eq. 1.91 becomes:

M2 = 2α

′∑n 6=0

ai−nα

in. (1.100)

Above two equations are called classical mass-shell conditions and one can conclude from

them the mass of a string from its oscillations. So on the off chance that we consider the

motion of a string, it resembles a particle, at that point this particle can have a mass

which we can expressed in terms of oscillation modes.

1.4 Relativistic Quantum Strings

For quantum relativistic strings, one converts the classical fields to quantum operators

which satisfy the condition of canonical quantization. The parameters which appear in

24


classical solutions become creation and annihilation operators and the classical solutions

become solutions of the operator equation. One can read the spectrum of the quantum

relativistic strings by applying creation operators to the vacuum state. Once all this

done, we would finally be able to do the quantization - quantize X i(τ,σ) because only

this has the independent degrees of freedom.

We define canonical momentum density Πτi for which the corresponded conserved cur-

rent would be worldsheet momentum in the similar way as we defined momentum density

in Eq. 1.96 for which the corresponded conserved current was spacetime momentum.

Πτi =

12πα′ ∂τX i. (1.101)

We have found the canonical momentum so now promote all the classical field to operators

by imposing a canonical quantization condition at a given timeslice.

[X i(τ,σ), X j(τ,σ′)]= 0

[Πi(τ,σ),Π j(τ,σ′)]= 0

[X i(τ,σ),Π j(τ,σ′)]= ιδi jδ(σ−σ′).

(1.102)

In conclusion, all oscillation modes xi, pi,αin, αi

n become operators. When we plug-in

these mode expansions into Eq. 1.102, we get:

[xi, p j]= ιδi j (1.103)

[αin,α j

m]= [αin, α j

m]= nδi jδn+m,0, (1.104)

while all other commutators vanish. αi−n / αi−n and αin / αi

n can be related to the creation

(ai−n)† / (ai−n)† and annihilation operators ain / ai

n respectively.

For n > 0, creation and annihilation operators are related as

(ai−n)† = 1p

nαi−n, (ai

−n)† = 1pnαi−n (creation operators) (1.105)

(ain)= 1p

nαi

n, ain = 1p

nαi

n (annihilation operators) (1.106)

25


where 1/p

n is just normalization factor. We will generally use the notation of α instead

of a just for the sake of convenience.

Essentially we have done the quantization of the string: determined the Heisenberg

equations, its solutions which can be characterized by the Eq. 1.89 and 1.90 and the

constant of integration in these equations satisfy the commutation relations.

Now we can work out the spectrum of the quantum relativistic strings. Vacuum state

(ground state) is defined when we quantize the string. The lowest state is called the

“oscillator vacuum" state. So the vacuum are the states which are annihilated by all the

annihilated operators. Therefore, the ground state satisfies:

αin |0, pµ⟩ = 0, n > 0.

αin |0, pµ⟩ = 0, n > 0.

(1.107)

Here we are taking the vacuum to be in a spacetime momentum eigenstate.

Similarly, for closed strings one can built the excited states from creation operators

(αi−n, αi−n with n > 0):

αi1−n1αi2−n2

αi3−n3 ...α j1−m1α

j2−m2αj3−m3 |0, pµ⟩ . (1.108)

We would like to define the oscillation number operator for the sake of convenience and

here the order of operators is crucial:

N in = 1

nαi−nα

in ⇒ nN i

n =αi−nα

in

N in = 1

nαi−nα

in ⇒ nN i

n = αi−nα

in,

(1.109)

where n is the frequency of nth oscillator and there is no summation on i and on m. Now

the level matching condition (Eq. 1.92 ) for closed strings can be written as:

∑n 6=0

nN in = ∑

n 6=0nN i

n, (1.110)

and the mass shell condition (Eq. 1.100) in quantum version can be written as:

M2 = 2α′

D−1∑i=2

∑n 6=0

n(N in + N i

n)+a0, (1.111)

26


where D stands for spacetime dimensions and a0 is the ground state energy for closed

string which is given by

a0 = 2α′

D−1∑i=2

∞∑n=1

n = 2(D−2)α′ ζ(−1)=−D−2

6α′ , (1.112)

where ζ(−1)≡ 1/12. Hence, Eq. 1.111 becomes:

M2 = 2α′

D−1∑i=2

∑n 6=0

n(N in + N i

n)− D−26α′ . (1.113)

For open string, only single set of oscillation is required therefore choose αi−n and the

typical states can be written as

αi1−n1αi2−n2

... |0, pµ⟩ . (1.114)

Also the quantum version of Eq. 1.99 is given by:

M2 = 1α′

D−1∑i=2

∑n 6=0

nN in +ac, (1.115)

where ac is the ground state energy for the open string and it’s value can be calculated

by

ac =−D−224α′ . (1.116)

Hence, Eq. 1.115 becomes:

M2 = 1α′

D−1∑i=2

∑n 6=0

nN in −

D−224α′ . (1.117)

Each of Eq. 1.108 and Eq. 1.114 describe a state of a string for closed and open strings

respectively. The state of a string oscillates in the particular oscillation modes and then

moves in a spacetime. Such object looks like a point particle from distant. Thus, each

state of a string corresponds to a spacetime particle whose mass can be calculated by Eq.

1.113 and Eq. 1.117 accordingly.

Spectrum of open strings:

1. The oscillation vacuum state on the worldsheet is defined as |0, pµ⟩ which implies

that N im = 0 for all i,m. The lowest mode is just the vacuum, therefore there is no

27


oscillators and this should describe a spacetime scalar particle because other than

momentum there is no other quantum number. The mass of this scalar particle can

be calculated from Eq. 1.117 which is

M2 =−D−224α′ . (1.118)

When we deal with the spacetime dimensions above 2 we encounter negative

mass and we called this particle tachyon. Hence, the ground state is tachyonic.

Negative square mass ordinarily reveals that the system has instability and not in

the ground state. Here I would like to clear that we are in the ground state of the

worldsheet but not in the ground state of the spacetime. This infers that the system

is not in the lowest state when the string is propagating in the flat Minkowski

spacetime. It’s not a major issue, it simply means that we have not discovered the

right ground state for the spacetime. It doesn’t mean the theory is conflicting.

2. Now for the 1st excited state, we apply αi−1 on this worldsheet ground state:

αi−1 |0, pµ⟩ , (1.119)

corresponding to the value of n = 1, the Eq. 1.117 becomes:

M2 = 1α′ −

D−224α′ = 26−D

24α′ . (1.120)

The interesting thing in Eq. 1.119 is the spacetime index ‘i’ which implies that

this state transform as a vector under SO(D−2) because in the light-cone gauge,

only the subgroup SO(D−2) of Lorentz symmetries is manifest. If something is

propagating in Minkowski spacetime then it has to fall into the representations of

the Lorentz group. Therefore, all string excitations (as they are viewed as spacetime

particles) must fall into the representations of the full Lorentz group.

Recall a result that Lorentz vector particles in D spacetime dimension should have

D −1 independent modes if the particle is massive and have D −2 independent

modes if the particle is massless. In our case, the 1st excited state is a vector with

28


only D−2 independent modes therefore for the consistency of Lorentz symmetries

it should be massless. So, Eq. 1.120 becomes:

M2 = 0⇒ D = 26. (1.121)

This results is known as the critical spacetime dimension of string theory. For

D = 26, particle should be massless and we actually find such particles like photon.

For D 6= 26 in case of massless particles, Lorentz symmetry is lost implies that

the particle never fall into a representation of a Lorentz group. As, being stated,

Lorentz symmetry isn’t kept up at the quantum level however it is preserved in

classical action. This simply means that this cannot be the right, hence D must be

equal to 26 for massless particles in the 1st excitation of the string. This produces

the field of a massless vector field Aµ.

3. Once we fixed the spacetime dimension to be 26, the other higher excitations are

all massive. The massive 2nd excited state is represented as

αi−1α

j−1 |0, pµ⟩ , αi

−2 |0, p⟩ . (1.122)

So just to condense story for the open string, we discover the tachyon, locate the massless

vector and after that discover infinite number of massive particles.

Now we study the spectrum of closed strings:

1. The oscillation vacuum state on the worldsheet is also defined as |0, pµ⟩ which

implies that N im = N i

m = 0 for all i,m. The ground state illustrates that this must

be a spacetime scalar particle whose mass can be calculated from Eq. 1.113 which

is

M2 =−D−26α′ . (1.123)

Again we find the tachyon for any spacetime dimension more than 2. Hence, the

ground state is tachyonic for closed strings.

2. Now the second lowest (1st excitation) mode using level matching condition, we

have

αi−1α

j−1 |0, pµ⟩ , (1.124)

29


corresponding to the value of n = 1, the Eq. 1.113 becomes:

M2 = 26−D6α′ . (1.125)

Representations of the Lorentz group for massless particles again give us D = 26

spacetime dimensions. The same story happens, and we reach at the point where

M2 again become massless.

The above state (Eq. 1.124) can be further decompose into following:

26∑i=2

αi−1α

i−1 |0, pµ⟩ (trace part). (1.126)

Due to the same index, it gives us the trace part which produces a scalar field Φ.

Now the other decomposition is

e i jαi−1α

j−1 |0, pµ⟩ (symmetric traceless part). (1.127)

This is precisely the generalization of spin-2 representation. In conclusion this

represents that the massless particle has a spin-2 and we called that particle

graviton. Hence, the symmetric traceless part gives graviton field Gµν. Now we

have to take the antisymmetric part which is follows as:

bi jαi−1α

j−1 |0, pµ⟩ (antisymmetric traceless part). (1.128)

The anti-symmetric part gives us Bµν (Kalb-Ramon, or B-field).

3. Similarly, the other higher excitations are also all massive.

In summary, what we described so far are only for the bosonic string but this can be

generalized to fermions also and this would be called as superstring and the theory is

called superstring theory which we will see in the next section.

1.5 Relativistic Superstrings

Superstrings are needed because they contain spacetime fermions which are necessary

in any theory that width is hoping to describe the nature and it’s spectrum does not

30


contain tachyon which implies that it can describe D-branes which are stable. We had

Xµ(τ,σ) field on the worldsheet in our bosonic string so for fermions we have to introduce

a new kind of field on the worldsheet ψµ(τ,σ). Now when we write a Nambu-Goto action

we have to add the fermionic field in it but that will be very complicated therefore we will

write a different kind of action, we’ll have an action that will be Nambu-Goto for Xµ(τ,σ)

fields and the action for ψµ(τ,σ) fields. The Xµ has the interpretation of embedding the

string into our normal space so people try to think of the ψµ as embedding the string

some superspace. As Xµ(τ,σ) are commuting variables classically but in quantum theory

they become commutators which need not to be zero. Similarly, ψµ(τ,σ) would be anti-

commuting variables classically and become anti-commutators quantum mechanically

which sometimes not be zero.

Write down the Nambu-Goto action in light-cone gauge for superstrings:

S = 14πα′

∫dτ

∫ π

0dσ

( .X j .

X j − X j′ X j′)+Sψ, (1.129)

where α′ is a slope parameter which is related to string’s tension as α′ = 1/2πT0~c, j

represents the coordinate in light-cone gauge and Sψ is the Dirac action for a fermion

that lives in 2D world is defined as:

Sψ = 12π

∫dτ

∫ π

0dσ

(ψ

j1(∂τ+∂σ)ψ j

1 +ψj2(∂τ+∂σ)ψ j

2

). (1.130)

Let us write down the Polyakov action here

SP = 14πα′

∫d2σ

p−g gαβGµν

∂αXµ∂βXν+ ιψµ

γβ∂αψν

, (1.131)

where γµ are Dirac spinors (real) which are defined as:

γ1 =

0 −1

1 0

, γ2 =

0 1

1 0

. (1.132)

In order to find out the equations of motion and the boundary conditions for fermionic

fields let’s fluctuate the ψµ fields in action Sψ we get,

(∂τ+∂σ)ψ j1 = 0, (∂τ−∂σ)ψ j

2 = 0 (equations of motion) (1.133)

31


ψj1(τ,σ∗)δψ j

1(τ,σ∗)−ψ j2(τ,σ∗)δψ j

2(τ,σ∗)= 0, (boundary conditions) (1.134)

holds for all times. By Eq. 1.133 we imply that ψ j1 and ψ

j2 are right-moving and left-

moving respectively:

ψj1(τ,σ)=Ψ j

1(τ−σ), (1.135)

ψj2(τ,σ)=Ψ j

2(τ+σ). (1.136)

If we impose boundary conditions Eq. 1.134 at the end points we get,

ψj1(τ,σ)=±ψ j

2(τ,σ), (1.137)

where we can choose the sign. Since both fields are quadratic in the action Sψ so

their signs can be changed without physical consequence. Therefore, at σ∗ = 0 we

conventionally choose

ψi1(τ,0)=ψi

2(τ,0). (1.138)

Once we chose this arbitrariness at one end then at the other end the choice of sign is

physically relevant without changing the condition (Eq. 1.137):

ψi1(τ,π)=±ψi

2(τ,π). (1.139)

⇒ψi1(τ,π)=ψi

2(τ,π), (Ramond (R) sector)

=−ψi2(τ,π). (Neveu-Schwarz (NS) sector) (1.140)

Let’s define the fermionic field Ψ j all over the interval σ ∈ [−π,π] irrespective of just at

the end points:

Ψ j(τ,σ)≡

ψ

j1(τ,σ) σ ∈ [0,π] (periodic fermion),

ψj2(τ,−σ) σ ∈ [−π,0] (antiperiodic fermion).

Hence, the boundary conditions become modified as:

Ψj1(τ,σ)

∣∣∣∣σ=0

=Ψ j2(τ,σ)

∣∣∣∣σ=0

(1.141)

Ψj1(τ,σ)

∣∣∣∣σ=π

=+Ψ j2(τ,σ)

∣∣∣∣σ=π

(Ramond boundary condition), (1.142)

Ψj1(τ,σ)

∣∣∣∣σ=π

=−Ψ j2(τ,σ)

∣∣∣∣σ=π

(Neveu-Schwarz boundary condition). (1.143)

32


For open strings, the solution of the equation of motion (Eq. 1.133) after Ramond

boundary condition (Eq. 1.142) in terms of oscillator expansion is given by:

Ψj1(τ,σ)=

pα′ ∑

n∈Zs j

ne−ιn(τ−σ). (1.144)

Ψj2(τ,σ)=

pα′ ∑

n∈Zs j

ne−ιn(τ+σ). (1.145)

When applying Neveu-Schwarz boundary condition (Eq. 1.143), the solution becomes:

Ψj1(τ,σ)=

pα′ ∑

n∈Z+ 12

b jne−ιn(τ−σ), (1.146)

Ψj2(τ,σ)=

pα′ ∑

n∈Z+ 12

b jne−ιn(τ+σ), (1.147)

where n ∈Z+ 12 guarantees that Ψ j is anticommuting such that

eιn(σ+2π) =−eιnσ. (1.148)

Neveu-Schwarz fermion changes its sign when we choose σ→ σ+2π that is why we

chose this value of σ.

For closed strings, right-moving and left-moving are independent periodic function

of 2π just like in bosonic string. For right movers we have

Ψj1(τ,σ)=

pα′ ∑

n∈Zs j

ne−ιn(τ−σ) (Ramond sector), (1.149)

Ψj1(τ,σ)=

pα′ ∑

n∈Z+ 12

b jne−ιn(τ−σ) (Neveu-Schwarz sector). (1.150)

Similarly, we have two possibilities for left movers Ψ j2. Therefore, we have four sectors:

(R,R), (R,NS), (NS,NS), (NS,R).

1.5.1 Quantization of superstrings

Just like in the quantization of quantum relativistic strings, we would do the same here;

converts the fermionic fields to quantum operators which satisfy the anticommutation

relation such as Ψµ

A(τ,σ),Ψµ

B(τ,σ′)= 2πα′ηµνδ(σ−σ′). (1.151)

33


The expansion coefficients become creation and annihilation operators. Negatively moded

coefficients would be creation operators while the positively moded ones are annihilation

operators, the same as we defined in bosonic string quantization.

1.5.1.1 Open superstring quantization

For open strings, expansion operators satisfy the following anticommutation relation:

For R sector:s j

n, skp= δn+p,0δ

jk, (1.152)

For NS sector:b j

n,bkp= δn+p,0δ

jk. (1.153)

Now we can work out the spectrum of NS sector. Here I am writing the direct

result of mass-shell condition but you can derive this by following the same technique as

we did previously: apply creation operators to the ground state and look at its spectrum

and reach to the mass-shell condition.

M2 = 1α′

( ∑n 6=0

αj−nα

jn +

∑r∈Z+ 1

2

rb j−rb j

r −(D−2)

16

). (1.154)

For ground state, Eq. 1.154 becomes:

M2 =− (D−2)16α′ . (1.155)

Hence, the ground state is again tachyonic for D > 2. For 1st excited state, apply b j−1 on

the ground state:

b j−1/2 |0⟩ , (1.156)

corresponding to this value of n = 1, the Eq. 1.154 becomes:

M2 = (10−D)16α′ . (1.157)

Here the 1st excited state is a vector with only D −2 independent modes exactly the

same as 1st excited state of quantum open string therefore for the consistency of Lorentz

symmetries it should be massless. So, Eq. 1.157 becomes

M2 = 0⇒ D = 10. (1.158)

34


Now the mass-shell condition for R sector is given as:

M2 = 1α′

∑n≥1

(α

j−nα

jn +ns j

−ns jn). (1.159)

This relation implies that all the Ramond ground states are massless and all excited

states are massive. For Ramond sector, there would be equal number of bosonic and

fermionic states in the excited states which is a signal of supersymmetry on the world-

sheet. In conclusion, Ramond sector has world-sheet supersymmetry.

1.5.1.2 Closed superstring quantization

In principle close superstrings is very easy once we have the clear picture of the open

superstrings. Closed superstrings have four sectors to open superstrings have two sectors.

Ramond sector of the open superstring gives us fermions while NS sector gives us bosons.

Four sectors of closed superstrings are:

(R,R), (R,NS), (NS,R), (NS,NS). (1.160)

(NS,NS) and (R,R) sectors give spacetime bosons while from (R,NS) and (NS,R) sectors,

spacetime fermions emerge. We need an operator whose value on states tells us about the

nature of the states, either bosonic or fermionic therefore we define an operator called

(−1)F , where F stands for fermion number. If the state is bosonic then this operator has

value +1 on the states and −1 if the state is fermionic. With the help of this operator we

can truncate the NS sector. The reason for doing truncation is that there are tachyonic

states in NS sector but not in Ramond sector. So in order to get supersymmetry in states

we must truncate the above four sectors. The set of states with (−1)F =+1 comprises

the NS+ sector and with (−1)F =−1 comprises NS− sector. The NS+ sector contains the

massless states and throws away the tachyonic states whereas the NS− sector contains

a tachyon. Now the R+ sector is defined as the set of R states with (−1)F =+1 and with

(−1)F =+1, R− sector is defined. It can easily be shown that these two sectors (R+ and

R−) contain the same number of states we define these just for the convention. The main

purpose is to define NS+ and NS−.

35


Above discussion expresses that by combining additively the set of states from the R−and NS+ sectors, open superstrings completely characterized and has a supersymmetric

spectrum. Therefore, closed strings are generally acquired by combining multiplicatively

right-moving and left-moving parts of open superstrings. So we defined right and left

sectors in such a way:

right sector :

NS+R+

, left sector :

NS+R−

. (1.161)

By combining multiplicatively these two sectors lead to one kind of closed superstring

theory called IIA which has the following sectors:

type IIA superstring: (R−,R+), (R−,NS+), (NS+,R+), (NS+,NS+). (1.162)

The type IIA superstring has no tachyons and its massless states are obtained by

combining the massless states of the various sectors which produces following fields:

IIA spectrum: Gµν, Bµν, Φ, Aµ, Aµνρ. (1.163)

Here Aµ is a Maxwell field and Aµνρ is three-index antisymmetric gauge field. These

two are produced by massless (R−,R+) bosons while the first three fields (graviton,

Kalb-Ramond and scaler) are produced by massless (NS+,NS+) bosons.

When the take the same type of Ramond sectors type IIB superstring arises :

right sector :

NS+R−

, left sector :

NS+R−

. (1.164)

which has the following sectors:

type IIB superstring: (R−,R−), (R−,NS+), (NS+,R−), (NS+,NS+). (1.165)

These produce following fields when apply to the ground state:

IIB spectrum: Gµν, Bµν, Φ, χ, Aµν, Aµνρσ. (1.166)

Due to the massless R-R bosons in the type IIB theory, antisymmetric gauge field Aµνρσ

with four indices, scalar field χ, a Kalb-Ramond field Aµν arises and the first three fields

are produced by massless (NS+,NS+) bosons.

36

CH

AP

TE

R

2D-BRANES AND T-DUALITIES

This chapter aims to highlight some important concepts related to D-branes:

how they arise in string theory, how the associated gauge fields arise on their

world volumes, form of the action that describes them and finally discuss the

T-dualities of closed and open strings.

2.1 Gauge fields on D-branes

D-branes provide surfaces to which open strings attach to. In a d dimensional space,

consider a p+1 dimensional D-brane which is denoted as a Dp brane. To specify these

‘hyperplanes’ we need (d− p) linear conditions. To see why this is the case, let us look at

some results we already know: In three spatial dimensions where d = 3, a 2 brane (p = 2)

is a plane. To specify it, we need (d−p = 3−2= 1) one linear condition. For instance, x = 0

specifies the y− z plane. Similarly, a piece of wire along the z axis (p = 1) is specified

by (d − p = 3−1 = 2) two conditions: x = 0 and y = 0. To sum it up, we need as many

conditions as there are spatial coordinates normal to the brane. Sp, come back to the

Dp-brane.

Let us use xµ to denote spacetime coordinates and split these coordinates into two

37

CHAPTER 2. D-BRANES AND T-DUALITIES

groups. In the first group, there are the coordinates tangential to the brane which clearly

are time and p spatial coordinates since the brane is p-dimensional. The other (d− p)

coordinates are normal to the brane. The location of the brane, we know, is specified by

the values of the coordinates normal to the brane. Let us denote these in the following

manner:

xa = xa, a = p+1, ...,d. (2.1)

Since we are looking at open strings that end on the Dp-brane, the coordinates normal

to the brane must satisfy the Dirichlet boundary conditions, that is:

X a(σ,τ)|σ=0 = X a(σ,τ)|σ=π = xa, a = p+1, ...,d. (2.2)

where X a denote the string end-point coordinates. The string coordinates X a are referred

to as the DIR coordinates with the obvious reference to Dirichlet boundary conditions.

The string is free to move along directions tangential to the brane, the coordinates of

the endpoints satisfy the following equation:

X j′(σ,τ)|σ=0 = X j′(σ,τ)|σ=π = 0, j = 0,1, ..., p. (2.3)

Since these satisfy Neumann boundary conditions, these are referred to as the NM

coordinates. To summarize, we can now split the string coordinates as

X0, X1, ..., X p︸︷︷︸NM coordinates

X p+1, X p+2, ..., X d︸︷︷︸DIR coordinates

. (2.4)

In terms of light-cone coordinates, these are written as:

X+, X−, X i︸︷︷︸NM

X a︸︷︷︸DIR

, i = 2, ..., p a = p+1, ...d. (2.5)

Here index ‘i’ represents the states/coordinates along the directions tangent to the brane,

whereas a-type for the directions normal to the brane.

There are fields corresponding to each coordinate which can be viewed after applying

Fourier transformation upon them. Hence, the fields live on the Dp-brane. In order to

move forward, first figure out the dimensionality of the Dp-brane. The dimensions of

Dp-brane are the same as the dimensions of the coordinates tangential to the brane

38


which are one for x+, one for x− and p−1 for xi, i = 2, ..., p. Hence, the dimensions of

Dp-brane is p+1.

Now we start looking for the fields that live on Dp-branes. Let begin with the ground

states which are tachyonic, therefore tachyonic field is one of them and its a Lorentz

scaler. After this, let’s move to the excited states where we have two cases: the first in

which the oscillator arises from a coordinate tangent to the brane and the other in which

oscillator arises from a coordinate normal to the brane. In the former case, there are

p−1 of these states which is equal to the spacetime dimensions of the brane minus 2.

The reason for doing minus two is just simple because the creation operator has an index

i which runs from 2 to p form p−1 states and they transform as a Lorentz vector on the

brane. We conclude that these are photon states and the associated field which lives on

the brane is a Maxwell gauge field. While in later case, there are (d− p) states living

on the brane and the index a is simply a counting lable not Lorentz index. Hence, for

each direction normal to the Dp-brane we get a massless scaler field.

As a summary, we have two very important results:

1. A Dp-brane has a Maxwell field on its world-volume.

2. A Dp-brane has a massless scalar for each direction normal to it.

2.2 Parallel Dp-branes and open strings

We can repeat the above analysis for the case where a string is stretched between two

different branes, that is, with one end point on each. Based on that, to avoid digression, I

would directly write down the expression for mass-shell condition (M2) which turns out

as:

M2 =( xa

2 − xa1

2πα′

)2+ 1α′

(−1+

∞∑n=1

p∑i=2

nai†n ai

n +∞∑j=1

d∑a=p+1

jai†j ai

j

). (2.6)

Here the branes under consideration are assumed to be parallel and have the same

dimensions (see Fig. 4.1). Moreover, xa = xa1 specifies the location of the first Dp-brane

39


and xa = xa2 of the second one. The ground state now has

M2 = −1α′ +

( xa2 − xa

1

2πα′

)2. (2.7)

Hence, if the two branes are separated by∣∣xa

2 − xa1∣∣= 2π

pα′ or more, the ground state is

FIGURE 2.1. Parallel D2-branes.

no more tachyonic. A tachyonic ground state corresponds to an instability of the D-brane.

It can be shown that it leads to a decay of the D-brane that is why tachyons are a source

of instability, thus consider to be avoidable.

The first excited states arise in two ways as before. From the third part of the second

term of the Eq. 2.6 we get d− p states with ( xa2−xa

12πα′ )2. Again, since ‘a’ is not a Lorentz

index on the brane, these are d− p massive scalar fields. States with the same mass

also arise from the second part of the second term. This time, these indices are Lorentz

indices on the brane, so there are p−1 of these states. However, unlike the previous case,

these are not massless.

When the separation between two branes goes to zero, there are 4 open string sectors:

strings can either have one end point on each of the branes or both end points on the

same brane. The quantum mechanical formalism of open strings in the presence of two

parallel Dp-branes has four sectors. Each of the four sectors of a string on two parallel

D-branes are shown in the Fig. 4.1. Consequently, we now have four massless gauge

fields. Different configurations of strings ending on a stack of D-branes can be thought of

as having resulted from the interaction of some other string configurations. For example

40


if for instance we have a string that has one end point on say brane 1 and the other on

brane 3 then this is like a string ending on brane 1 at one end and brane 2 on the other

joining with another string ending on brane 2 on one end and brane 3 on the other (see

Fig. 4.2). This rearrangement of strings represents how the states on each brane are

FIGURE 2.2. Interaction of D-branes.

allowed to interact with those on the other. We therefore have gauge fields that interact

with each other.

For 2 parallel D-branes we have four sectors of open strings so for N coincident

D-branes, we have N2 open string sectors and therefore N2 massless gauge fields. This

is the content of a U(N) gauge field theory. In fact, at low energies, U(1) becomes

unimportant and we are left with SU(N). This is because geometrically, U(1) deals

with the collective coordinates of the brane and in low energy regime in which we are

interested in, the corresponding modes are not excited 1 from here on. Another thing to

note here is that by introducing brane separation, we have a way to give mass to our

gauge fields. Hence, this is the analog of the Higgs mechanism in the string picture.1The U(1) mode represents the free motion of the brane. We will only be concerned with the modes

that couple to gravity and since the rest will be discarded so we can safely start ignoring U(1).

41


2.3 String and D-brane charges

When a point particle interacts with Maxwell field, it carries electric charge. Similarly

when strings incorporates with Kalb-Ramond field they carry a new kind of charge -

string charge. String charge can be envisioned as a current flowing along the string for

the strings whose one end is attached to one brane and the other with another brane.

Strings can end on D-branes and it does not violate law of conservation of string charge

because endpoints of the string carry electric charge and the resulting electric field lines

on a D-brane carry string charge.

A charged point particle couples to a Maxwell field. It traces out one-dimensional

worldline and has just one tangent vector. The Maxwell field Aµ also carried one index.

The interaction of a charged point particle with a Maxwell field is given by the following

lagrangian:

LP = q∫

Aµdxµ

dτdτ. (2.8)

This index matching approach to predicting candidate lagrangians is extremely impor-

tant when writing down actions for multi-dimensional things such as branes and strings.

This is exactly what we will employ here. As we know that a 2-dimensional worldsheet

or worldvolume is traced by a string. Naturally, since it has two vectors that span the

sheet, we would expect there to be a two form field to multiply the tangent vectors to

give a Lorentz scalar, or in more relevant terms, the lagrangian. Hence, we should have

something of the form:

−∫

dτdσ∂Xµ

∂τ

∂Xν

∂σBµν(X (τ,σ)), (2.9)

where Bµν is a two-form called the Kalb-Ramond field. It is called an electric coupling

because it is the natural generalization of the electric coupling of a point particle to

a Maxwell field. Hence, we can say that the string carries electric Kalb-Ramond

charge. It can be shown that, to ensure re-parametrization invariance of the string

action, Bµν must be antisymmetric. For the sake of completeness we will mention here

the corresponding field strength Zµνρ which is a generalization of the Maxwell field

42


strength Fµν.

Zµνρ ≡ ∂µBνρ+∂νBρµ+∂ρBµν. (2.10)

Since Bµν is the only thing that will be used in this thesis, the details of some of the

other accompanying concepts including Zµνρ are skipped and the interested reader is

referred to [2].

After dealing with the generalization of the strings, let us now move on to branes.

A Dp-brane has a world volume which is p+1 dimensional. One would then expects

branes to couple to p+1 forms. These are called Ramond-Ramond (RR) forms. Indeed

the coupling term in the action is given on the following:

−∫

dτdσ1...dσk∂Xµ

∂τ

∂Xµ1

∂σ1 ...∂Xµk

∂σk Yµµ1...µk (X (τ,σ1, ...σk)). (2.11)

For point particle, electric charge can be evaluated from Maxwell equation which relates

the electric current and the electromagnetic field such as:

∂Fµν

∂xν= jµ. (2.12)

The generalization of Eq. 2.12 for the strings is given as:

1κ2

∂Zµνρ

∂xρ= jµν, (2.13)

where κ is just a constant which balances the dimensions of the equation. From above

two equations, one can conclude that, like point particles, strings and branes also

carry charges. It is as a result of charge and energy conservation that charged branes,

for example, cannot decay if there are no lighter decay products that carry charge.

Hence, charged branes are more stable than uncharged branes. Since we will always be

interested in studying only the stable structures, we will only be concerned with strings

and branes that carry charges and couple to the corresponding fields. Here I would like

to state the result directly without going in detailed derivation that string endpoints are

oppositely electrically charged [2].

43


2.4 T-dualities

String theory offers a lot of extra dimensions. We must find a way to deal with these

extra dimensions in our theories. One of the ways is to compactify these dimensions.

This compactification, amazingly, leads to a symmetry: If a spatial dimension is curled

up into a circle of radius R then the scenario is physically equivalent to a scenario where

the radius is ∝ 1R . There is a duality symmetry which relates these to each other. If

the space is being compactified into a torus then this compactification is called toroidal.

Therefore, this symmetry is called T-duality where T stands for toroidal. Duality in

physics is when two seemingly different physical systems have a dictionary that allows

one to map all the physical properties of one system on to the other and vice versa.

2.4.1 Closed strings and T-duality

In order to introduce the terminology of T-duality, let us write the most generalized form

of the zero modes (α250 and α25

0 ) expansion for the closed string (generalization of Eq.

1.89)

Xµ(τ,σ)=(

xµ

2+ xµ

2

)+

(−ι

√α′

2(αµ0 +α

µ

0)τ+√α′

2(αµ0 −α

µ

0)σ)

+oscillating terms. (2.14)

The corresponding spacetime momentum of the string is given as: (From Eq. 1.94)

pµ = 1p2α′ (αµ0 + α

µ

0). (2.15)

If we move around the string such that σ→σ+2π, the oscillator terms are periodic and

we have

Xµ(τ,σ)→ Xµ(τ,σ)+2π

√α′

2(αµ0 − α

µ

0). (2.16)

If there is no compactification then αµ

0 = αµ0 , the second term of above equation vanishes

and we get the same result as previous. Due to this, Eq. 2.15 simplifies as

pµ = 1p2α′ (2αµ0)=

√2α′ α

µ

0 . (2.17)

⇒αµ

0 = αµ0 =√α′

2pµ. (2.18)

44


To introduce the notion of T-duality, in order to do so let us assume the string

(bosonic one) which is propagating in a 26 spacetime dimensions. When one of the spatial

dimension is being compactified in such a way as it forms a circle of radius R. Let’s

call that compactified dimension X25(τ,σ) therefore its expansion has to be modified in

order to incorporate the periodic boundary condition (Eq. 2.16) while the expansion of

the coordinates Xµ, for µ = 0, ...,24, does not change as compared to the expansion in

flat Minkowski 26-dimensional space. The momentum eigenvalue is quantized along

that direction, p25, which is being compactified. As we know that in quantum mechanics,

wavefunction contains the factor eιp25x25

and we have x25 which is increased by 2πR shows

that when you going around it once, it’s wavefunction should return to its original value.

Since the circle has discrete Fourier modes the target space momentum is quantized.

From this, we conclude that the momentum in the 25 direction is quantized and it takes

the values

p25 = KR

, K ∈Z. (2.19)

Here K is called the Kaluza-Klein excitation number. The identification x ∼ x+2πR

had two impacts. Before compactification, the momentum operator has a continuous

spectrum but once the compactification is done (circle is made), the momentum gets

quantized, so we lost the states that don’t fulfill the quantization condition. Second, we

get some new states - the winding states that wind around the recently made circle.

Subsequently we both lost and gained some states [2]!

However, from Eq. 2.15 for µ= 25 we get:

α250 + α25

0 = p25p2α′ . (2.20)

From Eq. 2.19, above equation becomes:

α250 + α25

0 = KR

p2α′ . (2.21)

Now the periodic boundary conditions for closed strings for that compacted coordinate

which parametrized by σ of range 2π becomes modified as

(periodic boundary condition) X25(τ,σ+2π)= X25(τ,σ)+2πRW , W ∈Z, (2.22)

45


where W stands for winding number: the number of times the string wraps around the

circle (see Fig. abc). Now by comparing Eq. 2.22 with Eq. 2.16, we get

α250 − α25

0 =√

2α′ RW . (2.23)

Solving Eq. 4.1 and Eq. 2.23 simultaneously, one may get

α250 =

(KR

+ RWα′

)√α′

2, (2.24)

α250 =

(KR

− RWα′

)√α′

2. (2.25)

One can use these two above expressions to compute the mass formula as

M2 =−pµpµ = K2

R2 + W2R2

α′2 + 2α′

(NR +NL −2

), (2.26)

where NR and NL represent the total number of levels on the right and left moving sides

respectively. There is a term in above expression which gives the momentum states of

the string corresponding to the Kaluza-Klein excitation number and another new term

which rises the winding states. For this situation, according to T-duality if we have a

system that transforms under the following mapping:

R → R = α′

R. (2.27)

then it leaves the mass-shell expression invariant provided that the string winding

number (W) is exchanged with the Kaluza-Klein excitation number (K).

The uniformity between a circle of radius R and R is a straightforward implication

that ordinary geometric ideas and intuitions might break down in string theory at the

string scale. The swapping of K and W signifies that excitations of momentum states in

one system relate to the excitations of winding-mode states in the dual system and the

converse is true [5].

Following is the transformation known as T-duality, under which the physical quantities

such as energy-momentum tensor and correlated functions are invariant.

α0 →α0 and α0 →−α0, (2.28)

46


Hence, Eq. 2.22 can be equivalently written as:

X25(σ,τ)= X25(σ,τ)+2πRW . (2.29)

Here the x coordinate which is responsible for the parametrization of a circle with peri-

odicity 2πR is replaced by a x coordinate through which the dual circle is parametrized

with periodicity 2πR. The corresponding momentum is p25 =W /R.

In conclusion, we summarize the generic behavior of the spectrum (Eq. 2.26) for

different values of R.

• As R goes to infinity, the winding states get massive and the momentum states

form a continuum.

• For small R, the reverse is true. As R goes to zero, the momentum states get

massive and the winding states form a continuum.

2.4.2 Open strings and T-duality

A characteristic thing to ask now is that what happens to the open strings when a

T-duality transformation is applied on them. Open strings have no winding modes when

they compactified on a circle of radius R. Although, thinking about winding number of

open strings is not a wise concept because topologically an open string can simply be

contracted to a point. Since the winding modes were critical in the closed strings which

relate two theories by T-duality but in case of open strings one should not anticipate that

open strings transform the same way as closed strings did [5].

In order to find out the T-dual of an open string let us first write the mode expansion

of open strings which we have derived from Neumann boundary conditions in Chapter 1:

Xµ(τ,σ)= xµ+vµτ+ ιp

2α′ ∑n 6=0


and the string spacetime momentum is related as:

pµ = l2π

vµα′ . (2.31)

47


For the sake of easiness we set α′ = a/2 and l = 1. Hence, Eq. 2.30 becomes:

Xµ(τ,σ)= xµ+ pµτ+ ι ∑n 6=0


It is convenient to write the above equation in terms of left and right moving parts such

as

Xµ

L(τ+σ)= xµ+ x′µ

2+ pµ(τ+σ)

2+ ι

2

∑n 6=0

1nαµne−ιn(τ+σ), (2.33)

and

Xµ

R(τ−σ)= xµ− x′µ

2+ pµ(τ−σ)

2+ ι

2

∑n 6=0

1nαµne−ιn(τ−σ), (2.34)

where x′µ is an arbitrary number which vanishes out when the usual open string

coordinate is being made by adding above two equations.

Now comes to compactification. Let us once again compactified one dimension on a

circle of radius R and then applying a T-duality transformation which is

Xµ

L → Xµ

L and Xµ

R →−Xµ

R . (2.35)

Under this transformation we would get the dual string coordinates in the 25 direction

such as

X25(τ,σ)= X25L − X25

R = x′25 + p25σ+ ∑n 6=0

1nα25

n e−ιnτ sinnσ. (2.36)

Here the noticeable thing is that the zero-mode expansion is independent of τ from which

the momentum term arises. This clearly states that we don’t have momentum in the 25

direction. We also see that the oscillator terms die out at the endpoints σ= 0,π which

implies that the endpoints do not move in the X25 direction. This means that T-duality

maps Neumann boundary condition (∂σX ≡ 0) into Dirichlet boundary condition (∂τX ≡ 0)

and vice versa in the relevant directions [6].

The compacted dual string coordinates at the end points are:

X25(τ,σ)|σ=0 = x′ and X25(τ,σ)|σ=π = x′+ p25π. (2.37)

Here I used the term dual string coordinate, the word dual implies that we moved

from the system R → R =α′/R. Therefore, by using Eq. 2.19 and Eq. 2.27 for the dual

48


radius. Eq. 2.37 becomes

X25(τ,σ)|σ=0 = x′ and X25(τ,σ)|σ=π = x′+2πKR. (2.38)

This implies that the string wraps the dual circle K times. Since, the end points of the

string are not allowed to move (they are fixed) by the Dirichlet boundary conditions

therefore the winding mode is stable. Hence, this string cannot unwind without breaking.

In summary, when we compactified one of the 26 dimensions of the bosonic string

onto a circle of radius R with Neumann boundary conditions they would have momentum

but no winding modes but once the T-duality transformation is done they would have

winding modes but no momentum in the dual circular direction with Dirichlet boundary

conditions. From Eq. 2.38 one can conclude that the end points of dual open string are

attached to the hyperplane (X25 = x′25) called Dirichlet-brane or in short D-brane and

they can wrap integer number of times around the circle [5].

49

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3ADS SPACE AND CONFORMAL FIELD THEORY

In this chapter, first we would look at the basics that underlie the first half of

the AdS/CFT duality. To avoid digression, some details will be skipped. For the

derivations of the results, the interested reader is referred to [7]. Afterwards,

we will begin by looking at some essential properties of conformal field theories in flat

space in an attempt to develop an understanding of these theories before discussing the

AdS/CFT correspondence.

3.1 Anti de Siter Space

3.1.1 Introduction

An Anti de Sitter space is a space of Lorentzian signature (+1,−1,−1,−1,−1, ...,−1)

analogue of the Lobachevski space, space with a constant negative curvature and has

a signature (+1,−1,−1,−1,−1, ...,−1,+1), and a solution to the vacuum Einstein Field

equations with a cosmological constant. It is an n dimensional Riemannian manifold

whose metric is not positive definite. The scalar curvature of an AdS space is by definition

a negative constant. The proof is as follows:

50

CHAPTER 3. ADS SPACE AND CONFORMAL FIELD THEORY

Recall that the vacuum Einstein equation with a cosmological constant is given by:

Rµν− 12

gµνR = 12Λgµν. (3.1)

By taking the trace on both sides:

gµνRµν− 12

gµνR = 12Λgµνgµν, (3.2)

⇒ Rµµ−

12

nR = 12

nΛ, (3.3)

⇒ R =− nn−2

Λ, (3.4)

where R denotes the Ricci scalar and n is the dimension of the space. We can see that we

have a spacetime that has a negative curvature in the absence of matter. Next, we use

the expression for R in Eq. 3.1 to get the following:

Rµν = 12

gµν(−Λn

n−2

)+Λgµν,

Rµν =− Λ

n−2gµν.

(3.5)

Hence we have a case where the Ricci tensor is proportional to the metric tensor. More-

over, an AdS space is a maximally symmetric manifold which implies that it is

invariant under translations (homogenous) and invariant under rotations (isotropic).

Mathematically, it can be shown that it translates into the following condition:

Rµνρσ = Rn(n−1)

(gµρ gνσ− gνρ gµσ

), (3.6)

where Rµνρσ denotes the Riemann tensor.

3.1.2 Geometry of AdSn+1 space

Consider an n+1 dimensional AdS space and embedded it on n+2 Lobachevski manifold

with coordinates yµ = (y0, y1, y2, ..., yn, yn+1) and a metric with scale factor L2 is defined

as:

ds2 ≡ (d y0)2 −n∑

i=1(d yi)2 + (d yn+1)2 = L2. (3.7)

51


This metric is called a conformally flat metric. To prove this, let us introduce po-

lar/stereographic coordinates (xa)= (x1, x2, ..., xn+1) on AdS space as follows:

y0 = L1+ xaxa

1− xaxa ,

yi = L2xi

1− xaxa , i = 1,2, ...,n+1(3.8)

To compute metric:

d y0 = dL1+ xaxa

1− xaxa +4Lxadxa

(1− xaxa)2 ,

dyi = dL2xi

1− xaxa + 2L(1− xaxa)2

(1− xaxa)δi

k +2xixkdxk.

(3.9)

Using this,

ds2 =− 4L2

(1− xaxa)2 dx2, (3.10)

where dx2 =∑ni=1(dxi)2 − (dxn+1)2. Hence, the metric is conformally flat:

gµν =− 4L2

(1− xaxa)2ηµν. (3.11)

From this, one can compute the christoffels symbols of this metric:

Γµνσ =

12

gµα(∂νgσα+∂σgνα−∂αgνσ

)= 1

2

(4xν

1− xaxaδµσ+

4xσ1− xaxaδ

µν−

4xµ

1− xaxaδνσ

).

(3.12)

It is an easy task to evaluate Riemann tensor by using christoffel symbols:

Rµνσα = 1L2

(gναgµα− gναgµα

). (3.13)

This is just the Eq. 3.6 which implies thatthe embedded space is the maximally symmet-

ric.

The Ricci tensor is given by

Rνα = nL2 gνα. (3.14)

By tracing above equation, one can get Ricci scaler as:

R = n(n+1)L2 (3.15)

52


This implies that (n+1) dimensional space solves the vacuum Einstein equation only if

the cosmological constant satisfies

Λ=−n(n−1)L2 < 0. (3.16)

Thus we have verified that our n+1 dimensional space is indeed an AdS space.

As we defined AdS space as the locus of points which satisfies following relation:

y2 = (y0)2 −n∑

i=1(yi)2 + (yn+1)2 = L2. (3.17)

The transformations under which this expression remains invariant form a SO(n,2)

group. This implies that SO(n,2) is the isometry group of the AdSn+1.

3.1.3 Boundary of AdSn+1 space

First, define the boundary of the AdSn+1 space as the sets of points yµ with the conditions

that yµ→∞ and yµ ∈ AdSn+1 then define the new set of variables such that yµ = R yµ,

with R →∞. In terms of new coordinates, Eq. 3.17 becomes:

R2

( y0)2 −n∑

i=1( yi)2 + ( yn+1)2

= L2,

( y0)2 −n∑

i=1( yi)2 + ( yn+1)2 = L2

R2 ,

( y0)2 −n∑

i=1( yi)2 + ( yn+1)2 = 0. (3.18)

since 1R2 → 0 as R →∞.

Now note that if yµ is the point on the boundary of the AdSn+1, then the point

t yµ also lies at the boundary as it satisfies Eq. 3.18. This means that we can scale the

coordiante yµ in Eq. 3.18 to get

( y0)2 + ( yn+1)2 =n∑

i=1( yi)2 = 1. (3.19)

53


This shows that the boundary of the AdSn+1 space is S1 ×Sn−1 which implies that at

the boundary we would have a dimensions less than by one depicted as following:

AdSn+1 = S1 ×Sn

At the boundary:

AdSn = S1 ×Sn−1

3.1.4 Example: AdS2+1 space

It is easy to start with simple example of an AdS space, AdS2+1 and the results then can

then be generalized. To do so, we follow the Lobachevski embedding prescription and

proceed as follows:

Define the locus of points

x2 + y2 −w2 − z2 =−L2 (3.20)

Consider the following change of variables:

w = Lcosh(µ)sin(t)

z = Lcosh(µ)cos(t),

where µ ∈ [0,∞) and t ∈ [0,2π).

The metric now becomes

ds2 = dx2 +d y2 −dw2 −dz2 = dx2 +d y2 −L2 sinh2 (µ)dµ2 −L2 cosh2 (µ)dt2. (3.21)

Consider another change of variables for x and y:

x = Lsinh(µ)cos(φ)

y= Lsinh(µ)sin(φ),

where φ ∈ [0,2π).

Thus, Eq. 3.21 becomes:

ds2 =−L2 cosh2 (µ)dt2 +L2dµ2 +L2 sinh2 (µ)dφ2. (3.22)

54


Now define a new coordinate: sinh(µ)= r/l, and the metric transforms to

ds2 = (L2 + r2)dt2 +(1+ r2

L2

)−1dr2 + r2dφ2. (3.23)

From this, our discussion of AdS2+1 space has been completed.

3.2 Conformal Field Theory

3.2.1 Introduction

Conformal field theories are the quantum theory of fields which are invariant under

conformal transformations. Intuitively, conformal transformations are those transforma-

tions that leave the angles between the vectors invariant. One such transformation is

the scaling:~x → a~x. The theories that are conformally invariant are the ones that do not

have a preferred scale.

To study the conformal field theories, we must first study the conformal group. In the

next section, a detailed review of the conformal group is given.

3.2.2 The Conformal Group

The conformal group is the group of transformations under which the metric remains

preserve up to an arbitrary scale factor, gµν(x) →Λ2(x)gµν(x). It includes not only the

Poincaré group but the inversion symmetry xµ→ xµ/x2 as well [8].

The conformal group of Minkowski space is generated by the Poincaré transformations

such as:

xµ→λxµ, (3.24)

and the special conformal transformations

xµ→ xµ+aµx2

1+2xνaν+a2x2 . (3.25)

55


3.2.3 The Conformal Algebra

For the sake of completeness, we express the Lie algebra of the conformal group; confor-

mal algebra. Before this, we will denote the generators of conformal transformations as

follows:(for translations) Pµ =−ι∂µ,

(for Lorentz rotations) Mµν =−(xµPν− xνPµ),

(for scaling transformations) D =−ιxµ∂µ,

(for special conformal transformations) Kµ =−2xµD+ x2Pµ.

(3.26)

They obey the conformal algebra:

[Mµν,Pρ]=−ι(gµρPν− gνρPµ),

[Mµν, Mρσ]=−ιgµρMνσ±permutations,

[Mµν,Kρ]=−ι(gµρKν− gνρKµ),

[Mµν,D]= 0,

[D,Kµ]= ιKµ,

[D,Pµ]=−ιPµ,

[Pµ,Kν]= 2ι(Mµν− gµνD),

(3.27)

with all other commutators vanishing.

In the special case of d = 2 the conformal group become infinite dimensional. We will

discuss its aspects later in this chapter.

3.2.4 Conformal Group in d> 3

In order to determine the conformal group for this case, let us first write the dimension

of the algebra by counting the total number of generators N, which is clear from the

indices on the generators, (from Eq. 3.26). Hence,

N = d+1+ d(d+1)2

+d,

N = (d+2)(d+1)2

.(3.28)

56


From above, one can see that the algebra defined in Eq. 3.27 is similar to SO(d,2) algebra

(with signature −,+,+, ...,+,−) with generators Jab(a,b =−1,0,1...,d−1) by defining

Jµν = Mµν,

Jµd = 12

(Kµ−Pµ),

Jµ(d+1) =12

(Kµ+Pµ),

J(d+1)d = D.

(3.29)

They also satisfy the usual SO group commutation relation:

[Jab, Jmn]= ι(ηanJbm +ηbmJan −ηamJbn −ηbnJan) (3.30)

Hence, the Conformal group corresponds to the SO algebra in d dimensions where d > 3.

3.2.5 Conformal Group in d= 2

Earlier, we had restricted our discussion of the Conformal Group to d > 3. Now let us

look at the Conformal Group in d = 2. One may ask why we could not generalize the

treatment in the previous question to d = 2. It turns out, as we will notice in a moment,

that the conformal group is infinitely large in d = 2 and our earlier treatment does not

make that obvious. The globally defined transformations in the d > 3 case, it should be

kept in mind, are the same as in the d = 2 case.

For d = 2, the signature of the metric is (−1,1) and the conformal transformations are

nothing but the Cauchy-Riemann equations [9]. The generators of this transformations

are then given by:

jn =−zn+1 ∂

∂z,

jn =−zn+1 ∂

∂z,

(3.31)

where z is some parameter of the the function which is the solution of Cauchy-Riemann

equations. These generators also follow the following commutation relations:

[ ja, jb]= (a−b) ja+b,

[ ja, jb]= (a−b) ja+b,

[ ja, jn]= 0.

(3.32)

57


This is called Witt algebra. Since, n ∈Z therefore there is an infinite number of linearly

independent generators which leads us to the important fact that: In 2d, the conformal

group is infinite dimensional.

The globally defined transformations on the Riemann sphere S2 'C∪ ∞ are ones that

are well defined at z = 0 and z =∞ but not all the generators are well-defined [9].

Following are the well-defined generators for z = 0:

jn =−zn+1 ∂

∂z, non-singular (non-continuous) only for n ≥−1 at z = 0. (3.33)

For z =∞, let us perform the transformation z =−1/b and study b → 0,

jn =−(− 1

b

)n−1 ∂

∂b, non-singular only for n ≤+1 at z =∞, (3.34)

where ∂/∂z = (−b)2∂/∂b. Similarly, we get for jn.

From the form of these generators, we see here that the global transformations

are generated by j−1, j0, j1 and j−1, j0, j1. In order to see what these transformations

represent, we notice the following:

1. j−1 =−∂/∂z and j−1 =−∂/∂z are the translational generators.

2. j0 + j0 =−r∂/∂r, where z = reiφ, is the 2d dilation generator.

3. ι( j0 − j0)=−∂/∂φ generates rotations.

4. j1, j1 are the generators of special conformal transformations (Eq. 3.25).

Hence, this is the group SL(C,2)/Z2 ≈ SO(3,1). Therefore, the Conformal group corre-

sponds to the SL(C,2)/Z2 ≈ SO(3,1) algebra in d dimensions where d = 2.

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4D-BRANES AND ADS/CFT CORRESPONDENCE

A fter examined the rudiments, we are now able to understand the main topic

to this thesis: AdS/CFT correspondence. Primarily, we will try to motivate how

this correspondence comes about to provide a background for the things in the

following chapters. Details of the AdS/CFT dictionary will be skipped and the interested

reader is referred to [10] to study them.

4.1 The Correspondence

This correspondence states that quantum gravity in interior of the space-time is equiva-

lent to a certain quantum field theory defined on the boundary of that space-time. By

equivalent we mean that they can describe the same physics and hence there is a map

between operators in the conformal field theory and fields in supergravity.

Consider a stack of N parallel, coincident D3-branes which are extended along a

(3+1) dimensional plane in a 10 dimensional space-time. In this background, there are

two types of string excitations: the open string excitations, as we studied in the previous

chapter, correspond to the D3-branes while the closed string excitations are of the 10

dimensional ambient space. The interaction term between the closed strings and open

59

CHAPTER 4. D-BRANES AND ADS/CFT CORRESPONDENCE

strings can be ignored at low energies. Hence, the low energy limit of a stack of D3-branes

in 10 dimensions is described by two independent decoupled systems.

In order to derive this correspondence, first we loot at the observation that in string

theory black p-branes, generalizations of black holes in p spatial dimensions, are the

solutions of supergravity in the low energy limit of string theory are same as D-branes

[11]. Hence, D-branes act as a curved background where closed strings propagate which

implies that D-branes are described by supergravity solutions in 10 dimensions at low

energy.

So, we start with the solutions of supergravity equations and see how D3-branes

effect the 10 dimensional ambient spacetime at low energies. The metric is given by:

ds2 = H− 12ηµνdxµdxν+H

12 (dr2 + r2dΩ2

5), (4.1)

where ηµν is the Minkowski 4 dimensional spacetime metric along the branes and

H = 1+ R4

r4 , R4 = 4πgsNl4s . (4.2)

Here R is the radius of the curvature of the AdS space and gs is the string coupling.

R/ls must be larger than (4πgsN1/4) in order to use the supergravity approximation of

string theory. This simply implies that in supergravity picture we are in the following

limit:

gsN >> 1. (4.3)

Generally, the metric (Eq. 4.1) is not fully correct from the perspective of an observer

who is at r =∞ in the low energy limit when an object is moving towards the horizon

r → 0. According to that observer, any object moving towards r = 0 (the near horizon

region) would appear to have lower and lower energy regardless of its ‘actual’ energy

because of red shift [11]. However, that observer also sees low energy excitations arising

from the standard large wavelength excitations, that is, energies that are already low

even before the red shift. So, to determine the correct metric in the region around r = 0,

let us introduce a new coordinate, for the sake of our convenience, z = R2/r. The correct

metric becomes:

dss = R2ηµνdxµdxν+dz2

z2 +R2dΩ25, (4.4)

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which is the AdS5 metric times S5 metric. Hence, this describes the space AdS5 ×S5.

Now, as we learned in Chapter 2 that the open string excitations on D-branes are

described by a U(N)→ SU(N) theory. It can be shown that at low energies, the complete

description is given by an N = 4, SU(N) supersymmetric theory. In our case, we have

D3-branes which are extended along (3+1) dimensions. Hence, open string excitations

are described by N = 4, U(N) super Yang-Mills theory in 4 dimensional minkowski

space.

Conclusion:

At low energy limit, the two decoupled systems; open string excitations due to N

D3-branes are described by N = 4, U(N) super Yang-Mills theory in 4 dimensional

minkowski space is equivalent to the closed string exciations which are well described

by type IIB supergravity 1 in AdS5 ×S5 dimensional flat space.

Thus, N = 4, U(N) super Yang-Mills theory in 4 dimensional minkowski space is

equivalent to type IIB supergravity in AdS5 ×S5. This is the AdS/CFT conjecture.

4.2 From Symmetry Groups

There are 15 number of generators 2 for N = 4 dimensions which define conformal

lie algebra. The generators of these transformations satisfy the SO(4,2) commutation

relations which is also the isometric group of AdS5 as shown in previous chapter.

Moreover, N = 4, SU(N) has an R-symmetry 3 group SO(6) which matches the isometry

group of S5. As a result, the symmetry groups of the two theories match providing further

evidence of a correspondence between the two.

The AdS/CFT correspondence provides us with a powerful tool to study gauge theories

which are difficult to study on their own using theories of supergravity. In fact, in any1Type IIA superstring theory at the limit in which the string coupling goes to infinity becomes a new

theory called M-theory which has 11-dimensions.2N = (d+2)(d+1)

23An internal symmetry under which supercharges Q iα rotates. It turns into a global symmetry when

its algebra is represented on fields, under which fields rotate.

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given situation, we can make use of the duality to study a particular problem using

either one of the two theories, whichever is easier in the particular context.

4.3 The ’t Hooft Large N Limit

In gravity theory N and gs represents the number of D3-branes and string coupling res-

pectively. The more N means the more D3-branes and each D-brane has a corresponding

gauge field as we discussed in Chapter 2 which implies that there would be lots of gauge

fields for large N. While in gauge theory, N and gY M represents the rank of SU(N) and

gauge coupling respectively. Larger N results in a larger number of gluons which would

make a chain in which there is a strong interaction between gluon pairs (color and

anti-color) in gauge theory as seen in below Fig 4.1.

FIGURE 4.1. Interaction between chain of pair of gluons.

As we know from our previous knowledge that we can make closed string from

two open strings as we seen from Fig. 4.2 therefore, the two open strings splitting

interaction gY M is equal to one closed string splitting interaction governed by gs coupling.

Mathematically, these are related as:

4πgsN = g2Y M N =λ (4.5)

From Eq. 4.3, we get:

gsN >> 1, or g2Y M N >> 1. (4.6)

The relation between string’s coupling gs and gravitational constant is given as:

GN = 8π6 g2sα

′2. (4.7)

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FIGURE 4.2. Closed string interaction from two open string splitting interac-tions, therefore g2

Y M = gs.

As N approaches to ∞, gravity coupling becomes weak gs → 0 provided λ>> 1 and

constant. This shows that gravity theory becomes weakly coupled at large N. This is

all we can see from Eqs. 4.5 and 4.7. But in gauge theories, ’t Hooft showed that gauge

theories with dirac fields have expansion parameter known as ’t Hooft coupling λ= g2Y M N.

He considered the perturbative expansion in λ and make it fixed such that λ<< 1. This

is known as the ’t hooft limit. Hence, gauge theory is strongly coupled at large N. But

we see from Eqs. (4.3 and 4.5) we require λ >> 1 and fixed instead. Since these two

descriptions are valid in opposite regimes, λ >> 1 for supergravity in AdS5 ×S5 and

λ<< 1 for gauge perturbation theory , therefore we conclude that AdS/CFT is a duality.

Conclusion:

At large N gauge theory is strongly coupled but the dual gravity theory is weakly

coupled which is easy to study. It has been a long standing problem to study strongly

coupled gauge theories. If understood, newer and better insights into phenomena such

as quark confinement would become possible. The usual approach through pertrubation

theory fails here precisely because the coupling constants are large and higher order

Feynman diagrams contribute more and more. The strong-weak duality proposed by the

AdS/CFT correspondence offers a tractable way to study phenomena that are described

by strongly coupled gauge theories including quark confinement and chiral symmetry

breaking [12]. Using the correspondence, we are also able to see how an analysis based

on supergravity leads to insights about quark confinement/deconfinement transitions,

zeeman splitting and level crossing in the dual gauge theory, thus demonstrating the

power of the AdS/CFT conjecture.

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4.4 The Large λ Limit

Once the ’t Hooft limit is set then the only parameter left is λ. So, it is exceptionally

enlightening to establish the meaning of an expansion around large λ. From prior

knowledge we know that for large λ, gravity picture dominates. Let us recall the string

action (Polyakov action) as we discussed in Chapter 1:

SP =−T0

2

∫d2σ

p−g gαβ∂αXµ∂βXνGµν. (4.8)

Here T0 is the string’s tension, Gµν is the spacetime metric which is AdS5×S5 in our case

defined in Eq. 4.1 and gαβ is the worldsheet metric. Experimentally, we don’t measure

T0 but slope parameter; Regge slope α′ which is related to string’s tension as:

α′ = 12πT0~c

. (4.9)

In natural units, string’s tension becomes:

T0 = 12πα′ (4.10)

= 12πl2

s, (4.11)

where ls is the length of the string. Hence, Eq. 4.8 becomes in AdS5 ×S5 metric form:

SP =− R2

4πα′

∫d2σ

p−g gαβ∂αXµ∂βXνGµν, (4.12)

where R is the radius of the curvature of the AdS space.

For correspondence, let us apply low energy limit as:

E → 0 (4.13)

⇒ ls → 0 (4.14)

⇒α′ → 0 (4.15)

After inserting the value of R from Eq. 4.2, the overall coupling constant becomes:

R2

4πα′ =√

gsN4π

(4.16)

R2

4πα′ =√

λ

4π. (4.17)

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Now, Eq. 4.12 becomes:

SP =−√

λ

4π

∫d2σ

p−g gαβ∂αXµ∂βXνGµν, (4.18)

It is clear from Eq. 4.17, that role of α′ in string theory has been exchanged by 1/pλ [13].

4.5 Summary

D3-branes in 10 dimensional spacetime:

N = 4 SYM ⇔ Type IIB SUGRA in AdS5 ×S5 space

From Symmetry Group:

N = 4 has SO(6) group ⇔ Isometry of S5

For all N:

g2Y M = gs ⇔ R4 = 4πgsNα′2

’t Hooft Limit:

Strongly coupled ⇔ Weakly coupled

Large λ Limit:1pλ

expansion ⇔ α′ expansion

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CH

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5ADS/CFT CORRELATION FUNCTIONS AND

HOLOGRAPHY

This chapter is concerned to the studies of correlation functions in the AdS/CFT

correspondence. Correlation functions correspond to physical observables and

govern the dynamics of the theory. In the perspective of duality, their importance

is vital as one can brainstorm about the dynamics of strongly coupled gauge theories by

performing calculations on the gravity side. We will also see the mapping between the

correlation functions of local operators in CFT to the fields in gravity theory.

5.1 Correlation Functions

I would like to add a digression here; the operator O in CFT is dual to some field Φ in bulk

theory and the boundary value of gravity field behaves as a source φ for a CFT operator

O . Let us work in Euclidean space for the sake of simplicity. The correlation functions

are defined by the Euclidean path integrals. The generating functional of gravity theory

must be the same to the generating functional of gauge theory. Otherwise, then there

would be no equivalence. Mathematically, it is summed by the following statement of the

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CHAPTER 5. ADS/CFT CORRELATION FUNCTIONS AND HOLOGRAPHY

AdS/CFT correspondence:

Zgravity[Φ|∂AdS =φ0]= ZCFT[φ], (5.1)

Zgravity[Φ|∂AdS =φ0]=⟨

e∫

dd xφi0(x)O i(x)

⟩. (5.2)

Here φ0 is the gravity field at AdS boundary and the index i represents the sum over all

the gravity fields in gravity theory, and corresponding all local operators in gauge theory.

In CFT, correlation functions are calculated from generating functionals as:

⟨O1(x1),O2(x2)...On(xn)⟩ = δn

δφ10(x1)δφ2

0(x2)...δφn0 (xn)

ZCFT[φ0]∣∣∣∣φi

0=0. (5.3)

5.1.1 Correlation Functions in CFT

The CFT operator or gauge operator O obeys following conformal algebra:

[D,O ]=−ι∆O (5.4)

Here ∆ represents the eigenvalue of dilatation operator and is called the scaling dimen-

sion of O . Conformal generators can built complete representation by acting on O (x). Eq.

5.4 says that we have O (x)→ a∆O (ax) under this rescaling x → ax.

Generally, correlation function for n primaries is given as:

⟨O1(ax1),O2(ax2)...On(axn)⟩ = a−∆1−∆2−...−∆n⟨O1(x1),O2(x2)...On(xn)⟩. (5.5)

5.1.1.1 Two-point Functions

The two-point function (propagator) is defined as:

G(x1, x2)= ⟨φ1(x1)φ2(x2)⟩ (5.6)

This function satisfies following differential equation:

[z(x1)∂1 +∆1∂z(x1)+ z(x2)∂2 +∆2∂z(x2)]G(x1, x2)= 0, (5.7)

where z is some parameter of the function which is the solution of Cauchy-Riemann

equations as we describes in Chapter 3. Interested reader may find the complete working

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of the solution of this equation in [14]. Thus, the complete solution is given by:

G(x1, x2)= B12

|x1 − x2|−2∆ , ∆=∆1 =∆2. (5.8)

Here B12 is just a coefficient and has no physical meaning, it can be set to 1 by normal-

ization of primary fields.

5.1.1.2 Three-point Functions

Just like two-point function, the three-point function allowed by conformal invariance is

given by:

⟨O1(x1)O2(x2)O3(x3)⟩ = B123

|x1 − x2|∆1+∆2−∆3 |x2 − x3|∆2+∆3−∆1 |x3 − x1|∆3+∆1−∆2. (5.9)

Here B123 is called operator product expansion (OPE). Since we have already fixed the

normalizations in the two-point function, therefore B123 has a real physical prediction of

the theory.

One can get all the data of a CFT from the scaling dimensions ∆i and B123 coefficients

of three-point functions. Thus, higher correlators can be determined if one know the

three-point function in just three points.

5.1.1.3 Four-point Functions

The four-point function is difficult to calculate because it is not completely conformally

invariant but under equal external weights ∆1 =∆2 =∆3 =∆4 =∆, which implies that it

is highly constrained function. The generic form is given as:

⟨O1(x1)O2(x2)O3(x3)O3(x4)⟩ = f (m,n)|x1 − x2|−2∆|x3 − x4|−2∆. (5.10)

Here f (m,n) is a function of

m = (x1 − x2)2(x3 − x4)2

(x1 − x3)2(x2 − x4)2 , n = (x1 − x4)2(x2 − x3)2

(x1 − x3)2(x2 − x4)2 , (5.11)

which is the only independent invariant function under the conformal group.

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5.1.2 Correlators Functions in Gravity Theory

The generating functional of gravity theory must be the same to the generating functional

of gauge theory by AdS/CFT dictionary as depicted in Eq. 5.1:

Zgravity[Φ|∂AdS =φ0]= ZCFT[φ]. (5.12)

So, we’ll calculate the correlation functions from AdS side and see whether there are

same to CFT’s or not.

Generally, it is very difficult to calculate Zgravity because we do not know the full

quantum gravity but in the semi-classical limit: gs → 0 and α′ → 0. The value of gen-

erating functional depends on the boundary conditions of the fields that defined on

the boundary of AdS, so we need to identify its boundary conditions. After all these

approximations we have

Zgravity[Φ|∂AdS =φ0]∼ e−N2SC[φ] × (Quantum Corrections). (5.13)

Here SC represents the on-shell classical gravity action which is defined as for massive

scaler fields as:

SC = N2∫

dx4dzz5

z2(∂φ)2 +m2R2φ2 +φ3 + ...

(5.14)

The boundary condition for the fields as we approach the AdS boundary z → 0

φ= z−∆+nφ0(x), (5.15)

where n is the number of dimensions in AdS.

Gauge correlators pop-out while computing the classical action in AdS as a functional

of the boundary conditions. I will not do the whole calculations here. Interested reader

may find the proper workings in Witten’s orginal paper [15].

5.1.3 Summary

Correlation function in gauge theory turns out to the same as the correlation functions

in bulk theory within the semi-classical limits.

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5.2 Holographic Principle

A quantum gravity system in (d+1)-dimensional spacetime can be fully described by

the degrees of freedom living on the d-dimensional boundary of this spacetime. This is

the holographic principle. In other words, a gravity theory in a bulk is describable by

a boundary theory. If the bulk is AdS then the holographic correspondence would be

AdS/CFT correspondence; tells us how much the detailed information in the bulk can be

stored completely at the boundary of a gauge theory. The precise relation between both

theories is given by [15].

From Eqs. 5.1 and 5.13, we get:

Zgravity[Φ|∂AdS]= e−N2SC[φ] × (Quantum Corrections)= ZCFT[φ]=⟨

e∫

dd xφi0(x)O i(x)

⟩.

(5.16)

This implies that the AdS/CFT correspondence is a realization of the holographic princi-

ple.

Let us take this into an alternative way. By holographic principle, one can extract

every bit of information of the bulk by the dual CFT which is not possible by any local

field theory. It contains an area’s worth of degrees of freedom, avoiding the redundancy

of a local description. Quantitatively, one can check holographic principle by computing

the CFT’s degrees of freedom N which does not exceed the boundary area A. To do this,

consider Bekenstein-Hawking’s entropy formula which states that entropy of a black

hole is proportional to event horizon’s area A.

SBH = A4GN

. (5.17)

In CFT, number of degrees of freedom is proportional to the square of gauge fields

described by U(N) gauge theory as we learned from Chapter 2.

Ndof ≈N2

∈3 , (5.18)

where ∈ represents the parameter of the spacetime slicing. Hence, Eq. 5.17 becomes:

SBH ≈ N2

∈3 . (5.19)

70


In AdS, the area of the bulk can be calculated from Einstein-Hilbert action as∫ Rd3x

pg ≈ L3

∈3 , (dt = 0,dz = 0) (5.20)

According to this, Eq. 5.17 becomes:

SBH ≈ L3

4GN ∈3 . (5.21)

From Eqs. 5.19 and 5.21, we get

N2 = L3

4GN. (5.22)

Area of the boundary=Volume of the space. (5.23)

5.3 Conclusion

The strong-weak duality proposed by the AdS/CFT correspondence offers a tractable way

to study strongly coupled gauge theories just by studying weakly coupled bulk theory

which is easy to understand within semi-classical limits, thus demonstrating the power

of the AdS/CFT conjecture. Using the correspondence, we were able to study correlation

functions of these two theories which turn out to be the same through which we studied

holographic principle and found that AdS/CFT correspondence is a realization of the

holographic principle.

71

BIBLIOGRAPHY

[1] J.W. van Holten, arXiv:hep-th/0201124.

[2] B. Zwiebach, “A First Course in String Theory", Cambridge University Press, Cam-

bridge, 2004.

[3] Polchinski, J. G. “String theory", Cambridge University Press, Cambridge, 2007.

[4] David Tong, arXiv:hep-th/0908.0333.

[5] Katrin Becker, Melanie Becker and John H. Schwarz, “String Theory and M-Theory

A Modern Introduction", Cambridge University Press, Cambridge, 2006.

[6] C.V. Johnson, “D-Branes", Cambridge University Press, Cambridge, 2002.

[7] Jens L. Petersen, arXiv:hep-th/9902131.

[8] Ofer Aharony, Steven S. Gubser, Juan Maldacena, Hirosi Ooguri, Yaron Oz, “Large

N field theories, string theory and gravity", Physics Reports, Volume 323, Issues

3-4, 2000.

[9] R. Blumenhagen and E. Plauschinn, “Introduction to Conformal Field Theory",

Springer-Verlag, Berlin Heidelberg, 2009.

[10] Elias Kiritsis, “String Theory in a Nutshell", Princeton University Press, New Jersey,

2007.

[11] Horatiu Nastase, “Introduction to the AdS/CFT Correspondence", Cambridge: Cam-

bridge University Press, 2015.

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BIBLIOGRAPHY

[12] V. G. Filev and R. C. Raskov, “Magnetic Catalysis of Chiral Symmetry Breaking: A

Holographic Prospective", Adv. High Energy Phys. 2010.

[13] Eric D’Hoker, and Daniel Z. Freedman, “Supersymmetric Gauge Theories and the

AdS/CFT Correspondence", TASI 2001 Lecture Notes.

[14] A. N. Schellekens, “Introduction to Conformal Field Theory",

Fortschritte Der Physik/Progress of Physics, 44(8), 605-705.

https://doi.org/10.1002/prop.2190440802

[15] Edward Witten, “Anti de Sitter Space and Holography", Advances in Theoretical

and Mathematical Physics 2(2): 253-291, 1998.

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d-branes and ads/cft...d-branes and ads/cft by junaid saif khan department of physics lahore...

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