STUDIES ON THE ADSICFT CORRESPONDENCE
Wolfgang Muck
Diplom-Physiker
Martin-Luther-Universitat Halle-Wittenberg, Germany, 1 995
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@ Wolfgang Muck 1999 SIMON FRASER UNIVERSITY
November 1999
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Abstract
This thesis summarizes original research on the topic of the "AdS/CFT correspondence."
This correspondence, which was first conjectured by Maldacena [Adx Theor. Math. Phys.
2,231 (1998)j and subsequently formulated by Gubser, Klebanov and Polyakov [Phys. Lett.
B 428, 105 (1998)l and by Witten [Adv. Theor. Mrrtlz. Phys. 2,253 (1998)j, relates field theories
on ( t i + I)-dimensional anti-de Sitter (Ads) spaces and conformal field theories (CFk) in d
dimensions with each other. Its main prediction is that the correlation functions of certain
quantum CFTs are determined by the dynamics of classical field theories on Ads spaces.
Starting from a correspondence formula provided by the authors above, several CFT
correlation functions are calculated and agreement with the forms dictated by conformal
invariance is found. The necessary renormalization is carried out in the "E-prescription."
Details of renormalization and the breaking of conformal symmetries in special cases
are investigated by means of the example of the scalar field. The "asymptotic prescrip-
tion" is used to prove a suggestion by Klebanov and Witten [hep- t h / 9 90 5 1041 about the
treatment of irregular boundary conditions valid to all orders of perturbation theory.
The treatment of Ads gravity, which enables the calculation of correlation functions of
CFT energy momentum tensors, is carried out in the time slicing formalism. The calcu-
lated two-point functions and Weyl anomalies agree with results known from pure CFI'
considerations.
The Wess Zumino model on Ads4 is discussed as an example containing supersymme-
try. It is shown that the model yields the correlation functions of conformal fields belonging
to a d = 3, N = 1 superconformal multiplet.
To Paola, Z U ~ O can read the text of this thesis,
but probably will not zmierstand much of its meaning;
to Max, who zuozild have understood the meaning,
but cannot read it anymore.
Acknowledgments
I would like to express my deepest gratitude to the following people, who have helped and
accompanied me during my time as a Ph-D. student at Simon Fraser University:
First, I would not be in this position today, had it not been for the immense love and
care I received from my wife, Paola Travascio. May God bless her.
Secondly, I am indebted to my senior supervisor, Dr. K. S. Viswanathan, for making me
interested in the subject of my research. My progress was always boosted by his relevant
scientific advice, whereas his patience and sincere cordiality made working under his hand
a pleasure.
Thirdly, my thanks go to the members of my supervisory committee, professors A. Das,
A. Zhi tni tsky and H. Trottier, for their stimulating discussions.
Last, but not least, my life in Canada would have been hardly pleasant without the
many nice people I have found here: My office pals and friends Andrew DeBenedictis,
Eldon Emberly, Jiirgen Wendland (who let me write this thesis on his computer), and I
had a good time transforming a nearly windowless hole into the nicest student office of
the department. I had many cheerful conversations about important and not so important
(such as physics) parts of life with my fellow students Kam Kallio, Rick Drociuk, Dan
Vernon and Peter Matlock (to name only a few) as well as with the friendly faculty and
staff of SFU's physics department.
Finally, Paola and I vastly enjoyed spending time with our good friends Ted and Nicole,
James G. and Sarah, James S. and Paula, Radek and Mara, Cristina and Jonas, Madhu and
Mischa, Vito and Heather, and (finally one not named in a couple) Immo. Thank you all!
Contents
Approval ii
... Abstract 111
Dedication iv
Acknowledgments v
Contents vi
1 Introduction 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 A Short Guide for the Reader 1
1.2 ~Maldacena's Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
. . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Correspondence Formula 5
2 First Steps 8
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 The Geometry of Adsd+, 9
2.1.1 Construction and Coordinate System . . . . . . . . . . . . . . . . . . 9
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Thesymmetry Group 11
. . . . . . . . . . . . . . . 2.2 Basics of Conformal Field Theory in d Dimensions 14
. . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Conformal Symmetry 14
. . . . . . . . . . . . . . . 2.2.2 Representations of the Conformal Algebra 15
. . . . . . . . . . . . . . . . . . 2.2.3 Restrictions on Correlation Functions 16
2.2.4 Properties of the Energy Momentum Tensor . . . . . . . . . . . . . . 18
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Motivation for a Duality 21
CONTENTS vii
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 An AdS/CFr Dictionary 22
3 Exploring the Correspondence -- The Scalar Field 24
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 The~.Prescription 25
. . . . . . . . . . . . . . . . . 3.1.1 Review of the Green's Function Method 26
. . . . . . . . . . . . . . . . . . . . . 3.1.2 The Free Scalar Field on Adsd, 27
3.1.3 TheSpecialCasesa=~z . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 The Asymptotic Prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
3.2.1 Regular Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . 34
. . . . . . . . . . . . . . . . . . . . . . 3.2.2 Irregular Boundary Conditions 36
. . . . . . . . . . . . . . 3.3 Ads Interactions and Higher-Point CFT Correlators 39
3.3.1 General Method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
. . . . . . . . . . . . . . . 3.3.2 First Order tnteractions of the Scalar Field 41
4 CorreIation Functions for Spinor and Vector Fields 43
4 The Free Massive Vector Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
. . . . . . . . . . . . . . . . . . . . . . 4.1.1 Solution of the Field Equations 34
4.1.2 CFT Two-Point Function for Spin-1 Fields . . . . . . . . . . . . . . . . 47
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Free Dirac Field 28
4.2.1 Solution of the Dirac Equation . . . . . . . . . . . . . . . . . . . . . . 48
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 TheCasenr > 0 51
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.3 TheCasenr<O 53
4.2.4 TheCasem=O . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
42.5 CFT Two-Point Function for Spinors . . . . . . . . . . . . . . . . . . . 54
4.3 First Order Interactions Between Spinor and Gauge Fields . . . .. . . . . . 56
5 Ads Gravitons and CFT Energy Momentum Tensor 60
5.1 Solution of the Linearized Einstein Equation . . . . . . . . . . . . . . . . . . 61
5.2 CFT Two-Point Function of Energy Momentum Tensors . . . . . . . . . . . . 64
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 GeneralFormalism 64
5.2.2 Cancellation of Non-Local Divergent Terms for d > 2 . . . . . . . . . 65
5.2.3 Two-Point Function for Odd d . . . . . . . . . . . . . . . . . . . . . . 66
. . . . . . . . . . . . . . . . . . . . 5.2.4 Two-Point Function for Even d > 2 67
... CONTENTS vlu
. . . . . . . . . . . . . . . . . . . . . . . 5.2.5 Two-Point Function for d = 2 68
. . . . . . . . . . . . . . . . . 5.2.6 Dimensional Regularization for Even d 69
. . . . . . . . . . . . . . . . . . . . . . 5.3 Counter Terms and the Weyl Anomaly 70
5.3.1 The Weyl Anomaly in Conformal Field Theories . . . . . . . . . . . . 70
5.3.2 Scale Invariance and its Breaking by Non-Covariant Counter Terms 71
5.3.3 General Formalism for the Calculation of Counter Terms . . . . . . . 73
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 d = 2 74
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 d = 4 75
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.6 d = 6 76
6 The Wess-Zumino Model on Ads4 78
. . . . . . . . . . . . . . . . . . . . . . . 6.1 The N = 1 Supersymmetry Algebra 79
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Superconformal Algebra 80
. . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Construction of Ads Superspace 81
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 The Scalar Superfield 85
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 The Wess-Zumino Model 87
7 Conclusions and Outlook 90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Summary 90
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Future Work 92
Appendices 93
A Fields on Curved Manifolds 93
A.1 Covariant Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
A.2 SymmehyTransformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
B Various Integrals 96
B. l Fourier Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
. . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2 Feynman Parameterization 98
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3 Other Integrals 98
C Time Slicing Formalism 100
CONTENTS
D Spinor Grassmannian Variables 103 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.1 Basics 103
. . . . . . . . . . . . . . . . . . . . . . . . . D.2 Calculation of the Killing Spinor 104
Bibliography 107
Chapter 1
Introduction
1.1 A Short Guide for the Reader
Writing this thesis certainly was a difficult task, but it is appreciated by the author that
reading it will not be an easy one, either. In order to help the reader to find his or her
way through the many formulae, a brief orientation shall be given in this first section as to
which material is in the thesis and how to read it.
This thesis summarizes original research carried out by the author in collaboration with
his supervisor, Dr. K. S. Viswanathan, on the topic of the "AdS/CFT correspondence". This
is a new and very exciting topic - barely two years old - and it is only natural that a
whole army of theoretical high energy physicists are working on it. Therefore, the material
covered by recent research is already quite extensive, and it would be a formidable task
to review all of it in this thesis. However, the reader will be referred to relevant related
literature in the course of the text- Moreover, the lectures and reviews [145,45,133,44,150,
I I give an excellent sununary of many aspects of the AdS/CR correspondence which are
not covered in this thesis such as a more detailed discussion of the string theory context.
As a historical introduction, Maldacena's conjecture is summarized in section 2.2. Then,
section 7.3 introduces the correspondence formula, upon which most of the thesis material
is based. Moreover, its interpretation and expected results are discussed in general terms.
Chapter 2 provides some background information about Anti-de Sitter space and con-
formal field theories and motivates the correspondence by finding a relation between Ads
fields and conformal fields living on the Ads horizon. The reader who is somewhat famil-
CHAPTER I . IIWRODUCTION
iar with the mechanism of the AdS/Cm correspondence can certainly skip chapter 2 and
use it as a reference in the remaining chapters, which contain the bulk part of the author's
research.
Chapter 3 explores the AdS/CFT correspondence using the example of the scalar field.
Thus, the scalar field serves as a testing ground for the AdS/CFI' correspondence for-
mula. Although being the simplest example, important details such as the assignment
of boundary values, regularization and renormalization of the effective action and the ap-
pearance of non-covariant counter terms shall be discussed. These details illustrate the
general technique used in all cases throughout this thesis- Moreover, two different regu-
larization schemes shall be explored. Hence, chapter 3 is essential for the understanding of
the details of the mechanism of the AdS/CFT correspondence.
Chapter 4 contains calculations for spinor and vector fields. Except some interesting
new details in the spinor case, the methods explained in chapter 3 are used. Thus, chapter 4
can be skipped by the reader not interested in these cases.
The reading of chapter 5 is recommended to every reader. It discusses the gravitational
field on Ads space, which is the most important case, since its boundary value couples to
the CFT energy momentum tensor. The two-point function of CFT energy momentum ten-
sors shall be derived in detail, as well as the counter terms in the renormalization scheme.
If the number d of dimensions is even, non-covariant counter terms occur and break the
conformal invariance, which gives rise to the well-known Weyl anomaly.
Chapter 6 deals with the simplest case involving supersymmetry: the Wess-Zumino
model on Ads4. The N = 1 supersymmetric model is constructed using a superspace
formalism. Thus, chapter 6 is mostly of pedagogical value. The author learned much
about calculating with supersymmetry, and the reader unaccustomed to it might as well.
Then, there are the appendices.
Appendix A, besides being a reference for notations, is aimed at those readers who feel
uneasy with the conventions used in fieid theories on curved manifolds.
Appendix B provides a list of integals used in chapters 3,4 and 5 and thus is merely a
reference section.
Appendix C reviews the time slicing formalism, also called initial value formulation
of gravity, which is used in chapter 5. It is recommended that the non-expert in general
relativity read this appendix before proceeding with chapter 5.
Finally, appendix D reviews the basic concepts about Grassmannian spinor variables
and contains technical calculations for chapter 6. It is needed only in connection with
chapter 6.
Some general remarks about style are due at this point. The main part of the thesis,
chapters 3-6, is reasonably self contained, although the calculations are not always pre-
sented in detail. Nevertheless, the reader should be able to follow the general idea and
either tntst the presented results or rederive them along the lines given in the text. Con-
sidering the length of some calculations, this way of presenting them seemed the only one
reasonable. Mathematical symbols are mostly explained in the text, and the meaning of the
remaining ones (like J g in an integral) should be unambiguous horn their context.
The author wishes happy and successful reading.
1.2 Maldacena's Conjecture
The starting point of some exciting developments over the past two years, summarized
under the phrase "AdS/CE.T correspondence," was a conjecture by Maldacena about the
duality of classical type LIB supergravity on Adsj x S' and N = 4, U ( N ) super Yang-Mills
theory in the large N Limit. As a more historical introduction to the subject, this section shall
summarize the initial developments leading to the AdS/CFT correspondence formula.
Consider the extremal black 3-brane solution of type IIB supergravity [94] (cf. also the
reviews [47,1] and the book [136]), namely
where df2: is the angular metric of the unit five-sphere s5, XI, denotes the world volume
coordinates of the 3-brane, and the p-branes are located at r = 0. The function f (r) is given
by
Moreover, the characteristic length 1 is related to the number N of coincident 3-branes by
the equation [81,101]
CHAPTER I . LNTRODUCTlON 4
where K is the gravitational constant of 10-dimensional supergravity [I&, 144,96,100,86,
871.
The interest in the supergravity solution (1.1) containing 3-branes arose from work by
Gubser, Klebanov and Tseytlin [81, 83, 80, 1011, who calculated the absorption cross sec-
tions for various particles falling into non-dilatonic p-branes. By comparison with the clas-
sical absorption cross section for the corresponding gravitational background they found
that the results agreed exactly for absorption by 3-branes, whereas there was only a scal-
ing agreement for 2- and 5-branes. This result led them to conclude that scattering from
extremal 3-branes is unitary and well described by perturbative string theory Moreover,
in supergra~ity, the extremal 3-brane solution is the only RR-charged solution which is
non-singular [MI.
Since the supergravity theory in question is the low energy limit of type IIB superstring
theory [113,78,79], the gravitational constant K is given in terms of the string coupling g,t,
and the Regge slope d and reads (1361
Combining equations (1.3) and (1.4) yields
On the other hand, a stack of N coincident D3-branes in type IXB superstring theory has
a decoupled .U = 4, U ( N ) Yang-Mills theory associated with it, whose coupling constant
is related to the string coupling by [18]
Therefore, equation (1.5) becomes
where one can identify the 't Hooft coupling, &,N [149]. As discussed in 11361, one can
trust the classical solution (1.1), if the characteristic length I is much larger than the string
length I, = fi [143], because then the geometry appears smooth on the string scale. In
other words, the 't Hooft coupling should be large.
CHAPTER 1. LNTRODUC77ON 5
Maldacena [I141 took this limit even further. With a glimpse on equations (1.2) and (1.7)
he demanded the 't Hooft coupling to be large compared with the value of r measured in
units of the string length. Then the constant 1 in equation
respect to the r dependent term, and the metric (1.1) becomes
Then, defining xo = 12/r, the metric (1.8) takes the form
(1.2) can be neglected with
The form (1.9) of the metric reveals that the near horizon geometry of the extremal su-
pergravity solution (1.1) is Ads5 x s5 (cf. section 2.1.1 for a construction of Anti-de Sitter
space). The length scale 1 plays the role of both, the radius of the five-sphere s5 and the
"radius" of Adss. Moreover, it shows that the metric (1.1) is not singular for r + 0. There
is no kink, but a sphere S5 of finite radius 1.
Building upon these facts, Maldacena made the much cited conjecture that in the large
N limit classical type IIB supergravity on Adss x S' (with I given by equation (1.7)) is dual
to fl = 4, d = 3 + 1 super Yang-Mills theory with gauge group U ( N ) . This new form
of duality has some exciting features: First, it is non-perturbative in the string coupling
g,!,, and thus also in the Yang-Mills coupling g y ~ . Secondly, it is a strong-weak coupling
duality in the sense that, when the 't Hooft coupling &,N becomes large, perturbative
super Yang-MilIs calculations break down, but just in this limit can the classical type IIB
supergravity solution be trusted. Lastly, it is a classicalquantum duality, because classi-
cal supergravity is conjectured to be dual to a quantum gauge theory. In fact, quantum
supergravity effects are suppressed by powers of 1/ N [114].
1.3 The Correspondence Formula
It did not take long until an explicit formulation of Maldacena's conjecture was found.
Gubser, Klebanov and Polyakov (821 and Witten [I601 independentiy proposed to identify
the classical on-shell supergravity action, expressed in terms of given boundary values,
with the effective action of super Yang-Mills theory, where the supergravity boundary val-
CHAPTER I . M R O D U C n O N
ues play the roles of generating currents. Moreover, Witten suggested that via this iden-
tification m y field theory action on (d + l)-dimensional anti-de Sitter space gives rise to
an effective action of a field theory on the d-dimensional horizon of Anti-de Sitter space.
Most importantly, this field theory on the horizon must be a conformal field theory, be-
cause the Ads symmetries act as conformal symmetries on the Ads horizon (cf. chapter 2).
This duality has since been called the AdS/CFT comespondmce.
The general correspondence formula is [I601
where the functional integral on the left hand side is over all fields Y whose asymptotic
boundary values are Yo,' and O denotes the conformal operators of the boundary confor-
mal field theory.
In the classical limit, which will be considered exclusively throughout this thesis, the
functional integral on the left hand side of equation (1.10) becomes redundant, and the
correspondence formula can be given in the simple form (82,1601
where Ihds is the classical on-shell action of an Ads field theory, expressed in terms of the
field boundary values Yo, and WCFT is the CFT effective action with generating currents,
given by minus the logarithm of the right hand side of equation (1.10). However, one must
expect IAdS to be divergent as it stands, because of the divergence of the Ads metric (1.9) on
the Ads horizon, xo = 0. Thus, in order to extract the physically relevant information, the
on-shell action has to be renormalized by adding counter terms, which cancel the infinities.
After defining the renormalized, finite action by
where idiv stands for the local counter terms, one identifies IAdsan with the C f l effective
action. Thus, the meaningful correspondence formula is
'The exact meaning of this statement shall be clarified in chapter 3.
CHAPTER I . h'VTRODUCTION
Given a field theory action on Ads space and a suitable regularization method,' it is
straightforward to calculate the renormalized on-shell action lAdS,fin. On the other hand,
the CFT effective action
WCFT[yO] = - ln exp d d x c ? ~ ~ ( 1 ) contains all information about the conformal field theory living on the Ads horizon, in that
all correlation functions of its operators O can be obtained in a standard fashion. Thus, the
AdS/CFT correspondence formula (1.13) provides for the most amazing fact that the prop-
erties of certain conformal field theories can be obtained from seemingly unrelated theo-
ries, namely field theories on Ads spaces. Moreover, any field theory on Ads space, which
includes gravity,3 has a corresponding counter part CFT, whose action might not even be
known. Thus, the AdS/CFT correspondence might be an invaluable tool for formulating
non-trivial CFTs in various dimensions, although studies of this aspect have not proceeded
very far yet.
As a conclusion of this section, two basic, but important aspects related to the corre-
spondence fomula shall be discussed briefly- First, why can one be sure that WCFT is the
generating functional of a conforrrral field theory? As already mentioned, the Ads symme-
tries act as conformal symmetries on the Ads horizon. Thus, by virtue of the invariance of
the Ads action IAdS under Ads symmetries, WCFT is invariant under conformal transfor-
mations, as long as the counter terms do not break these symmetries. This case, which is
the generic one, is given when all counter terms are cmnrialrf. Hence, the CFT correlation
functions will obey all restrictions imposed by conformal invariance (cf. section 2.2.3). This
will be explicitly shown in the calculations in chapters 3 and 4.
The famous exception appears when Idiv contains at least one non-covariant term,
which inevitably breaks some of the Ads symmetries. Therefore, also the conformal sym-
metry of the C R effective action will be broken. The interesting and encouraging fact
about this is that the breaking of conformal invariance is a strong signature of the quantum
character of the CFT and tells about the anomalies in the algebra of quantum conformal
operators. The most notable example is the Weyl anomaly, which is discussed in chapter 5,
section 5.3.
'Two possible regularization methods will fx discussed in detail in chapter 3. 'The boundary values of gravitons couple to the energy momentum tensor, which is part of every CFT.
Chapter 2
First Steps
The AdS/CFT correspondence relates field theories on (d t 1 )-dimensional Ads spaces
and d-dimensional CFTs with each other. The aim of this chapter is to provide some back-
ground for the reader about both sides of this correspondence, namely Anti-de Sitter space
and conformal field theory, and give a motivation for the correspondence.
Information on the geometrical properties of Ads spaces can be found in most ad-
vanced text books on General Relativity, and Weinberg's book [I571 has been found most
suitable in the present context. Field theories on Ads spaces have been considered first
by Fransdal et al. [66,67, 70, 68,9] and by Avis, Isham and Storey (171. A very important
development was the systematic calculation of various propagators by Burgess, Liitken,
Allen, Jacobsen and Turyn [29,3,4,5,151]- Supersymmetric extensions of Ads spaces and
supersymrnetric models have been considered in [99,161,97,152,69,26,135,28,88].
On the other hand, conformal field theories in dimensions d > 2 have enjoyed a grow-
ing attention after the great successes of the case d = 2. Earlier studies can be found in
[113,56], and more recent relevant references are [131,62,51,63]-
The review of Ads space, which is presented in section 2.1, focuses on two points.
First, an Ads space is explicitly constructed. This provides the necessary expressions for
later calculations. Secondly, the Ads symmetries are found, and the symmetry algebra is
represented in a form which reveals its isomorphism to the conformal algebra.
The review of the basics of CFTs in section 2.2 first recalls the definition of conformal
transformations and the explicit expressions for conformal transformations of Euclidean
space. Secondly, the expressions for the symmetry algebra operators acting on quasi-
CHAPTER 2. FIRST STEPS 9
primary conformal fields are given. Quasi-primary conformal fields are important, because
they form the basic field content of any ClT. Conformal invariance severely restricts the
form of correlation functions in CFTs. Some simple, but important examples are reviewed
in section 2.2.3. This mainly serves the purpose of providing expressions with which the
results of later calculations can be compared. Finally, some basic properties of the energy
momentum tensor and expected Ward identities are given in section 2.2.4.
Section 2.3 discusses a simple analytical relation between classical Ads fields and quasi-
primary conformal fields living on the Ads horizon, whereas in section 2.4 group theoreti-
cal arguments provide an AdS/CFT "dictionary," which relates the masses of various Ads
fields with the conformal dimensions of their corresponding boundary CFT operators.
I t should be pointed out that the notations and conventions used throughout this chap-
ter are mostly standard. Nevertheless, it is recommended that the reader first consult ap-
pendix A in order to (again) become familiar with the basic concepts of fields on general
curved manifolds.
2.1 The Geometry of AdSd+l
2.1.1 Construction and Coordinate System
As is well known [157j, an Ads space is a maximally symmetric space and can thus be
represented as a hyperboloid embedded into a higher dimensional Minkowski space. The
following considerations apply to Ads spaces with Euclidean signature, but many results,
notably those regarding the symmetry algebra, can be straightforwardly camed over to
the general case. Let the dimensions of the Ads and the embedding Minkowski spaces be
d i 1 and ii + 2, respectively, and let the embedding be defined by
where 1 is the "radius" of the hyperboloid, and A, B = - 1.0,. . . , d. The Minkowski metric
tensor is given by
'1-1-1 = -1, qpv = bPY and q - l P = 0
(with p = 0,1,. . . , d). The metric
CHA M E R 2. F B T STEPS 10
readily represents the AdS metric, if one takes the coordinates yp as Ads coordinates and
defines y-' via equation (2.1).
It is useful to introduce a new set of Ads coordinates by
The domain of the new variables is given by 0 < xo < x, x*
the Ads metric (2.3) takes the form
(2-4)
E IR (i = 1,. . . , d) . Moreover,
Obviously, Adsdrl is an open space, i-e. it does not possess a boundary. However, it is
useful to consider the boundary of the coordinate patch, namely the conformally compact-
ified Euclidean space given by xo = 0 plus the single point xo = x, as a pseudo boundary,
which will be called the Ads horizon.
In the sequel the components xi shall be denoted coIlectiveIy by the vector x and the
abbreviations
will be used. Moreover, no distinction shall be made between upper and lower indices
when refemng to the coordinates xp, i-e. xp = x,.
For completeness the expressions for the affine connections, the curvature tensor, Ricci
tensor and curvature scalar are provided here:
Moreover, using the tetrads
the spin connections (A.3) are found to be
CHAPTER 2. FIRST STEPS 11
2.1.2 The Symmetry Group
The definition (2.1) of anti-de Sitter space is invariant under transformations of the embed-
ding Minkowski space of the form (y') A = Ri\g where the (d + 2) x (d + 2) matrix R satisfies RrqR = and R-ll > 0. The group of such matrices consists of two subsets, one
being the Lie group SO (d + 1, I ) , whereas the other can be represented by 1 x S O ( d + 1,l)
with an inversion 1.
First, the transformations R E S O ( d + 1 , l ) shall be discussed. Using an infinitesimal
coordinate transformation, 72 = 1 + M, one has
where the generators ( ~ ~ ~ 1 % = $ 1 7 ~ ~ - 6CqCB form the standard basis of the so(d + 1,l)
algebra and satisfy the commutation relations
One can introduce a corlfornml basis of so(d + 1,l) by defining
Hence, the algebra element M of equation (2.22) takes the form
where the new parameters are given by
Then, one can show using equations (2.4) and (2.12) that the infinitesimal change of the
coordinates x' under a symmetry transformation is given by
It is easily checked that dxp is a Killing vector.
CHAPTER 2. FIRST STEPS 12
Moreover, it is found from equations (2.13) and (2.14) that the elements D, Pi, Ki and Lii
of the conformal basis of so(d + I, 1) satisfy the commutation relations
Equations (2.18) are the standard commutation relations of the conformal algebra, hence
the name confornral bnsis. Thus, this explicitly demonstrates the isomorphism of the confor-
mal algebra and so(d + 1, l ) .
Turning to the second set of symmetries, one encounters the inversion i, whose action
on the coordinates yl' of the embedding Minkowski space can be defined by the matrix
Obviously,
ZZ= 1, (2.20)
which must hold in every representation. Using equation (2.4), the transformation induced
on the . rp coordinates is found to be
Following the general argument given in appendix A.2,
be accompanied by a transformation of the local Lorenh
equation (A.9) and reads
the coordinate inversion has to
frame, which is obtained using
Equation (2.22) is also the matrix representation of the inversion acting on a spin-1 field.
No confusion should arise from using the same symbol Z for both, the embedding space
and Lorenk frame transformation matrices.
CHAPTER 2. FIRST STEPS 13
For the transformation of spinor fields one also needs a spinor representation S(Z). As
in the standard Lorentz case [XI it is obtained by demanding the invariance of the spin
matrices,
Equations (2.23) and (2.20) aiways have a solution
Then, one finds
Hence, using S(Z) as spinor representation for the inversion, the covariant matrices rP transform as a pseudo vector.
On the other hand, one could try to impose the stronger condition that also the covari-
ant gamma matrices be invariant, i.e.
which entails that equation (2.23) is valid and that f p transforms as a vector under the
inversion. I t turns out that the stronger condition (2.26) only has a solution respecting the
condition (2.20), i f ( d + 1) is even. In these cases one can define
where are the gamma matrices spanning the Clifford algebra for the embedding space
metric q One can show that the solution to equations (2.26) and (2.20), -unique up to a
change of sign, is given by
This expression has been given only for completeness and will not be used for calculations.
CHAPTER 2. FIRSTSTEPS
2.2 Basics of Conformal Field Theory in d Dimensions
2.2.1 Conformal Symmetry
Let g,, be the metric tensor of some ddimensional manifold with respect to some coordi-
nates xi . A transformation x + xf(x), under which the metric tensor changes as
is called a conformal transformation. Conformal transformations are a generalization of
ordinary symmetry trans formations, in that every symmetry trans forma tion sa tisfies equa-
tion (2.29) with A(x) = l .
Given such a transformation x -P xf, one can define the transformation matrix
which satisfies
Obviously, the conformal transformations form a group.
A very important case, and the one most studied, is the case of conformal symmetry
on a flat manifold, which will be assumed to be Euclidean here. The generalization to
pseudo-Euclidean cases is straightforward. Hence, let g,, = bii Then, for d > 2, the
transformations satisfying equation (2.29) are (431: ~- I dilatations: i i = c x ' ,
I i translations: x = xi + b', . . I rotations: P=R,xJ,
. xf inversion:' 2' = -
x* '
as well as any combination of the above. It is assumed that c > 0 and R € 0 ( d ) .
'For the inversion to be defined on the whole of !+it", one has to consider the confomally compactified
Euclidean space !R' U (00).
CHAPTER 2. FIRST STEPS 15
It is easy to see by a direct comparison with equation (2.17) that the infinitesimal ver-
sions of the transformations (2.32) are identical with the symmetry transformations of
AdsLi+, restricted to its horizon, xo = 0. Hence, the same is hue for the finite transfor-
mations connected to the identity. Furthermore, it means that the symmetry operators
generating the dilatations, translations, rotations and SCTs are D, Pi, Lii and Kit respec-
tively, which satisfy the commutation relations (2.18). Moreover, also the inversion is iden-
tical to the Ads inversion formula (2.21) restricted to xo = 0. Thus, the Ads symmetry
transformations directly correspond to conformal transformations of the Ads horizon.
2.2.2 Representations of the Conformal Algebra
The representations of the conformal algebra relevant to conformal field theory act on car-
rier spaces spanned by quasi-primary conformal fields and their descendants. These no-
tions will b e defined in the sequel. A field Y (x) transforming under a conformal transfor-
mation x -+ x' as
where r [R( x)] denotes the appropriate matrix representation of the transformation (2.30).
is called a qzmsi-printmy field of scnlbzg or conforntnl dimension A [113]. As can easily be
checked, the derivatives of a quasi-primary field are not themselves quasi-primary fields
and are called secotldn y as well as descendmts of the quasi-primary field.
As in section 2-2-1, the focus of the present discussion will lie on the conformal trans-
formations of flat space with gij = bii- First, consider the transformations connected to
the identity. Using infinitesimal transformations, one can in a standard fashion determine
from equation (2.33) the representations of the group generators Pi, L;,, D and K;. As Pi and
L,, form a Poincare sub-algebra of the conformal algebra, one finds as expected
CHAPTER 2. FIRSTSTEPS 16
where T ( L , , ) is an irreducible matrix representation of Lii of the appropriate spin. Simi-
larly, one obtains for D and Ki [43]
It is easy to show that these operators satisfy the commutation relations (2.18).
Secondly, the inversion will be considered, which is of particular importance not only
for simplification purposes, but also for the representaticn of correlation functions [56].
The associated transformation matrix, which is given by equation (2.30). but will be called
Z in this case, is found to be
This is also the spin-1 representation of 2, whereas the spin-1 /2 representation is
Note that in this equation yi is a Euclidean gamma matrix in d dimensions, unlike y, of
section 2.1.2.
2.2.3 Restrictions on Correlation Functions
The presence of conformal symmetry in a field theory imposes tight constraints on the
form correlation functions of conformal fields could possibly take. Typically, the non-
coincidence terms of two- and three-point functions are e s sen t i a l~ fixed (up to a constant),
whereas 11-point functions with n 2 4 can depend on arbitrary functions of the so-called
011-lmn~rouic ratios. The latter are combinations of the field positions, which are invariant
under conformal transformations. For 11 = 4, there are two independent an-harmonic ra-
tios, namely [33]
where xi, = Ixi - x i [ - The number of independent an-hamonic ratios rapidly grows with
! I , more specifically, there are n ( n - 3) / 2 of them.
CHAPTER 2. FlRST STEPS 17
The purpose of this section is to review some key expressions, which are needed for
the confirmation of the form of correlation functions derived via the AdS/CFI' correspon-
dence. The reader is referred to the original work [56, 131,511 and to the textbook [13] for
more details.
Two-point functions of operators Ou (u is the spin index of the field operator) are con-
fined by conformal invariance to the form [56]
Three-point functions can be written in the general form (131)
where the dii are given by
t *s3 is a homogeneous tensor sa tidying
for all R,
The variable 9 can be chosen to be 0 or I, whichever is more convenient.
If there are only scalar fields, the general form (2.38) reads
C H A M E R 2. FIRST STEPS
where the Aii are given by equation (2.39) with q = 0.
Especially interesting are correlation functions involving conserved currents I' and the
energy momentum tensor Ti, (cf. section 2.2.4), which have conformal dimensions d - 1
and d, respectively. The special interest arises from the fact that these operators obey con-
servation laws, namely
aili = 0 and ai7; = 0. (2.31)
Equation (2.37) readily yields the two-point functions of conserved currents and the energy
momentum tensor. While one uses A = d - 1 and f [Z] = 1 (cf. equation (2.35)) in the case
of conserved currents, for the energy momentum tensor one uses A = d and
which is the representation of the inversion for symmetric and traceless tensors.
For three-point functions of the form ( Vi02U3) one can use equation (2.38) with [q = 2, A, = 11 - 1 and q = 1. Moreover, the conservation law (2-41) requires that [I311
Similar statements can be made about three-point functions involving Ti, [131].
The form of n-point functions with n > 4 is not uniquely fixed by conformal invariance,
but involves arbitrary functions of the an-harmonic ratios. In order to give an example, the
general form of a four-point function involving onIy scalar fields is [43]
where L = xf=l Air and f is an arbitrary function.
2.2.4 Properties of the Energy Momentum Tensor
Let I be the action functional of a particular field theory defined on a manifold R. Then,
the energy nronzenttrnr tensor Ti; of this field theory is often defined via the variation of the
action under small metric changes:2
:The relation to its canonical definition is described in (431.
CHA M E R 2. FIRST STEPS
or, equivalently
In a quantum field theory, T,, is of course an operator.
There are standard properties of the energy momentum tensor, which shall be reviewed
now. First, covariance of the action implies its invariance under isomorphisms, x' = x + ~ ( x ) . Since
equation (2.45a) implies the conservation law
Secondly, consider a Weyl rescaling
If the action 1 is invariant under any such Weyl rescaling, equation (2.45a) implies the
tracelessness of the energy momentum tensor,
The inverse statement is true as well. Moreover, from equation (2.29) follows that, if an
action is invariant under Weyl rescahgs, then it is also conformally invariant. However,
the inverse is not true, because, while A is arbitrary in equation (2.47), it is not so in equation
(2.29). Hence, tracelessness of the energy momentum tensor implies conformal invariance,
but conformal invariance does not necessarily mean that the energy momentum tensor is
traceless.
Classically, the existence of symmetries leads to the existence of conserved currents \A, which satisfy on-shell
This is Noether's theorem, which is textbook knowledge [43]. It is easy to see from equa-
tions (2.46) and (2.48) that the conserved currents corresponding to translational and scale . .
invariance on a flat manifold are 7') and Ti'xi, respectively.
CHAPTER 2. FIRST STEPS
Quantum mechanically, the analogue of conserved currents are Ward identities. Since
also this is textbook knowledge, only the general formula shall be given. Let Gz be the
generator of the symmetry connected with the conserved current I:, acting on the field 4 according to the formula
Here w, is an infinitesimal parameter. Then, the corresponding Ward identity reads [43]
The validity of a Ward identity depends upon the fact that both, the action and the func-
tional integral measure of the quantum theory, are invariant under the corresponding
transformation. There are cases, in which only the action is invariant, but the functional
integral measure is not. In these cases one says that the Ward identity is anomalous, since
the classical symmetry is broken at the quantum level.
Inserting the classically conserved currents 7"' and ~ ' i x , , which correspond to trans-
lational and scale symmetry, respectively, into equation (2.49) and using equations (2.34).
one finds the Ward identities [43]
and
As is well known [46, and references therein], the Ward identity (2.51) is anomalous, if the
dimension d is even. The corresponding so-called W q l auonmly is expressed in
In the context of the AdS/Cm correspondence, the Weyl anomaly will be discussed and
calcuIated in chapter 5, section 5.3.
C H A M E R 2. FIRST STEPS
2.3 Motivation for a Duality
This section serves as a bridge over the gap between the basics of Ads spaces and CFTs on
one side and the explicit formula for the AdS/CFT correspondence on the other side. It is
built upon two observations.
First, a brief look at sections 2.1.2 and 2.2-1 reveals the relation between the Ads symme-
tries and conformal symmetries. It rests upon the fact that the symmetry algebra of Adsd+ I
and the conforrnal algebra are both isomorphic to so(d + 1, I), which was derived explic-
itly in section 2-1.2. This means that to every symmetry of AdSa+l corresponds a conformal
symmetry acting on flat d-dimensional space and vice versam3 Moreover, a direct compar-
ison of equations (2.17), (2-21) (with 1 = 1) and (2.32) reveals that the flat d-dimensional
space actually is the horizon of Adsd+
Secondly, there is a relation between fields on Adsd,* and conformal fields living on
the Ads horizon. As an example consider a free scalar field on AdSd,I (cf. section 3.1.2),
which can be written in the form
where the constant A depends on the mass of the scalar field. Performing the symmetry
transformation x' = cx, which yields
and using @'(x') = @ ( x ) one finds
Hence, @" is a quasi-primary field of conformal dimension d - A. It naturally couples to a
conforrnal operator 0 of scaling dimension A via the integral I d% Go (x) O(x) . Moreover,
the form of equation (2.52) suggests that &, and thus also 0, live on the Ads horizon.
This establishes a unique relation between the Ads scalar field @(x) and the quasi-primary
conformal field O(x) via a bulk-boundary propagator. Similar relations can be found for
fields of any spin.
"rom the isomorphism of the algebras follows only the retation between symmetries connected to the
identity in a group theoretical sense. The relation for the inversion has to be checked separately.
C H A M E R 2. FLRSTSTEPS
2.4 An AdSICFT Dictionary
The simple argument of the last section, which provided an analytical relation between
classical Ads fields and quasi-primary conformal operators living on the Ads horizon, can
be generalized using group theoretical methods. h fact, this was one of the first questions
investigated after Mddacena's conjecture.
By virtue of the isomorphism of the AdSd+l symmetry group and the conformal group
in d dimensions, these groups possess identical irreducible representations, which are char-
acterized by a set of quantum numbers (the eigenvalues of the Casimir operators) [XI.
Moreover, to such an irreducible representation there exist a classical Ads field' and a
quasi-primary conformal field which transform according to said representation. Thus,
the quantum numbers of the irreducible representation determine the appropriate param-
eters of both, the Ads field (like spin and mass) and of the quasi-primary conformal field
(like spin and conformal dimension) or, put differently, the appropriate Ads and Cm pa-
rameters are related to each other. As is obvious from the coupling, J & @o(x)O(x), the
Ads and CFT fields must be of equal spin. Therefore, these relations form a kind of "dictio-
nary" between the masses of Ads fields and the conformal dimensions of their CFT counter
parts.
For example, the mass parameter nl of an Ads scalar field and the conformal dimension
A of the corresponding quasi-primary scalar field satisfy [82,54,160]
Obviously, there are two solutions for A, namely
but unitarity of the irreducible representation (541 restricts the conformal dimension to
Therefore, for m2 > 1 - d2/4, only the + sign in equation (2.55) is possible, whereas both
choices are allowed for - d 2 / 4 5 nz2 < 1 - d2/4. Hence, it seems that in the latter case
"By analogy with the Poincare group [XI, a first order equation of motion probably exists only for =me
fields, nameIy the ones with a low spin.
there exist two quasi-primary scalar fields which correspond to the same Ads field. This
ambiguity has its counter part on the Ads side. If -d2/1 < nr' 5 1 - d 2 / 4 , then there exist
two kinds of boundary conditions [26,28] for a scalar field on Ads space, called regular and
irreglrlnr, which correspond to the choice of the + and - signs, respectively
As it turns out, the correspondence formula (1.13) is valid only for the treatment of
regular boundary conditions (cf. chapter 3 for more details). A method suitable for irreg-
ular boundary conditions has been proposed by Klebanov and Witten [102], proven to be
correct in the case of scalar fields by the author of this thesis [I251 (cf. section 3.2-2) and
applied to spinor fields by Rashkov [140].
Group theoretical studies [54, 61, 531 have provided further "dictionary entries" for
fields in various dimensions. These relations between the the mass parameters of Ads
fields and the conformal dimensions of the corresponding quasi-primary fields have been
confirmed by the calculations of ClT two-point functions for scalar fields [82,160,11,121,
651, Dirac spinor fields [91,120,140], massless [65,30] and massive vector fields [120,110,
1171, gravitons [106, 14, 1221, Rarita-Schwinger fields (33, 155, 103, 1161, antisymmetric p-
form fields [13,111,112] and massive symmetric tensor fields [137].
Further studies of the representations of the symmetry algebras as well as their su-
persymmetric extensions in the context of the AdS/CFT correspondence can be found in
(55,57,60,59, 84, 85,6,7,8,58,126]. Supersyrnrnetry, which combines quasi-primary fields
of different conformal dimensions into superconformal multiplets, makes the considera-
tion of irregular boundary conditions necessary (cf. chapter 6 for an example).
Chapter 3
Exploring the Correspondence - The Scalar Field
The scalar field, being the simplest of fields, will be used as a trial ground for the AdS/CFT
correspondence. As indicated in section 1.3, the Ads field theory action needs to be renor-
malized, and there are two commonly used regularization schemes. Of course, physical re-
sults must not depend on the regularization scheme chosen, whereas counter terms might
depend on it. In this chapter, the two commonly used regularization schemes, the E- and
asymptotic prescriptions, shall be applied to scalar field theory on Adsdtl. Considering
them in detail will not only demonstrate the agreement of the finite physical results, which
represent the C R correlation functions, but also the advantages and disadvantages of ei-
ther scheme. These can be summarized as follows: The E-prescription, which is explained
in detail in section 3.1, formulates a standard Dirichlet boundary value problem and thus
can employ standard methods for its solution. Furthermore, the covariant counter terms
can be read off directly from the divergences of the un-renormalized action and do not
contribute to the finite result. Hence, the finite result can be calculated by simply ignoring
the divergent terms. The main disadvantage of the E-prescription is that it includes only
Ads fields obeying regular boundary conditions (cf. section 3.2). On the other hand, the
asymptotic prescription can be used to formulate the AdS/CR correspondence for Ads
fields obeying regular and irregular boundary conditions. This is demonstrated in detail
in section 3.2. Obtaining the finite result is more subtle than in the E-prescription, since the
CHAPTER 3. EXPLORLNG THE CORRESPONDENCE - THE SCALAR HELD 25
leading counter term contributes to it, but the requirement of general covariance leads to
the correct result.
In the literature, two-point functions of conformal scalar fields have been calculated
from Ads scalar field theory for regular boundary conditions in 182, 160, 11, 121,65, 148,
24,1181, and for irregular boundary conditions in [102,125,118].
Typically, regularization and renormalization are necessary only for the calculation of
the CFT two-point functions. Section 3.3 explains in principle the calculation of CFT three-
and higher point functions from interacting Ads field theories and considers as simple
examples the first order interactions of scalar fields.
3.1 The ePrescription
The E-prescription for regularizing the on-shell Ads action can be formulated in the fol-
lowing procedure:
1. Instead of using the Ads horizon x o = 0 (cf. section 2.1), introduce a cut-off boundary
.ro = E. Then, formulate and solve the field theory following from the Ads action as
a Dirichlet boundary value problem at the cut-off boundary.
2. The conformal boundary value Go (cf. section 2.3) is related to the Dirichlet boundary
value Gi by a field rescaling of the form
In order to find the exponent A, determine the asymptotic behaviour of the field so-
lutions for the limit E --+ 0.
3. Express the on-shell action in terms of #o and select the €-independent term, which
represents the physical result- Terms which diverge as e + 0 are compensated by
counter terms and can thus be ignored. The remaining terms vanish in this limit and
do not contribute to the finite result.
In the following sections, this procedure is carried out explicitly for the scalar field. In
order to give a self-contained presentation, the Green's function method, which is used
for solving the Dirichlet boundary value problem, is reviewed in section 3.1.1. Only the
CHAPTER 3. EXPLORLNG THE CORRESPONDENCE - T H E SCALAR FIELD 26
terms leading to the ClT two-point function need regularization, and they can be obtained
by considering the free scalar field. This is done in section 3.1.2 for scalar fields with a
generic mass, whereas section 3.1.3 considers the special cases non included in the generic
treatment.
3.1.1 Review of the Green's Function Method
The starting point is the action for an interacting, massive scalar field living on a manifold
f2, which has a boundary dR,
where dx stands for the invariant volume measure of integration over the manifold indi-
cated at the integral sign, and lint contains the interaction terms. The equation of motion
derived from the action (3.1) is
Defining the Green's function as a solution of the equation
with the boundary condition
the solution to the equation of motion (3.2) satisfymg the Dirichlet boundary condition
@( x) / ,, = #a (x) can be written in the general form
'The functional variation is done covariantly, cf. (191.
CHAFER 3. EXPLORING THE CORRESPONDENCE - THE SCALAR HELD 27
where
In equation (3.5b). n p denotes the normal vector to the boundary aR pointing outwards of
a. Substituting equations (3.5) back into the action (3.1) and integrating by parts, one ob-
tains the on-shell action as a functional of the boundary value @a,
Here, lint and B ( x ) are to be understood as perturbative series in #('I, which in turn de-
pends on the boundary value #a via equation ( 3 3 ) .
3.1.2 The Free Scalar Field on AdSd+l
Following the general method outlined in section 3.1.1, the on-shell action for a free mas-
sive scalar field on ( d + 1)-dimensional anti-de Sitter space with a cut-off boundary will
now be calculated. Hence, one can identify R = AdSd+l with the metric (2.5)' and the
boundary af2 is characterized by xo = r plus the single point XJ = m.
The first step is the calculation of the Dirichlet Green's function. The massive wave
equation
has the linearly independent solutions
k is a momentum d-vector, and k = Ik(. I, and K, denote the standard modified Bessel
functions of order a. These modes can now be used to construct the Green's function in a
CHAPTER 3. EXPLORLNG THE CORRESPONDENCE - T H E SCALAR FIELD 28
standard fashion- First, consider
which satisfies equation (3.4~1). Go satisfies the boundary condition for xo = 0 instead of
xo = r. However, it has significance for the treatment of interactions, where the E -+ 0 limit
can be taken right away, and for the asymptotic prescription in section 3.2. Furthermore,
it can be compared to results from older literature- The expression (3.8) can be integrated
and gives
where F denotes the hypergeometric function,
(y' denotes the vector (-yo, y)), and
The expression (3.9) coincides with the ones found b y Burgess and Liitken (291 and Allen
and Jacobsen (31 after using transformation formulae for the hypergeometric function (751.
I t is now time to incorporate the boundary condition (3.4b) at .ro = E . One can easily
change equation (3.8) to accommodate this, namely by adding a suitable term. One finds
that
is the Green's function asked for. Obviously, lim G, = Go. It does not seem possible to s-0
perform the momentum integration, but this is not even necessary. In fact, one arrives at
giving diS(x - y) for xo = E. Substituting equation (3.12) into the expression (32%) for the
free field, one obtains
CHAPTER 3. EXPLORLNG THE CORRESPONDENCE - THE SCALAR FIELD 29
where 4, (k) is the Fourier transform of the boundary value &(x)- Note that the invariant
integral measure on af2 is dr = ddxe-'' and n' = (-E,O).
Before proceeding to the calculation of the CTT two-point function it is necessary to
make the connection between the boundary value 4, and a conformal field living at xo = 0.
Ln order to do this, one should consider the E + 0 limit of expression (3.13). Substituting
the leading order behaviour of Ka(ke) for small c, one can integrate and finds (see equation
(B.4))
where A = d / 2 + a. This expression is similar to equation (2.52), but one must define
in order to ensure that e0 is a field of conformal dimension d - A (cf. equation (2.53)). The
factor a has been included for later convenience.
The last step in this section is the calculation of the CFT two-point function of scalar
operators, which is obtained from the Ads on-shell action. Substituting equation (3.12)
into the general formula (3.6) yields
p) = e-Qol(k) { (q - a) + k, a ln [(k~)'K,(k)] ( k ) (3.16)
This should be expressed in terms of Oo, which was defined in equation (3.15). Moreover,
one can isolate the terms, which diverge in the limit E -+ 0. As it turns out, all divergent
terms represent local expressions in terms of 4, and its derivatives. Hence, they can be
cancelled by adding local covariant surface terms as counter terms to the action and will
thus be ignored.
The finite result is contained in the second term in the braces in equation (3.16). It is
found by considering the series expansion of zaKa(z) for small z, which is [7512
'This formula is valid only in the caseof generic, non-integer a. The cases a = tr involve logarithms, but the
finite result does not change. This is a result of the fact that lim K,(z) = K,(z) and is demonstrated explicitly a-.n
in section 3.1 -3.
CHAPTER 3. EXPLORING THE CORRESPONDENCE - THE SCALAR FIELD 30
where the notation ( a ) = T(a + j ) / T ( R ) has been used. Proceeding to expand the logarithm
in equation (3.16) one then finds
Here, the interesting term is the leading non-analytic one, which is of order zZa. Substitut-
ing this term and equation (3.15) into equation (3.16) and using the formula (B.2) yields the
finite result
respondence
This formula
Thus, the CFT two-point function of scalar operators, as obtained From the AdS/Cm cor-
formula (1.13), reads
2ca (~(x)~(Y)) = I x - y l z d (3.20)
represents the two-point function of scalar operators of conformal dimension
A in agreement with the general form dictated by conformal invariance, cf. equation (2.37).
On the other hand, terms in equation (3.18) of order zZn with rt < a will contribute
divergent terms, but as they are local (k2 can be replaced by -A), they can be cancelled by
adding local counter terms. All other terms do not contribute in the limit E + 0.
Obviously, the two-point function (3.20) displays short distance singularities, because
of 2 4 > d . The treatment of these by considering the correlator as a distribution has been
proposed in [12], whereas the use of local counterterms (which were neglected here) has
been suggested in 131 1. According to this procedure, the infrared cut-off parameter E acts as
an ultraviolet regulator for the Cm short distance singularities. A different kind of integral
regularization has been used in (1231.
3.1.3 The Special Cases a = n
The derivations of CFr two-point functions in section 3.1.2 involved the series expansion of
the modified Bessel functions K,(z). As mentioned in a footnote on page 29, the expansion
(3.17) is valid only for generic, non-integer a. It is characterized by two series, the leading
one starting in powers z-", the other, sub-leading one, in za. Moreover, the first term of
the sub-leading series was responsible for the finite result. It will be shown in this section
CHAPTER 3. EXPtORING THE CORRESPONDENCE - 773E SCALAR FIELD 31
that the non-local part of the finite result is the same for integer a, thereby justifying the
generic approach taken so far. However, a logarithmically divergent term is generated as
well, which has to be cancelled by a non-covariant counter term breaking the conformal
invariance.
In the cases of integer a = n the distinction between the two series by powers breaks
down, as the sub-leading series would be contained in the leading series. However, the
expansion of Kn (z) is 175)
which shows that the sub-leading series is characterized by a logarithm. Here, @(x) de-
notes the standard psi function [75],
Hence, one can write
where only the leading terms of the two series have been written down and the ellipsis
denotes all other term.
The CFT two-point function of scalar fields was obtained by evaluating the integral (cf.
equation (3.16))
ln [ ( k ~ ) " Kn (ke)]
and discarding the divergent local terms in the result as well as terms of higher powers in
E . Using equation (3.22) one finds
d 4(- l ) ' ~ + ~ z- In [znKn ( z ) ] = ( f ) 2 n l n ? + . . . ,
dz [r(n>I2 2
where the ellipsis denotes terms of even powers in z, which yield local terms in the integral
(3.23), as well as sub-leading terms with a logarithm, which can be neglected in the E: -+ 0
CHAPTER 3. EXPLORING THE CORRESPONDENCE - 7WE SCALAR FIELD 32
limit. Hence, the integral
should be considered. The term containing in E yields a divergent, but local contribution
in the E + 0 Limit. It must be cancelled by adding a counter term, but this counter term
cannot be covariant because of the logarithm. As explained in section 1.3, this leads to
the breaking of conformal invariance in the renormalized action, i.e. in the CFT effective
action. In the context of the AdS/CFT correspondence, this symmetry breaking was first
observed by Aref'eva and Volovich [12]. Hence, the conformal symmetry of the resulting
CFT two-point function will also be broken. However, it is easy to see that this will not
affect the finite distance two-point function, because the change of the counter term under
a dilatation is a local expression.
The interesting term in equation (3.24) is
Y;
4(-l)"+l lx - yll-i [ r ( ~ ) ] q 2 4 i (5)2 '1 Jdkkf i2n~i-l(klx - yl) Ink,
0
which after integration becomes [75, formula 6.7711~
The last fraction in this expression has to be defined by analytic continuation. Thus, one
finds that only the term
yields a contribution to the final result, which becomes
3~ regularizing factor e-d2 in the integrand can be used to ensure the existence of the integral, cf. ap- pendix B.1.
CHAPTER 3. EXPLORLNG THE CORRPOADENCE - THE SCALAR RELD 33
where c, was defined in equation (3.10). This agrees with the expression for generic a.
Hence, one can conclude that the generic approach yields the correct finite distance CFT
two-point function in all cases. The appearance of the nontovariant counter term indi-
cates that conformal invariance is broken, but this affects only local terms in the two-point
function, which have not been considered.
3.2 The Asymptotic Prescription
The asymptotic prescription for regularizing the Ads action is somewhat more elegant
than the E-prescription, in that it does not require two representations for the f ree field
solution and directly extends to the treatment of interactions. The procedure can be stated
as follows:
1. Solve the field equations and express the solutions in terms of some conformal fields,
which are assumed to live on the horizon.
2. For small -1-0, the field solutions can be expressed as a sum of two power series in ro.
The appearance of the sub-leading series is very important, in particular its leading
term must be determined.
3. Express the on-shell action in terms of the asymptotically expanded field solution,
isolate divergent terms and re-express them using covariant expressions. These co-
variant terms are cancelled by counter terms. Then, the remaining finite result yields
the CF.T two-point function.
The other very nice feature of the asymptotic prescription is the possibility to include
irregular boundary conditions. Besides a solution of the form (3.14), which is related to the
solution found in the €-prescription, there should be another solution, because the equation
of motion is second order. I t is certainly not difficult to show that it is given by
Following the argument of section 2.3, the function f is a conformal field of scaling dimen-
sion d / 2 + a, which couples to a boundary field operator of dimension d / 2 - a. A solution
@(x) of the form (3.26) is said to obey an irregular boundary condition.
CHAPTER 3. EXPLORDVG THE CORRESPONDENCE - THE SCALAR FIELD 34
The fields obeying regular boundary conditions give rise to CFT correlation functions of
operators with conformal dimensions A restricted by d 2 d / 2 . Hence, the use of irregular
boundary conditions might help to obtain correlation functions for operators with scaling
dimensions A < d / 2 . However, the generalization does not extend to all values of A. First,
as can be seen from formula (BA), a Fourier transform of the integral kernel in equation
(3.26) does not exist, if a > d / 2 . Secondly, the notion of field quantization restricts the
possible values of a to a < 1. This restriction stems from both, the finiteness of the energy
of the Ads field (261 and the requirement of unitarity of the quantum algebra of conformal
operators [%, 1021.
The interacting scalar field obeying regular boundary conditions shall be reconsidered
in section 3.2.1 using the asymptotic prescription of the AdS/CFT correspondence, but in
view of the restriction on a for irregular boundary conditions only the cases 0 5 a < 1
will be considered. A generalization to larger values of a is possible and involves more
counter terms. The treatment of irregular boundary conditions shall be demonstrated in
section 3.2.2.
3.2.1 Regular Boundary Conditions
Starting from the action (3.1)' the solution of the equation of motion (3.2) is obtained as
(3.27)
where B ( y) and G (x, y ) are given by equations (3.3) and (3.9), respectively. Moreover, the
free field solution @ O ) is
The integral kernels, which are called bulk-Do~lndnry propagators, are given by
where c=& is given by equation (3.20). Their Fourier transforms are obtained using equa-
tion (B.3) and read
C H A m R 3. EXPLORING 7HE CORREPONDENCE - THE SCALAR FIELD
Equations (3.30) and (3.28) imply that the boundary functions @y' and are related by
Considering equations (3.28) and (3.30). the series expansion of the modified Bessel
function, equation (3.17), shows that the free field @ ( O ) can be written as a sum of two d / Z - a d / ? + a series, whose leading powers are xo and xo , respectively. Thus, one finds by direct
comparison with equations (3.28) and (3.30) that the small xo behaviour of # ( O ) is
xo-+O 4-a ( 0 ) :-a ( 0 ) @'"(x) x & (x) i- x(j @+ (x), (3.32)
where sub-leading terms have been dropped. Moreover, the Green's function (3.9) goes
like
Hence, the interaction contributes only to the #- part of the asymptotic boundary be-
haviour, i.e. one can write
lo -0 #(x) = x: (X) + (x) ,
where
Identical relations hold for the Fourier transformed expressions.
Now consider the on-shell action, treated as a functional of the regular boundary values
&- . Integrating equation (3.1) by parts yields
The first term must be regularized, which is done by writing
CHAPTER 3. EXPLORLNG W E CORRSPONDENCE - THE SCALAR FlEtD 36
where the ellipses indicate contributions from sub-leading terms and other terms which
vanish for xo = 0. The first term in the last line is cancelled by a covariant counter term.
Hence, the renormalized on-shell action is
where equations (3.3k), (3.34b), (3.31), (3.27) and (3.28) have been used. The first term in
equation (3.35) is given by
and yields the correct two-point function of scalar operators of conformal dimension A =
ri /2 + a in agreement with the E-prescription. The other two terms in equation (3-35) agree
with the general formula (3.6) and have to be expressed as a perturbative series in terms of
4'')- However, by virtue of equations (3.28) and (3.31b) this yields a perturbative series in
terms of the boundary function #I-.
3.2.2 Irregular Boundary Conditions
The treatment of irregular boundary conditions follows an idea by Klebanov and Witten
[102]. It is straightforward to see that expressing the action Ifin as a functional of and
using the correspondence formula (1.13) leads to a CTT two-point function with the wrong
sign. However, consider the expression
Using equation (3.31) and the expression
CHAPTER 3. EXPLOiUlVG THE CORRESPONDENCE - TNE SCALAR FIELD
one finds
or, after an inverse Fourier transformation,
This expression holds to any order in perturbation theory and shows first that @+ can be
regarded as the conjugate field of @- and secondly that the functional
has a minimum with respect to a variation of #-. Klebanov and Witten's idea [I021 is to
consider the minimum value of the functional 1, which becomes a functional of #+ only:
as the action functional to be used in the correspondence formula (1.13). They showed that
the two-point function of scalar conformal operators thus obtained comes out correctly.
In the following their result shall be confirmed and interactions included. The mini-
mum of the functional J is easiest found from equations (3.35) and (3.38), giving
Here equations (3.34b), (3.31), ( 3 . 3 4 ~ ) ~ (3.30) and (3.27) have been used. The first term in
equation (3.39) can be inversely Fourier transformed using equation (B.2), which yields
'This just amounts to a Legendre transformation.
CHAPTER 3. EXPLORfNG THE: CORRESPONDENCE - THE SCALAR HELD 38
This yields the correct two-point function, if one uses the correspondence formula5
The second and fourth terms in equation (3.39) can be combined by defining the Green's
function
This modified Green's function G also satisfies equation (3.4a), because the second term in
equation (3.42) does not contribute to the discontinuity. Moreover, using equations (3.8)
and (3.30) one finds
which differs from equation (3.8) only by interchanging a and -a. Hence, the result (3.9)
can be taken over, yielding
Thus, inserting equation (3.42) into equation (3.39) yields
Moreover, one can see from equation (3.43) that, for small xo, G behaves as
Hence, writing
'The factor a occurs also for regular boundary conditions, since it was absorbed into 40 in equation (3.15).
C H A m R 3. EXPLORAG THE CORRESPONDENCE - THE SCALAR FIELD 39
the interaction contributes only to @-. This in turn means that, expressing lint and B as
a perturbative series and using equation (3.46), the functional 1 is naturally expressed in
terms of the irregular boundary value @+. Moreover, it has the expected form, in that it
is obtained from equation (3.35) by replacing a with -a and #- with a+. An important
point is that the Green's function G must be used for the calculation of internal lines.
Finally, by a calculation similar to that of the derivation of equation (3.37) one finds
sT[@tl = 2aQ- ( x ) . ( W t ( x )
This is a final confirmation that the fields @- and @+ are conjugate to each other.
The analysis of this section proves the correctness of Klebanov and Witten's proposal
to any order in perturbation theory.
3.3 Ads Interactions and Higher-Point CFT Correlators
3.3.1 General Method
The previous sections explained the calculation of CFT two-point functions by means of
the scalar field and focused on the necessary regularization of the on-shell action- As a
formal result, the renormalized generating functional for the AdS/CFT correspondence
was found in equations (3.35) and (3.44), where the interaction terms are identical to those
in equation (3.6) and did not have to be renormalized. Moreover, these terms are to be
expressed as perturbative series in terms of the free field solution, which is naturally con-
nected via a bulk-boundary propagator to the prescribed boundary values. Hence, the
interaction terms in the on-shell action give rise to connected Cm n-point functions, where
11 > 2. Witten [I601 found that the individual terms of the perturbative series can be nicely
represented by graphs resembling the familiar Feynrnan diagrams. Examples of a first and
second order graph are shown in figure 3.1.
In the literature, CFT three- and four-point functions from first order interactions in
Ads field theory have been calculated for the first time by the author of this thesis [I211
(cf. section 3.3.2) for the scalar field. Subsequently, three-point fimctions of many other
fields have been obtained in [a, 106,30,73,105,95,16,108,20]. A comparison with a per-
turbative ClT calculation can be found in [42]. Particularly interesting in connection with
CHAPTER 3. EXPLORING THE CORRESPONDENCE - W E SCALAR FIELD 40
Figure 3.1: Example of Witten diagrams for a) a three-point and b) a second order four-point
function. The rim of the circIes represents the boundary af2 on which the CFT fields live. The
position of the conformal fields So is indicated by the crosses. The interior of the circles represents
the bulk R, where the interactions take place. Hence, the interaction points, which are integrated
over, are represented by dots, lines from the rim into the interior are bulk-boundary propagators,
whereas interior lines behveen the interaction points are bulk Green's functions.
the work presented in this thesis is [16], which considers the three-point function of gravi-
tons.6 This particular three-point function contains information about the central charge in
the operator product expansion of CFT energy momentum tensors, which is shown to be
in agreement with earlier results for the two-point function.
A second and even more interesting group of articles deals with four-point functions
containing the exchange of Ads fields. These calculations are important, because the func-
tional form of CFT four- and higher-point functions is only restricted, but not determined
by conformal invariance (cf. section 2.2.3). Thus, they contain non-trivial information about
the particular CFT under consideration.
Pioneered by Liu and Tseytlin [109, 1071, the understanding of the treatment of Ads
exchange diagrams advanced significantly due to a series of papers by Freedman, d8Hoker
et al. [64,37,38,39,40,41]. Other work can be found in [27,32, 142, 981. The logarithmic
short distance behaviour found for four-point functions has been interpreted in [21j in
terms of anomalous scaling dimensions.
'The work on gravitons by the author is presented in chapter 5, but does not include three-point functions.
CHAPTER 3. EXPLORING T H E CORRESPONDENCE - TNE SCALAR FIELD 41
3.3.2 First Order Interactions of the Scalar Field
In the following, first order interactions between scalar fields are considered. These rep-
resent the simplest examples of interactions and are useful for an exploration of the tech-
niques needed for the calculation of more complicated graphs involving other fields. Al-
though starting for a general contact interaction between 11 scalar fields, the focus will lie
on r l = 3 and n = 4. The results obey the conditions imposed by conformal invariance (cf.
section 2.2) and thus support the validity of the AdS/CFT correspondence.
Consider an interaction term
where the scalar fields 4, have masses nr;. This form is a generalization of the interactions
considered in [121], in that it allows for interactions between different fields.
In evahating this expression as a function of the boundary values one realizes that
there are no divergent terms in the E + 0 limit. Hence, one can use equations (3.14) and
(3.15). According to the general formalism, the first order term of the action becomes
with the integral
where the abbreviation
has been
(3.49) the
used. Using the correspondence formula (1.13) one can read off from equation
first order contribution to the ,I-point functions of CFT operators O;,
After a Feynman parameterization (cf. equation (B.5)) and performing the y integration
using the expressions in appendix B.3, equation (3.50) takes the form
r ( ( d ) ) r " I" =
2 n:=, vi) /n i=l (dy i~p ' - ' ) s(z:=l~i- 1)
CHAPTER 3. EXPLOR.llVG THE CORRFSPOAIDENCE - THE SCALAR FIELD 42
Here, x,, = xi - x,. Moreover, introducing new integration variables by 7 1 = PI and Yi = Pi (i 2 2), the integration over is trivial and leads to
it seems hopeless to try to perform the remaining integrals for general n, but one can
do it for n = 3, whereas for n = 4 one can make enough progress in order to show that the
resulting CFT correlation function has the expected form. For n = 3 the integrations over
P2 and P3 yield
Proceeding to tz = 4, the special case Ai = A for i = 1,2,3,4 shall be chosen for
simplicity. Then, equation (3.52) yields after integration over P4 and P3
A change of integration variable and the introduction of the harmonic ratios [43] rl and C
by
x13xll 7 - X 1 z . Y ~ p=- X12X34 e q = - and C = - X23Xz.1 1 1 4 x 2 3 1 1 3x24
finally leads to the expression
r(2A - d / 2 ) 2ni I4 = j d z F (A, d;Zd; 1 - ('l +02 - - 4 sinh2 i) n XG i d (3.54)
r(2A) (r l@, ( ~ 4 ~ t tL i<;
Equations (3.53) and (3.a) are of the form expected for CFI' correlation functions, as
can be seen by direct comparison with the respective expressions in section 2.2. While the
form of the three-point function was determined uniquely up to a coefficient, the four-point
function still had some freedom in form of a function f (q , C). Equation (3.54) determines
this function uniquely (to first order) for the particular case considered.
Chapter 4
Correlation Functions for Spinor and
Vector Fields
The techniques for the calculation of CFT correlation functions from the AdS/CFT corre-
spondence were explained in detail in chapter 3 using the example of the scalar field. In the
present chapter, massive vector and spinor fields are considered. Besides providing further
examples, these fields are important ingredients in physically relevant theories, although
not the only ones.
The E-prescription shall be used to regulate the relevant Ads actions in sections 4.1
and 4.2, in which the free vector and spinor fields, respectively, are considered. The mass
parameters will assumed to be generic. The correctness of this approach for the calculation
of CFT two-point functions was justified in section 3-1.3.
The first order interaction between spinor and gauge fields is considered in section 4.3.
It will be shown that the resulting three-point function obeys the standard Ward identity
d a t e d to the gauge symmetry.
CHAPTER 4. CORRELATION FUNCTIONS FOR SPaVOR AND VECTOR FIELDS 44
4.1 The Free Massive Vector Field
4.1.1 Solution of the Field Equations
The starting point is the action
with the usual expression for the field strength
The equation of motion derived from the action (4.1) is
which implies the subsidiary condition
Within the representation of anti-de Sitter space given by the metric (2.5) one can use
equations (4.2) and (4.3) to obtain an equation for Ao, which is
Introducing r i12 = n12 - d + 1, one realizes that equation (4.4) resembles the equation of
motion of the free scalar field with mass parameter r i l , equation (3.7). Hence, its solution
can be taken over and reads
with
The coefficient no(k) will be determined later from the boundary conditions.
Proceeding to the components A,, it is useful to introduce spin-l fields with Lorentz
indices (cf. appendix A.l) by
CHAMER 4. CORRELATION FUi'VC77ONS FOR SPLZVOR AND VECTOR FlEtDS 45
Then, the equation of motion for Ai, as obtained from equations (4.2) and (4.3), is found to
be
The solution of the homogeneous part of equation (4.8) can be taken over from the scalar
field, whereas the inhomogeneous equation is solved by making a good guess as to which
form its solution should have. The result is
One has now to impose the subsidiary condition (4.3), which in terms of the spin-1
fields reads
Inserting equations (4.5) and (4.9) into equation (4.10) yields
which determines a0 in the generic case of massive vector fields, but leaves it undetermined
in the massless case. In order to find a prescription, which is valid for both cases, the
Dirichlet boundary conditions shall first be imposed on the fields Ai. For this purpose it is
useful to write
Setting xo = E in equation (4.9) one then finds
E b iKe + k; [b& + i a o - ~ d + l ] = ~-!A,-,i(k),
k (4.13)
where the argument kc of the modified Bessel functions has been omitted, and A, i (k ) denotes the Fourier transform of the Dirichlet boundary value of the field Aim One can
determine bi and no from equation (4.13) by identifying the first term on the left hand side
with the right hand side and demanding that the second term on the left hand side be zero.
This yields
CHAPTER 4. CORRELAl7ON FL/NCTIONS FOR SPRVOR AND VECTOR FIELDS 46
Finally, imposing the subsidiary condition by substituting equations (4.12) and (4.15) into
equation (4.11) one finds the missing coefficient
where a functional relation of the modified Bessel functions has been used to rearrange the
denominator, and the constant A = it + d / 2 has been introduced.
Proceeding analogously to the scalar field, the next step is to make a connection be-
tween the boundary values and conformal fields Aori living on the horizon. Substitut-
ing equations (4.3, (4.12), (4.14), (4.15) and (4.16) into equation (4.9) and replacing K,(k€)
by the leading order term of its series expansion (3.17) for the limit E: + 0, one finds
where the constant cz is given by
Similarly, taking the limit in equation (4-5) yields
Equations (4.17) and (4.18) can be combined to
- kcad .*-r
A p ( x ) - $-' J ddy ~ c , i ( y ) I' (X - y), A - 1 (x- y)2* "
where I is the spin-1 representation of the coordinate inversion defined in equation (2.22).
Equation (3.19) suggests that the conformal horizon field should be defined by
and has conformal dimension d - d. Hence, it couples to a spin-1 operator of confomal
dimension A.
CHAPTER 4. CORRELATION FUNCnONS FOR SPZNOR AND VECTOR FIELDS 47
4.1.2 CFT Two-Point Function for Spin-1 Fields
As a last point in this section the ClT two-point function of conformal spin-1 fields Ji shall
be calculated using the AdS/CFT correspondence formula (1.13). Thus, the action (4.1)
has to be evaluated on-shell and expressed in terms of the boundary values AOpi. After an
integration by parts and using equation (4.2) ;he action (4.1) takes the value
where Foi = aoAi - aiAo contains the interesting part. Using the solutions (4.5) and (4.9)
with the coefficients obtained in equations (4.12), (4.14), (4-15) and (4.16) one finds
One now takes the limit E -t 0 by substituting the relevant terms of the series expansion
of the modified Bessel functions, equation (3.17). into equation (4.22). Experience from the
sczlar field tells that the relevant terms are proportional to k2"bii and k2"-'kikj. One obtains
these by keeping only the leading order terms for the denominators in equation (4.22) and
using the appropriate terms for the numerators. In particular. the kZa term from equation
(3.17) is needed only for Ka- in the numerator of equation (4.22). One finds
where the ellipsis denotes all other terms representing either local divergent terms. which
are cancelled by adding appropriate counter terms to the action. or terms of higher order
in E which vanish in the limit E -+ 0. Performing the integrals in equation (4.23) using
formula (B.2) and inserting the result into equation (4.21) yields
C H A M E R 3. CORRELATION FUNCl7ONS FOR SPaVOR AND VECTOR FIELDS 48
where Z is given by equation (2.35). To equation (4.24) the necessary counter terms must
be added, leading to the renormalized result
where equation (4.20) has been used.
Finally, using the AdS/CFT correspondence of the form
one can read off from equation (4.25) the finite distance two-point function of the operators
li as
Formula (4.27) is in agreement with the form expected by conformal invariance, equation
(2.33, for the two-point function of spin-1 operators of conformal dimension A.
In view of the fact that the integrals in equation (4.23) have to combine such as to give
exactly the expression for Zij in equation (4.27), the correctness of the result is a non-trivial
check of the E-prescription- Moreover, the result coincides for the massless case with the
one obtained in (651.
4.2 The Free Dirac Field
4.2.1 Solution of the Dirac Equation
The investigation of the AdS/CFT correspondence for spinor fields starts with the standard
Dirac action supplemented with a surface term,
The presence of the surface term was initially justified by the fact that the Dirac action
is zero on-shell and thus would not generate any CTT correlation functions (911. A very
elegant argument for its existence was later given by Henneaux 1901, who argued that the
surface term is necessary in the variational principle, because - as will be shown later -
not all the boundary values are fixed. This is similar to the well known Gibbons-Hawking
CHAPTER 4. CORRELATION FUNCTIONS FOR SPLNOR AND VECTOR FlELDS 49
term [74] suppIementing the Hilbert-Einstein action of gravity (cf- section 5). H e ~ e a u x ' s
method was applied by Rashkov [I411 to the case of Rarita-Schwinger fields.
The equation of motion for I) derived from the action (4.28) is the Dirac equation
where the matrices yp are the Euclidean gamma matrices, as described in appendix A.1.
Similarly, the equation of motion for $ is
Acting with ypa, on equation (4.29) one obtains the second order differential equation
The solution of equation (4.31), which does not diverge for + x, is obtained in a similar
fashion to the scalar and vector cases and is given by
where the spinors n' satisfy yon= = &a=. The expression (4.32) is in general not a solution
of the Dirac equation (4.29). in fact, substituting it into equation (4.29) one finds that the
spinors a- and n - must be related by
The next task is to impose boundary conditions on the solution (4.32). However, there
is a major difference to the scalar and vector cases. The origin of this difference lies in
the nature of the differential equations, which serve as the equations of motion for the
fields. The scalar case (cf. section 3.1) and vector case (cf. section 4.1) involve second order
differential equations. Hence, it is possible to impose two sets of boundary data on a given
solution, namely the field and its derivative. In the AdS/CFT context one demands instead
of the latter that the field be well behaved in the volume of Adsdtl, i-e. for 10 -+ m. This
yields a unique solution to the Dirichlet problem. On the other hand, the Dirac equation
(4.29) is a first order differential equation. The xo + cm behaviour of the solutions of the
CHAFER 4. CORRELATlON FUNCTIONS FOR SPUVOR AND VECTOR FIELDS -50
Dirac equation is crucial from the Ads field theory point of view and cannot be abandoned.
Hence, only half of the general solutions are available for fitting the boundary data, which
means that only half the components of the spinor cl, can be prescribed on the boundary,
the other half being fixed on-shell by a relation, which will be determined in a moment.
Moreover, this other half remains arbitrary off-shell, which means that the supplementary
boundary term in equation (4.28) is needed in order to make the variational principle well
defined.
This result is important also from a CFT point of view. Considering the boundary term
of the action (4.28) one realizes that, if the entire boundary spinor were prescribed, then
there would be only a contact term in the CFT two-point function. The trade-off is that the
correspondence formula yields only correlators for spinors, which have half the number of
components as the field +. This means that the boundary spinors are Weyl or Dirac spinors
for 1i even or odd, respectively.
Letting xo = E in equation (4.32) yields
where aE(k) is the Fourier transform of the
modified Bessel functions has been omitted.
two ways, namely by
boundary spinor and the argument kc of the
We can determine ai from equation (1.34) in
. + I kiy @, (k) a i ( k ) = - i c - ~ - - . K"1it
where @: = +(I k yo) #,. Substituting equation (4.36) into equation (4.35) one finds that - the boundary values JIz and $; are related by
The question as to which of the functiors @$ should be used as boundary data is in gen-
eral not a matter of choice, but is dictated by the E + 0 limit. Here thrw cases must be
CHAPTER 4. CORREtATION FUNCnONS FOR SPZNOR AND VECTOR ElELDS 51
distinguished. If m > 0, K,- I diverges slower than K,, 4 for E + 0 and thus @,i + 0, :
if +; is fixed. This is in agreement with the condition found in [91]. On the other hand,
$: cannot be prescribed for nl > 0, as $; would then diverge. The case nz < 0 is just vice
versa. For a1 = 0, K- 4 = K 4 and hence one can prescribe either of the functions +!J$. These
three cases shall now be considered separately.
4.2.2 The Case m > 0
Inserting equations (4.36) and (4.33) into equation (4.32) yields the final form of the solution
of the Dirac equation,
in order to find the connection between the boundary data +, and a conformal field +a living on the horizon, the limit E: + 0 should be taken in equation (4.38)- Using again the
leading order term of equation (3.17), one finds
where
One can solve in a similar fashion the equation of motion (4.30) for the conjugate spinor
and obtains
where $: = $,f (1 f yo). Again a relation between the components of the boundary
spinor is found and reads
Moreover, the E + 0 limit of the solution (4.40) is found to be
CHAPTER 4. CORRELA77ON FUNC77ONS FOR SPINOR AND VECTOR FlELDS 52
Equations (4.39) and (4.42) suggest the definitions
The boundary fields 90 and 6,' both have conformal dimension d / 2 - nl and thus
couple to CFI' operators of conformal dimension d / 2 + nz. The two-point function of these
operators can be found from the AdS/CFT correspondence formula, equation (1.13). In-
serting the solutions of the equations of motion into the action (4.28), the bulk term van-
ishes and the surface term can be written a s
Using the relations (4.37) and (4.41) one then finds
The expansion in powers of E is easiest carried out using the identity
Then, from equation (3.18) one finds
Substituting the significant term into equation (4.45) and carrying out the momentum in-
tegration using equation (8.2) yields
The renormalized action ifin is identified with the CFT effective action in the presence of
spinor sources +i and $:. However, reading off the CFI' two-point function of quasi-
primary spinors is a bit more subtle than the scalar case and will be explained in sec-
tion 4.2.5.
CHAPTER 4. CORRELATION FUNCTIONS FOR SPLNOR AND VECTOR FIELDS 53
4.2.3 The Case m < 0
The case rri < 0 differs from the case nr > 0 only in that the expression (4.11) acquires a
minus sign, i.e.
Moreover, the relations (4.37) and (1.41) must be inverted, since $: and I$; are the inde-
pendent fields now. Thus, using K , ( r ) = K-,(z),
Substituting equations (4.48) and (4.49) into equation (4.47) leads to
which after renormalization and the momentum integration becomes
4.2.4 The Case m = 0
The case nr = 0 is special for two reasons. First, it is a limiting case for both the previously
considered cases ni > 0 and rrl < 0. Hence, the CFT correlation functions of these two cases
must go to the same limit as nr + 0. Comparing equations (4.46) and (4.51) one already
sees that this is the case. Secondly, because K: (z) = fie-', the crucial integrals can be - calculated exactly without resorting to the expansion (3.17). This is an additional check of
the derivation with the expansion.
Consider for example equation (1.38) with nr = 0. Carrying out the momentum inte-
gra tion yields
whose E i 0 limit agrees with equation (4.39).
moreo over, it is straightforward to check the result (4.46) for n r = 0.
CHAPTER 4. CORRELA77ON FUNCnONS FOR S P N O R AND VECTOR FIELDS 54
4.2.5 C f l TwbPoint Function for Spinors
Having found the on-shell vaIue of the Ads action for spinors in equations (4.46) and (4.51).
the next step is to use it for the calculation of CFT spinor two-point functions. Here, two
cases must be distinguished depending on whether the dimension d is even or odd. This
distinction derives from the dimensionality of spinors, which is 21d/*1 [35].
The Cases with Odd d
If d is odd, then the horizon spinors have only half the number of components of the bulk
spinors. Hence, the independent boundary components form a Dirac spinor on the horizon
and couple to a conformal Dirac spinor. Therefore, the AdS/CFT correspondence formula
(1 -13) takes the form
where the superscript sign is chosen according to the sign of m, and I,!J~= and 6;' denote
the independent components of the boundary spinors only. The choice of the sign of m is
of no significance. The matrices y, can be given in the representation [35]
where < denote the Euclidean gamma matrices in d dimensions.
Now, the results (4.46) and (4.51) can be combined in the expression
Thus, because the functional derivative is only with respect to the independent compo-
nents I&= and Go', the resulting two-point function is
This is the expected two-point function for Dirac spinors of conformal dimension d / 2 + I m I, as can be seen by direct comparison with equation (2.37).
CHAf"7-ER 4. CORRELA77ON FUNCTIONS FOR SPINOR AND VECTOR FIELDS 55
The Cases with Even d
If d is even, then the bulk and boundary spinors have the same number of components.
This means that the matrices yi also form the Euclidean gamma matrices in d dimensions.
Moreover, the matrix yo is proportional to the chirality matrix of the d-dimensional Clifford
algebra. In the chiral representation, yo still takes the form as in equation (4.54). Then, the
relation (yo, yi) = 0 implies that yi must be of the form
where y,-- and yFC are some matrices. The important point here is that the diagonal
elements are zero.
In order to calculate CFT two-point functions, one must use the AdS/Cm correspon-
dence formula in the form
where the bulk Dirac fields and +z have masses lrrr 1 and - Inrl, respectively. The inde-
pendent boundary values couple to the Weyl spinors x=. The CFT two-point functions of
these Weyl spinors are obtained from equations (4.58) and (4.55) as
However, combining these correlation functions using equation (4.57) yields
Again, this is in agreement with the form expected from conformal invariance.
Generally, the correlation functions (4.56) and (4.59) display short distance singularities,
which should be cured using the methods available in the literature [12,31,123].
CHAPTER 4. CORRELATlON FUNCTIONS FOR SPINOR AND VEflOR FIELDS 56
4.3 First Order Interactions Between Spinor and Gauge Fields
Calculating the first order interaction between the spinor and massIess vector fields serves
two purposes- First, it provides another detail of the AdS/CFT correspondence in form of
the vector-spinor-spinor three-point function. Secondly, a check of the Ward identity cor-
responding to the gauge invariance will reveal that no supplementary surface term of the
order of the gauge coupling is needed. Of course, this is in agreement with the variational
principle.
The starting point is the action for minimally coupled spinor and gauge fields supple-
mented by the spinor surface term,
The following analysis will be done for m > 0.
The equations of motion derived from (4.60) are
and its conjugate
Following the method outlined in section 3.1.1, the gauge field is split into its free part A(')
and the remainder A( ' ) . Substituting (1.62) into (4.60) and using the equation of motion for
FIo1, one finds
Most importantly, all bulk terms vanish. Moreover, the Dirichlet boundary condition for
the fields Ai implies that the vector Green's function vanishes on the boundary. Hence, A::,)
and consequently the second term in equation (4.64) are zero. The first term only yields the
two-point function for the conserved currents I , which has been calculated in section 4.1.2.
However, the last term will give the twu-point function for the spinors and the three-point
function coupling I and the spinors. This surprising fact comes about as follows. Going
CHAM'ER 4. CORRELA77ON FUNCTIONS FOR SPLNOR AND VECTOR FELLS 57
back to the derivation of the spinor two-point function, one realizes that it was generated
by the relations (4.37) and (4.41) between the + and - components of the spinors on the
boundary. These relations will be altered by the presence of the interaction. Writing
where S(x, y) is the spinor Green's function defined by
one finds using equation (4.37)
where the argument kc of the modified Bessel functions has been omitted. Similarly, one
writes for the conjugate field
Thus, using equation (4.41)
Subs ti tu ting equations (4.67) and (4.70) into the spinor surface
finds that the contribution to the action of first order in q is
term in the form (4.44), one
- +;l)(x) ($Lo'+ (x) - *Lo)- ( x ) ) ] + C q q 2 ) .
'The relation between 5 and S is of no importance here.
CIlAPTER 4. CORREL.ATION FUNCTIONS FOR SPINOR AND VECTOR mELDS 58
On the other hand, from equations (4.69) and (4.66) one can obtain
and
respectively. Inserting equations (4.65b), (3.73), (4.68b) and (4.72) into equation (4.71) one
then finds
Remarkably, equation (4.74) is identical to the minimal coupling term. It is determined
by a bulk integral, which means that the bulk behaviour of the fields can be used. Substi-
tuting equations (4.17). (4.18) (with A = d - I), (4.39) and (4.42) into equation (4.74). the
tedious direct calculation involves Feynman parameterization of the denominator (cf. ap-
pendix 8.2)' integrals of the type described in appendix 8.3 and heavy numerator algebra.
The result is
where x,t, = x, - xo.
The calculation can be simplified significantly by making use of the coordinate inver-
sion. Transforming all fields with the appropriate representation matrix, it folIows from
equation (2.25) that 1'(11 = - 1"). The transformed expression I f ( ' ) is much easier to calcu-
late than I ( ' ) , if one inverts with respect to the position of the vector field.
After further algebra one finds that equation (4.75) can be rewritten in the form
- iqFnt T(d/2) 1 ( I , ( x z ) x ~ ( x I ) z - ( x ~ ) ) = r vx13i ( 6 j k - 2
~ r 2 ( d - 1 + 2nz) 4,x$J2x~;' 1 1 3 (4.76)
1 - Yo x ( - ) [ ( d - z)#+ + (Zm + 1)bk1] 1. Xi3 *k
CNAPTER 3. CORRELA77ON FUNCnONS FOR S P m O R AND VECTOR FIELDS 59
Using the same arguments as in section 4.2.5 one can conclude that the CFT vector-
spinor-spinor three point function involving Dirac spinors is
where fi are the appropriate gamma matrices in d dimensions. This equation is in agree-
ment with formulas (2.38) and (2.43), which represent the form expected from conformal
invariance.
The final step is to check the Ward identity related to gauge invariance (cf. the general
Ward identity (2.49))
From equation (4.77) one finds
Comparing equations (4.79) and (4.56) (or (4.59)) with equation (4.78) one sees that the
Ward identity is satisfied. This result is significant, since it implies that, to first order in q,
no extra supplementary surface term is required in the action for interacting fields.
Chapter 5
Ads Gravitons and CFT Energy
Momentum Tensor
Ads gravitons play a special role in the AdS/CFT correspondence. Their boundary values
couple to the ClT energy momentum tensor, which is a very important quantity in confor-
mal field theories. Its properties in operator product expansions include the existence of
central charges, which in turn are closely related to the well known Weyl anomaly.
The work presented in this chapter considers two aspects of the gravitational field in
the AdS/CFT correspondence. First, in section 5.1 linearized gravity is considered, whose
solution is used in section 5.2 to obtain the two-point function of CFT energy momentum
tensors. Generally, the rr th order term in the expansion of the gravity action around the Ads
background (2.5) in powers of metric perturbations will yield the n-point function of the
CFT energy momentum tensor. Here, only the two-point function shall be calculated using
the E-prescription. It was first obtained by Liu and Tseytlin [106], who used the asymptotic
prescription. The three-point function was calculated by Arutyunov and Frolov [16].
Secondly, section 5.3 deals with the counter terms which are necessary to renormalize
the gravity action. These are calculated for an arbitrary boundary metric without the need
to linearize around a given background. Non-covariant counter terms appear for even
dimensions d and break the global scale invariance of the action. The Weyl anomaly, which
is associated with this symmetry breaking, is first discussed in general terms and then
calculated for d = 2,3,6.
CHAPTER 5. ADS GRAVTTONS A N D CFT ENERGY MOMENTUM TENSOR 61
Ln the context of the AdS/CFT correspondence, the Weyl anomaly was first derived
by Henningson and Skenderis [92,93j. Further discussions, such as the Weyl anomaly in
higher derivative gravity and the connection of the AdS/CFT counter terms to counter
terms used to regularize energy momentum tensors in General Relativity, can be found in
[129, '10,77,2,153,130,50,128,23,124,134,115].
The starting point of the consideration of the gravitational field is the gravitational
action without matter fields:
It consists of the Hilbert-Einstein action with a cosmological constant A = -d (d - I ) / (212),
the Gibbons-Hawking term and a cosmological boundary term which was justified in [I061
in order to cancel the bulk volume singularity. It can also be obtained using a Hamiltonian
formalism [14,15,127].
The nature of the problem strongly suggests to use the time slicing formalism [119],
which is also called the initial value formulation of gravity [156]. The reader is referred to
appendix C, where the principles and useful formulae of this formalism are summarized.
5.1 Solution of the Linearized Einstein Equation
The beauty of the time slicing formalism lies in the fact that the non-physical degrees of
freedom in Einstein's equation can be removed by simply choosing handy expressions for
the lapse function n and the shift vectors IZ'. In linearized gravity, which introduces the
concept of gravitons (or gravitational waves) [la], all expressions are expanded in powers
of perturbations around a given background. Thus, it is handy to choose r l and ni identical
to their respective background expressions. As before, the length scale 1 will be chosen
equal to 1 throughout this and the following section. With the Ads background metic
(2.5) this directly translates into
L IZ = - and n' = 0.
Xo
Then, the remaining physical degrees of freedom are the metric components gij, which will
be expanded around the background. Writing as usual
CHAPTER 5. ADS GRAVITONS AND CFT ENERGY MOMENTUM TENSOR
where gj, = xi2b i i is the background metric, the expression hi = gkhti shall be used as
the graviton field. Henceforth Latin indices will be raised and lowered with the Euclidean
metric.
The next step is to linearize the equation of motion and the two constraints. First,
linearizing the constraint equations (C.13) and (C.14) yields
where A = a,& and
respectively. Secondly, the equation of motion (C.16). linearized and suitably combined
with equations (5.1) and (5.5) leads to the equation
The equations (5.4), (5.5) and (5.6) are identical with the ones found by Arutyunov and
Frolov [16].
It is now possible to solve equations (5.4), (5.5) and (5.6) and impose the Dirichlet
boundary condition on the graviton fields 8;. Comparing the trace of equation (5.6) with
the constraint (5.4) yields the simple equation for I t ,
whose solution is given by
Solving the constraint (5.5) one then obtains
Substituting equations (5.8) and (5.9) into equation (5.4) yields
CHAPTER 5. ADS GRAVlTONS AND CIT EhrERGY MOMENTUM TENSOR 63
Thus, writing
and
the trace and divergence of hi have been found as functions of their boundary values a =
I I ( ~ ~ = ~ and ni = il,hjlx,,=,.
The next step begins with substituting the solutions (5.8) and (5.9) into the equation of
motion (5.6), which yields the inhomogeneous differential equation
The homogeneous part of equation (5.13) is identical with the equation of motion of a
massless scalar field, whose solution can be taken over from section 3.1.2. For the particular
solution one can make the ansatz
and obtains
Taking the trace and divergence of the particular solution (5.14) and comparing with equa-
tions (5.8) and (5.9) one finds h ( p ) = Ir and d j k ( p ) j = a,hj, respectively, which means that
the homogeneous solution of equation (5.13) must be traceless and transversal. Combining
the homogeneous and inhomogeneous solutions and using the relations (5.15) and (5.10)
one finally obtains the free graviton field in radiation gauge:
CHAPTER 5. ADS GRAWONS AND CFT ENERGY MOUENTUM TENSOR 64
where k; ( k) is the Fourier transform of the boundary data h; 1, =,. Moreover, setting xo = E
in equation (5.16) yields the coefficient
which is the traceless transversal part of hi in momentum space [16].
5.2 CFT Two-Point Function of Energy Momentum Tensors
5.2.1 General Formalism
In order to calculate the two-point function of the CFT energy momentum tensor, the action
(5.1) must be expanded around the Ads background. Terms of first order in the gravitons
Ii", will not appear on-shell, whereas terms of second order yield the two-point function.
Higher order terms naturally lead to higher order CFT correlation functions, but will not
be considered here.
Substituting Einstein's equation (C.15) and the gauge conditions (5.2) into the action
(5.1) and expanding to second order in hi yields the on-shell action
It shows that the normal derivative of the field on the boundary, dolt; l x , = E , contains all the
information needed.
Before continuing with the explicit calculation, it is useful to introduce the projection
operators
which are the transversal and traceless transversal projection operators, respectively. One
can easily show from equation (5.17) that hi is given by
CHAPTER 5. ADS GRAVTTONS AND CFT ENERGY MOMENTUM TENSOR 65
Hence, from equation (5.16) one finds
Inserting equation (5.22) into the action (5.18) one obtains
11; ( k ) . (5.23) (d - l)k2
This expression must be regularized and renormalized, because there are terms divergent
in the E -+ 0 limit. Then, the finite action Ih is identified with the CFT effective action W
in the presence of sources k;,
In this case, 9;; = 6;;. but this will be generalized in section 5.3.
For the regularization, three cases must be distinguished. This distinction stems from
the fact that for even d the index of the modified Bessel function appearing in equation
(5.23) is an integer, and thus the expansion must be treated as in section 3.1.3. Moreover, the
case d = 2 is special, because the last term in equation (5.23), which is divergent for d > 2,
becomes finite and contributes to the two-point function. The cases of odd d, even d > 2
and li = 2 will be considered separately in sections 5.2.3, 5.2.4 and 5.2.5, respectively, and
will be unified in section 5.2.6 using dimensional regularization. However, before starting
to calculate the CFT two-point functions, the focus in section 5.2.2 will lie on potentially
problematic divergent terms for d > 2.
5.2.2 Cancellation of Non-Local Divergent Terms for d > 2
Before turning to the renormalized finite action, it is important to consider some of the
divergent terms in the action (5.23). One realizes that for d > 2 the last term, which is
proportional to cZFd, is divergent and contains a non-local contribution from the factor
k' k;hJkc in the projection opera tors. Non-local divergent terms a prohibited, since they
cannot be cancelled by counter terms, which are necessarily local. Hence, there must be a
contribution horn the first term which cancels this non-local divergent term.
Looking at the expansion of the modified Bessel function, which starts like
za K. (i) = 2'- I r ( a ) [I + z2 4(1 - a) + - - I ,
CHAPTER 5. ADS GRAVITONS AND 6FT ENERGY MOMENTUM TENSOR 66
one obtains
Inserting equation
tional to are
(5.25) into the action (5.23) one finds that
(5.25)
the divergent terms propor-
where equation (5.20) has been used. From the definition (5.19) of the transversal projection
operators one realizes that the terms containing FkjKki cancel. The other t e r n contain at
least one factor kZ, which cancels the k' in the denominator, thus making the expression
local. Moreover, all other divergent terms, which have not been written out in equation
(5.26), stem from terms in the expansion (5.25) which contain at least two powers of k2,
thus again yielding only local expressions. Hence, all divergent terms are local and can be
cancelled by adding appropriate boundary counter terms which are discussed in detail in
section 5.3.
5.2.3 Two-Point Function for Odd d
For d odd, the index of the modified Bessel function in equation (5.23) is not an integer.
Therefore, the expansion (3.17) can be used, where the interesting term yielding the finite
result is again the leading non-analytic one. Hence, inserting the expansion (3.17) into
equation (5.23) one finds after regularization and renormalization
From equations (5.27) and (5.24) one can directly read off the two-point function CFT of
energy momentum tensors in momentum space as
which is transverse and traceless as expected.
CHAPTER 5. ADS GRAVrTONS A N D CFT ENERGY MOMENTUM TENSOR 67
The inverse Fourier transform of equation (5.28) is obtained using the integral (B.2),
which yields
where the position space expression of
analysis one finds
the projection operator must be used. After further
where Z(x ) is the spin-1 representation of the inversion, given by equation (2.35), and C J , ~
was defined in equation (3.10).
Finally, a comparison with equations (2.37) and (2.42) reveals that the two-point func-
tion (5.30) has the exact form expected from conformal invariance.
5.2.4 Two-Point Function for Even d > 2
For d even, the index of the modified Bessel function is an integer. Hence, one cannot
use the generic expansion formula (3.17), but must resort to the expansion (3.21). As in
section 3.1.3, the non-local finite terms of the action are obtained from the leading term
with a logarithm. Hence, one finds from equation (5.23)
where A is some constant which cannot easily be given as an expression for arbitrary even
~f. From equation (5.31) one can read off the CFT two-point function in momentum space,
As for odd ti, this two-point function is traceless and transversal as expected.
One can proceed to caIculate the inverse Fourier transform of the two-point function
(5.32). For simplicity, only the non-local contributions are considered here. Thus, the inter-
esting term contains Ink and is treated in a similar fashion as in section 3.1.3 yielding the
result
CHAMER 5. ADS GRAVI'TONS AND CFT ENERGY MOMENTUM TENSOR 68
which is identical to equation (5.29). Thus, the non-local part of the C E two-point function
for even d is identical to equation (5.30), and one could argue that one might have used
dimensional regularization to obtain this result - setting d = 2 ( n + E ) , using the generic
expansion formula and letting E -P 0 at the end. In addition, there are local contributions,
which can be obtained via a careful consideration of the short-distance singularity. The
necessary details can be found in [131].
5.2.5 Two-Point Function for d = 2
This case is very special, not because of the expansion of the modified Bessel function, but
because the traceless transversal projector P(', vanishes identically in d = 2. This is easily
seen from equation (5.20). Hence, one can read off the two-point function from equation
(5.23), which yields
A very important observation from equation (5.34) is that
i.e. the two-point function is transversal, but not traceless. This indicates the breaking
of the Ward identity (2.51) and directly relates to the existence of the conformal anomaly
which will be discussed in section 5.3.4.
The non-local position space two-point function is obtained by inverse Fourier trans-
forming equation (5.34). This is done using dimensional regularization, d = 2 + 2 ~ , which
yields
Then, using x-*' zs 1 - c ln x2 one finds the standard result [51]
where c = 24n denotes the central charge of the Virasoro algebra. Notice that one must
re-introduce the gravitational constant into the final result-
CHAPTER 5. ADS G R A W O N S AND CFT ENERGY MOMENTUM TENSOR 69
5.2.6 Dimensional Regularization for Even d
The obvious agreement between the results for even d > 2, equation (537), with the generic
expression (5.29) suggests the use of dimensional regularization in order to formally unify
the calculations. For completeness, the dimensional regularization shall be carried out
now in momentum space. It will turn out that for d = 2 the expressions exactly agree,
whereas for larger w e n d there occurs an ambiguous term, which does not have physical
significance.
Thus, let d = 2(n + E > in equation (5.28), which can be rewritten as
Expanding equation (5.38) in a Laurent series in E and discarding the divergent term pro-
portional to 1 / ~ , one obtains the renormalized two-point function
Notice the presence of the last term in the braces, which stems from the expansion of the
projector Pi:. In order to make a connection with the previous results, equation (5.39) should be com-
pared with equations (5.32) and (5.31). First, one observes that (5.39) is not traceless, in
contrast to equation (5.32). Hence, for d > 2, the disagree, but the difference is not
significant, because it can be modified at will by the addition of a local term to the CFT
effective action (for d = 4 such a term is proportional to I d k ~ g ~ ~ ) [131]. Ln other words,
the counter terms used in the two methods are different, and this difference amounts to the
insignificant difference of the finite results.
Secondly, for n = 1, equation (5.39) agrees exactly with equation (5.34) (remember
P:', = 0 for d = 2). This is as expected, because the trace, equation (5.35). is related to
the coefficient of the unique term of the conformal anomaly (cf. section 5.3.4), which is
a physical quantity. Moreover, the counter term in dimensional regularization, which is
given by equation (5.26), agrees up to the usual insignificant term -- I/c with the counter
term used with the exact expansion of the modified Bessel function (notice E: # e).
CHAPTER 5. ADS GRAVITONS AND CFT E ' G Y MOMENTUM TENSOR 70
5.3 Counter Terms and the Weyl Anomaly
5.3.1 The Weyl Anomaly in Conformal Field Theories
Conformal or Weyl anomalies play a significant role in modem theoretical physics, be it in
high energy physics, cosmology, or statistical mechanics. They naturally appear in theo-
ries whose action functionals are invariant under local re-scalings of the mekic, g,,(x) -t
a(x)g,, (x). For example, for if = 2 the Weyl anomaly is closely related to the central charge
c of the Virasoro algebra [I%]
To be precise, the Weyl anomaly takes the form (431
with c = D - 26 for bosonic string theory [138], where D is the number of space-time
dimensions. For the critical dimension, D = 26, bosonic string theory is anomaly free.
Similar arguments hold for fermionic strings, whose critical dimension is D = 10 [139].
The reader is referred to the review (461 for a somewhat historical account of the devel-
opments and for a more comprehensive list of references on this subject.
There are many ways to calculate the Weyl anomaly (see again [46] for references).
Perhaps the most fundamental, Fujikawa's method (71,721 attributes the lack of quantum
Weyl invariance in a classically Weyl invariant theory to the non-invariance of the measure
in the functional integral of the quantum theory.
Let W denote the effective action of a quantum field theory, defined by
Defining an infinitesimal Weyl rescaling by bgi i (x) = 2A(x)gii (x), the Weyl anomaly is
given by
On general grounds [25,36], the Weyl anomaly occurs only, if the dimension d is even,
and it is of the form
CHAPTER 5. ADS GRAVITONS AND CFT ENERGY MOMENTUM TENSOR 72
where Ed is the Euler density and Id is a conformal invariant. The last term is insignifi-
cant, because it can be changed at will by the addition of a finite local counter term to the
effective action [36, 1311.
For d = 2, one has the well known E-, = R, and there are no further conformal invari-
ants, i.e. b = 0 in equation (5.42).
For d = 4, one finds
where Cijkc is the Weyl tensor. The coefficients n and b have been computed by many
authors and are given by [22]
where 114, l r ~ and n v are the number of free scalar, Dirac spinor and vector fields in the
theory, respectively.
For ti = 6, I4 contains three algebraically independent terms, which have been given in
reference [25].
Within the AdS/CFT correspondence, one expects to find conformal anomalies for even
ii, because the boundary CFT is supposed to be a quantum theory. Put converseIy, the ex-
istence of conformal anomalies in the AdS/CFT correspondence proves that the boundary
CET really is a quantum theory, since the conformal anomaly does not appear in cIassica1
theories. Having found the explicit expressions for the anomaly, one can then constrain the
possible field content of the boundary CFT, e.g. by using equation (5.43) for d = 4.
5.3.2 Scale Invariance and its Breaking by Non-Covariant Counter Terms
The AdS/CFT correspondence, as formulated in section 1.3 and used in chapters 3 and 4,
relates CFI's on d-dimensional conformally cornpactified Euclidean spaces with field theo-
ries on (d + 1 )-dimensional Ads spaces. Strictly speaking, the case of gravitons considered
in section 5.1 is already a generalization, in that it deals with an Einstein space R with
negative cosmological constant, which differs only slightly from the Ads space used as
CHAPTER 5. ADS GRAWONS AND CFT ENERGY MOMEhrTUM TENSOR 72
the background. However, this generalization presents no problem - on the contrary, it
is natural to extend the AdS/CFr correspondence on the "Ads" side to arbitrary Einstein
spaces R with negative cosmological constants [160]. Such a space R possesses a horizon
manifold 312, on which it determines a conformal structure. For simplicity, 3f2 is assumed
to have the topology of a &sphere in the sequel. Then, in generalization of equation (2.5),
there is a set of coordinaies on R for which the metric takes the form [52,76]
where g i , ( x , 0) = g i i ( x ) is the horizon metric. For the following considerations it is useful
to use the dimensionless variable p = 4 / l2 , i-e. ds2 takes the form
1' 1 . - ds' = - (dp) ' + -2,; (x, p)dx'dxl.
4 9 P
Besides any coordinate symmetries, the metric (5.45) is invariant under
which ccnstitutes a global rescaling of the horizon af2.
Obviously, the gravity action on the manifold R is invariant under the rescaling (5.46).
Hence, the AdS/Cm correspondence implies the scale invariance of the CFT effective ac-
tion, if it were not for non-covariant divergent terms, which have to be cancelled by non-
covariant counter terms. Such divergent terms have the form
where Jd, ddx f i L C is itself scale invariant, and the cut-off boundary is characterized by
p = E . Hence, for the scale transformation (5.46) with a = 1 + 2 A one finds
which, because of I = ffi, + Idiv and 61 = 0, implies
Equation (5.48)
to the necessity
shows that the scale invariance of the CFT effective action is broken due
to renormalize with a non-covariant counter term. The right hand side of
CHAPTER 5- ADS GRAVKFONS AND CFT ENERGY MOhIENT[/M TENSOR 73
equation (5.48) is the integrated Weyl anomaly. Within the A d S / C n correspondence, Ih
is identified with the effective action W of the boundary CFT: Hence, comparing equations
(5.48) and (5.41) yields the anomaly
where I' is continuous, but otherwise arbitrary. This determines the significant terms of
the Weyl anomaly (5.42).
5.3.3 General Formalism for the Calculation of Counter Terms
As for the calculation of the two-point function of CE.T energy momentum tensors, the
starting point is the gravity action (5.1). Again the time slicing formalism shall be used,
but no expansion around a given background will be carried out. The derivation involves
only the fact that the Einstein manifold R is locally asymptotic Ads, i.e. can be given the
metric (5.45). Thus, the results will be valid for any prescribed metric on the boundary df2. Let p = XO be the "time" coordinate of R. In order to utilize the advantages of the time
slicing formalism, one can impose the gauge choices
After isolating the leading behaviour of gi, for small p (which can be found from the equa-
tion of motion or by simply using equation (5.45)) by defining
the equation of motion (C-16) becomes
Here, R,, = Ri, is the Ricci tensor of the time slice hypersurface. After raising an index
with the metric g ' ~ one realizes that it is handy to define the quantity
In fact, equation (5.52) becomes
1 ' ~ : + ( d - 2)8; + 06; - p (20jt + ~ q ) = 0.
CHAPTER 5. ADS GRAVITONS A N D CFT ENERGY MO-.MENTUM TENSOR 74
Similarly, rewriting the constraints (C.13) and (C.14) using equations (5.50), (5.51) and (5.53)
leads to the equations
and
respectively.
In the AdS/CFT correspondence the on-shell value of the action (5.1) as a functional of
prescribed boundary values g,, has to be calculated, where the boundary is given by p = E .
First, the on-shell action is easily found to be
In order to find the singular terms in the limit E -+ 0, one can differentiate equation (5.57)
with respect to E, which leads to
The trace of the equation of motion (5.54) has been used in order to simplify this expression.
One can find the singular terms by calculating 0 from equations (5.54), (5.55) and (5.56) as a
power series in o, keeping only terms of order smaller than ~ i . Thus, for odd d the singular
terms in equation (5.58) are proportional to powers E - " + ! . On the other hand, for even d
equation (5.58) contains a term proportional to 1 / ~ , which yields a corresponding term
proportional to In E in [. As explained in section 5.3.2, this logarithmic divergence is the
source of the Weyl anomaly in the regularized finite action. The counter terms shall now
be calculated for the cases d = 2,4,6.
There is not really much to do for d = 2. In fact, the divergent term in equation (5.58) is
obtained from the leading order solution for 8. Using the constraint (5.55) one finds
CHAPTER 5. ADS GKAVITONS AND CFT ElVERGY MOIMENTUM TENSOR 75
Hence, the divergent term in the action is
By comparing equations (5.47), (5.48) and (5.60) one obtains the well known Weyl
anomaly for d = 2,
Comparing this to the standard result (5.40) one finds the central charge c = 241 n, which
is in agreement with the result from the two-point function (cf. sectio~ 5.2.5, where 1 = 1).
Starting from the constraint (5.55) one finds
Here, the leading order behaviour of the term in parentheses is sufficient. The equation of
motion (5.54) gives
which in turn yields
Hence, one finds
One can read off from equation (5.63) the Weyl anomaly for d = 4,
which in turn yields the coeffidents a and b (cf. equation (5.42))
Cl3MTE.R 5. ADS CRAWTONS AND CFT ENERGY MOMENTUM TENSOR 76
Again, this can be compared with the calculation of the two-point function in sec-
tion 5.2.4. Erdmenger and Osbom (511 gave the general expression
where Gj,, was given in equation (2.42). Moreover, the constants CT and b are related by
CT = 6 4 0 b / d . On the other hand, from equation (5.30) one can read off CT = SO/$,
which yields b = 1 / 8 in agreement with equation (5.65) for I = 1.
Finally, the expressions for a and b shall be compared to the expectations from the large
N limit of N = 4 super Yang-Mills theory. For this purpose, the gravitational constant
must be re-introduced. Using q o = 2 d I 2 l 4 / ~ (cf. equation (1.3)), one has to multiply
everything with
where
is the volume of the compactification sphere s5. Thus, one finds
On the other hand, for the particle content of N = 4 super Yang-Mills theory [89],
n+ = 6, n$ = 2 and n v = 1 (times N~ - 1 for the adjoint representation of the SU(N)
gauge group), one expects from equation (5.43)
This is in agreement with equation (5.66) in the large N limit, which supports Maldacena's
conjecture about the duality between the large N limits of type W supergravity on Adss x
5' and Af = 4 super Yang-Mills theory (cf. section 1.2).
The constraint (5.55) yields
CEZAPTER 5. ADS GRAWONS AND CFT ENERGY MOMEN7UM TENSOR 77
where the term in parentheses has to be calculated up to order E. Starting from the equation
of motion (5.54) one obtains
which in turn yields
The quantities B and t$' can be found by differentiating the equation of motion (5.54) with
respect to p, leading to
The missing quantity R( is given by
where the constraint (5.56) has been used. Then, taking the trace of equation (5.71) yields
Thus, substituting everything back into equation (5.68) leads to
Finally substituting equations (5.67) and (5.73) into equation (5.58) yields the result
Again, the conformal anomaly A6 can be read off from equation (5.74), and the result is
in agreement with the expression found in [92]. Moreover, it agrees with a result derived
using Weyl co-cycles [25] up to an irrelevant total derivative term.
Chapter 6
The Wess-Zumino Model on Ads4
It is certainly wise to start with simple examples when exploring new terrain. Hence,
being the simplest supersymmetric field theory, the Wess-Zurnino model is considered in
this chapter. The Wess-Zumino model was originally constructed for Minkowski space-
time [158,159], but has been generalized to Ads4 in [97,26,28]. Moreover, it is a choice of
simplicity to pick Ads4, i.e. d = 3, because the superalgebra does not contain additional
bosonic operators (cf. section 6.1 and [152]).
The work in this chapter is presented in a reasonably self-contained fashion.' The
reader is referred to appendix D.1 for the necessary background about spinor Grassman-
nian variables. In section 6.1 the N = 1 supersymmetric grading of SO (4,l) is constructed,
which is, in section 6.2, compared to the N = 1 superconformal algebra in d = 3. Based on
this superalgebra, the relevant superspace is constructed in section 6.3 with the focus on
a space-time covariant description. The somewhat lengthy calculation of Killing spinors
has been put into appendix D.2. Then, scalar and chiral superfields, their components and
supersymmetry transformations are considered in section 6.4. Finally, section 6.5 contains
the construction of the free, massive Wess-Zumino model. Its supersymmetric action will
be derived from the superspace formalism. The conformal dimensions of the correspond-
ing boundary operators are read off from the mass parameters and obey the conditions for
the conformal dimensions of the d = 3 super-Poincar6 multiplet. This provides a good
indication that the Ads/= correspondence is valid also for supersymmetric theories.
'The main reason for not simply using the available literature is a pedagogical one: The author learned a
good deal more about supersymmetry.
CNAPTER 6. TNE WESS-0 MODEL ON ADS4
6.1 The J i f = 1 Supersymmetry Algebra
In order to find a supersymmetric extension of Ads space, a superalgebra containing the
Ads symmetry algebra in its bosonic part must be constructed. As shown in section 2-1.2,
the symmetry algebra of Euclidean Ads4 is so(4,l). Its generators MAB satisfy the com-
mutation relations (2.13). Starting with these relations it is straightforward to construct the
(complex) N = 1 grading of so(4.1).
First, one needs a spin-; representation of the symmetry generators MAB. The use
of the Minkowski fivespace for the construction of Ads space (cf. section 2.1 .I) suggests
the introduction of 4 x 4 gamma matrices ?A satisfying (PA, pB) = 2vA& Then, the spin
matrices in five dimensions are SAB = f [yA, yB J, and they form the spin-$ representation of
the algebra elements MAB. The gamma matrices of the four-dimensional Euclidean Lorentz
frame of Ads4 are given by
satisfying (y,, yb) = 26,6, (a, b = 0,1, 2,3)- Secondly fermionic generators Qa are introduced (a = 1,2,3,4), which forin the odd
part of the superalgebra and transform as so(4,l) spinors, i-e.
Finally, the superalgebra closes with the anti-commutator (see appendix D.1 for matters
of notation)
Two remarks are due at this point. First, the validity of equation (6.3) is conditional upon
the fact that thefivedimsionnl Minkowski algebra is graded. For higher dimensions (eg.
Ads5) one would have to introduce additional bosonic operators to obtain closure of all
Jacobi identities [152]. Secondly, the equations (2.13), (6.2) and (6.3) define the complex
superalgebra B (0 /2) , whose real form is osp(1,4) [351. Unfortunately, osp(l,4) does not
contain so(4,l) in its even part, which means in other words that no Majorana spinors exist
for the Minkowski five-space. However, osp( l,4) contains so (3,2), which is the symmetry
group of Ads4 with Minkowski signature. Resorting to a Wick rotation at the end to make
the results valid, this fact shall be ignored and the analysis carried out formally.
C'HAP'ER 6. THE WESS-ZUMlNO MODEL ON Am
6.2 Superconformal Algebra
The discussion in section 2.1.2 showed the isomorphism between the Ads symmetry alge-
bra and the conformal algebra generating the conformal transformations of the Ads hori-
zon. In this section, the isomorphism between their N = 1 supersymmetric extensions will
be demonstrated.
The N = 1 grading of the conformal algebra (2.18) is well known in the literature [152],
but a direct comparison with the superalgebra constructed in section 6.1 seems useful. This
is done by choosing a particular representation of the matrices yA. Choosing
where ci are the Pauli spin matrices and 1 is the 2 x 2 unit matrix, one easily finds from the
definition (2.14) the spinor representations of the conformal basis elements, which are
Splitting the spinor operator Qa into two 2tomponent spinors,
equation (6.2) leads to the commutators
[K;,qa] = ( u ; s ) ~ , [K;, sa] = 0.
Furthermore, the charge conjugation matrix 5 has the form
ClHPTER 6. THE WESS-ZUMINO MODEL ON ADS4 81
where c is the charge conjugation matrix in three dimensions. Hence, using the identity
equations (6.3) and (6.5) yield the anti-commutators
{qa, q p ) = 2 ( ~ c - l ) a@ Pi,
aS {sa, s@} = -2 (BC-') Ki,
aS Iqa, so} = ( 2 ~ - ' D - d'lij) -
Equations (2.18), (6.6) and (6.8) form the N = 1 superconformal algebra in three dimen-
sions [152].
This section shall be dosed by indicating the relations which the superconformal sym-
metry imposes on the conformal dimensions of the fields of a given super multiplet Obvi-
ously, the operators Lip Pi and qa form the three-dimensional N = 1 Poincare superalge-
bra. Therefore, consider for simplicity a scalar super-Poincare multiplet consisting of the
scalar fields O and 3 and the spinor field X, which satisfy the supersymmetry relations
Imposing conformal symmetry- on the multiplet means that the scaling dimensions of the
fields must satisfy
This relation is obtained by acting with the commutator [D, q] on the fields 0 and x and
using equations (2.34). Notice that the spinor operator s is expressed in terms of K; and q
and thus does not introduce new fields into the mu1 tiplet.
6.3 Construction of Ads Superspace
In order to obtain the (N = 1) supersymmetric extension of Ads4, Grassmannian coordi-
nates 8P must be introduced in addition to the Ads coordinates x p . Then, one postulates
CHAPTER 6. THE WESS--0 MODEL 0NADS4 82
that the symmetry algebra of the superspace is given by the graded Lie algebra constructed
in section 6.1. Thus, the inLinitesimal super-coordinate transformations can be determined
from the knowledge of the superalgebra. The method to be used has been described by
Zurnino [I611 and applies to any group Q with a subgroup 31.
In the case at hand, an element g E G is uniquely represented by
where e is some Grassmannian spinor variable, whose relation to the coordinate spinor 6 will be defined later, and h(x) E 31 = S0(4,1) is a function of the coordinates xp. Then,
by virtue of the group axioms one can write
and consider the transformations + and x + x' as induced by the group element
go. In the following the abbreviations O = {Q and M = $ d B M A s will be used. In the
case of go E 71, i.e. an even transformation, equation (6.12) takes the form eMeeh (x) =
ee'eM'h(x), where M' and 0' are determined by the BakerCampbeLl-Hausdorff formula.
For infinitesimal M one finds M' = M and 8' = 0 + [M, O]. By definition, the even part
eMh(x) = I z ( x ' ) yields equation (2.17) and thus does not contain new information, whereas
the odd part yields a h e a r transformation of the Grassmannian spinor variable, namely
On the other hand, for odd transformations, go = eR with the abbreviation R = EQ,
one writes
Then, with R infinitesimal, one finds
and
CHAPTER 6. THE WESS-ZUMZlVO MODEL ON Am
where Zumino's notation (1611
1 A Y = Y, X A Y = [X ,Y] , x2 A Y = [X,[X,Y]], etc.
has been used. Equations (6.15) and (6.16) are evaluated explicitly by using the anti-
commutator (6.3) and various Fiem identities listed in appendix D.1, leading to
and
respectively. It turns out that by defining the Grassmannian superspace coordinates by
and calculating their transformation laws from equation (6.17) one obtains
which is considerably easier than equation (6.17). The coordinate 6 still is an SO(4,l)
spinor, i-e. it transforms under even transformations as
Moreover, equation (6.18) becomes
which determines 6xp via equations (2.15) and (2.17). Hence, expressing M in terms of the
conformal basis using equation (6.7) yields the transformation parameters
CHAPTER 6. THE WESS--0 MODEL ON ADS4
Thus, the supersymmetry transformation Sxp is given by equation (2.17) using the param-
eters of equation (6.23).
Although this solves the problem of finding the superspace transformations, a space-
time covariant formulation would be much more desirable. Such a formulation involves
the Killing spinors which are calculated in appendix D.2. In fact, it is easy to show from
equation (D.2) that the quantity Q A ~ ~ A - ' ; \ is a Killing vector for any spinors fi and 2. On
the other hand, dx' is a Killing vector and can thus be expressed in such a form. A direct
comparison using equations (2.17), (6.23), (D-ll), (D.12) and (D.15) shows that
Equations (6.20) and (6.24) represent the supersymmetry transformation of the Ads super-
space in a space-time covariant form. It is with this form that one can hope to effectively
carry out the calculations involving superfields. Moreover, it will aliow these formal re-
sults to be carried over to the Lorentzian signature case, where Majorana spinors exist.
Notice that dxp in equation (6.24) is generically complex for Euclidean signature, because
no Majorana spinors exist in this case.
To conclude this section, the invariant integral measure for integration over the Adsl
superspace shall be found. First, one observes that the bosonic part dx = d4x (xo)-' is
in itself invariant under any variable transformation, i-e. also under the supersymmetry
transformation (6.24). For the fermionic part of the integral measure, make the ansatz
d8 = d46 p ( 6 ) and demand that it be invariant under the transformation 6 - b = 6 + b6,
where 66 is given by equation (6.20). From equation (6.20) follows that
Multiplying equation (6.25) with p(b) and expanding to terms Linear in t' yields the equa-
tion
whose solution up to a multiplicative constant is
CHAPTER 6. 7HE WESS--0 MODEL ON Am 85
Because p(0) is a scalar, d4Op(6) is also invariant under the bosonic transformation (621).
Hence, the expression
is the invariant superspace integration measure.
6.4 The Scalar Superfield
In this section the scalar superfield shall be expanded in powers of the Grassmannian vari-
ables and the transformation laws for the individual components derived. Then, a chirality
condition will be imposed and +he transformation rules for the components of the chiral
superfield found.
Osp(l,4) superfields have already been treated in the literature some time ago [99,97].
However, it should be noted that the present derivation differs in some points from these.
Keck [99] coupled a spinor field directly to the SO(4,l) spinor variable ( of section 6.3F
thereby demanding that the spinor field too be an SO(4,l) instead of a Lorentz spinor,
which is unconventional from a field theory point of view. On the other hand, Ivanov and
Sorin [97] considered the Killing spinor 8 (see appendix D.2) as the independent Grassman-
nian variable, which can directly be coupled to a Lorentz spinor field. However, the com-
plicated transformation rule for 8 under supersymmetry transformations is a minor draw-
back of their very complete formulation, which suggests to consider the SO(4, l) spinor 6 as the independent superspace variable and realize the coupling to Lorentz spinor fields
via a matrix A (x), which is calculated in appendix D.2. This treatment seems to combine
the nice features of both references, [99] and [97]. In addition, it yields the Killing spinor 0
as a side product.
To start, a suitable, but absolutely general expansion of the scalar superfield is given by
where the components A, F, G and E are scalars, t/q and @2 are Lorentz spinors and VP is
a vector field. - - - - - - -
'In his case they were S0(3,2) spinors due to the different signature.
The supersymmetry transformation formulae for the components are obtained by cal-
cula ting
taking into account the transformation rules (6.20) and (6.24), the Killing spinor properties
(D.15) and (D.16), as well as the identities (D.7) and (D.8). Introducing the Killing spinor
L = A - Z, the results are
In order to obtain a chiral superfield, the c h i d projection operators 1 1
L ( 1 4 ) and R = - ( l+ i j? l ) 2 2
are introduced and the chirality condition (shown for a left handed c h i d superfield)
is imposed, where @ is some Dirac spinor field. Consistency of the transformation formulae
(6.29) then demands the relations
c 'HAFER 6. TEE WESS-ZUMLNO MODEL ON A- 87
so that only A, LJ, and F remain as independent components. Their supersymmetry trans-
formations are obtained from equations (6.29) and (6.32) and are given by
In the case of a right handed superfield, L has to be replaced with R and i with -i in
equations (6.32) and (6.33).
6.5 The Wess-Zumino Model
The Wess-Zumino multiplet contains scalar fields A, B, F and G and a Dirac spinor field J1 and can be thought of consisting of a left and right handed scalar chiral multiplet. Hence,
their supersymmetry transformations are easiest found by considering chiral superfields.
Therefore, define
and use the projection operators L and R to obtain the left and right handed components
of I/J, respectively Then, the left and right handed c h i d superfields are given by
respectively. From equations (6.33) follow the transformation rules of the chid field com-
ponents, which are
CHAPTER 6. l7dE WESS--0 MODEL ON ADS4 88
To find the supersymmetry transformations of AR, FR and R+, simply replace L with R in
equation (6.37).
For the Wess-Zurnino model one also needs "conjugate" superfields, which are intro-
duced by defining
where the Killing spinor 0, = (en), has been used and the dots indicate terms similar to
those in equations (6.35) and (6.36).
A manifestly supersymmetric action is then given by the expression
which describes the non-interacting Wess-Zumino model with a mass term. After inserting
the integration measure (6.26), the Berezin integration can be performed and the result re-
expressed in terms of the fields A, B, F and G. Hence, one obtains (up to a multiplicative
constant and surface terms, which have been dropped)
Solving the equations of motion for the auxiliary fields F and G gives
F = ( m - l ) A and G = - ( r n + l ) B . (6.41)
Similar relations hold for and G. Substituting equation (6.41) back into the action (6.40)
yields the onshell supersymmetric action [26,28]
CHAPTER 6. THE W E S S - 7 7 0 MODEL ON ADS& 89
The mass parameter m describes the mass of the fermion Q. Moreover, for rn = 0 the scalar
fields A and B are conformally coupled. The bulk action Sbulk has to be accompanied for
the AdS/CFT correspondence by a surface term derivable from the variational principle
[90], as well as counter terms.
It seems straightforward to read off from equation (6.42) the conformal dimensions of
the boundary operators coupling to the boundary values of A, B and @. However, care
must be taken when spedfying boundary conditions. According to [26,28], one must use
the irregular boundary conditions for one of the scalar fields, if lm 1 < 1 (cf. section 3.2),
whereas for irnl 3 $ all fields are assigned regular boundary conditions. Thus, the con-
formal dimensions of the boundary operators corresponding to the fields A, B and @ are
given by
where the plus and minus signs correspond to assigning the regular and irregular bound-
ary conditions, respectively. Consider the case rn 2 0. For rn < 0 only the roles of A and B
interchange. Comparing the values (6.43) with equation (6.10) one can identify the bound-
ary fields corresponding to the Ads fields A, B, and J, with the primary conformal fields
0, 3 and X, respectively. Moreover, if nr < $, the irregular boundary condition must be
used for B in order to make this identification.
Chapter 7
Conclusions and Outlook
7.1 Summary
It is time to summarize the results presented in this thesis and to indicate clearly the con-
tributions of the author's research to the progress made in the field.
The AdS/CFT correspondence is an active field of research, which was sparked by
Maldacena's conjecture about the duality of type IIB supergravity on AdSx S5 and Af =
4, d = 4, S U ( N ) super Yang-Mills theory in the large N limit [114]. Subsequently, the
AdS/CFT correspondence formula [82, 1601 provided an explicit tool for calculations of
CFI' correlation functions from classical field theories on Ads spaces.
This thesis is the result of the research of its author on the subject of the AdS/CFC cor-
respondence, which was started just after the AdS/Cn correspondence formula appeared
in the literature. The initial goal was to explore the correspondence by calculating Cm
correlation functions of various quasi-primary operators. This goal was achieved with the
calculation of the two-point functions for the scalar, spinor and vector fields as well as for
the C R energy momentum tensor and with the consideration of several first order inter-
actions. It was shown that the non-contact CFT correlation functions thus calculated obey
the restrictions imposed by conformal invariance. Moreover, the validity of the AdS/CFT
dictionary (cf. section 2.4) was confirmed.
The main achievement in this research was the consistent exploration and application
of the E-prescription as a regularization procedure which deals with the infinities on the
Ads horizon. The use of the c-prescription was explained in detail in chapter 3. Regular-
CHAPTER 7. CONCLUSIONS AND OUTLOOK 91
ization leads to the breaking of conformal invariance, as was discussed in chapter 3 and
used in section 5.3 in order to calculate the Weyl anomaly. Moreover, it must be empha-
sized that for a long time the E-prescription was the only regularization procedure for the
AdS/CET correspondence known to be entirely accurate. To be precise, a subtlety, which
was eliminated later by the correct identification of counter terms, led to a wrong coeffi-
cient in the two-point function for massive scalars [65]. Therefore, it had to be suspected
that some of the earlier results from the asymptotic prescription would suffer from the
same subtlety, and an independent confirmation by means of the E-prescription seemed
appropriate. This was the case for the spinor field, the gravitons and the Weyt anomaly,
although none of these suffered from the subtlety. On the other hand, the treatments of
massive Ads vector fields and of the interactions considered in sections 3.3 and 4.3 were
entirely novel.
It can be said safely that the calculations of CFT correlation functions and of the Weyl
anomaly confirm the validity of the AdS/CFT correspondence, i.e. a classical field theory
on Adsd+ is dual to some d-dimensional CR. Not only do the correlation functions obey
the restrictions imposed by conformal invariance, even the details of renormalization re-
semble older CFT calculations (cf. the calculation of the two-point function of CFT energy
momentum tensors in section 5.2).
In order to give a complete picture, the thesis also contains the asymptotic prescription,
which had been used for many calculations in the literature even before it was consistently
forrnuIated by Klebanov and Witten [102]. This prescription is essential for formulating the
AdS/CFT correspondence for Ads fields with irregular boundary conditions. The regular
and irregular boundary data appear to be a pair of canonically conjugate fields with respect
to the renormalized effective action and its Legendre transform, respectively. The method
for including irregular boundary conditions proposed by Klebanov and Witten [I021 was
proven to be exact to all orders in perturbation theory (cf. section 3.2).
There is no doubt that the consideration of the Wess-Zumino model on Ads4 in chap-
ter 6 is only of pedagogical value, since it was considered long ago by Breitenlohner and
Freedman [26], and the results (the validity of the AdS/CFT dictionary for supersymrnet-
ric theories) could have been obtained from their paper. Thus, it should be of value as a
supersymmetric example of the AdS/CFT correspondence to newcomers in the field and
as a non-textbook exercise for calculating with Grassmannian variables.
CHAPTER 7. CONCLUSIONS AND O[ITLOOK
7.2 Future Work
It is certainly impossible to mention all interesting aspects of the AdS/CFI' correspondence
which could still be studied. The following is intended as a selection of topics directly
related to this thesis.
As explained in chapter 3, the E-prescription does not yield CFT correlation functions
for all quasi-primary fields satisfymg the unitarity condition. For the case of the scalar
field, the method necessary to include the rest was analyzed in section 3.2 and involves
the asymptotic prescription. Although it can be anticipated that this method would work
equally well for other fields, only the case of spinor fields has been investigated to date
[140]. A systematic investigation of the asymptotic prescription for other fields includ-
ing the treatment of irregular boundary conditions when appropriate is still lacking. This
would be particularly interesting in the cases of fields with gauge invariances, like the
massless vector field and the graviton. Pure gauge degrees of freedom might have an un-
desirable asymptotic fall-off. For example, the solution for the graviton given in section 5.1
contains a term -- xi. Moreover, this solution is related by a gauge transformation [I221 to
the one found by Liu and Tseytlin [1061, who used the asymptotic prescription and where
such an asymptotic behaviour does not occur.
The results for the Weyl anomaly obtained in section 5.3 suggest a second topic for
investigation. The expressions for the Weyl anomaly found for various even dimensions
depend on only one parameter, namely the Ads radius I. Ln contrast, general CFT' consider-
ations allow for as many coefficients as there are conformal invariants in a given dimension
plus one for the Euler density term (cf. equation (5.42)). The fact that the AdS/CFI' calcula-
tion uniquely relates the coefficients of the individual terms in the Weyl anomaly with each
other means that the CFTs obtained via the AdS/CFT correspondence are in general very
special (except for d = 2). Thus, it would be worthwhile to investigate and characterize
these CFTs.
Appendix A
Fields on Curved Manifolds
A.l Covariant Derivatives
In this section the basic notions of covariant derivatives of fields of arbitrary spin on a
general torsionless manifold shall be recalled. Here the notion of spin relates to the repre-
sentation of the field with respect to the local Lorentz group, which is an invariance group
in the tetrad formalism. In this formalism, a vector field Ap is related to a spin-i field A*
by A p = e:A".
Let g,, be the metric tensor of the space-time with respect to some coordinates xp, and
let e: be the tetrads ("vielbeiu") such that g,, (x) = e t (x)e$(x) qob, where qab is the metric
tensor of the local Lorentz frame.
The covariant derivative of a field Y of arbitrary spin is defined as
where cupab = -wpba are the spin connections and r(Mab) is the appropriate matrix repre-
sentation of the generator M,b of the local Lorentz group. For a spin-1 field
and thus one has
APPENDLX A. FIELDS ON CURVED MANIFOLDS 94
The spin connections are defined by demanding the covariant derivative of the tetrads to
be zero, i.e.
which yields
The spin-$ representation of the Lorenh generators is given by
1 T ( W b ) = Sub = 4 [Y*, ~ b ] ,
where y,, are the Gamma matrices satisfying
Hence, the covariant derivative of spin-$ fields is given by
Finally, the covariant Gamma and spin matrices defined by
1 u b r, = e> yff and t,.,, = - [r,, r,] = epe,SQb 4
are used often in the main text.
A.2 Symmetry Transformations
This section contains a brief review of the transformation rules of scalar and spinor fields
under coordinate symmetry transformations in a symmetric curved space time.
Assume that the metric g,, admits coordinate symmetry transformations, i.e. there are
transformations x + x', which leave g,, invariant,
Infinitesimal transformations 2' = x' c <' are generated by a set of Killing vectors Lp, which must satisfy
APPENDLY A. FIELDS ON CURVED MANIFOLDS
in order for equation (A.7) to hold.
In the tetrad formalism a transformation of the local Lorentz frame must be performed
in addition to the coordinate transformation such that also the tetrads remain invariant.
Thus, one writes
where the rotation matrix 72 satisfies R T q ~ = q. Moreover, equation (A.9) should be
regarded as the definition of R. In the case of an infinitesimal transformation generated by a Killing vector C, one has
,U - - 6; + UPb and finds from equation (A.9)
The transformation rule for a field Y of arbitrary spin under a coordinate symmetry
transformation is then given by
Y1(.') = r (R) Y (x), (A.11)
where f (R) is the appropriate matrix representation of the Lorentz transformation R. Hence, a scalar field transforms under an infinitesimal coordinate transformation as
whereas the transformation of a spin-f field is given by
Equation (A.13) can be readily generalized to fields with arbitrary spin.
Appendix B
Various Integrals
B . l Fourier Integrals
This section provides the expressions of d-dimensional Fourier integrals, which are often
needed in the main text. Use shall be made of the integral table (751 to derive them, and it
w i l l be assumed that the necessary conditions for the existence of the integrals are satisfied.
The convention x = I xl is being used.
Integrals of the Form 1 & e-lk*' f ( k )
Using spherical coordinates, one can write the above integral in the form (for d > 2)
where
is the area of a unit sphere in n dimensions. Then, the 8 integration can be carried out using
[75, formula 3.915 5.1, which yields
ddk - d / 2 5% 1 e-ik-x f (k) =
( 2 4 4 2 / d k k i J d , ( k x ) f ( k ) .
5 - 0
It can be shown that this result holds also in the cases d = 1 and d = 2.
A PPENDLX B- VARIOUS INTEGRALS
Integrals of the Form 1 & e-ik-xkB
As an integral of this type in not defined as it stands due to the large k behaviour of the
in tegrand, the regularized integral
shall be considered and the limit p + 0 be taken at the end. This will be possible for x > 0,
which is sufficient for the intended applications of the formula. First, one carries out the
angular integration using equation (B.1) and obtains
.r 1-d/2 J dk k i - ~ -, (kx) e-pk2. (2n)dl' -
0
Then, use (75, formula 6.631 1.1 to find (provided J + > 0)
where @ (a; c; z ) is the degenerate hypergeometric function. Since Q, (a ; c; 0) = 1, the inte-
gral is divergent for x = 0 in the limit p + 0. However, for x > 0 one can use the leading
order term of the asymptotic expansion of Q, (a; c; - z) for large z [147], which yields
One notices that the expression vanishes for p = 212. This represents the intuitive fact that
k' can be replaced by -A.
Integrals of the Form / & e-"~'kYKB (kn)
In order to evaluate this integral, one first performs the angular integration using equation
(B. 1). This yields
APPEND= B.
Then, one uses
VARIOUS LMEGRALS
[75, formula 6.576 3.1 to obtain (provided y + d > P )
In the special case y = i$ one can use the identity [75, formula 9.221 1.1 and finds
The case y = - f 3 is of course restricted to < d / 2 .
B.2 Feynman Parameterization
This section recalls the standard formula for the so-called Feynman parameterization of
integrals [132]. It is needed for the evaluation of interaction graphs and reads
Notice that the range of integration usually is from 0 to 1, but it can be extended to cr; due
to the presence of the 6 function. The sums and products run from
B.3 Other Integrals X -rptv- 1
Integrals of the Form / d x (9 + n2)p
0
Inhoducing a new integration variable y by x = ny one finds
Finally, using [75, formula 3.251 2.1 one obtains (provided -v < p
APPENDIX B. VARIOUS LNTEGRA LS
.C
Integrals of the Form ] ddx (6 + a ' ) ~
Since this integral only depends on x = 1x1, one can use spherical coordinates and perform
the angular integration to obtain
Then, using equation (B.6) leads to
Integrals of the Form ddx x i x j f ( x ) J It shall be assumed that the function f ( x ) has properties, which ensure that this integral is
defined. I t is easy to see that the integral is zero for i # j. Moreover, it must have the same
value for every i. Hence, the only form satisfymg these constraints is
where the angular integration can be carried out.
Integrals of the Form ddx xix;xj# f ( x ) J Again, this integral is symmetric with respect to any interchange of indices. Moreover,
taking the trace of any two indices, one obtains an integral of the form (8.8). Hence, the
only form satisfying these constraints is
Integrals involving more coordinates can be obtained in a similar recursive fashion.
Appendix C
Time Slicing Formalism
This appendix shall begin with a review of basic geometric relations for immersed hy-
persurfaces [29]. Let a hypersurface be defined by the functions ~ ' ( x ' ) , (p = 0,. . . d ,
i = 1,. . . d) and let g,, and gii be the metric tensors of the embedding manifold and the hy-
persurface, respectively. The tangents aiX' and the normal Np of the hypersurface satisfy
the following orthogonality relations:
g,, aiXpaiXY = gij,
aiXpNp = 0,
NPNp = 1.
In the sequel a tilde will be used to label quantities relating to the (ti + 1)-dimensional
space time manifold, whereas those relating to the hypersurface remain unadorned. Notice
that the symbol D denotes a covariant derivative with respect to whatever indices may
follow. Then, there are the equations of Gauss and Weingarten, which define the second
fundamental form Hii of the hypersurface,
The second fundamental form describes the extrinsic curvature of the hypersurface and is
related to the intrinsic curvature by another equation of Gauss,
APPENDU( C. TIME SLICING FORMALJSM
Furthermore, it satisfies the equation of Codazzi,
In the time slicing formalism [156,119] one considers the bundle of immersed hypersur-
faces defined by XO = const., whose tangent vectors are given by aiXo = 0 and a i X p = 6;
( p = 1,. . . d ) . One conveniently splits up the metric as (shown here for Euclidean signa-
ture)
whose inverse is given by
and whose determinant is g = n2g. The quantities n and N' are called the lapse function
and shift vector, respectively. The normal vector N p satisfying equations (C.2) and (C.3) is
given by
(C. 10)
where the sign has been chosen such that the normal points outwards on the boundary
( 1 1 > 0 without loss of generality). Then, one obtains the second fundamental form from
the equation of Gauss (C.4) as
1
where the prime denotes a derivative with respect to the time coordinate (Xo).
The advantage of the time slicing formalism is that one removes the diffeomorphism
invariance in Einstein's equation by specifying the lapse function n and shift vector nr
and thus obtains an equation of motion as well as constraints for the physical degrees of
freedom gij. Consider Einstein's equation without matter fields,
APPENDIX C. TIME SLICING FORMALISM
Multiplying it with N"Nv and using the equation of Gauss (C.6) as well as the relation
(C.3) one obtains the first constraint,
where H = H:. Similarly, multiplying with N'aiXY, using the equation of Codazzi (C.7)
and the relation (C.2) yields the second constraint,
Finally, rewriting equation (C.12) in the form
and projecting out its tangential components leads to the equation of motion
(C. 16)
Appendix D
Spinor Grassmannian Variables
D.1 Basics
This appendix will summarize the notations and useful formulae for Grassmannian spinor
variables. Covariant conventions shall be used, which are independent of the signature
of the "pseudo"-Euclidean metric. Hence, notions involving complex conjugates will not
appear. Moreover, the presentation will concentrate on D = 1 and D = 5, whose Clif-
ford algebras have irreducible representations of dimension 4. Information on the Clifford
algebras and their representations can be found in [35].
A spinor 8 has components Oa (a = 1,2,3,4), which are Grassmannian variabIes, i.e.
for any two spinors 8 and q they satisfy
Spinor matrices usually carry an upper and a lower index, such as 6;, (y,): etc. HOW-
ever, spinor indices can be lowered and raised with the charge conjugation matrix and its
inverse, respectively:
For D = 4,5 the charge conjugation matrix is anti-symmetric. The scalar product of two
spinors can now be defined by
Now consider D = 5, and in view of the convention in the main text adorn all quantities
with a hat.
The vector space of 4 x 4 matrices with only lower indices is spanned by 16 matrices,
which can conveniently be chosen to be [135]
the anti-symmetric charge conjugation matrix t,
the 5 anti-symmetric matrices PA) and
the 10 symmetric matrices (eSAB) = C[yA, PB]/4-
The symmetry properties of the latter two follow directly from
Moreover, using this basis the following matrix identity holds:
The identity (D.6) leads to various Fierz identities. Moreover, products involving two or
more identical spinor factors can be simplified. In particular, one finds
Equation (D.8) implies that products involving three and four factors 0 can always be writ-
ten in the forms (8e ) (&) and A ( @ ) ~ , respectively. Products containing more than four
factors 8 are zero because of the Grassmannian property (D.1).
D.2 Calculation of the Killing Spinor
In this appendix the matrix A (x) shall be calculated, which relates the Lorentz and SO(4, l )
spinors with each other. It will turn out that the Lorentz spinor derived from the SO(d + 1.1) spinor 0 automatically is a Killing spinor.
According to equation (6.21) 8 is an SO(4,l) spinor. However, the spinor field @(x)
conventionally is a Lorentz spino:, i.e. it transforms as a spinor under rotations of the local
APPENDLX D. SPLNOR GRASSMANNLAN VARIABLES 105
Lorentz frame (cf. appendix A.2). Hence, a matrix A ( x ) is introduced such that the scalar
product $A ( x ) ~ ( x ) can be formed. The matrix A ( x ) can be calculated using the knowledge
of the transformation laws under the SO(4 , l ) symmetries. Thus,
where be, 6 x p and i5@ are given by equations (6.21), (2.17) and (A.13), respectively. As the
parameters bi, c and wi; are independent, equation (D.9) yields the following system of
equations for A ( x ) :
S ( L ~ ; ) A (x) - A (x)Si; + (.rid, - xjdi) A (x) = 0.
The solution of equations (D.10) is not unique, but any solution will suffice. A solution of
equations (D. 10) is
A ( x ) =
I t is also useful to know
n - ' ( x ) =
Consider the spinor
0~ A - ' 0, one finds
x' [ I - ~ S ( D ) ] - - [ l + ~ S ( D ) ] + -S(pi)- (D. 11)
2 2 6 Jxa
the inverse A -' ( x ) , which is easily found to be
(D. 12)
( 0 ~ ),, which by construction is a Lorentz spinor. Since &j =
where C-* is the inverse of the charge conjugation matrix C for Lorentz spinors. Equation
(D. 13) yields C = A ' ZA , which leads to
C = -C. (D. 14)
APPENDLX D. SPLNOR G R A S S W N L A N VARLABLES
Finally it can be checked explicitly from equation (D.11) that
and
which shows that
is a Killing spinor.
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