AdS/CFT, Black Holes, And Fuzzballs · conserved R-R charges in string theory and investigation of microscopic structure of black holes. We will discuss these constructions in the

AdS/CFT, Black Holes, And Fuzzballs

by

Ida G. Zadeh

A thesis submitted in conformity with the requirementsfor the degree of Doctor of Philosophy

Graduate Department of PhysicsUniversity of Toronto

Copyright c© 2013 by Ida G. Zadeh

Abstract

AdS/CFT, Black Holes, And Fuzzballs

Ida G. Zadeh

Doctor of Philosophy

Graduate Department of Physics

University of Toronto

2013

In this thesis we investigate two different aspects of the AdS/CFT correspondence. We

first investigate the holographic AdS/CMT correspondence. Gravitational backgrounds

in d + 2 dimensions have been proposed as holographic duals to Lifshitz-like theories

describing critical phenomena in d + 1 dimensions with critical exponent z ≥ 1. We

numerically explore a dilaton-Einstein-Maxwell model admitting such backgrounds as

solutions. We show how to embed these solutions into AdS space for a range of values

of z and d.

We next investigate the AdS3/CFT2 correspondence and focus on the microscopic

CFT description of the D1-D5 system on T 4 × S1. In the context of the fuzzball pro-

gramme, we investigate deforming the CFT away from the orbifold point and study

lifting of the low-lying string states. We start by considering general 2D orbifold CFTs

of the formMN/SN , withM a target space manifold and SN the symmetric group. The

Lunin-Mathur covering space technique provides a way to compute correlators in these

orbifold theories, and we generalize this technique in two ways. First, we consider exci-

tations of twist operators by modes of fields that are not twisted by that operator, and

show how to account for these excitations when computing correlation functions in the

covering space. Second, we consider non-twist sector operators and show how to include

the effects of these insertions in the covering space.

Using the generalization of the Lunin-Mathur symmetric orbifold technology and con-

ii

formal perturbation theory, we initiate a program to compute the anomalous dimensions

of low-lying string states in the D1-D5 superconformal field theory. Our method entails

finding four-point functions involving a string operator O of interest and the deformation

operator, taking coincidence limits to identify which other operators mix with O, sub-

tracting conformal families of these operators, and computing their mixing coefficients.

We find evidence of operator mixing at first order in the deformation parameter, which

means that the string state acquires an anomalous dimension. After diagonalization this

will mean that anomalous dimensions of some string states in the D1-D5 SCFT must

decrease away from the orbifold point while others increase.

Finally, we summarize our results and discuss some future directions of research.

iii

Dedication

To my parents Ali and Maryam

iv

Acknowledgements

The following is a list of people to whom I would like to express my gratitude.

My advisor, Prof. Amanda Peet, for help in sparking ideas, and for suggestions,

guidance, encouragement, and support.

Prof. Amanda Peet, Prof. Ben Burrington, and Dr. Gaetano Bertoldi, for productive

discussions, rewarding and fun collaborations, and interesting courses.

Prof. Ben Burrington, especially for inspiring discussions and valuable suggestions.

Prof. Samir Mathur, for stimulating discussions, insightful comments, and for hospi-

tality during my visits to The Ohio State University.

Prof. Erich Poppitz, for helpful comments and advice.

My committee members, Prof. Erich Poppitz, Prof. Peter Krieger, and Prof. David

Bailey.

Fellow student Daniel O’Keeffe, for helpful discussions and comments.

Prof. Werner Israel at the University of Victoria, for his support.

The staff of the Physics department, especially Teresa Baptista, Krystyna Biel, Helen

Iyer, Carrie Meston, Julian Comanean, and Steven Butterworth.

My partner Alijon. My mother Maryam, and my sister Anahita. Special thanks to

my father Ali for his constant encouragement, support, and sense of humour.

v

Contents

1 Introduction 1

1.1 String theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Black holes in string theory . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

1.4 AdS/CMT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.5 Fuzzball proposal and black hole information paradox . . . . . . . . . . . 13

1.5.1 Mathur’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.5.2 Fuzzball conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5.3 Two-charge and three-charge fuzzballs . . . . . . . . . . . . . . . 19

1.5.4 AdS3/CFT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.5.5 D1-D5 CFT at the orbifold point . . . . . . . . . . . . . . . . . . 23

1.5.6 Moving away from the orbifold point . . . . . . . . . . . . . . . . 26

1.6 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2 Lifshitz black brane thermodynamics in higher dimensions 30

2.1 Analysis of the model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.1 Reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.2 Perturbation theory at the horizon . . . . . . . . . . . . . . . . . 37

2.1.3 Perturbation theory at r =∞: AdS asymptotics . . . . . . . . . . 39

2.1.4 Other considerations, and setup for numerics . . . . . . . . . . . . 41

vi

2.2 Results and discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3 Twist-nontwist correlators in MN/SN orbifold CFTs 51

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

3.2 The Lunin-Mathur technique, and generalizations . . . . . . . . . . . . . 54

3.2.1 Lunin-Mathur . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

3.2.2 Generalization to the non twist sector. . . . . . . . . . . . . . . . 60

3.3 Example calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

3.3.1 Excitations orthogonal to twist directions . . . . . . . . . . . . . . 65

3.3.2 Non-twist operator insertions . . . . . . . . . . . . . . . . . . . . 71

3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4 String states mixing in the D1-D5 CFT near the orbifold point 84

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

4.2 Perturbing the D1-D5 SCFT . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.2.1 The D1-D5 superconformal field theory . . . . . . . . . . . . . . . 86

4.2.2 Cocycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

4.2.3 Deformation operator . . . . . . . . . . . . . . . . . . . . . . . . . 95

4.2.4 Conformal perturbation theory . . . . . . . . . . . . . . . . . . . 99

4.2.5 Four-point functions and factorization channels . . . . . . . . . . 105

4.3 Dilaton warm-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.3.1 Four-point function . . . . . . . . . . . . . . . . . . . . . . . . . . 108

4.3.2 Mapping from the base to the cover . . . . . . . . . . . . . . . . . 110

4.3.3 Summing over images . . . . . . . . . . . . . . . . . . . . . . . . . 114

4.3.4 Lack of operator mixing . . . . . . . . . . . . . . . . . . . . . . . 115

4.4 Lifting of a string state . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

4.4.1 Passing to the covering surface . . . . . . . . . . . . . . . . . . . 118

4.4.2 Image sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

vii

4.4.3 Coincidence limit and operator mixing . . . . . . . . . . . . . . . 121

4.4.4 Conformal family subtraction and mixing coefficients . . . . . . . 121

4.5 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

5 Conclusions 130

Bibliography 134

viii

List of Figures

1.1 Ensemble of fuzzball microstates . . . . . . . . . . . . . . . . . . . . . . . 18

2.1 The plots of ln (4Gd+2 s) versus ln(LT ) for fixed µ = 1. . . . . . . . . . . 45

2.2 The plot of ln (c(z, d)) as a function of ln(z) for fixed value of µ = 1 . . . 47

3.1 Diagram of the three-fold cover of the base space . . . . . . . . . . . . . 58

ix

Chapter 1

Introduction

1.1 String theory

String theory provides a consistent theory of quantum gravity which in the low-energy

limit reduces to classical supergravity. Mathematical consistency requires string theory

to live in ten spacetime dimensions. Compactifying extra dimensions on small scale

compact manifolds provides a powerful tool for constructing lower dimensional effective

theories. Existence of higher dimensions in string theory has made it possible to construct

a wide variety of black holes and their extended counterparts which are localized in a

lower dimensional spacetime.

Fundamental ingredients of the theory are one-dimensional open and closed strings

whose oscillations give rise to the elementary particles of the universe. The motion of

a string in the ten-dimensional spacetime is described by the worldsheet theory. The

worldsheet is the two-dimensional surface that the string sweeps out during its motion.

Upon quantization of the classical worldsheet action one finds that the closed string

spectrum contains gravitons, as well as a host of other particles. The interactions given

by the worldsheet theory provide a unified picture of all the fundamental forces of nature.

The classical worldsheet theory is invariant under Weyl transformations. The quan-

1

Chapter 1. Introduction 2

tum theory, however, acquires Weyl anomalies. Let us consider the limit where the

radius of curvature of the background, R, is much larger than the string length scale ls:

lsR−1 1. We focus on massless states in this low-energy limit. We can then perform

perturbation theory in this limit and compute the beta functions. Requiring that the

quantum worldsheet theory be Weyl-invariant boils down to vanishing of the beta func-

tions. This yields Einstein’s equations in ten-dimensional spacetime and confirms that

classical gravity emerges as the low-energy effective description of string theory [1, 2].

The Newton’s constant in ten dimensions, G10, is given in terms of the string length and

the string coupling constant gs according to

16π G10 = (2π)7 l8s g2s . (1.1)

In addition to the fundamental strings, string theory has a class of building blocks

which are extended objects called Dp-branes. Dp-branes have (p + 1) spacetime dimen-

sions. They are objects on which the endpoints of open strings are located. Equations of

motion of the worldsheet theory allow for Dirichlet and Neumann boundary conditions for

the end points of open strings. Dirichlet boundary conditions violate Poincare symmetry

of the theory. In the presence of D-branes, Poincare invariance is broken spontaneously.

Open string endpoints satisfy Neumann boundary conditions in the (p + 1) spacetime

directions tangent to the brane and satisfy Dirichlet boundary conditions in the (9− p)

directions transverse to the branes. For this reason the branes are called Dirichlet branes

or D-branes. T-duality exchanges Neumann and Dirichlet boundary conditions. Thus,

performing T-duality in a direction tangential (perpendicular) to the Dp-brane results in

a Dp−1 (Dp+1)-brane.

The symmetry preserved by Dp-branes is SO(1, p) × SO(9 − p). The tension of D-


branes is related to gs and ls through the relation

τDp =1

(2π)p lp+1s gs

. (1.2)

The tension of the D-branes is proportional to the inverse of the coupling constant.

Therefore, they are non-perturbative objects. However, one can perform perturbation

theory in the presence of D-brane backgrounds by considering low-energy fluctuations of

open strings attached to the D-branes.

A great breakthrough was made by Polchinski in 1995 [3] by showing that Dp-branes

couple to the p-form fields of the R-R sector of superstring theory and therefore carry

R-R charges. This discovery set the stage for the construction of black holes that carry

conserved R-R charges in string theory and investigation of microscopic structure of black

holes. We will discuss these constructions in the next section.

1.2 Black holes in string theory

Astronomical observations have provided ample evidence of the existence of macroscopic

physical black holes in our universe. The evidence includes observations of Sagittarius

A∗ at the centre of our Milky Way galaxy, X-ray binary systems, and supermassive black

holes at the centre of other galaxies and clusters of galaxies [4, 5, 6]. Black holes have

been studied in theoretical physics for more than four decades. The laws of black hole

mechanics were formulated by Bardeen, Carter, and Hawking [7]. The analogy between

these laws and the laws of thermodynamics led Bekenstein to conjecture that the entropy

of a black hole is proportional to the surface area of its event horizon [8]. Semiclassical

analysis of Hawking [9, 10] showed that black holes are thermodynamic systems that

emit black body radiation. Hawking’s discovery resulted in the precise formulation of


black hole entropy

SBH =Ad

4Gd

, (1.3)

where Ad is the area of the event horizon, Gd is the d-dimensional Newton’s constant,

and ~ = c = kB = 1.

The breakthrough in understanding thermodynamic properties of black holes raised

two fundamental issues. First, equation (1.3) indicates that there are eSBH microscopic

degrees of freedom that contribute to the exponentially large value of the black hole

entropy. Second, Hawking’s results showed that black holes emit radiation through pair

production out of vacuum at the horizon. The emitted radiation does not reveal any

information about the matter that has made the black hole. These statements raised

challenging conundrums in black hole physics: what are the microscopic degrees of free-

dom of black holes? Where are these degrees of freedom located? Can we retrieve the

information about the matter that has fallen inside a black hole? We discuss the first

question in the remainder of this subsection. The second and the third questions will be

addressed in section 1.5.

String theory has provided a powerful framework for constructing black holes and

studying the statistical mechanical description of their entropy. The key point is that

string theory has extra dimensions and compactification of these dimensions enables us

to construct black holes in lower dimensions. The construction is done by wrapping the

building blocks of the theory, strings and Dp-branes, on the compactified space. Black

holes are bound states of such configurations.

String theory black holes on which we have the largest possible theoretical control

are BPS configurations. BPS black holes are supersymmetric systems for which the

Bogomolnyi-Prasad-Sommerfield (BPS) bound is saturated [11]. Supersymmetry protects

certain quantities of these black holes, including the statistical degeneracy of states,

against quantum corrections for any value of the string coupling constant. BPS black

holes constructed in string theory carry multiple conserved charges and are thus different


from astrophysical neutral black holes that we observe in the universe. Deviations from

the BPS configuration, for which there is still some theoretical control, have been studied

in great detail. The grand objective is to construct and understand microscopic structure

of neutral string theory black holes far away from the BPS bound.

To construct two-charge BPS black holes in five spacetime dimensions, we consider

wrapping D1 and D5 branes on the compact manifold S1×M4, whereM4 is T 4 or K3.

This is done such that the D1 branes are wrapped around S1. Using string dualities,

we can map this system to the system of fundamental strings carrying winding and

momentum charges. The degeneracy of the string states were evaluated in [12] for the

case M4 = K3, and the microscopic entropy of the system was found to be

Smicro = 4π√NwNp, (1.4)

where Nw is the winding charge, Np is the momentum charge.

The classical Bekenstein-Hawking entropy of this black hole vanishes. However, after

considering higher derivative corrections one finds that the black hole does indeed have

a microscopic horizon. The Bekenstein-Hawking entropy was then evaluated in [12, 13]

and found to match exactly the microscopic counting of the degrees of freedom (1.4).

Black holes with macroscopic horizons have also been constructed in string theory.

The first example was the three-charge BPS black hole in 5-dimensions constructed by

Strominger and Vafa (SV) [14]. This is done by wrapping N1 D1-branes on S1, N5 D5-

branes on S1×M4, and adding a gravitational wave moving along S1 carrying momentum

Nm/R, where R is the radius of the circle. The classical supergravity description is valid

for large values of charges and small value of the string loop coupling such that gsN1 1,

gsN5 1, and g2sNm 1. This 3-charge BPS black hole has a macroscopic horizon area.

The Bekenstein-Hawking entropy of the black hole is given by

SBH = 2π√N1N5Nm. (1.5)


SV [14] considered the system of D-branes and open strings stretched between the

branes. In the low-energy regime and in the limit where the size of M4 is much smaller

than the radius of S1 the worldvolume theory of the system of D-branes is described

by a (1+1)-dimensional sigma model that lives on S1. The subtleties and developments

of the worldvolume theory will be discussed later in subsections 1.5.4 and 1.5.5. SV

[14] computed the degeneracy of the BPS bound states of the sigma model in the weak

coupling limit for large values of charges. Since this quantity is protected from quantum

corrections, one expects that it is valid to extrapolate to the strong coupling limit where

the classical black hole with a macroscopic event horizon is the appropriate description.

This agreement was indeed observed in [14]: the microscopic entropy of the D-brane

system was evaulated to be Smicro = 2π√N1N5Nm, which exactly matches the black hole

entropy (1.5).

The analyses were extended to construct spinning three-charge BPS black holes

(BMPV) [15] and near-BPS black holes [16] in five spacetime dimensions and four-charge

BPS and near-BPS black holes in four dimensions [17, 18]. Agreements between string

theory microscopic counting and macroscopic classical black hole entropy were observed

in all cases. These remarkable results show that string theory explains the statistical

mechanical origin of the black hole entropy. The Bekenstein-Hawking entropy of a black

hole is described by open string excitations of the underlying D-brane system in the weak

coupling limit.

Das and Mathur [19] studied the near-BPS three-charge black hole and computed the

rate of Hawking emission of low-energy quanta from this system. They also computed the

rate of emission of low-energy quanta from the near-BPS bound states of the correspond-

ing D-brane configuration. They found striking agreement between the weak-coupling

string theory picture and the strong-coupling classical black hole computations despite

the fact that the non-renormalization theorems do not apply to this case.

The key point here is that the effective description of the D1-branes is not given by


multiple singly-wound strings wrapped around S1. D1-branes are in fact described by a

single long multi-wound string which is wrapped around S1. This phenomenon is referred

to as fractionation and was discovered in [20, 21]. The fractionation phenomenon asserts

that excitations of the D-brane/string configuration corresponding to the D1-D5-P black

hole carry fractional charges. The effective size of the bound states of the brane system

depends on the coupling constant and the number of the D-branes. For reviews on black

hole physics in the context of string theory we refer the reader to [22, 23].

Investigating D-branes, black hole entropy, and black hole information problem in

string theory has resulted in fundamental discoveries in theoretical physics in the past

two decades. These include the AdS/CFT correspondence and the fuzzball proposal. We

will discuss these topics in the remaining sections of this chapter.

1.3 AdS/CFT

Enigmas of black hole physics have stimulated novel discoveries in the past four decades.

String theory provides the framework to formulate and develop these advancements.

Bekenstein [8] discovered that the black hole entropy is proportional to the surface area

of its horizon and that the maximum possible amount of information contained in a

system of volume V is the information that can be stored stored in a black hole of

that size. These discoveries inspired one of the fundamental principles of physics: the

holographic principle. The principle was proposed by ’t Hooft [24] and Susskind [25].

According to the holographic principle, the number of degrees of freedom of quantum

gravity in a region of volume V is bounded by the number of degrees of freedom on the

surface area of this region in Planck units. This suggests that the degrees of freedom

of quantum gravity in d + 1 spacetime dimensions can be explained by the degrees of

freedom of a quantum field theory in d spacetime dimensions.

Since ’t Hooft’s and Susskind’s proposal was made, there has been a great interest


in understanding quantitatively the holographic principle in the context of string theory

which is our candidate theory of quantum gravity. AdS/CFT correspondence in string

theory is the realization of the holographic principle.

Initial observations of the correspondence were made by studying the entropy, temper-

ature, and absorption cross sections of D3-branes. These quantities were computed in the

low-energy worldvolume theory of the D3-branes and in the classical supergravity descrip-

tion of the branes and the results from the two theories we compared in [26, 27, 28, 29].

The worldvolume theory of a stack of N coincident D3-branes in the low-energy limit

is described by a (3+1)-dimensional N = 4 super Yang-Mills theory. This theory is a

superconformal field theory. It has sixteen supercharges and a SO(6) R-symmetry. The

gauge group of the theory is SU(N) and the coupling constant is g2YM = gs.

Alternatively, one can consider the low-energy, classical supergravity description of

the D3-branes which is the solution of the form

ds2 = H− 1

23 (r)

(− dt2 +

3∑µ=1

dx2µ

)+H

123 (r)

6∑i=1

dx2i , (1.6)

e2Φ = g2s , (1.7)

C4 = −g−1s

(H−1

3 (r)− 1)dt ∧ dx1 ∧ dx2 ∧ dx3, (1.8)

where xµ, µ ∈ 1, 2, 3 are coordinates along the brane, xi, i ∈ 4, · · · , 9 are coordinates

perpendicular to the brane, and r = (∑9

i=4 x2i )

1/2. Here Φ is the dilaton, C4 is the R-R

four-form potential, and H3 is a harmonic function of the form

H3(r) = 1 +(r3

r

)4

, r43 = 4π gsNα

′2. (1.9)

We now define the new coordinate u ≡ rα′

and take the decoupling limit proposed by

Maldacena [30]

α′ → 0, u =r

α′= fixed. (1.10)


The harmonic function in the decoupling limit reads

H3(r)→ 4π gsN

α′2u4, (1.11)

where the factor of 1 in (1.9) can be neglected compared to the large fraction. The metric

(1.6) in this limit is of the form

ds2 = α′(

u2

√4π gsN

(− dt2 +

3∑µ=1

dx2µ

)+√

4π gsNdu2

u2+√

4π gsN dΩ25

), (1.12)

where dΩ5 is the unit five-sphere metric. This metric has the geometry AdS5 × S5. The

radius of the sphere and the AdS radius are equal and are given by

R = (4π gsN)14 α′

12 . (1.13)

In the decoupling limit, physics on the AdS5 × S5 part of the D3-brane geometry (1.6)

decouples from the asymptotically flat region (r r3). It is the AdS5× S5 near horizon

region of the geometry which is singled out and contributes to physics in the decoupling

limit.

Maldacena [30] linked the gravity theory and the field theory together and proposed

the AdS/CFT correspondence: type IIB superstring theory on AdS5×S5 is equivalent to

(3+1)-dimensionalN = 4 super Yang-Mills theory. Under the AdS/CFT correspondence,

the states, operators, and correlation functions of the two theories are equivalent to each

other [31, 32]. The ’t Hooft coupling λ = gsN is kept fixed in the decoupling limit.

The ’t Hooft limit of the couplings corresponds to the limit where N → ∞, gs → 0

and λ being fixed. This corresponds to the weak coupling string perturbative expansion.

According to equation (1.13), the supergravity description holds for large values of the

’t Hooft coupling, when N → ∞ and gsN 1. In this limit, the bulk gravity theory is

weakly coupled whereas the dual field theory is strongly coupled. On the other hand, in


the limit λ 1 the field theory is weakly coupled and the dual string theory is strongly

coupled. This shows that AdS/CFT correspondence is a strong/weak duality.

AdS/CFT is a holographic correspondence because the degrees of freedom of the

ten-dimensional quantum theory of gravity are equivalent to degrees of freedom of the

four-dimensional dual quantum field theory living on a spacetime which is conformal to

the boundary of the AdS space [32]. For reviews on the AdS/CFT correspondence we

refer the reader to [33, 34].

Maldacena [30] considered string theory and M-theory on other spacetimes with ge-

ometries given by the product of an AdSd+1 space with a sphere. He defined the decou-

pling limit in each case and determined the dual conformal field theory in the large N

limit. M-theory on AdS7×S4 was conjectured to be dual to (5+1)-dimensionalN = (2, 0)

conformal field theory which describes the low-energy worldvolume theory of a stack of N

coincident M5 branes. Another example is the type IIB string theory on AdS3×S3×M4,

where M4 is either T 4 or K3, which is conjectured to correspond to (1+1)-dimensional

N = (4, 4) conformal field theory that describes the bound state of the D1-D5 system in

the low-energy limit.

Other examples of the AdS/CFT correspondence include the ABJM theory and the

higher spin holographic models. ABJM [35] considered a stack of N M2-branes and

conjectured that the (2 + 1)-dimensional N = 4 superconformal Chern-Simons-matter

field theory describing the low-energy regime of M2-branes is dual to M-theory on AdS4×

S7/Zk in the large N limit. Holographic vector models explore the duality between higher

spin gauge theories on AdS space developed by Vasiliev and conformal vector models in

(2+1)-dimensions [36, 37] and their generalizations to (1+1)-dimensions [38].

The AdS3/CFT2 correspondence has been proven a powerful tool in studying black

holes in string theory and investigating black hole microstates and black hole information

problem. We discuss some of the salient aspects of the correspondence in section 1.5.4. In

Chapter 4 of this thesis we use the AdS3/CFT2 correspondence to investigate the nature


of the microstates of the D1-D5 system. We study quantitatively the difference between

the string states which acquire quantum corrections versus supergravity states which are

protected against quantum corrections.

1.4 AdS/CMT

The AdS/CFT correspondence connects gravitational theories to non-gravitational the-

ories in lower dimensions. The strong/weak nature of the duality is the key point in

making it a pragmatic approach to study a variety of physical systems in regimes that

are typically unexplorable using conventional field theory techniques. Examples of the

AdS/CFT correspondence discussed in section 1.3 are special in the sense that they con-

tain a large number of symmetries. Superconformal field theories have been studied in

great detail in literature. Some cases considered are very symmetric theories [39]−[45]

whereas others have reduced symmetries [46]−[49]. All these theories however respect

Poincare symmetry as they are used to study systems of relativistic particle physics.

Holographic techniques have been extended to study other physical systems. One

of the interesting applications of the holographic principle is in the context of con-

densed matter theory (CMT) [50]−[61]. Poincare symmetry is not preserved in these

non-relativistic systems. The AdS/CMT correspondence has been intensively used to in-

vestigate properties of strongly coupled condensed matter systems. Moreover, the holo-

graphic principle suggests that these systems may exhibit stringy behaviour in some

regimes.

According to the holographic dictionary, black hole geometries with horizons are as-

sociated with field theories at finite temperature. U(1) gauge symmetries in gravity

correspond to conserved number operators in the field theory. In the context of holo-

graphic AdS/CMT, charged black holes describe field theories at finite temperature and

chemical potential [62, 63, 64].


There has been a great interest in studying quantum critical behaviour of condensed

matter systems using holographic models. At quantum critical points, condensed matter

systems show an anisotropic scaling symmetry of the form

t→ λzt, xi → λxi, z ≥ 1. (1.14)

Here z = 1 corresponds to the scaling symmetry of pure AdS space in the Poincare

patch. Lifshitz field theories describe quantum critical phenomena in condensed matter

systems with anisotropic scaling symmetry z ≥ 1. In the context of holographic models,

equation (1.14) suggests that the (d+2)-dimensional gravitational system dual to a (d+1)-

dimensional Lifshitz field theory has a spacetime metric of the form

ds2 = L2

(r2zdt2 + r2dxidxjδij +

dr2

r2

), i ∈ 1, 2...d. (1.15)

The scaling symmetry (1.14) is an isometry of the metric along with r → λ−1r.

Various phenomenological holographic models that admit the spacetime metric (1.15)

as a solution have been constructed [65, 66]. In Chapter 2 of this dissertation we consider

one such model, the Einstein-Maxwell-Dilaton model, and construct a particular class of

candidate holographic gravity duals to quantum field theories with Lifshitz symmetry.

The construction is done in general (d+2)-dimensional spacetime and was first presented

in [67]. We study embedding of the Lifshitz solutions in an asymptotically AdS space

and provide a UV completion. For review articles on holographic AdS/CMT we refer the

reader to [68, 69, 70].

The spacetime metric (1.15) does not contain a symmetry that mixes time and space.

Holographic models which describe condensed matter systems with the full Galilean scal-

ing symmetry have been constructed in [50, 51],[71]−[79]. Bottom-up phenomenological

models have been very helpful in investigating properties of condensed matter systems

in their strongly coupled regimes. However, it is also of great importance to develop


top-down models in which the gravitational setup is explicitly embedded in superstring

theory/M-theory and to study the genuine quantum field theory dual to this specific

embedding. Embedding of holographic models with Lifhshitz and Galilean symmetries

in 10 and 11-dimensional supergravity theories have been studied in [80]−[85].

There has been a great amount of progress and development done on holographic

AdS/CMT models since the research presented in Chapter 2 was published. Holographic

AdS/CMT models with broken translational symmetry have been recently developed.

Gravitational models dual to condensed matter systems with lattices were constructed in

[86, 87, 88]. Optical conductivity of the dual field theories was computed numerically and

the results were found to agree very well with the properties of cuprate superconductors.

Analytic models of holographic lattices have been explored in [89].

Another class of holographic AdS/CMT models with broken translational symmetry

have gravitational solutions with Bianchi symmetries [90, 91]. The dual gravitational

models are anisotropic but homogeneous and admit analytical solutions. A holographic

model describing metal-insulator quantum phase transition was constructed in [92] using

Bianchi geometries. Top-down holographic models with spontaneously broken transla-

tional symmetry at some critical temperature were developed in [93, 94, 95, 96].

1.5 Fuzzball proposal and black hole information para-

dox

In his semi-analytic calculations performed in in 1975, Hawking [9, 10] considered quan-

tum field theory in a curved spacetime. He showed that pairs of particles are being

spontaneously created out of the vacuum in the black hole background. Each pair is

composed of two-particles which are entangled with each other. One member of the pair

has positive energy and the other has negative energy. For pairs created near the horizon

of the black hole, the particle which has negative energy falls into the black hole while


the particle with a positive energy leaves the horizon and climbs to asymptotic infinity.

The outgoing quanta of the pairs form the Hawking radiation.

Hawking’s discovery of pair creation and black hole radiation results in two main

problems in black hole physics: loss of unitarity and loss of black hole information.

Suppose for simplicity that the initial mass which forms the black hole is in a pure

state. For every Hawking pair, the outgoing quantum and the ingoing quantum form an

entangled state. If the black hole evaporates completely, there remain only the radiated

Hawking quanta which are entangled with no other quanta and form a mixed state. In

this process, the initial pure state of the black hole evolves into a mixed final state and

this violates unitarity of quantum mechanics.

Moreover, Hawking pairs are created out of vacuum fluctuations at the black hole

horizon and do not carry information about the matter which originally collapsed and

formed the black hole. Therefore, after the black hole evaporates completely, we are left

with Hawking radiation and the information of the initial state of the black hole is lost.

This problem is referred to as the black hole information paradox.

The black hole information paradox and the loss of unitarity have been two of the most

subtle puzzles of theoretical physics for almost the past four decades. The evolution of

black holes is very different from that of an ordinary hot body such as a burning baseball

dictionary, or a piece of coal. In the case of a burning piece of coal, the radiated quanta

are created by the constituent components of the coal and are interacting with the other

components and carry away the information of the initial state of the piece of coal.

At the early stages of the burning process, the radiated photons are entangled with

the components of the coal and this entanglement increases until the system reaches the

Page time, where half of the initial entropy of the hot body has been radiated [97]. After

passing the Page time, the entanglement decreases and at the end of the process there

remain the radiated photons which are only entangled with each other. The evolution of

the system is unitary and all the information of the initial state is retrieved at the end.


1.5.1 Mathur’s theorem

The evolution of black holes is of a different nature than that of normal hot bodies, in the

sense that the remaining Hawking quanta do not contain the information of the initial

state of the black hole. One might argue that there exist small subleading corrections

to Hawking’s results which potentially resolve both the unitarity problem and the in-

formation paradox. The suggested scenario is as follows: the initially collapsed matter

has exponentially small effects of on Hawking pairs. The exponentially large number

of the Hawking pairs compensate for the exponentially small values of these subleading

corrections. This is claimed to result in removing the entanglement between the radiated

quanta and the infalling quanta. The small corrections are also supposed to transfer the

information of the black hole to the Hawking pairs and extract all the information of the

initial state.

Mathur [98] showed that the small corrections cannot remove the entanglement be-

tween the members of the Hawking pairs. Mathur’s theorem asserts that Hawking’s

results are robust against subleading modifications [99, 100, 101]. The two members of

a Hawking pair form an entangled state which is of the form

eγ a† b†|0〉a|0〉b, (1.16)

where a is the quantum that falls into the black hole, b is the quantum that climbs to

infinity, and γ is a number of order unity. At a specific time t we may divide the black

hole system into three subsystems: the Hawking quanta radiated prior to time t (b),

the Hawking pair created at time t which consists of two members p(t) = (at, bt), and the

black hole interior which is composed of the initial matter that has collapsed and formed

the hole and the infalling members of the pairs (M, a).

Mathur’s theorem has two important results. First, under small corrections to Hawk-

ing’s leading results, the entanglement entropy of the Hawking pair p(t) with the rest of


the system is very small: S(p) < ε, where ε is the norm of the change to the initial state

and ε 1. This shows that the newly created pair is entangled very weakly with the

previously emitted quanta and the black hole. The second result is obtained from the

strong subadditivity of entanglement entropy

S(b+ bt

)+ S

(p(t)

)> S

(b)

+ S(at

). (1.17)

This results in the following inequality

S(b+ bt

)− S

(b)> ln 2− 2 ε, (1.18)

which shows that the entanglement of the Hawking radiation and the interior of the black

hole always increases after each pair production, even after the Page time, by at least an

amount of (ln 2 − 2 ε). This trend continues until the black hole evaporates completely.

Subleading corrections to the Hawking state cannot decrease the entanglement.

Mathur’s theorem proves that in order to remove the entanglement between the Hawk-

ing radiation and the rest of the black hole and obtain a pure final state, there needs to

be order unity changes to the state of the low energy modes at the black hole horizon. To

resolve the unitarity problem, the state of the Hawking pair has to become orthogonal

to the vacuum.

1.5.2 Fuzzball conjecture

The resolution to the black hole information paradox and unitarity problem in the context

of string theory was discovered by Lunin and Mathur [102, 103]. According to Mathur’s

theorem, order one changes to the state at the horizon are required to resolve the paradox.

In other words, one needs to find black hole hair at the horizon. However, the no-hair

theorem in four dimensions states that the black hole solution is the unique configuration


carrying the conserved charges and adding perturbations at the horizon do not result in

other possible configurations [104, 105].

Mathur and collaborators showed that this is not the case in string theory: string

theory black holes indeed have a plethora of non-perturbative hair. This relies on the

extra dimensions of string theory. The existence of the black hole hair changes the state

at the horizon by order unity and allows for removing the entanglement between the

radiation and the hole.

The fuzzball proposal asserts that quantum gravity effects are not restricted to a

region of Planckian scale at the centre of the black hole, but that they act on a region

with a scale of the order the size of the black hole horizon. According to this picture, the

ensemble of the large number of black hole hair is a fuzzy structure of quantum config-

urations inside a region of the size of the black hole horizon. This relies on parametric

enhancement by powers of N where N is the number of the quantum field theory ingre-

dients. Mathur and company coined the term fuzzballs to refer to these configurations

and presented the fuzzball proposal which shows how black hole information paradox is

resolved in string theory [106, 107].

The fuzzball proposal states that the microstates of a black hole are non-singular

horizonless geometries which are not spherically-symmetric, and horizonless geometries.

They carry the same conserved charges and have the same asymptotic behaviour as that

of the black hole. However, they differ from each other, and from the traditional structure

of the black hole, in a region of the size of the would-be black hole horizon.

The fuzzball programme shows how to construct fuzzball microstates. Generic fuzzball

microstates are complicated string theory configurations but some classes of microstates

can be represented by supergravity solutions. It is important to investigate the difference

between string microstates and supergravity microstates in order to better understand

the quantum nature of black hole microstates. This is one of the main focuses of this

thesis and will be discussed in Chapter 4.


Figure 1.1: The traditional geometry of a black hole is a coarse graining over an ex-ponentially large number of microstates. These microstates have the same asymptoticproperties as that of the black hole but differ from each other in a region of the size ofthe would-be black hole horizon. The microstates of a black hole are singularity-free,horizonless, and non-spherically symmetric configurations. The interior region of a blackhole with a singularity at the centre is now replaced by a nontrivial ensemble of fuzzballs.

Mathur’s conjecture has dramatically modernized our perception of spacetime. Ac-

cording to the fuzzball proposal, the conventional geometry of a black hole emerges as

a coarse grained average over its fuzzball microstates. This is schematically shown in

figure 1.1. The vacuum state at the horizon of a traditional black hole is viewed as a

non-trivial ensemble of fuzzball states.

Creation of Hawking pairs out of vacuum fluctuations is not the radiation mechanism

in this picture. Rather, radiated quanta are generated by constituent components of the

system. These radiated quanta interact with other components, form entangled states

with the degrees of freedom of the fuzzball ensemble, and carry the information of the

microstates. This is morally just like the process of burning of a hot piece of coal. Order

one corrections to the (would-be) horizon state solves the unitarity problem. The emitted

radiation contains all the information of the initial state and the information problem is

resolved.

Mathur’s proposal provides a complete picture of the dynamical formation, evolution,

and evaporation of black holes. It has been shown that a shell of collapsing matter


tunnels into the fuzzball microstates. Fuzzball density of states is eSbh , where Sbh is the

Bekenstein-Hawking entropy of the black hole. The exponentially large phase space of

fuzzballs can compete against the exponentially small tunnelling amplitudes and result

in the tunnelling of the shell. This process has been shown to happen on a time scale

shorter than that of the Hawking evaporation time [108]. In the context of the fuzzball

programme, traditional black holes are emergent phenomena which appear as a result of

coarse graining over the underlying microscopic string theory configuration. The fuzzball

proposal describes the full structure of the quantum system, both for degrees of freedom

inside and outside the would-be horizon.

It has been recently argued [109, 110] that the unitary evolution of Hawking radiation

requires that an infalling observer encounter high energy quanta at the horizon of an

old black hole and burn up. Hitting the firewall at the horizon contradicts the black

hole complementarity principle. Mathur and Turton proposed fuzzball complementarity

[111, 112] and addressed the infall of high energy observers (E kBT ) into the fuzzball

degrees of freedom. The key point is that the energetic observer excites many fuzzball

microstates and the evolution of these collective modes is described by evolution in a

classical black hole geometry. The firewall paradox has been studied in many different

contexts including quantum information theory [113, 114].

1.5.3 Two-charge and three-charge fuzzballs

One of the main objectives of the fuzzball programme is to construct the microstates

of a black hole and identify their properties. Fuzzball microstates are quantum config-

urations. As mentioned earlier, some microstates have classical geometric descriptions.

Supergravity techniques have been important tools in constructing and classifying these

microstates. Conformal Field theory (CFT) techniques have also played an important

role in identifying fuzzballs. We will discuss the role of CFTs in more detail in the next

subsection.


Let us consider the two-charge five dimensional extremal black hole in string theory

constructed by compactifying N5 D5 branes on T 4×S1 and N1 D1 branes on S1. A large

class of classical supergravity microstates have been found for this system, sufficient

to account for the entropy of the corresponding black hole. All of these geometries are

smooth, horizonless, and not spherically symmetric. These classical solutions are referred

to as microstate geometries. The microstate geometries of the two-charge black hole have

been constructed in [115, 116, 117, 118].

The moduli space of the solutions constructed in [115] is parametrized by a closed

curve in the four dimensional non-compact space. Geometric quantization was performed

in [119] to quantize the moduli space and count the number of the microstate geometries

represented by these solutions. The correct fraction of the entropy of the D1-D5 black

hole corresponding to microstate solutions of [115] were produced successfully under

geometric quantization.

Another important observation was made in [103] by evaluating the fuzzball surface

area, Afuzz, which is defined as the size of the region behind which microstate geometries

differ from each other. The result was that the fuzzball surface area reproduces the

Bekenstein-Hawking entropy : Afuzz/(4G) ∼ SBH .

For the three-charge extremal black holes in five dimensions and four-charge extremal

black holes in four dimensions, a large number of classes of microstate geometries have

been constructed [120]−[133]. All these solutions are smooth geometries and do not have

a horizon. Despite the large variety of solutions, unlike the case of two-charge extremal

black holes, microstate geometries constructed so far are not sufficient to reproduce the

black hole entropy: fuzzballs have a very quantum nature and some generic fuzzballs

may not be well represented by supergravity geometries.

In addition to the 3-charge fuzzballs mentioned above, a large number of microstate

geometries associated with black ring solutions with horizons and singularities were con-

structed in [134]−[137]. All the geometries found are smooth horizonless multi-centre


solutions with non-trivial topologies.

Microscopic geometries corresponding to non-extremal three-charge black holes con-

structed so far are the JMaRT solutions [138], the running bolt solutions [139], and the

more recent class of solutions given in [140]. Similar to the previous examples, these

microstate geometries do not have singularities or horizons. Emission from JMaRT non-

extremal fuzzballs was studied in [141].

In the near-decoupling limit, geometries of the JMaRT fuzzballs and the non-extremal

D1-D5-P black holes are composed of an outer asymptotically flat region, an inner asymp-

totically AdS region, and a neck region which connects the two outer and inner regions

together. The inner region of the non-extremal black hole is a BTZ black hole (locally

AdS3) and the inner region of JMaRT fuzzballs is global AdS3. JMaRT geometries pos-

sess ergoregions.

The rate of the ergoregion emission was evaluated in [142, 143, 144] and it was shown

that this rate matches exactly the rate of the Hawking radiation of the black hole. For

review articles on various aspects of the fuzzball programme and detailed description of

the extremal and non-extremal fuzzball geometries with different charges we refer the

reader to [145]−[152].

1.5.4 AdS3/CFT2

Consider D1-D5 system constructed by wrapping N5 D5 branes onM4 × S1 and N1 D1

branes on S1, where M4 is T 4 or K3. If the radius of the circle is much larger than the

volume of the torus, then the D1-D5 system in the low energy limit is described by a

(1+1)-dimensional N = (4, 4) conformal field theory that lives on S1. The AdS3/CFT2

correspondence asserts that superstring theory on AdS3 × S3 × T 4 is dual to the (1+1)-

dimensional CFT describing the D1-D5 system in the low-energy limit. The AdS/CFT

correspondence has proven to be a powerful tool to study black hole entropy and the

black hole information paradox and to investigate quantitatively the properties of the


fuzzball microstates [102, 103, 153]−[163].

As conjectured in [153]−[158], the 2-dimensional dual CFT possesses a position in its

moduli space at which the CFT is described by a (1+1)-dimensional sigma model with

target space (M4)N1N5/SN1N5 . This special point in the moduli space is referred to as

the orbifold point, i.e., the target space of the sigma model is N = N1N5 copies of M4

permuted by the action of the symmetric group. In the D1-D5 CFT literature, most work

(including [14]) has been done at the orbifold point where CFT is free and tractable.

In the context of Mathur’s fuzzball proposal, microstates of black holes are smooth

horizonless configurations which carry no entropy. Under the AdS/CFT correspondence,

these microstates are mapped into pure states of the CFT whose entropy is zero. The

traditional non-rotating black hole geometry in this picture corresponds to a thermody-

namic ensemble of the pure states. Traditional rotating black hole geometry is dual to a

thermal state carrying R-charges.

The dual CFT identifies all the fuzzball microstates. Finding CFT microstate duals

of generic fuzzball geometries is a very complicated task in general. However, the CFT

duals of some non-generic classes of microstate geometries have been identified [164, 165,

166, 167]. For the JMaRT non-extremal fuzzball solutions, the microstates of the dual

CFT at the orbifold point have been identified in [168]. The emission process in CFT

was studied in detail through constructing CFT vertex operators which relate the AdS

inner region to the flat outer region in the near-decoupling limit. The rate of emission

from the dual JMaRT CFT states was computed and found to match exactly with the

rate of the ergoregion emission from the corresponding JMaRT geometries.

The Hawking radiation from non-extremal non-rotating black holes, the superradiance

from non-extremal rotating black holes, and the ergoregion emission from non-extremal

fuzzball geometries were computed in [168] in both the supergravity and the CFT sides.

The deep result obtained is that the three different emission phenomena in the gravity

picture in fact correspond to the same CFT emission process: the microscopic CFT


description of black holes draws a unique picture of the emission process.

1.5.5 D1-D5 CFT at the orbifold point

The symmetric product sigma model of the D1-D5 brane system has been studied in great

detail in the past two decades. Orbifold CFT techniques provide a set of powerful tools

to compute physical quantities in the field theory side of the AdS3/CFT2 correspondence.

The results of the AdS5/CFT4 correspondence showed that the three-point functions of

chiral operators of the supergravity limit of type IIB string theory on AdS5 × S5 and

the three-point functions of the corresponding chiral operators of the dual N = 4 super

Yang-Mills theory are equal in the large N limit [169]. Spurred on by these results, there

was a keen interest in understanding if the same observation can be made in the case of

the AdS3/CFT2 correspondence.

There was also great interest in going beyond the supergravity approximation and

evaluating correlation functions of the superstring theory on AdS3×S3×M4 background.

The key underlying point is that the bulk theory corresponds to points in the moduli

space of the dual CFT which are different from the position of the symmetric product

sigma model. Therefore, matching of the correlation functions of chiral primary operators

would indicate that these quantities are protected as we move around in the moduli space.

Computations of three-point functions of chiral primary operators in AdS3/CFT2

have been performed for the type IIB supergravity and worldsheet superstring theory

and the results were compared to the correlators of the symmetric orbifold CFT [160,

170]−[176]. The correlation functions of the worldsheet theory and the orbifold CFT

matched precisely. For the supergravity three-point functions studied in [170] it was

found that although the overall form of the correlators agree with the CFT computations,

but the coefficients did not match.

The mismatch between the three-point functions computed in supergravity and orb-

ifold CFT were investigated in [176] and the apparent existing disagreement was resolved.


The solution relies on the fact that one needs to take into account operator mixing in

order to define the correct identification between the orbifold CFT chiral operators and

their dual supergravity chiral operators. The resolution to the puzzle applies to both

extremal and non-extremal three-point functions. Extremal correlation functions have

the property that the conformal dimension of the operator with the highest dimension

is equal to the sum of the conformal dimensions of the remaining operators in the corre-

lator. For non-extremal correlation functions there is non-trivial linear operator mixing

between single particle chiral primary operators of the theory. For extremal correlators

there exists mixing with multi particle chiral primaries at the leading order. It would

be interesting to further investigate operator mixing of chiral primary operators with

single and multi-particle primaries in the context of conformal deformation theory, which

studies deforming the CFT away from the orbifold toward the points with black hole

description in the moduli space [177].

Taking into account the appropriate operator mixing of chiral primaries, the exact

agreement between the extremal and non-extremal three-point functions computed in the

orbifold CFT, worldsheet theory of string theory, and supergravity theory is obtained

despite the fact that these theories correspond to distinct points in the moduli space.

This discovery strongly advocates for the existence of a non-renormalization theorem

for three-point functions of chiral primary operators in AdS3/CFT2. This theorem was

proved later in [178] for the extremal three-point functions as well as general extremal

n-point functions of chiral primary operators where n > 3. The proof was extended

more recently in [179] to the non-renormalization theorem for all three-point functions of

chiral primary operators in AdS3/CFT2. Moreover, non-renormalization of three-point

functions of half-chiral primaries of (1+1)-dimensional N = (4, 4) CFT and three-point

functions of chiral primaries of (1+1)-dimensional N = (0, 4) CFT were proved in this

work.

Some extremal four-point functions of chiral primary operators were computed in


the symmetric orbifold CFT and worldsheet string theory and the agreement between

the correlators were confirmed [180, 181, 182]. Recursion relations for some classes of

extremal n-point functions (n > 4) were also derived in these works and it was found

that the recursion relations match up to an overall factor.

A novel method for computing correlations functions in the symmetric product orb-

ifold CFT was developed by Lunin and Mathur in [171, 172]. The action of the symmetric

group introduces a new sector to the Hilbert space of the (1+1)-dimensional free CFT.

This sector is referred to as the twist sector. Twist operators belonging to the twist

sector permute various copies of the target space and impose new boundary conditions

on the fields such that they return to themselves only under the action of the symmetric

orbifold. The Lunin-Mathur (LM) technique was originally developed to evaluate general

correlation functions of the twist sector operators.

The key idea is to invent a map which takes us from the base space on which the

physics problem is originally defined to the covering surface of the base. The fields of

the CFT have normal periodic boundary conditions on the covering surface. Therefore

the bare twist operator, which carries no mode excitations, is mapped into the identity

operator on the covering surface. One then relates the path integral on the base space

involving twist operators to the path integral on the covering surface with no twist

operator insertions.

The LM technique provides an elegant tool to transform complicated correlation func-

tions of twist sector operators in the base space to computations of simple correlators

on the covering surface. In [171] the authors considered a bosonic (1+1)-dimensional

CFT and compute the two-point, three-point, and specific examples of four-point func-

tions of twist operators. The results were then extended to study the (1+1)-dimensional

N = (4, 4) CFT in [172]. They evaluated extremal and non-extremal three-point corre-

lation functions of the chiral primary operators and found agreement with the previous

results mentioned above.


We will discuss some aspects of the LM method in more detail in Chapter 3. As

already mentioned, the LM technique computes correlation functions of twist sector op-

erators of the symmetric orbifold CFT. The non-twist sector of the orbifold CFT contains

states which are not affected by the action of the symmetric group. In order to study

the interaction of the twist and the non-twist sectors, one is interested in computing

correlation functions which contain both types of operators. Moreover, since the non-

twist sector states do not contain the complications of the symmetric orbifolding action,

they provide interesting examples for investigating the nature of string states and super-

gravity states of the microscopic CFT. These statements motivate the generalization of

the LM technique to compute correlation functions which contain contributions from the

non-twist sector. This generalization is done in [183] and will be described in Chapter 3

of this dissertation.

CFT techniques have been very helpful in studying microscopic aspects of the fuzzball

proposal. One of the useful CFT techniques is the spectral flow transformation [184].

Spectral flow transformation is defined for the extended superconformal algebras. Under

a spectral flow, the generators of the algebra are transformed such that the new algebra

and the old algebra are isomorphic. Spectral flow transformations with both integer and

fractional parameters were applied in [167] to the twisted Ramond ground states of the

orbifold CFT in the base space. They identified the microscopic CFT description of the

family of all two-centred extremal microstate geometries of Bena and Warner [134].

1.5.6 Moving away from the orbifold point

Since the orbifold CFT and the supergravity theory sit at different points of the moduli

space of the D1-D5 CFT, in order to compare the states of the two theories we have to

consider those states that are not renormalized as we move around in the moduli space.

The two-point functions and three-point functions of these states are protected against

renormalization. Generic string states of the CFT, however, are not protected against


corrections. Physical quantities constructed out of these generic states are renormalized

as one moves across the moduli space.

In the context of the fuzzball programme, it is of great importance to analyze prop-

erties of the microscopic degrees of freedom of black holes and to investigate the micro-

scopic description of the dynamical formation and evolution of black holes. This requires

a deep understanding of both the microscopic CFT and the supergravity descriptions

of the D1-D5 system. The grand objective is to connect the two pictures together and

to describe quantitatively in the microscopic language the emergence of black holes as a

coarse grained average over the fuzzball microstates.

In Chapter 4 of this dissertation we consider some of the microscopic CFT aspects of

the fuzzball proposal which were studied recently in [177]. The goal is to better under-

stand the quantitative properties of the orbifold CFT states as the theory is deformed

away from the orbifold point toward the point which has a black hole description. This

deformation is done under the action of the marginal deformation operators of the theory

which belong to the twist sector. We are particularly interested in lifting of string states

of the CFT under the deformation. Also of great interest is to determine mixing of string

states away from the orbifold point.

Another aspect of the microscopic CFT which was studied in [185, 186] is concerned

with the evolution of the states of the CFT under the effect of the deformation operator.

In [185] the authors compute the effect of the deformation operator on the Ramond

vacuum of the CFT and find that the final state has the form of a squeezed state carrying

pairs of bosonic and fermionic excitations on the Ramond vacua of the twist sector

vacuum. The results were generalized in [186] to find the evolution of general excited

initial states of the CFT.


1.6 Outline of the thesis

This thesis is composed of two parts. In the first part, which contains Chapter 2, we

investigate the holographic AdS/CMT correspondence and explore holographic duals to

quantum field theories with Lifshitz scaling symmetry. In the second part, which includes

Chapters 3 and 4, we investigate the AdS3/CFT2 correspondence in the context of the

D1-D5 brane system and the fuzzball proposal. Chapters 2, 3, and 4 of this thesis are

based on research originally presented in papers [67], [183], and [177], respectively. In

the remainder of this subsection we first describe the contributions of the author of the

dissertation to these articles and then present the outline of the thesis.

The article “Lifshitz-like black brane thermodynamics in higher dimensions” [67] is

the first paper I co-authored during my PhD. I joined the collaboration after the concep-

tual development of this project. My contribution to this work was in form of performing

analytical computations and numerical analyses of the Lifshitz holographic model as well

as working on the draft of the article with my collaborators. In the second phase of my

PhD research, I co-authored articles “Operator mixing for string states in the D1-D5

CFT near the orbifold point” [183] and “Twist-nontwist correlators in MN/SN orbifold

CFTs” [177]. I played a key role in the development of a new research programme in the

microscopic string theory aspect of the fuzzball proposal in our research group. I per-

formed all of the subsequent CFT computations, for comparison with my collaborators,

and contributed significantly to writing the paper drafts.

The outline of this dissertation is as follows. In Chapter 2 we study an Einstein-

Maxwell-Dilaton model and construct a particular class of candidate holographic gravity

duals to Lifshitz field theories in general (d + 2)-dimensional spacetime. We determine

the embedding of these Lifshitz solutions in an asymptotically AdS geometry and study

the thermodynamic properties of our Lifshitz backgrounds.

In Chapters 3 and 4 we explore properties of the D1-D5 CFT as it is deformed away

from the orbifold point toward the point in the moduli space which has a gravity descrip-


tion. We investigate lifting of low-lying string states of the CFT under this deformation.

In Chapter 3 we study general (1+1)-dimensional symmetric product orbifold CFTs and

develop the generalization of the Lunin-Mathur covering space method in two different

ways. First, we determine how to compute correlation functions containing twist sector

operators excited by modes of fields that are not twisted by the twist operator on which

they act. Second, we show how to compute correlation functions which contain both

non-twist and twist sector operators. We work two examples, one using a simple bosonic

CFT, and one using the D1-D5 CFT at the orbifold point. We show that the resulting

correlators have the correct form for a 2D CFT.

In Chapter 4 we develop a method to compute the anomalous dimensions of the low-

lying string states of the D1-D5 CFT as it is deformed away from the orbifold point. We

use the generalized Lunin-Mathur technique developed in Chapter 3 and conformal per-

turbation theory to investigating lifting of the string states. The CFT is deformed away

from the orbifold point under the action of the exactly marginal deformation operators

belonging to the twist-2 sector. The procedure of computing the anomalous dimension

is done in an iterative way through computing the relevant mixing coefficients at each

iteration stage. We check our method by showing how the operator dual to the dilaton

does not participate in mixing that would change its conformal dimension, as expected.

Next, we complete the first stage of the iteration procedure for a low-lying string state

of the form ∂X∂X∂X∂X and compute its mixing coefficient.

In Chapter 5 we present a summary of our results and a discussion of possible direc-

tions for future work.

Chapter 2

Lifshitz black brane thermodynamics

in higher dimensions

In this chapter we investigate holographic duals to Lifshitz field theories in general d+2

dimensions. The contents of this chapter were first presented in [67].

There has been much effort put into describing quantum critical behaviour of con-

densed matter systems. Quantum critical systems typically exhibit a scaling symmetry

t→ λzt, xi → λxi, z ≥ 1. (2.1)

From a holographic standpoint, this suggests the form of the spacetime metric

ds2 = L2

(−r2zdt2 + r2dxidxjδij +

dr2

r2

), i ∈ 1, 2...d. (2.2)

Here we will be concerned with phenomenological models that admit the metric (2.2)

as a solution. Two such models with d = 2 have Lagrangians given by [65]

S ′ =1

16πG4

∫d4x√−g(R− 2Λ− 1

4G2 − c2

2A2

)(2.3)

30

Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 31

(we call this model S ′) and [66]

S =1

16πG4

∫d4x√−g(R− 2Λ− 2(∇φ)2 − e2αφG2

), (2.4)

(we call this model S) where in either case G = dA is a two form field strength. There

are also examples where R2 corrections support solutions with the metric (2.2) [187, 188,

189, 190, 191]; however, we will not consider these here.

There are several differences between the above models. The solution with metric

(2.2) of the model [65] possesses an exact Lifshitz isometry of the background, where

the model we consider S is only “Lifshitz-like”: the background constructed in [66] has

a logarithmically running dilaton, and so the full solution, including the other fields, is

not exactly Lifshitz rescaling invariant. Further, the model S has a bulk U(1) gauge

symmetry and admits an exact black brane solution that asymptotes to the metric (2.2),

which can also be generalized to higher dimensions [192]. Finding black brane solutions

for action S ′ has proven more difficult, and one often needs to resort to numeric methods

[193, 194, 195, 196], but not always [194, 197] (an analogous analytic statement for an R2

extension may be found in [198], and one should also see [199] where a certain extension

to this model admits an analytic black hole). For extensions and variations to the basic

model S ′, see [198, 200, 201], and for the holographically renormalized action, see [202].

Similar actions to S can be found in [203, 204, 205, 206, 207, 208, 209, 210, 211, 212,

213, 214], which either contain different matter fields (probe, or back-reacted), different

couplings, or some combination. However, here we will study the action S, also studied

in [66, 196, 215, 216], generalized to arbitrary dimension as in [192]

S =1

16πGd+2

∫dd+2x

√−g(R− 2Λ− 2(∇φ)2 − e2αφG2

). (2.5)

We will consider “UV completing” by embedding black branes into asymptotically AdS

space, generalizing to generic d the discussion of [216]. We will explore the thermody-


namics of these backgrounds analytically and numerically.

As described in the previous work [216], there are two scales in the problem T and

µ, and there are two regimes, T µ (Lifshitz-like) and T µ (AdS-like). Therefore,

one might anticipate a thermal instability during the transition between the two regimes.

However, in the d = 2 case, we found that there was no such instability, and so no

discontinuous phase transition. We thought that this could be related to a Coleman-

Mermin-Wagner theorem applied to the thermal vacua of a 2 + 1 dimensional theory.

However, here we find that there is no discontinuous phase transition, and the model

smoothly interpolates between the Lifshitz-like behaviour and the AdS-like behaviour,

regardless of d.

Finally, in the Lifshitz-like regime, we expect a relation of the form

4Gd+2s = c(z, d)(Lµ)d(T

µ

)d/z(2.6)

where s is the entropy density, T is the temperature, d is the number of spatial dimensions,

and z is the critical exponent. The coefficient c(z, d) is in some sense a measure of the

number of degrees of freedom of the system, and we find its behaviour as a function of d

and z numerically.

The rest of this chapter is organized as follows. In section 2.1 we reduce the model to

an effective radial model for arbitrary dimension d, and find that the equations of motion

reduce to a set of 4 first order ordinary differential equations. We comment on the proper

normalization of the charge density q. We compute the perturbative expansion at the

horizon and at the AdS asymptotic, and use this to show the relation

E =d

d+ 1(Ts+ µn) (2.7)

which is necessary by the scaling symmetry of AdS [216].

Finally, in section 2.2, we turn to numeric results. We set up the numerical integration,


and show how to efficiently parameterize the different regimes of T/µ with one piece

of data specified at the horizon. We then numerically integrate, and show that for

d = 2, 3, 4 · · · 9 that there is no discontinuous phase transition: the solutions smoothly

and monotonically interpolate between the two regimes T/µ 1 and T/µ 1. We then

find c(z, d) for a range of z and d, and discuss some of its qualitative behaviour.

2.1 Analysis of the model

2.1.1 Reduction

We wish to consider dilatonic black brane solutions to the equations of motion following

from the action

S =1

16πGd+2

∫dd+2x

√−g(R− 2Λ− 2(∇φ)2 − e2αφG2

), (2.8)

and reduce on the following Ansatz

ds2 = −e2A(r)dt2 + e2B(r)(dxidxjδij

)+ e2C(r)dr2,

φ = φ(r),

A = eG(r)dt. (2.9)

Here, i ∈ 1, 2, · · · d and G = dA. We reduce the d + 2 dimensional action to a one

dimensional action 1

S =1

16πGd+2

∫2dt

d∏i=1

dxi

∫drL1D (2.10)

1We keep track of normalization for later use.


where the one dimensional Lagrangian is given by

L1D = deA+dB−C∂A∂B +d(d− 1)

2eA+dB−C(∂B)2 + e−A+dB−C+2G+2αφ(∂G)2

−eA+dB−C(∂φ)2 − eA+dB+CΛ, (2.11)

and we have defined ∂ ≡ ∂∂r

to simplify notation. Note that when d = 2, this Lagrangian

agrees with the one investigated in the earlier work [216]. It can be verified that all

equations of motion associated with the action (2.8) are reproduced by (2.11), viewing

C as a Lagrange multiplier for the above action. The equation of motion for C imposes

the “zero energy” condition. Note that, in this model, we have two redundancies: r

coordinate changes and U(1) gauge changes. We will refer to the r diffeomorphisms as

“coordinate gauge” transformations to differentiate from the U(1) gauge changes.

As in the previous work [216], we see that there are many conserved quantities. These

are associated with the symmetries of the 1D action 1) (A,B,C, φ,G) → (A + dδ1, B −

δ1, C, φ,G + dδ1), 2) eG → eG + const, and (A,B,C, φ,G) → (A,B,C, φ + δ2, G − αδ2).

These are understood as 1) as a rescaling of the time coordinate, and the xi coordinates

that leaves dt∏

i dxi invariant, 2) the global part of the gauge symmetry associated with

A, and 3) a redefinition of the gauge coupling by shifting the dilaton (we will call this

the “global symmetry” in what follows). There is also the Hamiltonian constraint or,

equally, the equation of motion for C. These symmetries lead to the following first-order

ordinary differential equations

deA+dB−C∂A∂B +d(d− 1)


−eA+dB−C(∂φ)2 + eA+dB+CΛ = 0, (2.12)

deA+dB−C∂A− deA+dB−C∂B − 2de−A+dB−C+2G+2αφ∂G = D0, (2.13)

2eA+dB−C∂φ+ 2αe−A+dB−C+2G+2αφ∂G = P0, (2.14)

e−A+dB−C+G+2αφ∂G = Q. (2.15)


which are completely equivalent to the equations of motion reduced on our Ansatz. We

find the combinations D0 = αD0 +dP0, P0 = 12P0 more convenient, and so the equations

become



−eA+dB−C(∂φ)2 + eA+dB+CΛ = 0, (2.16)

d (2∂φ+ α (∂A− ∂B)) eA+dB−C = D0, (2.17)

eA+dB−C∂φ+ αe−A+dB−C+2G+2αφ∂G = P0, (2.18)

e−A+dB−C+G+2αφ∂G = Q. (2.19)

Note that P0 transforms under global U(1) gauge transformations, and we may use this

to set P0 = 0, which we will do for the bulk of the chapter.

There are two known one parameter families of solutions (for nonzero α). First, there

is the AdS black brane

A(r) = ln

(Lr

√1−

(rhr

)d+1),

B(r) = ln(Lr),

C(r) = ln

L

r

√1−

(rhr

)d+1

,

φ(r) = φb, A = gbdt. (2.20)

where Λ = −d(d− 1)/(2L2), and gb and φb are arbitrary constants. For this background,

the conserved quantities are Q = 0, D0 = d(d+ 1)Ldαrd+1h /2 and P0 = 0. Further,

the solution has T = LT = rh(d+ 1)/(4π) by reading off the periodicity of imaginary

time at the horizon. Further, using the area law for the entropy density, we find that

s = rdh/(4Gd+2), or in thermodynamic terms s = (4π)dT dLd/[(d+ 1)d4Gd+2

]. This shows

that the number of degrees of freedom in the field theory is linear in Ld/(4Gd+2).


There is also the Lifshitz black brane given by [66, 192]

A(r) = ln

(LLaLr

z

√1−

(rhr

)z+d), ,

B(r) = ln(Lr),

C(r) = ln

LL

r

√1−

(rhr

)z+d , ,

2αφ(r) = ln(r−2dΦ

),

G(r) = ln

(z − 1)LdaLrz+d(

1−(rhr

)d+z)

2Q

. (2.21)

The constants are given by

L2L = L2 (z + d)(z + d− 1)

d(d+ 1),

Φ =2(Q/Ld−1

)2(z + d− 1)

d(d+ 1)(z − 1), (2.22)

aL is arbitrary (and may be removed by time rescaling), and α =√

2d/(z − 1). The

conserved quantities are

P0 = 0, D0 =√d/ [2(z − 1)] rz+dh aL L

d d(z + d), and Q, (2.23)

where Q is arbitrary. Given that B(r) is normalized in the same way as the AdS black

brane, we read

s =rdh

4Gd+2

. (2.24)

Finally, we see that T = rzhaL(z + d)/(4π). While this is a dimension-free measure

of temperature, it is not clear what units to use. This is to be expected for a non

relativistic theory because energies and length scales are not interchangeable. However,

this ambiguity is only a constant, and so s ∝ T d/z. When we consider embedding these

black branes into AdS space the ambiguity is removed.


2.1.2 Perturbation theory at the horizon

We first expand the functions as “correction functions” that should go to constant values

for an AdS background. We, therefore, define

A(r) = ln(Lr) + A1(r), B(r) = ln(Lr),

C(r) = ln

(L

r

)+ C1(r), G(r) = ln(L) +G1(r). (2.25)

We further define dimensionless conserved quantities

Q = Ld−1Q, D0 = LdD0, P0 = LdP0 = 0 (2.26)

where we remind the reader that we will only use the P0 = 0 gauge. We have four first

order differential equations, however we only have three dynamical functions. Therefore,

we may eliminate one function using an algebraic expression, and we do so for the field

e−2αφ:

e−2αφ =1

2r2Q2αeA1+C1

(dr2d+2α(d+ 1)

(eA1+C1 − eA1−C1

)−2rd+1D0 − 4drd+1eG1αQ+ 2αeC1−A1(αQeG1)2

). (2.27)

Using this, the other differential equations may be written as

∂eA1 −eC1

(D0 + 2dαQeG1

)dαrd+2

= 0, (2.28)

∂eC1 −e2C1−2A1

(2αeC1α2Q2e2G1 − 2deA1rd+1αQeG1 − D0e

A1rd+1)

dαr2d+3= 0, (2.29)

∂eG1 − αdr2d+2

2αQrd+2

((d+ 1)

(eA1+C1 − eA1−C1

)(2.30)

−2rd+1(D0 + 2dαQeG1

)+ 2αeC1−A1α2Q2e2G1

)= 0.


These functions may be expanded around a regular horizon as

eA1 = a0

((r − rh)

12 + a1(r − rh)

32 + · · ·

),

eC1 =c0

(r − rh)12

+ c1(r − rh)12 · · · ,

eG1 = a0

(g0(r − rh) + g1(r − rh)2 + · · ·

). (2.31)

One may read that the dilaton goes to a constant. With P0 = 0, we find that the

equations of motion require

a0 =2c0D0

αdr2+dh

, a1 =α2d(d+ 1)2c4

0 − 2drh(α2 − 2)(d+ 1)c2

0 + r2h (α2d− 4− 6d)

8r3h

,

c0 = c0, c1 =c0 (3α2d(d+ 1)2c4

0 − 2drh(2 + 3α2)(d+ 1)c20 + r2

h (3α2d+ 4 + 6d))

8r3h

,

g0 =d((d+ 1)c2

0 − rh)rdh2c0Q

, (2.32)

g1 =d2

8Qc0rh

(rd−2h α2(d+ 1)3c6

0 − rd−1h α2(d+ 1)2c4

0

−rdh(α2 + 2)(d+ 1)c20 + rd+1

h

(α2 + 2

)).

This will provide initial conditions for the equations of motion when we numerically

integrate. From this expansion we can read the temperature and entropy density

T =r2ha0

4πLc0

=D0

αdrdh2πL, s =

rdh4Gd+2

. (2.33)

Note that, in the above expressions, c0, a0, and D0 are all implicitly functions of rh; they

are chosen so that the metric functions approach their correctly normalized AdS values.

For example, we find, using the above type of expansion that the Lifshitz black brane

satisfies c0 =√α2 + 2

√rh/(d+ 1) while the AdS black brane satisfies c0 =

√rh/(d+ 1).

We expect that these two limits bound the physical values of c0.


2.1.3 Perturbation theory at r =∞: AdS asymptotics

We want to expand the solution about infinity and require that it asymptotes to the pure

AdS solution. The equations of motion are:



−eA+dB−C(∂φ)2 + eA+dB+CΛ = 0, (2.34)

d (2∂φ+ α (∂A− ∂B)) eA+dB−C = D0, (2.35)

eA+dB−C∂φ+ αQeG = P0, (2.36)

e−A+dB−C+G+2αφ∂G = Q. (2.37)

We use the expansion:

A(r) = ln(Lr) + A1(r),

B(r) = ln(Lr),

C(r) = ln

(L

r

)+ C1(r), (2.38)

φ(r) = ln(Φb) + φ1(r),

G(r) = ln(gb) +G1(r),

where now A1(r), C1(r), G1(r), and φ1(r) are understood to be perturbative functions.

Inserting equations (2.38) in the equations of motion (2.34)-(2.37) and performing the

integrations gives (to leading order)

A1(r) = −D0 + 2dαQgb − 2dP0

d(d+ 1)αLdrd+1+

Q2

d(d− 1)Φ2αb L

2(d−1)r2d, (2.39)

C1(r) =D0 + 2dαQgb − 2dP0

d(d+ 1)αLdrd+1− 2Q2

(d− 1)(d+ 1)Φ2αb L

2(d−1)r2d, (2.40)

G1(r) = − Q

(d− 1)gbΦ2αb L

d−2rd−1, (2.41)

φ1(r) =αQgb − P0

(d+ 1)Ldrd+1− αQ2

2d(d− 1)Φ2αb L

2(d−1)r2d. (2.42)


The constants of integration of equations (2.36) and (2.37) (which would lead to con-

stant modes in (2.42) and (2.41) ) are absorbed in the boundary values of Φb and gb ,

respectively. The constant of integration of equation (2.35) is removed by rescaling the

time coordinate.

We next want to evaluate the energy density. We follow closely the previous discussion

for the d = 2 case [195], and generalize it here to arbitrary d. We use the background

subtraction technique in [217], where the total energy density is given by

E = − 1

8π

(Nt

(dK −d K0

)−Nµ

t pµν rν

), (2.43)

where Nt is the lapse function, Nµt is the shift vector, dK is the extrinsic curvature of the

d-dimensional spatial boundary slice inside the constant-t slice, dK0 is the d-dimensional

extrinsic curvature of the reference background, pµν is the momentum conjugate to the

time derivative of the metric on the constant-t slice, and r is the spatial unit vector

normal to the constant-r surface. The reference background we take is pure AdS in the

Poincare patch.

For our metric, Nt = eA, Nµt = 0, r = eC , and the energy density is

E = lim

(−1

8π

)eA(d

2∂r(e2B)e(d−2)B

(e−C − e−Cref

)) ∣∣∣∣r→∞

. (2.44)

For the pure AdS reference, we have eCref = Lr. The total energy density then reads

E = limr→∞

2d rd+1

16πGd+2LC1 (2.45)

where we have dropped terms that go to zero in the large r limit. Therefore, we read the

energy density to be

E =1

16πGd+2

2

(d+ 1)

(D0 + 2dαQgb)

αLd+1. (2.46)

Now we need to normalize the charge density Q. For this, we follow the discussion of


[218], where the authors use the action to normalize Q in such a way that it is conjugate

to the fixed field value eG at infinity. In our context, this reads

SG2 =Vd2

µgeomT

qgeom, (2.47)

where qgeom is a normalized charge density and µgeom is the potential difference between

the horizon and the boundary. For us, we are using a gauge where P0 = 0 and so the

potential at the horizon is 0, and so gb = µgeom for this gauge choice. We calculate SG2

and find

SG2 =1

16πGd+2

2VdLd+1

µgeomT

Q, (2.48)

and so we identify the charge Q as

Q =16πGd+2L

d+1

4qgeom. (2.49)

Inserting the scaled quantities 2.26 and using the relations 2.33 for the temperature

and entropy density along with the normalization of Q, we find that

E =d

d+ 1(Ts+ µn) , (2.50)

where we have changed to field theory quantities µgeom = µL2, qgeom = n/L2. This is the

expected relation for a conformal field theory.

2.1.4 Other considerations, and setup for numerics

Given the discussion of perturbation theory around the horizon, we find that the constant

c0 has the following window of allowed values

√rh

d+ 1< c0 <

√(2 + α2)

√rh

d+ 1(2.51)


where the lower limit gives the pure AdS black brane, and the upper limit gives the

pure Lifshitz black brane condition. Using this, we may efficiently parameterize all black

branes in terms of this one quantity, which we now explain.

First, we may rescale r by a constant: this will affect the location of the horizon, and

effectively allows us to fix the horizon to be at rh = 1. After doing so, we may use time

rescaling to set D0 to be any value we wish, and further we may use the global symmetry

associated with shifting the dilaton to move Q to be any value we wish. Therefore, we

may fix all of the horizon data except for c0 using symmetries. Then, c0 parameterizes

the different black branes, and the allowed region of c0 is given by the above range (2.51).

However, now the asymptotic values of the fields are non-canonical. This means that,

given a c0, we take these asymptotic values of the fields eA, eG, eφ to be the output values

of the numeric integration started at the horizon. These output values encode how to

use the time rescaling and the global symmetry to bring them to their canonical values.

Hence, these parameterize a given T and µ for a fixed rh = 1 black brane. Near the

limiting values of the range (2.51) we expect the ratio T/µ to go to zero (Lifshitz-like

regime) or to infinity (AdS regime), and so all possible ratios of T/µ are explored.

To find the generic black brane with temperature T and chemical potential µ, we

would simply find the appropriate black brane at rh = 1 with the same value of T/µ. We

would then use the scaling symmetry of AdS to adjust, say T , to its correct value. The

resulting black brane has the specified value of T as its temperature, µ as its chemical

potential, and the position of the horizon rh will be determined by the rescaling. These

will all be unique as long as the ratio T/µ occurs only once for the rh = 1 reference black

brane. The crucial question is then whether the graph of T/µ vs. c0 is monotonic for

the rh = 1 black branes. If it is, there is always a unique black brane given a particular

value of T and µ, and so there are no possible discontinuous phase transitions. Indeed,

we find this is the case. For the sake of clarity, instead of graphing the quantity T/µ as

a function of the somewhat esoteric quantity c0, we simply graph s vs. T for fixed µ. We


will see that it is monotonic, and interpolates between the Lifshitz-like scaling s ∝ T d/z

and the AdS behaviour s ∝ T d. Accordingly, there is no discontinuous phase transition.

Next, we recall a subtlety that emerged in [216] associated with the definition of

the chemical potential µ, and we recall the resolution here. First, the physical value

of eG1 is not strictly determined. We may use the global symmetry (A,B,C, φ,G) →

(A,B,C, φ+δ2, G−αδ2) to rescale this value to any value we wish. Because this is a global

symmetry that does not involve the metric, the stress energy tensor (and therefore the

geometry), is not determined by this number. In fact, only global symmetry invariants

can determine anything in the geometry (this was also noticed in [215] for extremal

solutions). We wish to consider µ as a scale in the theory, i.e. it is the scale at which

new particles can be added/excited, and so we expect this to correspond to some scale

in AdS; thus, it must be a global symmetry invariant.

We fix this ambiguity by requiring the gauge coupling exp(−αφ) to approach 1 at

r → ∞ by using the global symmetry that redefines φ. This is equivalent to requiring

that the gauge kinetic term go to a canonical value, which, for simplicity, we choose to

be 1. This is something to be expected: only fixing both the asymptotic value of eαφ and

the asymptotic value of eG1 will determine the geometry. Only one combination, a global

symmetry invariant, can possibly affect the geometry, because the metric is not involved

in this symmetry. Therefore, one can read the numeric values of the functions at infinity

and determine µ,

µ =eG1eαφ

eA1

∣∣∣∣∣r=∞

. (2.52)

One can see that this definition is time rescaling and global symmetry invariant. To get

some physical interpretation of this result, we consider the case where eA1 has already

been fixed to go to 1 at infinity. Then the above simply states that what we have done is

take some bare quantities qgeom and µgeom and combined them into the global symmetry

invariant qgeomµgeom =(eαφ|r=∞ qgeom

) (e−αφ|r=∞ µgeom

). The second expression is made

out of global symmetry invariants (e.g(eαφ|r=∞ qgeom

)). These, therefore, can influence


the geometry and set a scale.

With the above considerations, and the perturbative results near the horizon, we may

use the equations (2.28)-(2.30) and numerically integrate them from the horizon to a large

value of r, where the functions settle to their AdS asymptotic values. These values at

infinity furnish the required information to determine T and µ for the black brane.

2.2 Results and discussion

Here we discuss the results found from numerically integrating the equations of motion.

In Figure 2.1 we give log-log plots of the dimensionless entropy density (4Gd+2s) versus

the dimensionless temperature T for fixed µ = 1 and various dimensions.

As seen in Figure 2.1, the entropy density is smooth and monotonic in T for a wide

range of z, and there are no possible discontinuous phase transitions associated with

going from T µ to T µ. The same behaviour is observed for the other dimensions

2 ≤ d ≤ 9 not plotted in Figure 2.1. Furthermore, we observe the correct asymptotic

behaviour: the slope approaches d/z in the Lifshitz-like regime (T µ), and the slope

approaches d in the AdS regime (T µ).

It was thought in earlier work [216] that the behaviour in d = 2 might follow from

the relatively low dimension of the dual model. However, the results of Figure 2.1 show

that this is not the case. We see no evidence for a discontinuous phase transition or

thermal instability, regardless of the dimension. To be sure of these results required a

careful numerical examination of the dilaton at infinity to ensure that it was, indeed,

smooth and approaching a constant value, as is appropriate for AdS. This is because the

expression (2.27) for φ has explicit r dependence, and so rounding errors will eventually

occur for sufficiently large r. The powers of r involved are dimension-dependent, so the

rounding effect is more severe the higher the dimension.

Next, we turn to the question of measuring the number of degrees of freedom for


(a) d = 3. (b) d = 5.

(c) d = 7. (d) d = 9.

Figure 2.1: The plots of ln (4Gd+2s) versus ln(LT ) for fixed µ = 1. Figures (2.1a), (2.1b),(2.1c) and (2.1d) correspond to d = 3, 5, 7, and 9, respectively. The different curves ineach plot correspond to different values of α, with α = 4 green (solid), α = 2 magenta(long-dashed), α = 1 cyan (dot-dashed), and α = 0.75 blue (dashed). α is related to zvia α =

√2d/(z − 1).


the Lifshitz-like theories. We note first that the entropy density of the system in the

Lifshitz-like limit is of the form

4Gd+2s = c(z, d) (Lµ)d(T

µ

)d/z, (2.53)

where c(z, d) is a function of the dimension and the critical exponent. We have chosen

µ = 1 in the plots shown in Figure 2.1, and so the above expression reduces to 4Gd+2s =

c(z, d) T d/z. Therefore, if we fit the ln(T ) ln(µ) portion of the graphs in Figure 2.1

with a line, the intercept value of this line gives ln(c(z, d)). This coefficient is a direct

measure of the degrees of freedom of the theory. Figure 2.2 depicts the behaviour of

ln(c(z, d)) versus ln(z) for various dimensions.

We can get a qualitative picture of how these must behave in the large z limit. First,

note that as z becomes large, the slope of the T µ part of the graph, d/z, goes zero.

The intercept, therefore, approaches a fixed value, and so c(z, d) must asymptote to a

constant number. This is a trend we do observe in Figure 2.2. Therefore, c(d, z) only

depends on the dimension of the theory at large z.

We cannot compare the values of c(z, d) we found for different z and different d, since

they correspond to different theories. Nonetheless, it would be interesting to compare

the values of c(z, d) we have found here for model S with the same quantities in different

models. We leave this to future work.

In the following, we will compare the large z behaviour of the graphs of Figure 2.2

with expectations from the dual field theory, using a simplified model. We start by

considering a (2π`)d volume box of some medium, and consider the excitations about the

ground state. For such a case, the “legal” wave numbers may be written

~k =

n1/`

n2/`

...

. (2.54)


Figure 2.2: The plot of ln (c(z, d)) as a function of ln(z) for fixed value of µ = 1. Thedifferent curves correspond to different dimension with d = 3 black (solid), d = 5 brown(long-dash), d = 7 red (dot-dash), and d = 9 coral (dash). Curves for d = 2, 4, 6, 8 behavesimilarly. First of all, notice that the curves’ intercept at ln(z) = 0 is given by the valueln((4π)d/(d+ 1)d

), which is non-monotonic in d. Secondly, we find it interesting that the

tails become flat out at z →∞ and that the large z behaviour for various d is monotonicin d. Given these two facts, it may be no surprise that the curves ln(c(z, d)) for fixedd are generically non-monotonic for small values of z and that, in fact, they cross eachother.

Further we assume Lifshitz scaling symmetry and rotational symmetry in the spatial

dimensions (broken only by the presence of the “box”). The only consistent relation

between the energy of a state and the wave numbers above is

ω~n = (√k2)zµ1−zδ = (

√n2)zµ

1

(`µ)zδ (2.55)

where the powers of µ (an intensive parameter) appear by dimensional analysis. The

coefficient δ will in general be a function of z and d. For the time being we will leave this


implicit, because we expect that δ should remain order 1 as it contains factors of π and√

2. Indeed, µ is supposed to be the relevant scale for new physics, and a large δ would

make this identification suspect.

Proceeding as in [216], we assume that the occupation number of excitations above

the ground state is e−βωnF ′(e−βωn

). This is merely a convenience since the function F

remains unspecified. The only content is that the occupation number is some function

of the ratio of the temperature to the energy of the state. An important physical as-

sumption has gone in at this point. Essentially we have a bath of particles, with some

number density n. We are assuming very small temperatures so that this density may be

considered very large, i.e., even though we are exciting a certain number of them above

their ground state, the “same number” remain unexcited. Therefore, we are treating this

number density of particles as an inexhaustible bath. This essentially means that the

number of possible excitations is approximately infinite, and there is no effective chem-

ical potential for adding new ones. The thermodynamics is governed by some collective

mode, the “phonons”, rather than the constituents of the material, the “molecules.”

Taking the above functional form of the occupation numbers, we may write

E = − ∂

∂βF, F =

∑n

F(e−βωn

). (2.56)

Calculating F , and approximating the sum as an integral, we find

F = Ωd−1

∫nddn

nF(e−βµ

δ(`µ)z

nz)

=Ωd−1

z

(`µ)d

(δβµ)d/z

∫nd/z

dn

nF(e−n)

(2.57)

and define, therefore,

F

(d

z

)=

∫nd/z

dn

nF(e−n). (2.58)


Finally, we read that the energy is

E = Ωd−1d

z2

(`µ)d

(δµ)d/zT 1+d/z F

(d

z

). (2.59)

Note that the constant `d = Vd/(2π)d. Hence, we may take derivatives of the above by

T at constant volume in a straightforward way. Therefore

S =

∫V

dT1

T

(∂E

∂T

)V

= Ωd−1d+ z

z2

(`µ)d

(δµ)d/zF

(d

z

)T d/z. (2.60)

Happily, this exhibits the T d/z behaviour that we see in the T µ limit of the gravity

system. Using the above expressions, one can show the expected thermodynamic relation

E =d

d+ zTs (2.61)

where we have divided by the volume to give the expressions in terms of the energy density

E and entropy density s. The above energy that we have arrived at is the “dynamical”

energy: it cannot be directly compared with the energy in AdS. The absolute energy in

AdS will have a constant offset, due to the finite charge density and chemical potential

of the background. Since this is some fixed “zero point” energy for low temperatures,

it does not come into the above considerations in the effective field theory2. For a

direct comparison to the numerical setup we would need to subtract off the asymptotic

contribution: we do not do this here, and content ourselves with seeing the correct

dependence of s on T for T µ, which then implies the above expression up to an

additive constant.

Now let us recall that we have done the above for one type of particle. We must

2This is akin to ignoring the rest mass of a non relativistic material when computing the energy.


multiply this by the number of species to make a meaningful comparison to the gravity

setup. In fact, the measure of the number of species is given by something like NS ∼

Ld/(4Gd+2) for the gravity setup, so the total entropy above gets multiplied by this

amount. Finally, dividing by volume (2π`)d, we find that the total entropy density

should be

4Gd+2s = 4Gd+2S

(2π`)d

= δ(z, d)−d/z κ(z, d)Ωd−1

(2π)dd+ z

z2(Lµ)d F

(d

z

)(T

µ

)d/z(2.62)

where, on the right hand side, the function κ(z, d) measures the number of species exactly:

NS = κ(z, d)Ld/(4Gd+2).

One possible function of interest is a simple exponential suppression, F(x) = x, i.e.

a Boltzmann distribution. This is in some sense a “dilute phonon” limit, where the

excitation modes do not interact. In such a situation we find

4Gd+2s = δ(z, d)−d/zκ(z, d)Ωd−1

(2π)dd+ z

z2(Lµ)d Γ

(d

z

)(T

µ

)d/z, (2.63)

and the above is a good unitless measure of the entropy density. We can, of course,

combine δ and κ into one function that measures the number of degrees of freedom.

We are now in a position to make a qualitative comparison to the graphs in Figure

2.2. First, we assume that κ(z, d) and δ(z, d) are always order 1 numbers, as discussed

below equation (2.55). Hence, in the large z limit, these must asymptote to some fixed

value, depending only on d. Therefore, all that is left to analyze is the behaviour of the

factor Γ(dz

)(d+ z)/z2. In the large z limit Γ

(dz

)(d+ z)/z2 → 1

dand so we see that the

whole expression (2.63) goes to a z independent value, depending only on d. Therefore,

the graphs in figure 2.2 qualitatively agree with the large z behaviour of equation (2.63),

given the restriction on δ(z, d) and κ(z, d).

Chapter 3

Twist-nontwist correlators in

MN/SN orbifold CFTs

In the second part of this dissertation we investigate the AdS3/CFT2 correspondence

in the context of the D1-D5 brane system. We focus on the microscopic string theory

aspect of the fuzzball proposal and investigate quantitatively properties of the microstates

of the D1-D5 superconformal field theory. In this chapter we consider general (1+1)-

dimensional symmetric product orbifold CFTs and generalize the Lunin-Mathur covering

space technique to compute twist-nontwist correlators. This chapter is based on research

originally presented in [183].

3.1 Introduction

As discussed earlier, the dual CFT of the D1-D5 system is conjectured to possess a

position in its moduli space, known as the orbifold point, where it is an N = (4, 4)

supersymmetric sigma model with target space MN/SN [153, 155]. Here, M is either K3

or T 4, and SN is the symmetric group. The target space MN would simply be N copies

of the CFT with target space M , and the SN symmetry acts by permutating the copies.

This motivates us to study 2D orbifold CFTs [219, 220, 221], with particular attention

51

Chapter 3. Twist-nontwist correlators in MN/SN orbifold CFTs 52

paid to symmetric group orbifolds [222, 223, 160, 171, 172, 180, 224, 225]. Orbifold

models inherit much of the parent theory’s structure, and so orbifolds of free theories are

particularly simple to deal with. However, added complications arise from the new sectors

of the Hilbert space, the twisted sector states. Such states are configurations that return

to themselves up to an application of the orbifold symmetry, i.e. they are a new set of

allowed boundary conditions when the theory is put on a cylinder. New operators must

be associated with these new states, according to the state-operator mapping. However,

it seems at first glance that these operators are rather implicitly defined, only given by

a new set of boundary conditions on the states.

It is exactly this concern that Lunin and Mathur [171, 172] addressed for symmetric

group SN orbifolds. The basic idea in [171] is to “untwist” the boundary conditions by

using a locally conformal map. We refer to the space where the CFT is originally defined

as the base space, and the space to which it is mapped as the covering space. The covering

space is a multiple cover of the base space. The purpose of the map is to make the fields in

the covering space have simple boundary conditions, with the complications of boundary

conditions in the base space being absorbed into the map between the two spaces. In [172],

the analysis was extended to the case where the CFT contains fermions, concentrating on

the case for the orbifold point of the D1-D5 CFT. Including fermions slightly complicates

the picture in the covering space because of the conformal transformation properties of

the fermions. Further, [172] discussed how to excite the twist sector operators with

modes of fields that are twisted. They concentrated on excitations involving symmetry

currents in the theory, and showed how this operation is lifted to the covering surface.

The twisted currents have fractional modes with very low conformal dimension, a fact

which they exploited in [172] to construct super chiral primary operators in each twist

sector.

The orbifold point of the D1-D5 CFT provides a point where the field theory is

tractable. However, we may only compare to supergravity computations for protected


quantities. To study non-protected quantities, we must deform away from the orbifold

point by an exactly marginal operator. The operator of interest for the D1-D5 CFT is

in the twist-two sector of the theory, and so we must account for the interactions of the

twist sector with all other sectors of the theory. A relatively simple class of operators to

consider are those operators that are not affected by the orbifold symmetry: the non-twist

sector. Several supergravity modes, including the dilaton, belong to this class, although

these are protected. We explore interactions between the twist and non-twist sector of

the D1-D5 CFT for some simple operators in the next chapter using techniques developed

here. Of particular interest to us is to compute the change in conformal dimension of

non-protected operators when moving away from the orbifold point.

The immediate purpose of this chapter is to extend the Lunin-Mathur (LM) construc-

tion to deal with interactions between the twist and non-twist sectors. There are two

possible ways this can happen. First, consider excitations of twist operators by modes

that are not twisted by that operator. It is relatively easy to construct the non-SN -

invariant operators because the operator being appended to the twist operator shares

no OPE with it. However, these added excitations can be twisted by other operators

appearing in a correlator. So, while the excitations are not twisted by the operator they

excite, they may be twisted by other operators in the correlator. We will explain how to

account for these excitations in the covering space in the next section.

Our second extension is to find how to calculate correlators that contain both twist

and non-twist sector fields. For this, we note that the location of a non-twist operator will

have several images in the covering space. Each of these images has a concrete meaning in

terms of the fields defined on the base space, and imply that summing over insertions at

each image is the correct procedure. For each of these extensions, we perform a sample

calculation, and show that the result gives the correct form of a 2D CFT correlator

in terms of the base space information. We concentrate on 3-point functions in our

examples, but the procedures can be applied to four point functions, as we will show in


Chapter 4.

The remainder of this chapter is organized as follows. We summarize the Lunin-

Mathur covering space technique in section 3.2.1, setting up our generalization which we

discuss in section 3.2.2. We illustrate our method with two examples. In section 3.3.1 we

consider a free X CFT factor appended to an arbitrary CFT. We use the modes of the

X operator to excite a bare twist operator in non-twisted directions. We show how to

compute a 3-point correlator involving this operator, using the covering space, and show

that it has the correct form for a 2D CFT 3-point function. In section 3.3.2, we consider

the D1-D5 CFT near the orbifold point. We use some excited versions of the super chiral

primaries constructed in [159, 172], and add a non-twist sector excitation constructed

from the bosons. We show that, after summing over the images, the correlator is in fact

of the correct form. We conclude in section 3.4 with some interpretations of the method,

and a discussion of applications.

3.2 The Lunin-Mathur technique, and generalizations

3.2.1 Lunin-Mathur

Here we will discuss the Lunin-Mathur technique [171, 172] (see also [226] for more ex-

planation). The LM technology was originally developed for bosonic theories in [171] and

later generalized to theories with fermions in [172]. In these works, particular attention

was paid to the twist operators associated with cycles in the SN group, as these form a

set of basic building blocks for SN (all group elements of SN can be written as products

of cycles that do not share indices). Excitations along directions twisted by the operator

were also considered in [172], with the emphasis on how current operators act on the

twist sector fields. Here, we summarize the salient features of LM.

First, consider the role of twist fields. Twist fields change the boundary conditions

that the fundamental fields must satisfy. In particular, they consider the effects of the


non-SN -invariant twist operators σ(12···n), which twist the first n copies of the CFT. These

act on the fields as

Φ1 → Φ2 → Φ3 → · · · → Φn → Φ1 (3.1)

where by Φ we mean a field of arbitrary weight and statistics. Here, and in what follows,

we will suppress all indices other than the copy index because the SN permutation leaves

other indices unchanged. The full SN -invariant operator is generated by summing over

the SN “images” of the non-SN -invariant operators:

σn =

√n(N − n)!

n(N − n)!√N !

∑g

σg(12···n)g−1 =

√n(N − n)!

N !

∑c∈C[(12···n)]

σc. (3.2)

In the first equality we sum over all group elements g of the symmetric group SN . In

the second equality, we sum over the conjugacy class of (123 · · ·n), which we denote

C[(123 · · ·n)]. This second sum just sums over all possible distinct n-cycles.

We are ultimately concerned with the evaluation of correlators of the form

〈σn1σn2σn3 · · · 〉. (3.3)

In order to compute such correlators, it would be sufficient to understand the correlators

involving only the non-SN -invariant twist operators e.g.

〈σ(1,2,3···n1) σ(2,3,4,···n2+1) σ(8,9,10···n3+7) · · · 〉. (3.4)

because the correlator (3.3) is just a sum of such terms.

Finding a way to represent the correlators (3.4) was the primary goal in [171, 172].

The basic idea is to map the problem of multiple copies of fields with twisted boundary

conditions to a problem of one copy of fields with normal periodic boundary conditions.

This is accomplished with a locally conformal map. The base space we will parameterize


with the complex variables (z, z) and the covering space we will parameterize with the

complex variables (t, t).

Let us imagine that there are s distinct indices involved in the twists in (3.4). In

this case, we will pay attention only to these s copies of the fields: the other copies do

not interact with these twists, and this part of the correlator factorizes. Now focusing

on only this set of s copies, we consider an s-fold cover of the space. In the covering

space, we only have one copy of the fields, but because a generic point in the base space

corresponds to multiple points in the covering space, we actually have multiple copies

of the fields defined in the base space. Thus, the map from the covering space to the

base space induces the correct number of functions/fields in the base space. For the time

being we will restrict ourselves to bosonic fields Φ, and will consider the extension to

fermions later in this section.

Next, when circling a twist insertion in the base space the fields must map as (3.1).

Thus, starting with the field Φ1, and circling the insertion of σ(1,2,3,···n), we find that

the function does not come back to itself, but rather comes back to Φ2. Thus, in the

covering space, the contour must be open such that the single function Φ is different at

the endpoints. It must be that these endpoints in the covering space are mapped to the

same point in the base space. Further, we construct the map from the base space to the

covering space such that there are distinguished points in the covering space [171]. These

distinguished “ramified” points are where the map looks locally like

z − z0 = b(t− t0)ni + · · · . (3.5)

For each ni cycle twist insertion, we must have one such point. Such a point in the

covering space is where ni images of the base space come together, and it is these points

that enforce the boundary conditions (3.1). If we consider a contour around this point

in the covering surface, it actually winds around a point in the base space ni times. This


point in the base space is the location of the σni insertion.

Next, we note that any given point on the contour around the twist insertion must

have s total images. It is clear that we have identified ni of these near the distinguished

point in the covering space. The other s − ni images must be near other locations in

the covering space. These points are isolated “non-ramified” points, and so going once

around these points in the covering space correspond to going once around the twist

insertion in the base space. This works in the case that each twist insertion is a cycle:

for products of cycles, we just take the coincidence limit of the considerations here.

To help think about the map, we restrict our attention in the base space to a patch

that does not have any contours that go around the twist insertions: we call this patch

the simply connected patch. We then may take an arbitrary point in this patch, and

consider one of its images in the cover. We consider expanding this neighborhood until

it fills the simply connected patch; we consider the expansion of the neighborhood in

the covering space as well. This defines one image of the simply connected patch on the

covering surface. We may do this with the other image points as well, and find all s

copies of the simply connected patch.

To each of these patches, we assign a function Φi associated with it. To help identify

these patches, we consider the periodicity when going around a twist insertion. A given

patch will have a certain number of points that are images of the location of twist

operators. If we consider those that are non-ramified, this gives the location of operators

that do not twist the function defined by the patch in question. This information should

identify the patch uniquely.

To help visualize this better, we consider figure 3.1 for the example of 〈σ(12)σ(23)σ(321)〉.

We see the expanded images of the simply connected patch in figure 3.1b. If we examine

the “lower island” patch in figure 3.1b, we see that there is an isolated image of σ(12) in

this patch (marked with an “x”, surrounded by a red contour). Since the function in this

patch is not twisted by σ(12), this patch must be associated with Φ3. We may consider


!(321)

!(12)

!(23)

non-twist

1

(a) base space

!2

!1

!3

!

!

!

1

(b) covering space

Figure 3.1: Diagram of the three-fold cover of the base space. The inside of the “clover-leaf” in figure 3.1a is the simply connected patch, which has the three images shown infigure 3.1b. The twist operators have 3 distinct indices: 1, 2, and 3. In the base space,we have 3 functions Φi, which in the covering space has been mapped to one function Φon 3 patches. We further show the three images of a generic point (where we will laterinclude a non twist insertion), and the three images of infinity of the base space).

the other patches similarly.

In this way, the Lunin-Mathur technique has mapped the problem of twisted bound-

ary conditions to a problem in the covering space with normal boundary conditions.

Consider an arbitrary configuration of fields Φi in the base that satisfies the boundary

conditions imposed by the twist operators. We see that this must correspond to a unique

configuration of Φ in the covering space. The reverse is also true: every configuration on

the covering surface corresponds to a configuration on the base space that satisfies the

boundary conditions imposed by the twist fields. Therefore, in calculating a correlator

by integrating over all configurations Φi in the base space, one may instead compute a

correlator in the covering space, integrating over all configurations Φ: the path integrals

are related.

It is crucial to account for the change to the measure of the path integral. We must

consider this because the locally conformal map used is not a member of the class of


SL(2,C) anomaly-free maps of the Riemann sphere, and the CFT we are dealing with

has a non-zero central charge c. The map induces a metric on the covering space g, which

is scaled back to a reference metric g = eφgc. The change to the measure by this rescaling

of the metric is given by the exponential of the Liouville action [227]

SL =c

96π

∫d2t√−gc [∂µφ∂νφg

µνc + 2R(gc)φ] . (3.6)

This factor, coming from an anomaly, simply multiplies the correlators in the covering

space,

〈∏i

Oi〉base = eSL〈∏i

Oi〉cover (3.7)

To compute this factor, the Liouville action must be suitably regulated, as was done in

[171] to compute certain 3-point and 4-point functions. When the insertions Oi on the

covering surface are set to 1, this defines the “bare twist” correlation function.

For supersymmetric theories, one must consider lifting fermions to the cover as well

[172]. In these cases, the conformal transformation properties of the fermions play an

important role. In the vicinity of a ramified point, a fermion field transforms as

z = atn + · · · → ψ(t) =

(dz

dt

) 12

ψ(z) =(antn−1 + · · ·

) 12 ψ(z) (3.8)

where we consider the point to be at z = 0 in the base space and t = 0 in the covering

space for simplicity. To circle the point at t = 0, we take t → exp(2πi)t, or in the base

space, z → exp(2πin). The twist operator is of order n, and so if we circle it n times in

the base space, the field comes back to itself. Thus ψ(exp(2πin)z) = ψ(z). This means

that in (3.8), ψ(exp(2πi)t) = exp(2πi(n − 1)/2)ψ(t) in the covering space. When n is

odd, ψ(t) returns to itself. However, if n is even, ψ(t) returns to minus itself. In this

case, ψ(t) must be antiperiodic in the covering space to furnish a ψ(z) in the base space

that is periodic when circling z = 0 n times. To account for this boundary condition,


there must be a spin field S at the location of the ramified point in the covering space

to ensure the correct periodicity conditions.

Finally, we wish to consider a large N limit for applications in AdS/CFT. It was shown

in [171] that, due to combinatoric factors, the leading order in 1/N is given by the case

where the covering surface is a sphere. Here, and in what follows, we will concentrate on

these cases, although we believe that the techniques here and in the next section should

extend to other Riemann surfaces.

3.2.2 Generalization to the non twist sector.

The Lunin-Mathur technique was developed for twist sector operators, with twist sector

excitations. Here we will generalize the LM technology for non twist sector operators,

and non-twist sector excitations of twist sector operators.

First we consider excitations of the twist operators by modes that are not twisted

by the operator. In such a case, the field acting on the twist operator shares no OPE

with it, and so we can simply multiply the operators together. For example, if we have

a bare twist σ(12) and we wish to excite it with a mode of a bosonic X operator, α3,−11,

we simply write this as ∂X3σ(12). This is for the non-SN -invariant operator. The full

SN -invariant operator would involve summing over all images of the SN symmetry group,

which now acts on all of the indices, 1, 2, 3... i.e.

σ′2 = ∂X3σ(12) + ∂X1σ(32) + ∂X4σ(12) + ∂X1σ(42) + · · · . (3.9)

where we consider all SN permutations of the indices 1, 2, 3 · · ·N . Half will be repeated

operators of the same kind, because σ(ij) = σ(ji). We would like to figure out how to

compute correlators with such excitations. Again, it will be sufficient to consider only

correlators of the non-SN -invariant operators, and then sum to make an SN -invariant

1The first subscript denotes the copy; the second denotes the mode.


correlator.

The prescription is as follows. Imagine that we are considering a correlator involving

twist operators, of which only one has an excitation in a copy that does not involve its

twist indices. When we expand out the gauge invariant operators, we will have two cases

come up: either the twist directions of the other twist operator are along the direction

of the excitation, or they are not. For example, if we consider a 3-point correlator of σ′2

along with itself and σ3, there will be terms of the form

〈(∂X3σ(12))(∂X1σ(23))(σ(321))〉 or 〈(∂X4σ(12))(∂X4σ(23))(σ(321))〉. (3.10)

In the second case, the ∂X4 terms factorize, because it does not have any directions in

common with the twists in the fields.

However, in the first case, we see that X3 and X1 are directions that are associated

with twist directions, just not directions for the operator that they act on to excite. The

solution to this problem is rather simple. Recall that ∂X1 adds a boundary condition

for the field X1. The function X1 is associated with a particular patch in the cover.

Thus, to add the excitation of this field, we make an insertion in the covering space in

the patch associated with X1. The location of the insertion is at the image of the twist

σ(12) in this patch. In our diagram, 3.1b, we see that patch Φ1 (X1 in our example) has

a point associated with the twist operator σ(12) in the upper half of the diagram. At this

location, we make an insertion of

(dz

dt

)−1∣∣∣∣∣t=to,1

× ∂X(to,1) (3.11)

where to,1 is the image of the “o” point in the patch for X1, i.e. the circle in 3.1b in the

upper half of the diagram. This produces the correct boundary condition for the field X1

at the point marked with the “o” in the base space. Similarly, we must make an insertion


of (dz

dt

)−1∣∣∣∣∣t=tx,3

× ∂X(tx,3) (3.12)

where tx,3 is the image of the point marked with an “x” in patch for X3, i.e. the “x” in the

lower half of diagram 3.1b. We would then need to add all other symmetric combinations

as well. This process can be generalized to more complicated excitations.

The general prescription is simple to state. First, take the SN -invariant operator, and

expands this in terms of non-SN -invariant pieces. For each piece, see whether the twist

parts of the operators agree to make sure that the product of all the cycles is 1 in some

order. Note that (12)(23) = (321) while (23)(12) = (123), so σ(12) and σ(23) must fuse to

both σ(123) and σ(321). This was considered in [171] when considering 4-point functions.

The basic observation was that if one takes σ(123) in a path around the σ(12) insertion, it

becomes σ(213) = σ(321). Any excitations of these operators along directions not twisted by

other operators simply factorize. Any excitations along directions that become twisted

by other operators in the correlator are accounted for by operator insertions at the

appropriate image in the covering space. Those parts of the operator that describe

the excitations are mapped using the correct conformal transformation properties. For

example, a combination ∂X∂X would transform with an additional Schwarzian derivative

piece when mapping to the cover, while ∂∂X would map as

∂2X(z)→ (∂z/∂t)−2∂2X(t)− (∂z/∂t)−3(∂2z/∂t2)∂X(t). (3.13)

To consider an arbitrary operator Oσ2, where O describes some non-twist excitations,

one only needs to know how O transforms under finite conformal transformations.

Next, we consider non-twist insertions into the plane. Let us illustrate this with

another example. Consider a simple type of non-twist sector field

O0 =1√N

N∑κ=1

∂Xκ (3.14)


where again we suppress all indices except for the copy index. We may consider inserting

such an operator in a correlator with twist sector fields. The twist sector fields are made

from sums over conjugacy classes of operators, as before. These can be expanded into

separate non-SN -invariant contributions. We want to address how to compute these non-

SN -invariant correlators individually, after which we can sum these together to find the

correct SN invariant combination.

Without loss of generality, we may consider the twisted directions to be the first s

copies. There are other combinations with similar operators involving s copies of fields.

Some of these are just symmetric group images of the operators we are considering, and

can be accounted for with a combinatoric factor, while others must be summed over.

Thus, we are considering a case where only the first s fields have twisted boundary

conditions. Our non-twist sector operator can then be written as the sum of two pieces

O0 =1√N

s∑κ=1

∂Xκ +1√N

N∑κ=s+1

∂Xκ

= O0,‖ +O0,⊥ (3.15)

whereO0,‖ is the first s terms (copies along the twists), andO0,⊥ is the other terms (copies

not along the twists). All operators and excitations there of involving the (s + 1, ..., N)

copies appear in factorized correlators, and can be computed with extant LM technology.

This leaves us to compute a correlator involving O0,‖ and a set of twist operators with

possible excitations along the first s copies of the CFT, generically of the form

A = 〈O0,‖(z0)∏`

σn`(z`)〉 (3.16)

where the twist fields σn` have twist indices along the first s directions, and may have

excitations along the first s directions.

Now, we must lift the computation to the covering space, and come up with a covering

space interpretation of O0,‖(z0). We lift the excited twist operators as in the last example.


First note that the position of this operator z0 is a generic point, and so it is uplifted

to s points in the cover t0,i. Each point on the cover is associated with a patch, as in

diagram 3.1b. Next, we note that the operator 1√N

∑sκ=1 ∂Xκ(z0) has the interpretation

as a sum of states. Each of these states has boundary conditions on only one of the fields

X1(z0), ..., Xs(z0). Since each of these fields is lifted to a particular patch in the cover, we

see that we must make an operator insertion at only one of these points. At which point

must we make the insertion? The answer is simple: we put the insertion at each image

point t0,i, and add the terms, just as the operator O0,‖(z0) is a sum of terms, each placing

boundary conditions on different fields. We lift ∂X using its conformal transformation

properties, i.e. in the cover the insertion at the ith point is

(dz

dt

)−1∣∣∣∣∣t=t0,i

× ∂X(t0,i). (3.17)

In our example in diagram 3.1, we would make the above insertions at one of the points

marked with in the covering space. We would compute all three insertions, and then

add the contributions together.

Again, we would generalize this the same way as above. Take an operator Oi1···iq that

describes an non-SN -invariant piece of an SN -invariant operators∑

SN (ik)Oi1···iq , which

is in the non-twist sector. Then, to lift this to the cover, we would need to put an

operator insertion of various pieces of O in the cover, transforming each piece according

to its finite conformal transformation properties. For example, given a non-SN -invariant

operator ∂X1∂X2 (here 1, 2 are copy indices) we would put an image of (∂z/∂t)−1∂X

in patch 1 and in patch 2. This is because the operator ∂X1∂X2 is associated with a

state with an excitation in both the first X1 mode, and the first X2 mode. This is just

the same as in simple quantum mechanics: sums mean “or” while multiplication means

“and.”

Further, we can combine these techniques in a straightforward way. We have given an


interpretation for each of these kinds of operators in the covering space, and so combining

them simply combines the steps. We put appropriately transformed “image operators”

at the appropriate images of the point in the covering space. These can be found once

the map is known. This interpretation also sheds light on the meaning of the multiple

images in the covering space, both for the twist sector operators when they appear with

other twists, and also for the non-twist sector operators.

We have thus found how to compute non-twist sector operators using a generalization

to the Lunin-Mathur technique. In the next section we give some example calculations,

and show that results generated by this technique have the expected form for CFT

correlators.

3.3 Example calculations

3.3.1 Excitations orthogonal to twist directions

In the sections that follow, we show how to use our generalization of the LM technology

to compute 3-point functions. To make these computations, we will need the explicit

form of the conformal maps. Here, we restrict to the case where the covering surface is

the two-sphere, which corresponds to the leading order in 1/N = 1/(N1N5), as explained

in [171].

The first map that we consider is for a correlator of the form 〈Σ2Σ′2Σ3〉, involving two

twist 2 operators, and a twist 3 operator, as shown in figure 3.1. We consider the position

of the operator insertions to be z = a1, a2,∞ for the twist 2, 2, 3 insertions respectively.

Some of the images of these points in the covering surface must be ramified. Near these

points in the cover, the map is locally z − z0 = b(t − t0)n + · · · . Different copies of

the simply connected patch meet at ramified points in such a way as to give the correct

boundary conditions for the fields. For two twist 2 fields and one twist 3 field, the correct


map to use is

z − a1 = (a1 − a2)t2(2t− 3) (3.18)

which is correctly ramified at t = 0 for a twist 2 operator. Note that this allows us to

find the other image of z = 0 located at t = 3/2. We may consider the location of the

other twist 2 operator, and see that

z − a2 = (a1 − a2)(2t+ 1)(t− 1)2 (3.19)

which is again correctly ramified at t = 1 for a twist two operator. We see the other

image of z = a2 is located at t = −1/2. The third ramified point is clearly located at

t =∞, z =∞, again with the correct behavior z = 2(a1 − a2)t3 + · · · .

In what follows, it will be convenient to consider the twist operators at finite points.

We accomplish this with SL(2,C) transformations in the z and t plane. We map the

locations of the twist insertions as follows: z = a1 will map to t = 0 as before, z = a2

to t = 1 as before, but now we will map the location of the twist three operator to be

at z = b and its image in the covering surface will be t = ω. Performing the needed

SL(2,C) transformations in the z and t planes leads to the map

z − a1 = − t2([1 + 2ω]t− 3ω)(a1 − a2)(a1 − b)(ω − 1)2

t2([1 + 2ω]t− 3ω)(a1 − a2)(ω − 1)2 + (t− ω)3(a2 − b). (3.20)

Using translation invariance of the base space we set b = 0 and, using the SL(2,C)

invariance of the t plane, we set ω = −1. This gives a simplified map

z = − a1a2(t+ 1)3

4t2(t− 3)(a1 − a2)− (t+ 1)3a2

. (3.21)


which one can also write as

z − a1 = − 4t2(t− 3)(a1 − a2)a1

4t2(t− 3)(a1 − a2)− (t+ 1)3a2

(3.22)

z − a2 = − (t− 1)2(5t+ 1)(a1 − a2)a2

4t2(t− 3)(a1 − a2)− (t+ 1)3a2

(3.23)

showing the correct ramifications at (z = a1, t = 0), (z = a2, t = 1), (z = 0, t = −1).

To determine the 3-point function for the bare twists, coming from the Liouville term,

we may simply use the result of [171]

|C2,2,3|2 =1

3112c2

59c

(3.24)

and so for twist operators located at finite points, we have

〈σ(1,2)(a1) σ(2,3)(a2) σ(3,2,1)(b)〉 =|C2,2,3|2

|a1 − a2|2c72 |a1 − b|

2c9 |a2 − b|

2c9

(3.25)

where c is the central charge of one copy of the CFT.

For an example, we will consider a setup where there is a free X CFT as part of the

full CFT (and then, of course, there are N copies). Recall that the order of the three

twist operators are 2, 2, and 3. Although this tells us what twist sector the operators

are in, it does not tell us about the excitations. To be specific, we consider the operator

σ′2 already discussed:

σ′2 = ∂X3σ(12) + ∂X1σ(32) + ∂X4σ(12) + ∂X1σ(42) + · · · . (3.26)

Let us consider what happens when we sum over all permutations. Each occurrence of

σ(12) will get dressed with all possible ∂Xi where i is neither 1 nor 2. Thus, we may write


out the operator as

σ′2 =N∑

i,j=1i<j

N∑k=1k 6=i,j

∂Xkσ(ij). (3.27)

To normalize, we note that there are N − 2 terms in the sum over k, and(N2

)terms in

the sum over i, j. This implies that we should normalize the operator with a factor

σ′2 =1√

(N − 2)(N2

) N∑i,j=1i<j

N∑k=1k 6=i,j

∂Xkσ(ij). (3.28)

The 3-point function we wish to consider is

〈σ′2(a1)σ′2(a2)σ3(b)〉 (3.29)

where σ3 is just a bare twist three operator, i.e.

σ3 =1√

2(N3

) ∑3-cycles

σ(i,j,k) (3.30)

and we set the location of the twist three operator to be b = 0 using translation invariance.

Now we expand this in terms of non-SN -invariant pieces. First, we note that to have a

nonzero answer for the 3-point function, we must have a combination in the twist sectors

of σ(i,j)σ(j,k)σ(i,j,k) or σ(i,j)σ(j,k)σ(k,j,i). These two possibilities give the same contribution,

so we will simply account for them with a combinatoric factor of 2, and take the second

possibility.

Without loss of generality, we may choose i = 1, j = 2, which brings in a combinatoric

factor of(N2

). Then, there are

(N−2

1

)ways to assign the last index k. Thus, we find that

〈σ′2(a1)σ′2(a2)σ3(0)〉 = (3.31)

2

√(N − 3)!3!√

2√N !

⟨[ N∑i=1i 6=1,2

∂Xi(a1)σ(12)(a1)

][ N∑j=1j 6=2,3

∂Xj(a2)σ(23)(a2)

] [σ(321)(0)

]⟩


We see that we can break up the sums into two parts. The first part is where i, j are

not twisted by any operators in the correlator, i.e. i, j ≥ 4. The second part is where i, j

are twisted by operators in the correlator, i = 3, j = 1. The terms for i, j ≥ 4 result in a

combinatoric factor of N − 3, and so we find

〈σ′2(a1)σ′2(a2)σ3(0)〉 = (3.32)

2

√(N − 3)!3!√

2√N !

(⟨[∂X3(a1)σ(12)(a1)

] [∂X1(a2)σ(23)(a2)

] [σ(321)(0)

]⟩+(N − 3)

⟨σ(12)(a1)σ(23)(a2)σ(321)(0)

⟩〈∂X4(a1)∂X4(a2)〉

)(3.33)

The second term above clearly has the correct form of a 3-point function. It simply

gives an additional factor of 1/(a1−a2)2, which is what should happen: the holomorphic

weight h of the twist-two operators have both increased by 1. This affects the 1/(a1 −

a2)h1+h2−h3 terms, but not the terms of the form 1/(a1− 0)h1−h2+h3 , nor any of the other

antiholomorphic terms in the 3-point correlator. We can simply add the weight of the

∂X operator to that of the original operator because the ∂X in question shares no OPE

with the twist operator.

The other requires us to lift the computation to the covering surface. We lift the

computation as

⟨[∂X3(a1)σ(12)(a1)

] [∂X1(a2)σ(23)(a2)

] [σ(321)(0)

]⟩(3.34)

→

⟨[(∂z

∂t

)−1∣∣∣∣∣t=3

∂X(3)

](∂z∂t

)−1∣∣∣∣∣t=− 1

5

∂X

(−1

5

)⟩ (3.35)

where we have plugged in the explicit locations of the other images of a1 and a2 given

the map (3.22), or equivalently (3.23). Here we use a notation → to mean “lift to the

cover, and strip the Liouville action”. This is convenient and allows us to concentrate

on the CFT calculation in the cover. The above computation on the covering surface is


trivial, and gives

⟨∂z

∂t

∣∣∣∣t=3

∂X(3)∂z

∂t

∣∣∣∣t=− 1

5

∂X

(−1

5

)⟩=

16a2

9a1(a1 − a2)

16a1

225a2(a1 − a2)

1

(3− (−1/5))2

=1

81(a1 − a2)2(3.36)

This extra dressing, just like the orthogonal piece, changes the 3-point function to be of

the proper form. Combining everything, we find

〈σ′2(a1)σ′2(a2)σ3(0)〉 = 2

√(N − 3)!3!√

2√N !

|C2,2,3|2(

181

+N − 3)

(a1 − a2)c72

+2a

2c9

1 a2c9

2 (a1 − a2)c72 a

2c9

1 a2c9

2

. (3.37)

One final concern is whether we have properly normalized the twist two operators. The

normalization would be accomplished by mapping the 2-point functions to the covering

surface. However, normalization is actually already taken care of in this case because

〈σ′2(1)σ′2(0)〉 =1

(N − 2)(N2

) ⟨ N∑i,j=1i<j

N∑k=1k 6=i,j

∂Xk(1)σ(ij)(1)N∑

i,j=1i<j

N∑k′=1k′ 6=i′,j′

∂Xk′(0)σ(i′j′)(0)

⟩

=1

(N − 2)

⟨N∑k=3

∂Xk(1)σ(12)(1)N∑k′=3

∂Xk′(0)σ(12)(0)

⟩

=

⟨∂X3(1)σ(12)(1)∂X3(0)σ(12)(0)

⟩=

⟨σ(12)(1)σ(12)(0)

⟩. (3.38)

In other words, normalization of the operator is completely taken care of via the normal-

ization of the bare twists. This was accounted for in the original computations in [171]

which leads to (3.24), and so our result (3.37) is indeed correctly normalized.


3.3.2 Non-twist operator insertions

For our next 3-point point function, we will consider a slightly simpler map, but a more

complicated field content. We will be considering a correlation function of the type

⟨σ0 σ

′′2 σ′′2

⟩(3.39)

where σ0 is a non-twist insertion, and σ′′2 is a twist two sector field. The conformal map

that we will use works only for the case of two twist operators, with twist order n = 2.

The map for two twist-n fields, putting them at z = 0 and z =∞, is

z = btn. (3.40)

If we are interested in putting the locations of the operators at finite points, we may use

the map

z = atn

tn − (t− 1)n(3.41)

where the location of the twist operators is now z = 0 and z = a. It is easy to check that

z − a = a(t− 1)n

tn − (t− 1)n(3.42)

so the ramified points are at t = 0 and t = 1 in the covering space.

Next, we would like to consider a specific field theory for concreteness. For this, we

will use the D1-D5 CFT. The moduli space of this CFT is conjectured to have an orbifold

point, where the field content is that of a N = (4, 4) CFT with N = N1N5 copies. Here,

N1 and N5 denote the number of D1 and D5 branes respectively. The field content for

one copy is four real scalars X i, and four real fermions in both the left and right moving

sectors ψj, ψk.

The presence of fermions complicates the lift to the covering surface when the twist


is of even order. We will be considering n = 2, and so this is a concern for us. At the

location of the ramified points in the cover there are spin fields. To deal with these spin

fields, we will bosonize the fermions, and write the spin fields in terms of exponentials

of the bosons. In terms of these fields, we will have a total of six right moving fields

φi(z) and six left moving fields φi(z). The first four of these fields will correspond to

the original bosons in the theory, breaking the left and right moving parts into φ and φ.

The final two in each sector correspond to the bosonized fermions. We will follow the

notation of [172] for the bosonized fields, and introduce the following vectors

A = (1, i, 0, 0, 0, 0), B = (0, 0, 1, i, 0, 0)

c = (0, 0, 0, 0, 0, 1), d = (0, 0, 0, 0, 1, 0) (3.43)

e = (0, 0, 0, 0, 1,−1), f = (0, 0, 0, 0, 1, 1)

Clearly A·φ,A∗ ·φ,B ·φ,B∗ ·φ form a complete basis for constructing fields associated with

bosons, and c · φ and d · φ give a complete basis for discussing bosonized fermions. The

combinations f ·φ and e ·φ are what naturally appear in spin fields, while A ·φ and B ·φ

are what transform naturally under the SU(2)1 × SU(2)2 = SO(4) internal symmetry

of the four bosons. Here, and in what follows, we will concentrate on the holomorphic

sector of the theory. The antiholomorphic sector will follow similarly, without further

need for comment.

We can now explicitly state the non-twist insertion that we wish to consider. We

take, for simplicity,

O =1√NA ·

N∑i=1

∂φi (3.44)

which is clearly invariant under the permutation group SN , and is also properly normal-

ized with respect to factors of N .

Note that here and in what follows we will ignore the effects of cocycles. For the

computations at hand, this should be sufficient, as we can explicitly construct cocycles


such that the spin fields we will consider, exp(±if · φ/2), have no additional operator

dressing [177]. Thus, there should not be any additional phases in the computations

below. For further details, see Chapter 4.

Next, we will need to consider which operators in the twist 2 sector we wish to include.

We consider the left-moving part of a deformation operator and its conjugate, given in

[159] as one of an exhaustive list of all (1,1) primary operators. Writing this in the

notation of [172] in the covering space, we find

OA(12)(z)→ 1√2b5/8

(: (A · ∂φ)e−if ·φ/2 : + : (B · ∂φ)eif ·φ/2

): (3.45)

OA(12)(z)† → 1√2b5/8

(: (A∗ · ∂φ)eif ·φ/2 : + : (B∗ · ∂φ)e−if ·φ/2 :

). (3.46)

In this expression b is the leading term in the expansion

z = bt2 + · · · . (3.47)

We have used translation invariance in the base space to move the twist operator to z = 0,

and translation invariance in the covering space to have the location of the ramified point

at t = 0. Both OA(12) and O†A(12) are both primary operators of weight 1.

We wish to consider certain excitations of these fields in twist directions so that we

get a nontrivial 3-point function. We use the techniques of [172] to excite this twist field

using the bosons. First, we note that in the neighborhood of OA,(12) (which we put at

z = 0 for convenience) ∂φa1 and ∂φa2 for a = 1, 2, 3, 4 do not have well-defined periodicity

conditions. Instead, a more natural combination is ∂φa1 + exp (2πim/2)∂φa2. When m is

odd, the field is antiperiodic, and when m is even, it is periodic. This is just decomposing

the collection of fields φn into eigenvectors of the operation 1 → 2 → 1. From this, we


naturally define modes of these operators as

A∗ · α(12)−m/2 ≡

∮dz

2πiz1−1−m/2A∗ ·

(∂φ1(z) + e2πim/2∂φ2(z)

)(3.48)

where we can see that the integrand is single valued, and we have picked a certain

direction for the excitation. We lift the action of this current to the covering space as

[A∗ · α(12)−m/2, OA,(12)(z)†] =

∮dz

2πiz1−1−m/2A∗ ·

(∂φ1(z) + e2πim/2∂φ2(z)

)OA,(12)(0)†

→∮

dt

2πi

(dz

dt

)1−1

(z(t))−m/2(A∗ · ∂φ(t)

)(3.49)

× 1√2b5/8

(: (A∗ · ∂φ)eif/2·φ(0) : + : (B∗ · ∂φ)e−if/2·φ(0) :

).

Note that A∗ is orthogonal to all vectors except A, and this does not appear anywhere

in the twist operator under consideration. Hence, there are no singular terms, and the

above operator is normal ordered. We simply need to expand z(t)−m/2 and ∂φ(t) to the

appropriate orders, and find the pole term. It turns out that for the correlator that we

consider later, m = 2 is the first term that will give a nonzero 3-point function. For

m = 2, we find

[A∗ · α(12)−2/2, O

†A,(12)] (3.50)

→ b−1−5/8

√2

:

(A∗ ·

(∂2φ− b1

b∂φ))(

(A∗ · ∂φ)eif/2·φ + (B∗ · ∂φ)e−if/2·φ)

:

where we have expanded the map to second order as

z = bt2 + b1t3 + b2t

4 + · · · (3.51)

We have now constructed an operator that we know how to lift to the covering space.

However, we would like to check if this operator is a quasiprimary. To do so, we will need


to apply

L` =

∮dz

2πiz2−1+`T (z). (3.52)

To apply this in the covering space, recall that the stress tensor does not transform

tensorially, but rather transforms as

T (z)→(dz

dt

)−2 (T (t)− c

12z(t), t

), (3.53)

with z(t), t denoting the Schwarzian derivative. In the base space, the stress tensor is

just

T (z) = −1

2: ∂φa∂φa :, (3.54)

and c = 6 in the cover. Thus, to check if our operator is a quasiprimary, we lift to the

covering surface and compute

[L`, [A

∗ · α(12)−2/2 , O

†A,(12)]

]→ b−1−5/8

√2

∮dt

2πi

(dz

dt

)−1

(z(t))1+`

(−1

2: ∂φa∂φb(t) : δab −

1

2z(t), t

)× :

(A∗ ·

(∂2φ− b1

b∂φ))(

(A∗ · ∂φ)eif ·φ/2 + (B∗ · ∂φ)e−if ·φ/2)

(0) : . (3.55)

for ` ≥ 0. We note that the OPE we need to compute above starts at order 1/t3, due

to the orthogonality of A∗ with itself and with B∗. We must expand the functions of t

according to (3.51), and we find

(dz

dt

)−1

=1

2bt

(1− 3b1

2bt− 8b2b− 9b2

1

4b2t2 + · · ·

), (3.56)

and

(z(t))1+` = b1+`t2+2`

(1 +

(1 + `)b1

bt+

(1 + `)(2b2b+ b21`)

2b2t2 + · · ·

). (3.57)


Therefore, the leading order behavior of these terms put together is t1+2`. Putting this

together, we expand out

[L`, [A

∗ · α(12)−2/2, O

†A,(12)]

]→ b−1−5/8+`

2√

2

∮dt

2πit1+2`

(1− 3b1

2bt+ · · ·

)(1 +

(1 + `)b1

bt+ · · ·

)×(−1

2: ∂φa∂φb(t) : δab +

3

4t2+ · · ·

)× :

(A∗ ·

(∂2φ− b1

b∂φ))(

(A∗ · ∂φ)eif/2·φ + (B∗ · ∂φ)e−if/2·φ)

(0) : . (3.58)

We can expand the OPE as

−1

2δab : ∂φa∂φb(t) : :

(A∗ ·

(∂2φ− b1

b∂φ))(

(A∗ · ∂φ)eif/2·φ + (B∗ · ∂φ)e−if/2·φ)

:

=2

t3: A∗∂φeif/2·φ

((A∗ · ∂φ)eif/2·φ + (B∗ · ∂φ)e−if/2·φ

):

+1

t2:

[A∗ ·

([3 +

(f/2)2

2

]∂2φ−

[2 +

(f/2)2

2

]b1

b∂φ

)]×[(A∗ · ∂φ)eif/2·φ + (B∗ · ∂φ)e−if/2·φ

]:

+1

t∂

[(A∗ ·

(∂2φ− b1

b∂φ))(

(A∗ · ∂φ)eif/2·φ + (B∗ · ∂φ)e−if/2·φ)]

+ · · · (3.59)

with all operators at the location t = 0. Combining the leading order behavior of the

prefactors t1+2` with the leading order singularity of the OPE 1/t3, we see that the leading

term in the integral is t−2+2`. We can see that the ` ≥ 1 will give 0 in the contour, so

that this operator is not just quasiprimary, but is in fact (Virasoro) primary, as long as

it has a well defined conformal dimension. To determine this, we must consider ` = 0.

For this value of `, we see that the 3/(4t2) contribution from the Schwarzian must be

taken into account. Specializing to `=0, we find


[L0, [A

∗ · α(12)−2/2, O

†A,(12)]

]→ −b

−1−5/8

2√

2

b1

b: A∗∂φeif ·φ/2

((A∗ · ∂φ)eif ·φ/2 + (B∗ · ∂φ)e−if ·φ/2

): (3.60)

+b−1−5/8

2√

2:

[A∗ ·

([3 +

(f/2)2

2+ 3/4

]∂2φ−

[2 +

(f/2)2

2+ 3/4

]b1

b∂φ

)]×[(A∗ · ∂φ)eif ·φ/2 + (B∗ · ∂φ)e−if ·φ/2

]:

=

[3 + (f/2)2

2+ 3/4

]2

b−1−5/8

√2

:

[A∗ ·

(∂2φ− b1

b∂φ

)]×[(A∗ · ∂φ)eif ·φ/2 + (B∗ · ∂φ)e−if ·φ/2

]:

(3.61)

and so the b1/b terms have conspired to give us back the same operator, compare (3.50).

Plugging in f 2 = 2, we read off the conformal weight of this operator as h = 2. This is

expected for a weight-1 excitation working on a weight-1 field. Further, this shows that

this operator is in fact a Virasoro primary operator.

Finally, we are ready to compute the 3-point function

1√N

1(N2

)⟨(A·(∂φ1(a) + ∂φ2(a) + · · · )) (

OA,(12)(b)+· · ·) (

[α(12)−m/2, OA,(12)(0)†] + · · ·

)⟩(3.62)

where · · · denotes all of the other terms that lead to permutation invariant operators. The

extra factors of 1/√(

N2

)normalize the sum of twist two operators OA,(i,j) with respect

to N . Above, there are(N2

)total terms for the twist 2 sector operators. We will find

nonzero expectation values only when the total twist is zero. Hence, we would calculate(N2

)terms, all of which give the same contribution, canceling the normalization factors

coming from the twist two operators. We find

1√N

⟨(A · (∂φ1(a) + ∂φ2(a))

)OA,(12)(b)

[α

(12)−m/2, OA,(12)(0)†

]⟩. (3.63)


We have dropped the higher terms in the sum ∂φ1 + ∂φ2 + · · · because these factorize,

and do not give contributions. Note that the order of this interaction is 1/√N , which is

expected on general grounds from [171].

To lift this computation to the cover, as mentioned, we will need the map

z(t) = bt2

t2 − (t− 1)2= b

t2

2t− 1. (3.64)

This maps the first twist operator (at b) to t = 1, and the excited twist operator to t = 0.

For future reference, we expand around these points to find

z(t) = −b(t2 + 2t3 + · · ·

), z(t)− b = b

((t− 1)2 − 2(t− 1)3 + · · ·

)(3.65)

We need to find the two images of a in the covering space, which we call t±. These are

determined from the map, and so we solve

a = bt2±

2t± − 1(3.66)

to find

t± =a±

√a(a− b)b

. (3.67)

Further, because the operator ∂φ1(a) transforms under conformal mapping, we will need

to compute

∂z

∂t

∣∣∣∣t=t±

=2bt±(t± − 1)

(2t± − 1)2=

2a(a− b)bt±(t± − 1)

(3.68)

where in the second equality we have used the definition (3.66) to remove the terms

(2t± − 1)2. We now lift the 3-point function (3.62) to the covering space, to find

⟨(A ·(∂φ1(a) + ∂φ2(a)

))OA,(12)(b)

[α

(12)−2/2, OA,(12)(0)†

]⟩. (3.69)

We lift this computation to the covering surface, needing two images of the non-twist


operator, which we then add together. This gives

⟨(A · (∂φ1(a) + ∂φ2(a))

)OA,(12)(b)

[α

(12)−2/2, OA,(12)(0)†

]⟩→

⟨[A ·(∂z

∂t(t+)∂φ1(t+) +

∂z

∂t(t−)∂φ1(t−)

)]× 1√

2b5/8

[(A · ∂φ)e−if ·φ/2 + (B · ∂φ)eif ·φ/2

](1) (3.70)

×(−b)−1−5/8

√2

[A∗ ·

(∂2φ− b1

b∂φ

)(A∗ · ∂φ eif ·φ/2 +B∗ · ∂φ e−if ·φ/2

) ](0)

⟩.

We see from (3.65) that b1/b = 2. Finally, we must match factors of f · φ in exponents,

and we get only two contributions

=(−1)−13/8

b1+5/4

∑±

bt±(t± − 1)

2a(a− b)(3.71)

×

[⟨A · ∂φ(t±)[A · ∂φe−if ·φ/2(1)][A∗ · (∂2φ− 2∂φ)A∗ · ∂φeif ·φ/2(0)]

⟩

+

⟨A · ∂φ(t±)[B · ∂φeif ·φ/2(1)][A∗ · (∂2φ− 2∂φ)B∗ · ∂φe−if ·φ/2(0)]

⟩].

We see that the first expectation value has two types of contractions to remove ∂φa, while

the second has only one. These become

=(−1)−13/8

b1+5/4

∑±

bt±(t± − 1)

2a(a− b)(3.72)

×

[(A∗ · A)

(−2

t3±− 2−1

t2±

)(A∗ · A)

−1

(1)2〈e−if ·φ/2(1)eif ·φ/2(0)〉

+(A∗ · A)

(−1

t2±

)(A∗ · A)

(−2

13− 2−1

12

)〈e−if ·φ/2(1)eif ·φ/2(0)〉

+(A∗ · A)

(−2

t3±− 2−1

t2±

)(B∗ ·B)

−1

12〈eif ·φ/2(1)e−if ·φ/2(0)〉

].

Note that the second line above gives no contribution. Further, there are several combi-


nations of the form

t±(t± − 1)

t2±=

(t± − 1)

t±. (3.73)

Explicitly summing these contributions using (3.67), we find

(t+ − 1)

t++

(t− − 1)

t−= 0. (3.74)

This type of cancelation is what causes the m = 1 case to vanish, and is the reason that

we did not use this seemingly simpler calculation to illustrate our technique. Summing

the other terms gives the result

=4(−1)−13/8

b5/4a(a− b)

(t+ − 1

t2++t− − 1

t2−

). (3.75)

Evaluating the sum, after plugging in t+ and t−, we find that the 3-point function (3.69)

lifts to the remarkably simple result

⟨(A · (∂φ1(a) + ∂φ2(a))

)OA,(12)(b)

[α

(12)−2/2, OA,(12)(0)†

]⟩→ −8(−1)−13/8

b5/4a2. (3.76)

Of course this is only the lifted part of the computation. We must include the contribution

from the Liouville term as well. We read this from equation (3.18) of [171], recalling that

the bare twists have weight h = (c/24)(n− 1/n) = 3/8 for c = 6, n = 2. Hence, we must

dress the above computation with a factor of b−3/4. Our total 3 point function becomes

⟨A ·(∂φ1(a) + ∂φ2(a)

)OA,(12)(b)

[α

(12)−2/2, OA,(12)(0)†

]⟩∝ −8(−1)−13/8

b2a2(3.77)

up to the normalization factors from [171] which we have not included, and normalizations


of (3.44), (3.45) and (3.51), which we could compute by calculating the 2-point functions.

To faithfully include the normalization from [171], we would need to include what was

happening in the right moving sector as well, include the extra factors of b from the

right moving spin fields in the covering space. This would help cancel some of the

phase ambiguity in (−1)−1−5/8 because there would be a factor of (−b)−5/8: we would

take these phases to be opposite in direction and cancel to give an overall −1 factor.

However, what is important for us here is that the above expression exactly matches the

expected behavior for a 3-point function of quasi primary fields:

⟨A(a)B(b)C(0)

⟩=

CABC(a− b)hA+hB−hCahA+hC−hBbhA+hB−hC =

CABCa2b2

(3.78)

where hA = 1, hB = 1, hC = 2. Note that this behavior comes about only after we summed

over different images. Each term had contributions from the conformal transformation

properties of ∂φ and from the particular images t± that get mapped to the postion z = a.

These all come together to give a result that is meaningful in the base space.

3.4 Discussion

We now comment on our generalization of the LM covering space technique, some of its

features, and possible obstacles to overcome. First, we note that in the generalization

to the non twist sector operators we could compute the images of the non-twist field in

the covering space explicitly. However, for general maps, where more covers of the space

are required, we expect this straightforward approach to begin to run into difficulties.

The relative simplicity we observed was partially aided by the low order of the twists,

but mainly came from the low number of twist operators. If we had instead considered

twist-n fields, the corresponding map with the twists at z = 0, t = 0 and z = ∞, t = ∞

would be

z = tn. (3.79)


In this context, finding the images in the covering space is trivial. We would just find

the nth roots of the location of the non-twist insertion z = a. To find the images when

the operators are at finite positions, one just feeds the various values of a1/n through the

SL(2,C) maps. The real difficulty would come in when dealing with more complicated

maps involving multiple covers and additional twist fields. For a map involving s covers,

one would need to solve an sth order polynomial in order to deal with the positions

explicitly. This is obviously impossible in general. However, we do not need to know the

individual pieces, we only need the sum. One can imagine more sophisticated methods

being available to obtain results in these cases, relating sums of powers of solutions to

polynomial equations to various coefficients of the polynomial. This seems to be the

only way for the summation to make sense in the base space. This problem should also

be important for the generalization to non-twisted excitations as well, given that the

locations of the non-ramified images of the twist operator insertions are found by solving

polynomials of an appropriately lower order.

One issue that our examples did not deal with is the possibility that there are different

maps that give the correct ramifications in the covering space. A problem related to this

multiplicity of maps was recently considered in [180]. It would be interesting to see

how to incorporate information about different maps and different images into the same

formulation. For example, seeing whether summing over images in the covering space

yields sensible results in the base space for every map, or whether one must combine the

information in different maps to make a sensible result, or whether working at large N

limits the possibilities. If we consider the possibility that there are different maps, we

note that each of these maps will give a different way to sew the multiple images of the

simply connected patch together. These different ways to sew together will give different

“topologies” of field configurations that satisfy the correct boundary conditions. In this

way, the path integral may factorize into these distinct topologies. It could be that the

result found in [180] is some statement about summing over these distinct topologies, each


of which contributes once, but because of the special nature of the extremal correlators,

the contributions from each topology is the same. Further, the presence of multiple maps

may be related to the ambiguity in the order that various group elements are multiplied

to get the identity. Of course, all of these considerations may be more a statement about

how one is computing non-SN -invariant pieces, and then needing to sum results to make

SN -invariant correlators. Exploring these types of calculation should shed more light on

the general process. We will leave these questions to future work.

There is one well known system where the above techniques can be used unmolested:

moving away from the orbifold point of the D1-D5 CFT. For this, the theory needs to be

deformed, and the correct operator to add to the action lives in the twist-2 sector of the

theory. Super chiral primary operators are protected from perturbative changes to their

conformal dimensions. Some of these are light operators, and correspond to supergravity

modes, and some of these supergravity modes are in the non-twisted sector of the CFT.

For this reason, one may not simply ignore the interactions between the twist and non-

twist sectors, and hope that it becomes unimportant in the gravity limit. Further, we

would like to be able to track what happens to these modes when the perturbation is

turned on, as a start on bridging the gap between strong and weak coupling. We will use

the techniques developed here to begin to address these issues in the next chapter.

Chapter 4

String states mixing in the D1-D5

CFT near the orbifold point

In this chapter we investigate lifting of low-lying string states as the D1-D5 CFT is

marginally deformed away from the orbifold point and study mixing between these states

at first order in conformal perturbation theory. The content of this chapter is based on

research first presented in [177].

4.1 Introduction

So far, the majority of work in the fuzzball arena has been on developing the supergrav-

ity side of the story and perturbative corrections to that picture. The CFT side of the

story is less well developed, and it is this side to which we contribute here. Other recent

CFT developments in fuzzball physics include using worldsheet CFT to perturbatively

construct new microstate geometries for a class of D-brane boundstates known as super-

strata [133] and using D1-D5 CFT to perform spectral flow of integer and fractional type

on the Ramond vacua of the twist sector and find microstate geometries corresponding

to the double-centre Bena-Warner geometries [134, 164, 165, 166, 167].

From the microscopic string perspective, the degrees of freedom of the D1-D5 system

84

Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point85

are described by a conformal field theory (CFT) living on a symmetric product orbifold

space. For D1-branes and D5-branes wrapped on S1 and S1×T 4 respectively, the orbifold

is1 (T 4)N/SN , where N = N1N5. Since our eventual goal is to help connect CFT physics

with black hole physics, we are interested in what happens when the D1-D5 CFT is

deformed away from the orbifold point. Throughout, we work in the large-N limit where

genus-zero diagrams dominate the string path integral.

In section 4.2 we describe our method for identifying operator mixing among low-

lying string states in the D1-D5 CFT. We outline the ingredients of the CFT, identify a

suitable set of cocycles for the fermionic operators, and describe how conformal pertur-

bation theory gives anomalous dimensions. We pick a deformation operator belonging

to the twist-2 sector of the theory which is a singlet under R-symmetry and under the

internal SU(2)s corresponding to directions of the T 4. We then show how taking factor-

ization limits of four-point functions involving low-lying string states and the deformation

operator allows us to identify operator mixing and to calculate mixing coefficients.

In section 4.3, we see why the dilaton operator does not mix, at first order in pertur-

bation theory. This makes use of Lunin-Mathur (LM) technology for symmetric orbifolds

[171, 172] and the results of the previous chapter generalizing LM to the non-twist sector.

In particular, we identify a suitable map from the base space to the covering surface and

compute four-point functions. Of course, the lack of mixing we find is in accord with

expectations from non-renormalization theorems [179].

In section 4.4 we present our main results. First, we settle on a low-lying string state

of the form ∂X∂X∂X∂X. Then we discuss the technicalities of lifting the correlation

function computation up to the covering space and summing over images. We next

take the coincidence limit of the four-point function and show how to subtract conformal

families of descendants in order to find the coefficient of mixing with other (quasiprimary)

operators of suitable weights. Finally, we evaluate the precise mixing coefficient for our

1Note that we are modding out by the symmetric group here, not the cyclic group.


string state. Our most interesting qualitative result is that mixing does take place at

first order.

In section 4.5, we give a brief accessible summary of our results and discuss what

remains to be done in order to nail down the anomalous dimensions for low-lying string

states of interest.

4.2 Perturbing the D1-D5 SCFT

4.2.1 The D1-D5 superconformal field theory

The theory under consideration has a N = (4, 4) supersymmetry. It contains SU(2)L ×

SU(2)R R-symmetry and a SO(4)I ' SU(2)1 × SU(2)2 symmetry group which cor-

responds to the directions of the torus. (Technically the SO(4)I is broken by periodic

identifications of the T 4, but it still provides a useful organizational principle.) Each copy

of the target space is a free c = 6 CFT which has 4 real bosonic fields X1, X2, X3, X4,

4 real fermionic fields in the left-moving sector ψ1, ψ2, ψ3, ψ4 and 4 real fermionic fields

in the right-moving sector ψ1, ψ2, ψ3, ψ4.

The bosonic fields X i can be written as doublets of SU(2)1 and SU(2)2 [168]

XAA =1√2X i (σi)AA, (4.1)

where σ1, σ2, and σ3 are the Pauli spin matrices and σ4 = i I, with I being the iden-

tity matrix. The indices A and A correspond to the doublets of SU(2)1 and SU(2)2,

respectively. The four real fermions of the left-moving part can be combined to form

complex fermions which transform as doublets of the SU(2)L and SU(2)2: ψαA. The

index α corresponds to the SU(2)L doublet. The reality condition imposes the following


constraint on the complex fermions

(ψαA)†(z) = −εαβ εAB ψβB(z). (4.2)

Thus we count four complex fermions ψαA, with two linearly independent reality con-

ditions. This gives two independent complex fermions, which is the same as four real

fermions. We similarly combine the four right-moving real fermions into complex fermions

which transform as doublets of the SU(2)R and SU(2)2: ψαA, where the index α corre-

sponds to the SU(2)R doublet. Again, the reality condition

(ψαA)†(z) = −εαβ εAB ψβB(z) (4.3)

reduces the counting to two independent complex fermions, or four real fermions.

The generators of the superconformal algebra in the left-moving sector are the stress

energy tensor T , the four supercurrents GαA, and the SU(2)L R-symmetry current Ja

which are given by

T =1

4εAB εAB ∂X

AA ∂XBB +1

2εαβ εAB ψ

αA∂ψβB,

GαA =√

2εAB ψαA ∂XBA,

Ja =1

4εAB ψ

αA (σ∗a)βγ ψγB. (4.4)

The generators of the superconformal algebra in the right-moving sector are T , GαA, and

Ja and are given by similar expressions.

The OPE of bosonic and fermionic fields are

∂XAA(z1) ∂XBB(z2) ∼ εAB εAB

(z1 − z2)2, (4.5)

ψαA(z1)ψβB(z2) ∼ −εαβ εAB

z1 − z2

. (4.6)


The left-moving generators of the CFT form a closed OPE current algebra, and similarly

for the right-movers. The complete OPE algebra and the mode algebra may be found in

[168].

We introduce N copies of the above CFT, and orbifold by the symmetric group SN

which interchanges different copies of the CFT. The orbifolding introduces twist sector

states, and so in the CFT we have corresponding twist sector operators. These twist

operators mix the different copies of the CFT. As one circles once around a point with

a twist operator insertion σn ≡ σ(123···n), the n copies of the bosonic and fermionic fields

are mapped together such that

X i(1) → X i

(2) → X i(3) · · · → X i

(n) → X i(1), (4.7)

ψi(1) → ψi(2) → ψi(3) → · · · → ψi(n) → ψi(1). (4.8)

This action is by itself not SN invariant, because it singles out the first n indices. One

must in fact sum over the group orbit of the group element (123 · · ·n) in the full SN

group, or equivalently to introduce one operator for each member of the conjugacy class

of (123 · · ·n). In what follows, we will consider only one member of the conjugacy class

as a representative. The contributions from other members of the conjugacy class will

be accounted for with combinatoric factors later on.

An important property of the twist sector is that they contain operators with frac-

tional modes. The construction of these operators is explicitly described in [172]. Let us

consider, for example, the R-current Ja acting on the twist operator σn. The fractional

mode operator is defined

Ja−mn

(z)σn(0) ≡∮z=0

dz

2πi

n∑k=1

Ja(k)(z) z−mn e−2πi (k−1)m

n σn(0), (4.9)

where the subscript (k) corresponds to the k-th copy of the target space involved in the

twist. The fractional mode operator is invariant under the action of the symmetric group


because the integrand is periodic as one circles around the origin in the contour integral.

The operator Ja−m/n has a fractional holomorphic conformal weight h = m/n.

The fractional modes of the R-current (4.9) are used to construct super chiral and

anti-chiral primary operators of the twist sector. This is done by starting with a bare

twist, and then applying fractional R-current J+ modes to find a superconformal ancestor

[172]. Super chiral primaries have the same conformal weight and R-charge h = m, h = m

while super anti-chiral primaries satisfy h = −m, h = −m.

4.2.2 Cocycles

In the computations presented in the following sections we will frequently use the bosonized

representation of the fermionic fields introduced in [172]. The bosonized language allows

one to easily evaluate correlation functions which contain fermionic fields. When bosoniz-

ing multiple (complex) fermions, one must introduce multiple bosonic fields. These un-

related bosonic fields do not share an OPE, and so they commute as operators, while the

fermions that they represent must anticommute. This is a well known problem, and the

introduction of cocycle operators is needed to guarantee that fermions anticommute.

Here, we will write down an explicit set of cocycles that guarantees the anticommu-

tation of bosonized fermions in the D1-D5 CFT at the orbifold point. In the bosonized

language, we have two holomorphic scalar fields, φ5, φ6, and two anti-holomorphic scalar

fields, φ5, φ62. These fields have the corresponding momenta p5, p6, p5, p6 which satisfy

the algebra

[φi, pj] = iδij, [φi, pj] = iδij, (4.10)

where any of the pi and pi commute, and the holomorphic and anti-holomorphic sectors

commute.

Fermions are replaced by exponentials of bosonic fields in the bosonized language.

2The indices here are chosen to agree with those used in section 4 of [172]. We make no distinctionbetween the index being up or down.


Therefore, it is natural to consider operators of the form

Ck ei (k5φ5+k6φ6+k5φ5+k6φ6), (4.11)

where ki and kj are real numbers and Ck is the cocycle operator which explicitly depends

on ki and kj. We define the cocycle Ck to be of the form

Ck = eiπck , ck ≡(k5 k6 k5 k6

)M

p5

p6

p5

p6

, (4.12)

where M is a 4×4 matrix. Individual fermions are special operators for which one of the

ki or ki is 1, while the rest of them are 0. Using this, we may constrain the matrix M by

requiring fermions to anticommute. This gives that Mij −Mji = ±1 for i 6= j. Thus, the

antisymmetric part of M is fixed, up to the choice of six signs, and the symmetric part

is undetermined from such considerations. We define the matrix M to be of the form

M =

12

12−1

212

−12−1

212−1

2

12−1

2−1

2−1

2

−12

12

12

12

. (4.13)

We choose this so that operators of the form e±i12

(φ5+φ6) or e±i12

(φ5+φ6) are not dressed

with any extra operators.

Bosonizing the fermions

We now explicitly write the cocycles associated with the four complex fermions in the

holomorphic sector, ψαA(z), and the four complex fermions in the anti-holomorphic sec-


tor, ψαA(z). We impose the reality conditions (4.2) and (4.3) to reproduce the correct

number of two complex, or four real fermions in each sector. When applying the reality

conditions in the bosonized language, we must be careful to reverse the order of the

exponentiated scalar field and the cocycle when we take the Hermitian conjugate

(Ckeik·φ)† = e−ik·φ(Ck)

† = ei(−k)·φC−k. (4.14)

Using the matrix M defined above, we find

ψ+1 = eiπc e−iφ6 , ψ+2 = eiπc eiφ5 ,

ψ−2 = −eiφ6 e−iπc, ψ−1 = e−iφ5 e−iπc, (4.15)

where

eiπc ≡ eiπ2

(p5+p6−p5+p6). (4.16)

One can move all the cocycle operators to the left by using the commutation relations

(4.10), if so desired. Using similar considerations for the anti-holomorphic side, we find

ψ+1 = eiπc e−iφ6

, ψ+2 = eiπc eiφ5

,

ψ−2 = −eiφ6

e−iπc, ψ−1 = e−iφ5

e−iπc, (4.17)

where

eiπc ≡ eiπ2

(p5−p6−p5−p6). (4.18)

One can easily check that the assigned cocycles guarantee the anti-commutation of

fermions. Let us first consider bosonized fermions which contain singular terms in their

OPEs. The singular part is proportional to (z1− z2)−1, where z1 and z2 are the positions

of the fermion operators on the complex plane. Therefore, a minus sign arises when we

switch the order of the two operators. For those bosonized fermions which do not share


a singular OPE, passing the cocyle operators across the exponentiated scalars produces

the minus sign. For instance, we consider the OPE

ψ+1(z)ψ+2(w) = eiπ2

(p5+p6−p5+p6)e−iφ6(z) eiπ2

(p5+p6−p5+p6)eiφ5(w)

= e[−iφ6,iπ2p6] ei

π2

(p5+p6−p5+p6) eiπ2

(p5+p6−p5+p6)e−iφ6(z) eiφ5(w)

= e[−iφ6,iπ2p6] e[iπ

2p5,iφ5] ei

π2

(p5+p6−p5+p6)eiφ5(w) eiπ2

(p5+p6−p5+p6)e−iφ6(z)

= −ψ+2(w)ψ+1(z), (4.19)

where we have used the Baker-Campbell-Hausdorff formula and the commutation rela-

tions (4.10).

We noted earlier that the fermions in the holomorphic sector transform as doublets

of SU(2)L and SU(2)2 and that the fermions in the anti-holomorphic sector transform

as doublets of SU(2)R and SU(2)2. Using the prescription described above for assigning

cocyles to the bosonized fields, one can construct the SU(2) symmetry currents. Let us

consider the holomorphic sector first. Equation (4.4) gives the R-currents J+ = ψ+1ψ+2

and J− = −ψ−1ψ−2. Using (4.15) and (4.16), we can then evaluate the OPE of the R-

currents J± and the fermions in the context of the bosonized language. Let us evaluate

the OPEs for ψ±1 as an example. For ψ±1 we have

J+(z)ψ−1(w) = eiπ2

(p5+p6−p5+p6)e−iφ6eiπ2

(p5+p6−p5+p6)eiφ5(z) e−iφ5(w) e−iπ2

(p5+p6−p5+p6)

∼ eiπ2

(p5+p6−p5+p6)e−iφ6(w)

z − w=ψ+1(w)

z − w, (4.20)

and

J−(z)ψ+1(w) = e−iφ5 e−iπ2

(p5+p6−p5+p6)eiφ6e−iπ2

(p5+p6−p5+p6)(z) eiπ2

(p5+p6−p5+p6)e−iφ6(w)

∼ e−iφ5(w) e−iπ2

(p5+p6−p5+p6)

z − w=ψ−1(w)

z − w. (4.21)


Similar considerations hold for ψ±2. We thus obtain

[J+0 , ψ

−A] = ψ+A, [J−0 , ψ+A] = ψ−A. (4.22)

Therefore the bosonized fermions transform as a doublet of the SU(2)L symmetry, as

required. One can perform similar computations in the right-moving sector and find that

the assigned cocycles reproduce the appropriate SU(2)R algebra.

One can also easily check that the fermions transform as doublets of SU(2)2 in the

context of the bosonized language. Let us denote the associated currents with J a. We

have J + = ψ+1 ψ−1, J − = −ψ+2 ψ−2. Using the bosonized fields (4.15) and (4.17) and

performing similar computations as in (4.20) and (4.21), we find

[J +0 , ψ

α2] = ψα1, [J −0 , ψα1] = ψα2, (4.23)

Analogous considerations hold in the right-moving sector.

Spin fields

Our next task is to figure out how to make spin fields work in the Lunin-Mathur context.

The main ingredients are as follows; we refer the interested reader to [172] for further

details. First, we use Lunin-Mathur technology to map twist sector operators into the

covering space where they have normal boundary conditions. Second, fermions belonging

to the even-twist sector have antiperiodic boundary conditions in the cover. Third, to

take fermion boundary conditions into account correctly, spin fields must be inserted in

the cover.

We now turn to constructing the cocycles associated with the spin fields on the cov-

ering surface. For spin fields we have ki = ±1/2, ki = ±1/2. Let us consider the

holomorphic sector first. There are four spin fields in this sector Sα and SA. Following


the procedure described above, we find that

S+ ≡ eiπc ei2

(φ5−φ6), S− ≡ e−iπc e−i2

(φ5−φ6), (4.24)

where eiπc is defined in (4.16). In the above expressions we have moved the cocycles to

the left. For these spin fields we find that

[J+0 ,S−] = S+, [J−0 ,S+] = S−, (S+)† = S−. (4.25)

The assigned cocycles make the two spin fields Sα transform as a doublet of SU(2)L and

a singlet of SU(2)2.

For spin fields SA we find

S 1 ≡ eiπ2 e−

i2

(φ5+φ6), S 2 ≡ e−iπ2 e

i2

(φ5+φ6). (4.26)

We note that the structure of the cocycles we defined in (4.12) and (4.13) is such that

the cocycles associated with exponentials of the form e±i2

(φ5+φ6) are just 1. The rescaling

of the spin fields with the above phases guarantees that they satisfy the SU(2)2 algebra

[J +0 ,S 2] = S 1, [J −0 ,S 2] = S 2, (S 1)† = S 2. (4.27)

These two spin fields do not carry SU(2)L charges.

Analogous computations are done for the anti-holomorphic sector and we find that

the spin fields in this sector are given by

S+ ≡ eiπ2

(p5−p6−p5−p6)ei2(φ5−φ6), S− ≡ e−i

π2

(p5−p6−p5−p6)e−i2(φ5−φ6), (4.28)

S 1 ≡ e−iπ2 e−

i2(φ5+φ6), S 2 ≡ ei

π2 e

i2(φ5+φ6). (4.29)

Finally, we note the the fermionic zero modes act on the spin fields and map them


to other spin fields. It is straightforward to check that these modes satisfy the gamma

matrix algebra Γi,Γj = 2 δij. The action of fermion zero modes on the spin fields can

be evaluated using the bosonized language. For this purpose, and some later ones, we

now record the OPEs of the fermions and the spin fields. In the left-moving sector we

find that

ψ−10 S+(0) = e−i

π2 S 1(0), ψ−2

0 S+(0) = e−iπ2 S 2(0),

ψ+10 S−(0) = −S 1(0), ψ+2

0 S−(0) = −S 2(0),

ψ+20 S 1(0) = e+iπ

2 S+(0), ψ−20 S 1(0) = S−(0),

ψ+10 S 2(0) = e−i

π2 S+(0), ψ−1

0 S 2(0) = −S−(0). (4.30)

Similar considerations apply for the right-moving sector.

4.2.3 Deformation operator

The D1-D5 CFT at the orbifold point has four exact marginal deformation operators

in the twist sector which deform the CFT away from the orbifold point. These defor-

mation operators have conformal weights (h, h) = (1, 1). They are singlets under the

SU(2)L× SU(2)R R-symmetry and preserve the N = (4, 4) supersymmetry of the CFT.

The deformation operators are constructed by applying modes of supercurrent GαA−1/2

and GαB−1/2 to the super-chiral and anti-chiral primaries of the twist-2 sector, σββ2 , with

conformal dimension (1/2, 1/2). The left and right-moving parts of the four marginal

deformation operators carry indices of the doublet of SU(2)1 of the internal SOI(4)

symmetry and the operators transform as 3 + 1 of SU(2)1 [23].

We will consider the singlet component which preserves the global SU(2)1 symmetry:

Od ∝ εAB εαβ εαβ GαA− 1

2GαB− 1

2σββ2 . (4.31)


We now split the above twist operator into left-moving and right-moving parts σββ2 (z, z) =

σβ2 (z)σβ2 (z). It is shown in [185] (Appendix B) that G−A− 12

σ+2 ∝ G+A

− 12

σ−2 . Therefore,

G−A− 12

σ+2 is a singlet of SU(2)L by itself. The same reasoning holds for the right-moving

sector. This could also be seen by the discussion in [226] (§5.1) that a single application

of supercharges G−A− 12

on a super-chiral primary σ+n constructs Virasoro primaries which

are annihilated by J+0 and therefore are the top members of the SU(2)L multiplet. In

the twist-2 sector this construction creates a singlet under SU(2)L. The deformation

operator with charge zero under SU(2)L × SU(2)R and SU(2)1 is therefore of the form

Od = εAB G−A− 1

2

G−B− 12

σ++2 . (4.32)

The perturbation that we add to deform the symmetric product CFT away from the

orbifold point is

Sint = λ

∫d2zOd(z, z) + a.c., (4.33)

where a.c. refers to the anti-chiral fields acted on by the Hermitian conjugate supercurrent

modes.

The supercurrents are GαA =√

2ψαA∂XBAεAB, where the summation over an omit-

ted target space copy index is implicit. Inserting this in (4.32) we obtain

Od =√

2[(ψ−1∂X 21 − ψ−2∂X 11)− 1

2(ψ−1∂X 22 − ψ−2∂X 12)− 1

2

−(ψ−1∂X 22 − ψ−2∂X 12)− 12

(ψ−1∂X 21 − ψ−2∂X 11)− 12

]σ++

2 . (4.34)

The modes of the supercurrents in the left-moving sector are given by

GαAm =

∮dz

2πiGαA zh+m−1, (4.35)

where h = 3/2. The same procedure holds for the right-moving supercurrents GαAm .


In the twist-2 sector, the insertion of super (anti-)chiral primary σ±±2 operators in

the base space corresponds to the insertion of spin fields S±± on the covering surface

[172]. The operators σ±±2 that we are concerned with have conformal weight (1/2, 1/2)

and carry R-charge (±1/2,±1/2). We discussed the bosonized representation of fermions

and spin fields in subsection 4.2.2. Here we will use them to evaluate the deformation

operator. It is shown in [172] that the local normalization of the spin fields is given by

b−18 S± where b is specified by the map from the base space to the cover in the vicinity

of the insertion point of the spin field: (z − z0) ≈ b (t − t0)2 (1 + b1t + O(t2)). The

right-moving part definitions follows similarly. In cases where there are more than two

spin fields or there are fermionic fields acting on the spin fields in a correlation function,

one needs to be careful about the anticommutation relations of fermionic fields and apply

cocycles to impose the correct anticommutation properties. This is the raison d’etre for

subsection 4.2.2.

We will now construct Od. Using (4.35) we have:

G−A− 12

G−B− 12

σ++2 (0, 0) =

∮0

dz

2πi

∮0

dz

2πiz

32− 1

2−1 z

32− 1

2−1G−A(z) G−B(z)σ++

2 (0, 0)

→∮

0

dt

2πi

(dz

dt

)1− 32∮

0

dt

2πi

(dz

dt

)1− 32

G−A(t) G−B(t)

(1

b18 b

18

S++(0)

)=

∮0

dt

2πi

∮0

dt

2πi

1(2bt(1 + 3

2b1bt+O(t2))

) 12

1(2bt(1 + 3

2b1bt+O(t2))

) 12

×

× G−A(t) G−B(t)1

|b| 14S++(0)

≈ 2

2|b| 54

∮0

dt

2πi

∮0

dt

2πi

(1− 3

4b1bt+O(t2)

)t

12

(1− 3

4b1bt+O(t2)

)t

12

×

×(ψ−1∂X 2A − ψ−2∂X 1A

)(t)(ψ−1∂X 2B − ψ−2∂X 1B

)(t)S++(0, 0), (4.36)

where the arrow in the second line implies that we passed from the base to the covering

sphere using the map which is locally of the form z ≈ b t2 (1 + b1t + O(t2)). Here and

in what follows, we simplify notation by suppressing the contribution from the Liouville


action which is always present [171, 172]. Thus → is really an instruction to pass to the

covering surface, and suppress the Liouville contribution. We will put in this contribution

in at the end. The OPEs of fermions and spin fields are evaluated in (4.30) in the context

of the bosonized language. The bosonic fields ∂XAA do not share singular OPEs with

the spin fields. Equation (4.36) then reads

G−A− 12

G−B σ++2 (0, 0) =

1

|b| 54

∮0

dt

2πi

∮0

dt

2πi

1

t

1

t×(

: ∂X 2A∂X 2Be−iφ5(t)−iφ5(t) ei2

(−φ6+φ5−φ6+φ5)(0,0) :

+ : ∂X 2A∂X 1Be−iφ5(t)+iφ6(t) ei2

(−φ6+φ5−φ6+φ5)(0,0) :

+ : ∂X 1A∂X 2Be+iφ6(t)−iφ5(t) ei2

(−φ6+φ5−φ6+φ5)(0,0) :

+ : ∂X 1A∂X 1Be+iφ6(t)+iφ6(t) ei2

(−φ6+φ5−φ6+φ5)(0,0) :)

=1

|b| 54

(: ∂X 2A∂X 2Be

i2

(−φ6−φ5−φ6−φ5) : + : ∂X 2A∂X 1Bei2

(−φ6−φ5+φ6+φ5) :

+ : ∂X 1A∂X 2Bei2

(+φ6+φ5−φ6−φ5) : + : ∂X 1A∂X 1Bei2

(+φ6+φ5+φ6+φ5) :)

(0, 0),

(4.37)

up to an overall minus sign coming from moving the right sector bosonized fermions to

the right to evaluate their OPEs with the right-moving spin fields. The deformation

operator (4.34) is then found to be of the form

Od (0, 0) =1

|b| 54

(: e

i2

(−φ6−φ5−φ6−φ5)(∂X 21∂X 22 − ∂X 22∂X 21

):

+ : ei2

(−φ6−φ5+φ6+φ5)(∂X 21∂X 12 − ∂X 22∂X 11

):

+ : ei2

(+φ6+φ5−φ6−φ5)(∂X 11∂X 22 − ∂X 12∂X 21

):

+ : ei2

(+φ6+φ5+φ6+φ5)(∂X 11∂X 12 − ∂X 12∂X 11

):

)(0, 0). (4.38)

The complete deformation is λOd + a.c., where a.c. refers to the action of the Her-

mitian conjugated supercurrent modes on the super anti-chiral primary fields σ−−2 . The


super chiral and anti-chiral primary operators of the left and right-moving sectors form

doublets of SU(2)L and SU(2)R, respectively. As discussed in [163], we can write the

left-moving super anti-chiral operator as

σ−−2 (z0, z0) = J−0 (z) J −0 (z)σ++2 (z0, z0). (4.39)

The anti-chiral part of the deformation operator is of the form

G+A− 1

2

(z′) G+B− 1

2

(z′)σ−−2 = G+A− 1

2

(z′) G+B− 1

2

(z′) J−0 (z) J −0 (z)σ++2 (z0, z0). (4.40)

One can then pass J− and J − to the left of G+A and G+A which gives G−A and G−A

acting on σ++2 (z0, z0). Therefore the two chiral and anti-chiral terms are identical and

the full deformation operator is of the form λOd, where λ is a real number. This can be

checked explicitly in the context of the bosonized language noting that the spin fields S±

and S± carry SU(2)L and SU(2)R indices and taking into account εαβ and εαβ factors

when taking their Hermitian conjugates. One may be able to account for these signs in

a more explicit way with more complicated cocycles [228].

Lastly, we note that we work with SN -invariant operators in the symmetric product

orbifold, which are constructed by summing over the conjugacy classes. The summa-

tion brings a combinatorial factor (which depends on N and n) in the definition of the

operators in the twist-n sector. We will discuss this later.

4.2.4 Conformal perturbation theory

We consider a quasi-primary field φi in the 2-dimensional N = (4, 4) D1-D5 SCFT. The

two-point functions of φi and other quasi-primaries read

〈φi(z1, z1)φj(z2, z2)〉0 =δij

zhi12 zhi12

, (4.41)


where the subscript “0” refers to the unperturbed CFT, hi (hi) are the unperturbed

(anti-) holomorphic conformal weights of φi, and z12 ≡ z1 − z2. Let us first assume that

there is no degeneracy in the conformal weight of φi, i.e., if hi = hj, then φi = φj. We

first perform the perturbation theory under this assumption and then generalize to the

case where multiple fields can have the same conformal weight.

We start by adding a small perturbation to the action of the free SCFT:

δS =∑A

λA

∫d2zOd,A(z, z), (4.42)

where λA are the coupling constants of perturbation and Od,A(z, z) are the exact marginal

operators of the theory. We perform Kadanoff’s conformal perturbation theory and

evaluate the deformation of the conformal weights [229]. Two-point correlation functions

of quasi primary operators of a CFT are determined by their conformal weights. Thus,

their deformation gives the deformation of the conformal weights. There are two ways

to evaluate the change in the two-point function. First, one evaluates the derivative of

the two-point function (4.41) with respect to the coupling constant

〈φi(z1, z1) φj(z2, z2)〉λA =δij

z2hi(λA)12 z

2hi(λA)12

=δij

z2“hi+λA

∂hi(λA)

∂λA

”12 z

2

„hi+λA

∂hi(λA)

∂λA

«12

= e−2λA

„∂hi(λA)

∂λAln(z12)+

∂hi(λA)

∂λAln(z12)

«〈φi(z1, z1) φj(z2, z2)〉0

≈

(1− 2λA

(∂hi(λA)

∂λAln(z12) +

∂hi(λA)

∂λAln(z12)

))〈φi(z1, z1)φj(z2, z2)〉0, (4.43)

where hi(λA) = hi+λA∂hi(λA)∂λA

and hi(λA) = hi+λA∂hi(λA)∂λA

to the first order in perturbation


theory. We then have

∂

∂λA〈φi(z1, z1) φj(z2, z2)〉λA =(

−2∂hi(λA)

∂λAln(z12)− 2

∂hi(λA)

∂λAln(z12)

)〈φi(z1, z2) φj(z2, z2)〉0.(4.44)

If the derivatives of the conformal weights (hi(λA), hi(λA)) vanish to the first order, then

one moves to the second order in perturbation theory

∂2

∂λ2A

〈φi(z1, z1)φj(z2, z2)〉λA

=

(−2

∂2hi(λA)

∂λ2ln(z12)− 2

∂2hi(λA)

∂λ2ln(z12)

)〈φi(z1, z1)φ†j(z2, z2)〉0, (4.45)

and so on.

The second way of evaluating the change in the two-point function is to use the path

integral formulation of the theory

〈φi(z1, z1)φj(z2, z2)〉λA =

∫d[X,ψ] e−Sfree+λA

Rd2zOd,A(z,z) φi(z1, z1)φj(z2, z2)∫

d[X,ψ] e−Sfree+λARd2zOd,A(z,z)

=

∫d[X,ψ] e−Sfree

(1 + λA

∫d2zOd,A(z, z) +O(λ2

A))φi(z1, z1)φj(z2, z2)∫

d[X,ψ] e−Sfree

(1 + λA

∫d2zOd,A(z, z) +O(λ2

A)) , (4.46)

where we expanded the perturbative terms to the first order in perturbation theory in

the second line. The above equation reads

〈φi(z1, z1)φj(z2, z2)〉λA

=〈φi(z1, z1)φj(z2, z2)〉0 + λA

∫d2z〈φi(z1, z1)Od,A(z, z)φj(z2, z2)〉+O(λ2

A)

1 + λA∫d2z〈Od,A(z, z)〉+O(λ2

A)

= 〈φi(z1, z1)φj(z2, z2)〉0 + λA

∫d2z〈φi(z1, z1)Od,A(z, z)φj(z2, z2)〉+O(λ2

A),(4.47)


where 〈Od,A(z, z)〉 = 0. We therefore obtain

∂

∂λA〈φi(z1, z1)φj(z2, z2)〉λA =

∫d2z〈φi(z1, z1)Od,A(z, z)φj(z2, z2)〉. (4.48)

One then needs to evaluate the above three-point function

∫d2z〈φi(z1, z1)Od,A(z, z)φj(z2, z2)〉 = CiAj

∫d2z × (4.49)

× 1

(z1 − z)hi+1−hj (z − z2)hj+1−hi (z12)hi+hj−1 (z1 − z)hi+1−hj (z − z2)hj+1−hi (z12)hi+hj−1,

where CiAj are the structure constants, the index “A” corresponds to the deformation

operator. The integral on the right hand side of the above equation needs to be regularized

by putting cutoffs at the insertion points of the quasi-primary fields: |z − z1| > ε and

|z − z2| > ε. As discussed in [230, 231], one can make the SL(2,C) transformation

y(z) =z12z

z12 − z, (4.50)

which brings the above integral to the form

∫ |y|= 1ε

|y|= ε|z12|2

dydyeiπ(hj−hi+1−(hj−hj+1))

yhj−hi+1yhj−hi+1(4.51)

where we have chosen any branch cuts coming from possible fractional powers of zij so as

to align with the branch cuts coming from possible fractional powers of zij, and cancel.

We now make the following comment. The above integral is a two dimensional integral,

and the domain of integration is rotationally symmetric. This means that the angular

part of the integral is unconstrained. This immediately requires that

hj − hi + 1 = hj − hi + 1

hj − hj = hi − hi. (4.52)


i.e. that the spin of the fields φi and φj should match. We have not broken the rotation

group with the addition of weight (1, 1) operators to the action (they are spinless), and

so we should expect this to be a respected quantum number in the perturbed theory.

Now we see that there are two different possibilities: 1) hi = hj, hi = hj (hence, i = j,

or explicitly φi = φj given our assumption), and 2) hi 6= hj, hi 6= hj (i 6= j) [232]. For

hi = hj the integral reads

∫d2z〈φi(z1, z1)Od,A(z, z)φi†(z2, z2)〉 =∫d2z

CiAi

(z1 − z)1 (z − z2)1 (z12)2hi−1 (z1 − z)1 (z − z2)1 (z12)2hi−1=

1

z2hi12 z2hi

12

CiAi

∫d2z

|z12|2

|z1 − z|2 |z − z2|2. (4.53)

The SL(2,C) transformation (4.50) simplifies the integral in the third line of the above

equation

CiAi

z2hi12 z2hi

12

∫|z−z1|>ε,|z−z2|>ε

d2z|z12|2

|z1 − z|2 |z − z2|2=

CiAi

z2hi12 z2hi

12

∫ε<|y|<|z212/ε

d2y

|y|2(4.54)

=(

2πCiAi ln(z12) + 2πCiAi ln(z12)− 2πCiAi ln(ε2))〈φi(z1, z1)φi†(z2, z2)〉0.

The ε-dependent term which diverges in the limit ε → 0 has to be absorbed in the

renormalization of φi.

For hi 6= hj the integral (4.49) reads:

∫d2z〈φi(z1, z1)Od,A(z, z)φj(z2, z2)〉 = CiAj

∫d2z × (4.55)

× 1

(z1 − z)hi+1−hj (z − z2)hj+1−hi (z12)hi+hj−1 (z1 − z)hi+1−hj (z − z2)hj+1−hi (z12)hi+hj−1.


Performing the same SL(2,C) transformation (4.50) we obtain:

∫d2z〈φi(z1, z1)Od,A(z, z)φj(z2, z2)〉 =

|z12|2(hi−hj)

zhi+hj12 z

hi+hj12

δsi,sj CiAj

∫ε<|x|<|z212/ε

d2y

|y|2(hi−hj+1)

= −2π δsi,sj CiAj

(εdj−di

dj − di1

z2hj12 z

2hj12

+εdi−dj

di − dj1

z2hi12 z

2hi12

), (4.56)

where where si = hi − hi is the spin and di ≡ hi + hi is the scaling dimension of φi. In

the limit ε → 0 the two terms in the last line of the above equation either diverge or

vanish. Again, the cut-off dependent term is absorbed into the renormalization of φi.

Using (4.54) and (4.56) we can now find the appropriate wave function renormalization

to the first order

φi → φi + λπ ln(ε2)CiAi φi + λ 2π

∑j

δsi,sjεdj−di

dj − diCiAj φ

j. (4.57)

After subtracting the infinite parts, we shall compare the finite result (4.54) with what

we obtained earlier in (4.44) and find the anomalous dimension to first order:

∂hi∂λ

= −πCiAi,∂hi∂λ

= −πCiAi . (4.58)

This is Kadanoff’s deformation theory to the first order. The structure constants CiAi,

which are the coefficients of the logarithmic terms ln z12 and ln z12, determine the anoma-

lous dimension of φi to the first order in perturbation theory. For discussion of deforming

the D1-D5 CFT away from the orbifold point at second order in perturbation theory, see

[225].

So far we assumed that there is no degeneracy in the conformal weights of quasi

primaries φi. To relax this condition, we simply note that the form of the integrals

remains unchanged for the cases hi = hj, hi = hj or hi 6= hj, hi 6= hj. Therefore, this

does not affect the sector where hi 6= hj because this requirement means that the fields


are distinct: this part of the computation remains intact. The only modification is for

the hi = hj part of the computation, where the coefficient is now CiAj. To find the

correction to the anomalous dimensions, we would need to diagonalize CiAj in the entire

block of fields with the same conformal dimension. Rather than considering all operators

with the same conformal dimension, one may consider other preserved symmetries, like

R-symmetry, to restrict the search. We could also imagine trying to find the operators

iteratively, by taking a given operator φ1 and finding the operators that this mixes with

that have the same conformal weight φ2 · · ·φn. One would then find all the operators

that these operators mix with that have the same conformal dimension, and so on until

one finds the full set of fields that mix. We will further outline how one may attempt to

do this in our discussion, in section 4.5.

4.2.5 Four-point functions and factorization channels

Having constructed the deformation operator, we can start the computation of the

anomalous dimensions of some candidate states of the D1-D5 orbifold CFT. The states

that we consider belong to the non-twist sector of the theory.

Super chiral primary states of the orbifold CFT and their descendants under the

anomaly-free subalgebra of the superconformal algebra make the short multiplets of the

theory [23]. The perturbative orbifold CFT and the supergravity theory are appropriate

descriptions in different parts of the moduli space of the D1-D5 system. Therefore, if

we want to compare the states of the two theories, we have to consider those which are

protected against corrections as one moves in the moduli space. The states belonging to

the short multiplets of the orbifold CFT do not acquire corrections as one moves across

the moduli space. These states identify the supergravity modes in the near-horizon

geometry of the D1-D5 system. The energy and two- and three-point functions of the

members of the short multiplets are not renormalized [179].

We consider one of the protected states of the orbifold CFT in the next section. This


state corresponds to the dilaton in the dual supergravity description. We evaluate the

anomalous dimension of the state to the first order in conformal perturbation theory and

show that it vanishes, as expected. Then, in the following section, we will consider a non-

protected state of the orbifold CFT. We will investigate the corrections to the conformal

eight of our candidate state as we deform the CFT away from the orbifold point. We

first describe our method of calculation in the remaining of this section.

As discussed earlier, in section 4.2.4, Kadanoff’s deformation theory gives the anoma-

lous dimensions acquired by the states of a CFT under a small perturbation. To the first

order in conformal perturbation theory, the anomalous dimension of a state |φi〉 with

non-degenerate conformal weight (hi, hi) is given by (4.58)

∂hi∂λ

= −πCiAi,∂hi∂λ

= −πCiAi,

where CiAi are the structure constants corresponding to the three-point functions 〈φiOd φi†〉.

For the non-twist sector operators that we consider, this correlator vanishes because of

a group selection rule. For CFT states with degenerate conformal weights (such as our

candidate states), one needs to worry about potential first-order operator mixing be-

tween φi and other quasi-primary operators φj with the same conformal weight (hi, hi).

If there exists such mixing, one has to identify all the operators which φi mixes with,

evaluate CiAj, diagonalize the matrix of the structure constants, and find the change to

the conformal weight.

In order to investigate first-order mixing between operators of the same conformal

weight we evaluate the four-point function involving the operator under consideration,

its Hermitian conjugate, and two insertions of the deformation operator

⟨φi(z1, z1)Od(z2, z2)Od(z3, z3)φi†(z4, z4)

⟩. (4.59)

We take the coincidence limit as z1 → z2, z3 → z4 (or equivalently z1 → z3, z2 → z4) and


find the leading singular term and its coefficient. This singular limit signals intermediate

quasi-primary operator(s) which mix with φi at the first order. We evaluate the conformal

weight of the quasi-primary operator(s) and subtract their conformal families from the

leading order singular limit of the four-point function (there may be more than one

quasi-primary operators φk, with the same conformal dimension, which contribute to

this leading singularity. The sum over CkiA of these operators must give the coefficient

of the leading singular term). Each conformal family is composed of an ancestor quasi-

primary operator and all its descendants under the Virasoro algebra. After subtracting

these intermediate conformal families, we find the remaining leading-order singularity,

evaluate the conformal weight of the quasi-primaries of this singular limit, and subtract

their intermediate conformal families. We continue this procedure until we exhaust all

the intermediate conformal families which mix with φi at the first order in perturbation

theory.

We are interested in conformal families whose ancestors φj have the same conformal

weight as φi 3. These operators contribute to the anomalous dimension of φi. Quasi-

primaries which do not have the same conformal weight as φi contribute only to the

wave function renormalization at the first order in perturbation theory (4.57), but do not

change the conformal dimension.

Computing the four-point function (4.59) and taking its coincidence limits is a robust

way to compute the mixing coefficient of the set of all quasi-primaries which mix with

φi at the first order. There is a variety of building blocks in the N = (4, 4) D1-D5

orbifold CFT which we can use to construct candidate quasi-primaries which participate

in operator mixing.

3There are two different types of operators which can have the same conformal weight as φi: quasi-primary operators which are the ancestors of conformal families, and secondary operators which are thedescendants of quasi-primaries under the Virasoro algebra. As shown in section 4.2.3, it turns out thatone needs consider only contributions from quasi-primary (i.e., non-derivative) operators.


4.3 Dilaton warm-up

The AdS3/CFT2 correspondence [30] matches the 20 exactly marginal operators of the

N = (4, 4) orbifold CFT with the 20 near-horizon supergravity moduli [23]. Under the

correspondence, the six-dimensional dilaton in supergravity is identified with the exact

marginal operator∑N

κ=1 ∂xi(κ)∂xi (κ), where xi, i ∈ 1, 2, 3, 4 are the four real bosonic

fields and, as before, κ is the copy index. We refer to this operator as the dilaton operator.

Since the exactly marginal operators are protected from getting corrections, one expects

that they do not acquire an anomalous dimension as one moves away from the orbifold

point. We check this as a warm up calculation by evaluating the four point function

(4.59) for the dilaton operator.

4.3.1 Four-point function

We can write the dilaton operator in terms of the complex bosons XAA:

N∑κ=1

∂xi(κ)∂xi (κ) =N∑κ=1

−εAB εAB∂XAA(κ) ∂X

BB(κ) . (4.60)

The dilaton operator is self-conjugate. The four-point function which we would like to

compute is of the form:

⟨( N∑κ=1


BB(κ)

)(z1, z1) λOd(z2, z2)×

λOd(z3, z3)

( N∑κ′=1

−εA′B′ εA′B′∂XA′A′

(κ′) ∂XB′B′

(κ′)

)(z4, z4)

⟩. (4.61)

We use the translational invariance of the correlation function and shift the position of

the deformations operator at z3 to zero. Defining the new positions as a1 ≡ z1 − z3,


b ≡ z2 − z3, and a2 ≡ z4 − z3, we obtain

⟨( N∑κ=1


BB(κ)

)(a1, a1) λOd(b, b)×

λOd(0, 0)

( N∑κ′=1

−εA′B′ εA′B′∂XA′A′

(κ′) ∂XB′B′

(κ′)

)(a2, a2)

⟩. (4.62)

The deformation operator contains a twist-2 operator σ02 which permutes two copies of the

target space. Let us denote the two copies as κ = 1, 2. To find the SN invariant operator

we have to sum over the conjugacy classes of 2-cycles. This brings a combinatorial factor

in the definition of the deformation operator and will be taken into account at the end

of this section. For the moment we consider the SN non-invariant correlator for the

two copies κ = 1, 2, compute the four-point function, and take the coincidence limit

to study the operator mixing. The overall combinatorial coefficient does not affect the

arguments here, but obviously does have to be taken into account when evaluating the

corrections to the conformal weight. To make the notation compact we define φdil (κ) ≡


BB(κ) . The four-point function (4.62) is rewritten as

⟨ N∑κ=1

φdil (κ)(a1, a1) λOd(b, b) λOd(0, 0)N∑κ′=1

φdil (κ′)(a2, a2)⟩

=

⟨ 2∑κ=1

φdil (κ)(a1, a1) λOd(b, b) λOd(0, 0)2∑

κ′=1

φdil (κ′)(a2, a2)⟩

+

⟨ 2∑κ=1


φdil (κ′)(a2, a2)⟩

+

⟨ N∑κ=3


κ′=1

φdil (κ′)(a2, a2)⟩

+

⟨ N∑κ=3


φdil (κ′)(a2, a2)⟩. (4.63)

The correlation functions on the third and fourth line of the above equation vanish

because the deformation operators permute copies 1 and 2 of the target space. The


correlation function in the last line factorizes and we obtain

⟨ N∑κ=1


φdil (κ′)(a2, a2)⟩

=

⟨ 2∑κ=1


κ′=1

φdil (κ′)(a2, a2)⟩

+

⟨ N∑κ=3

φdil (κ)(a1, a1)N∑κ′=3

φdil (κ′)(a2, a2)⟩⟨λOd(b, b) λOd(0, 0)

⟩. (4.64)

The factorized correlation function in the last line provides no information about the

mixing between the dilaton operator and other operators of the CFT because it is not

singular in the coincidence limits of concern, e.g. a1 → b. Thus this term is of no interest

for our purpose. The four-point function in the second line is the one which contains the

information we are after.

4.3.2 Mapping from the base to the cover

Lunin-Mathur (LM) technology [171, 172] for symmetric orbifolds allows computation

of correlation functions involving twist-sector operators by lifting the correlator to the

covering surface. The insight resides in choosing appropriate maps which correctly lift the

ramified points – the places where twist operators are inserted – to the covering surface.

This ensures the right number of images in the cover both for twist and non-twist sector

operators. In particular, LM showed how to normalize the bare twist contribution to

the correlation function properly by evaluating the Liouville factor corresponding to the

conformal map. This procedure is necessary to get the boundary conditions right for the

ramified points. LM technology also makes clear how to regularize integrals, in essence

by cutting out disks.

In Chapter 3, we further developed LM technology for symmetric orbifolds to the

non-twist sector. This turned out to be a very natural extension of their methods. We

needed the generalization in order to be able to calculate correlation functions involving


both twist and non-twist operators. To illustrate the techniques, we worked through

two examples: (i) excitations of twist operators by modes of fields unaffected by twist

operators, and (ii) non-twist operators.

For the task at hand, we want to evaluate our four-point correlator obtained in the

last subsection by using generalized LM technology. The correlation function has two

insertions of twist-2 operators at z = b and z = 0. We will first write the map from the

base space to the covering sphere and then use the map to pass to the cover and compute

the correlator.

We wrote down the map from the base space to the covering surface for two insertions

of twist-n operators in Chapter 3. Here we have n = 2 and the map is of the form

z = bt2

2t− 1. (4.65)

In the vicinity of the two insertion points we have:

z → b, t→ 1, (z − b) ≈ b1(t− 1)2 + b′1(t− 1)3 +O((t− 1)4),

z → 0, t→ 0, z ≈ b0 t2 + b′0 t

3 +O(t4), (4.66)

where b1 = b and b0 = −b. A generic point in the base space ak has two images on the

covering surface which we refer to them as t±k and are given by

t±k =1

b

(ak ±

√ak(ak − b)

). (4.67)

In the vicinity of a generic point the map is of the form:

(z − ak) ≈ ξ±k (t− t±k) + ξ′±k (t− t±k)2 +O((t− t±k)3), (4.68)


where

ξ±k =

(dz

dt

) ∣∣∣∣t=t±k

=2 ak (ak − b)b t±k (t±k − 1)

. (4.69)

There are two different contributions to the four-point function (4.64): a bare-twist

part and a mode-insertion part. The bare-twist part contains the two insertions of σ02

twists, and is computed using the Liouville action. Using the Lunin-Mathur method, the

normalized two-point function is

〈σ02(b, b)σ0

2(0, 0)〉〈σ0

2(1, 1)σ02(0, 0)〉

= |b|−4hσ02 , (4.70)

where hσ02

= 3/8.

We next evaluate the mode insertion part of the correlator. In section 4.2.3 we

evaluated the complete deformation operator on the cover (4.38). We now lift the dilaton

operator explicitly to the covering surface. We have

αAA−1 (κ) αBB−1 (κ) =

∮ak

dz

2πi

∮ak

dz

2πi(z − ak)1−1−1 (z − ak)1−1−1 ∂XAA

(κ) (z) ∂XBB(κ) (z)

→∮t±k

dt

2πi

(dz

dt

)1−1

(z − ak)−1

∮t±k

dt

2πi

(dz

dt

)1−1

(z − ak)−1∂XAA(t) ∂XBB(t)

=

∮t±k

dt

2πi

∂XAA(t)[ξ±k(t− t±k)

(1 +O(t− t±k)

)] ∮t±k

dt

2πi

∂XBB(t)[ξ±k(t− t±k)

(1 +O(t− t±k)

)]=

1

|ξ±k|2∂XAA(t±k) ∂X

BB(t±k), (4.71)

where the arrow in the second line implies that we passed from the base to the cover

using the local map (4.68). The above computation has a relatively simple form because

the operator ∂X∂X does not need to be regulated, and transforms like a tensor under

conformal transformations.


The dilaton operator is then given by

φtdil(t±k, t±k) ≡−1

|ξ±k|2εAB εAB ∂X

AA(t±k) ∂XBB(t±k), (4.72)

where the superscript t denotes the fact that this is an operator in the covering surface.

Following the methodology presented in Chapter 3, the mode insertion part of the four-

point function is obtained by summing over the images of the non-twist insertions

∑t±1

∑t±2

⟨φtdil(t±1, t±1) λOtd(1, 1) λOtd(0, 0) φtdil(t±2, t±2)

⟩, (4.73)

where t±1 and t±2 are the images of a1 and a2, respectively, under the map (4.65). We

first evaluate the four-point function in the above equation and then sum over the images.

Using the dilaton operator (4.72), the deformation (4.38), the map coefficients (4.69),

and taking into account the bare twist contribution (4.70), we find

|b|−32


⟩=

λ2

4|a1|−2 |a2|−2 |a1 − b|−2 |a2 − b|−2

8|t±1|2 |t±1 − 1|2 |t±2|2| t±2 − 1|2

|t±1 − t±2|4

+2(t±1) (t±2 − 1) |t±1 − 1|2 |t±2|2

(t±1) (t±2 − 1) (t±1 − t±2)2+ 2

(t±1 − 1) (t±2) |t±1|2 |t±2 − 1|2

(t±1 − 1) (t±2) (t±1 − t±2)2

+2(t±1) (t±2 − 1) |t±1 − 1|2 |t±2|2

(t±1) (t±2 − 1) (t±1 − t±2)2+ 2

(t±1 − 1) (t±2) |t±1|2 |t±2 − 1|2

(t±1 − 1) (t±2) (t±1 − t±2)2

+2|t±1|2 |t±2 − 1|2

|t±1 − 1|2 |t±2|2+ 2|t±1 − 1|2 |t±2|2

|t±1|2 |t±2 − 1|2

−(t±1) (t±2 − 1) (t±1 − 1) (t±2)

(t±1 − 1) (t±2) (t±1) (t±2 − 1)− (t±1 − 1) (t±2) (t±1) (t±2 − 1)

(t±1) (t±2 − 1) (t±1 − 1) (t±2)

. (4.74)

We normalize the four point function by two-point functions as in Chapter 3:


⟩⟨φtdil(0, 0) φtdil(1, 1)

⟩ ⟨Otd(0, 0) Otd(1, 1)

⟩ . (4.75)


(The normalization of the bare twist contribution has been accounted for in (4.70)). The

two-point functions are found to be

⟨Otd(0, 0) Otd(1, 1)

⟩= 8, (4.76)

⟨φtdil(0, 0) φtdil(1, 1)

⟩= 4. (4.77)

In the above, and in the following, note that there are combinatoric normalization

factors coming from summing over conjugacy classes of 2-cycles. This gives the 3-point

function an overall factor of 1/√N and the 4-point function an overall factor of 1/N .

We have chosen to suppress these combinatoric factors here for notational clarity: it is

always clear where to put them back in afterwards. We use the two-point function as a

guide for how to normalize; for further details we refer the reader to the second example

in the previous chapter. After the dust settles, these N -dependent combinatoric factors

turn out not to alter the mixing coefficients, and so we may safely ignore them until we

evaluate anomalous dimensions.

4.3.3 Summing over images

We sum over the images of the insertion points of the non-twist operators on the covering

surface to compute the complete correlation function

∑t±1

∑t±2


⟩=⟨(

φtdil(t+1, t+1) + φtdil(t−1, t−1))λOtd(1, 1)×

λOtd(0, 0)(φtdil(t+2, t+2) + φtdil(t−2, t−2)

)⟩. (4.78)


Inserting the four-point function evaluated in the previous section (4.74), the complete

normalized four-point function reads

⟨ 2∑κ=1


κ′=1

φdil (κ′)(a2, a2)⟩

=

λ2

2

2

|a1 − a2|4 |b|4+

+2−1 (a1 − b)12 a

122

a121 (a2 − b)

12

(1 + a1 (a2−b)

(a1−b) a2

)(a1 − a2)2 b2

(a1 − b)12 a

122

a121 (a2 − b)

12

(1 + a1 (a2−b)

(a1−b) a2

)(a1 − a2)2 b2

+

+2−3

(1 + a1 (a2−b)

(a1−b) a2

)a

321 (a1 − b)

12 a

122 (a2 − b)

32

(a1 − b)12 a

122

a121 (a2 − b)

12

(1 + a1 (a2−b)

(a1−b) a2

)(a1 − a2)2 b2

+

+2−3 (a1 − b)12 a

122

a121 (a2 − b)

12

(1 + a1 (a2−b)

(a1−b) a2

)(a1 − a2)2 b2

(1 + a1 (a2−b)

(a1−b) a2

)a

321 (a1 − b)

12 a

122 (a2 − b)

32

+

+2−3 a− 1

21 (a1 − b)−

32 a− 3

22 (a2 − b)−

12 a

− 12

1 (a1 − b)−32 a− 3

22 (a2 − b)−

12 +

+2−3 a− 3

21 (a1 − b)−

12 a− 1

22 (a2 − b)−

32 a

− 32

1 (a1 − b)−12 a− 1

22 (a2 − b)−

32 +

−2−4 a− 1

21 (a1 − b)−

32 a− 3

22 (a2 − b)−

12 a

− 32

1 (a1 − b)−12 a− 1

22 (a2 − b)−

32 +

−2−4 a− 3

21 (a1 − b)−

12 a− 1

22 (a2 − b)−

32 a

− 12

1 (a1 − b)−32 a− 3

22 (a2 − b)−

12

. (4.79)

One may take this result and express it in terms of cross ratios, showing that it does in

fact fit the correct form for a four-point function of quasi-primary fields.

4.3.4 Lack of operator mixing

Having constructed the four-point correlation function, we can now take the coincidence

limit (a1, a1) → (0, 0) and (a2, a2) → (b, b), and consider the leading singularity. The

OPE of two quasi-primary operators O1 and O2 has the form [233]

O1(z, z)O2(0, 0) =∑p

∑k,k

Cp k,k12 zhp−h1−h2+K zhp−h1−h2+KOk,kp (0, 0), (4.80)


where Op is a quasi primary, k, k denotes a collection of indices ki and ki which corre-

spond to the descendant states, and K ≡∑

i ki, K ≡∑

i ki. The index p accounts for all

conformal families which participate in the OPE. Taking the coincidence limit, the holo-

morphic part of the four-point function scales as a−3/21 (a2 − b)−3/2. According to (4.80),

h1 = h2 = 1 in our case and the quasi-primary Op has conformal weight h = 1/2. All

the descendants of this conformal family have half-integer weights. Subleading singulari-

ties also have half-integer conformal weights. Singularities corresponding to mixing with

h = h = 1 quasi-primary operators are absent in the coincidence limits of the four-point

function. Thus CiAj = 0, where j corresponds to any weight (1,1) quasi-primary. As

discussed earlier, the structure constant CiAi also vanishes because of a group selection

rule since it corresponds to the insertion of only one twist-2 operator in the base. The

fact that CiAi = 0 and CiAj = 0 indicates that the dilaton operator does not acquire an

anomalous dimension at the first order in perturbation theory, as expected.

4.4 Lifting of a string state

In this section we consider a string state of the superconformal algebra which is not

protected against corrections as one deforms the theory away from the orbifold point.

We study operator mixing at the first order and analyze lifting of the string state. The

non-twist sector has low-lying string states which are lifted under deformation. Here we

address the interaction between the twist and the non-twist sector. The string state that

we consider has the general form

∂Xam (κ1) ∂X

bn (κ2) ∂X

ck (κ3) ∂X

dl (κ4) |0〉R, (4.81)

where κi corresponds to the copy of the target space and |0〉R is a Ramond-Ramond

ground state. We make a simple choice and set the modes n = m = k = l = −1, and

choose the excitations to belong to the same copy of the target space. Our string state


is given by

N∑κ=1

φst (κ) |0〉R ≡ δab δcd

N∑κ=1

∂Xa−1 (κ) ∂X

b−1 (κ) ∂X

c−1 (κ) ∂X

d−1 (κ) |0〉R. (4.82)

The state is a singlet under the internal SU(2)1 × SU(2)2.

Physical states of the D1-D5 system are in the Ramond sector where the fermions have

periodic boundary conditions around the circle S1. We use spectral flow transformation

[184] to relate the correlation functions in the Ramond sector to correlation functions in

the Neveu-Schwarz (NS) sector. The four-point correlation function that we would like

to evaluate is

R〈0|N∑κ=1

φst (κ) λOd(z, z) λOd(z′, z′)N∑κ′=1

φst (κ′)|0〉R. (4.83)

Let us choose the Ramond-Ramond ground state which has conformal weight (h, h) =

(1/4, 1/4) and R-charge (1/2, 1/2) under SU(2)L × SU(2)R. Under performing a spec-

tral flow transformation with parameter α = −1, this Ramond ground state is mapped

into the NS ground state. The bosons are not affected under the spectral flow. The

deformation operator is also mapped into itself as we will now show.

Operators of the CFT which could be written as pure exponentials in the context of

the bosonized language transform as

φ(z)→ z−αm φ(z), (4.84)

under the spectral flow with spectral flow parameter α [168, 185]. Here m is the charge

under SU(2)L. The same transformation holds for anti-holomorphic operators. Super

chiral and anti-chiral primary operators are represented by pure exponentials in the

bosonized language [172]. The deformation operator contains the super chiral primary

σ++2 with R-charge (1/2, 1/2) under SU(2)L×SU(2)R. The holomorphic part of σ++

2 thus

transforms as σ+2 (z)→ zα/2 σ+

2 (z) under the spectral flow. The supercurrents transform


as G−A(z)→ z−α/2G−A(z). We then have

G−A− 12

(z′)σ+2 (z) =

∮z

dz′

2πiG−A(z′)σ+

2 (z)→∮z

dz′

2πiz′−

α2 G−A(z′) z

α2 σ+

2 (z)

=

∮z

dz′

2πi

(1− α

2(z′ − z)z−1 + · · ·

)G−A(z′)σ+

2 (z)

= G−A− 12

(z′)σ+2 (z)− α

2z−1G−A

+ 12

(z′)σ+2 (z) + · · · , (4.85)

where “ · · · ” denotes higher positive modes of the supercurrent. Since positive modes of

the supercurrent annihilate super chiral primaries, only the first term in the third line of

the above equation is non-vanishing and the deformation operator is not affected under

the spectral flow.

Under the spectral flow with α = −1 the physical problem in the Ramond sector is

mapped into a computation in the NS sector. The four-point correlation function that

we evaluate is

NS〈0|N∑κ=1

φst (κ) λOd(z, z) λOd(z′, z′)N∑κ′=1

φst (κ′)|0〉NS. (4.86)

As discussed in subsection (4.3.1), four-point functions which factorize are of no interest

for our purpose because they have no information about operator mixing. Let us de-

note the two copy indices of the target space which are twisted under the deformation

operator as κ = 1, 2. Then the four-point function in which we are interested contains∑2κ=1 φst (κ′). We will next explicitly compute φst on the covering surface and then eval-

uate the correlation function.

4.4.1 Passing to the covering surface

The four-point correlation function (4.86) has two insertions of twist-2 operators in the

base space. The map from the base to the cover is thus the same as in the previous section

(4.65). We set the insertion points of the two deformations at z = 0 and z = b and the


insertion points of φst at z = a1 and z = a2, as before. The bare-twist contribution to the

correlation function is again the normalized two-point function of twist-2 operators (4.70)

and is given by |b|−3/2. To evaluate the mode insertion contribution to the four-point

function, we lift the operators to the covering sphere.

The operator φst = ∂Xa−1 (κ) ∂Xa,−1 (κ) ∂X

b−1 (κ) ∂Xb,−1 (κ)(z, z) is a quasi-primary op-

erator with conformal weight (2, 2). Passing to the covering surface, we obtain

∂Xa−1 (κ) ∂X

b−1 (κ) =

∮ak

dz1

2πi(z1 − ak)1−1−1 ∂Xa

(κ)(z1)

∮ak

dz2

2πi(z2 − ak)1−1−1 ∂Xb

(κ)(z2)

→∮t±k

dt12πi

(dz1

dt1

)1−1

(z1 − ak)−1 ∂Xa(t1)

∮t±k

dt22πi

(dz2

dt2

)1−1

(z2 − ak)−1 ∂Xb(t2)

=

∮t±k

dt12πi

∂Xa(t1)

ξ±k(t1 − t±k)(

1 +O(t1 − t±k)) ∮

t±k

dt22πi

∂Xb(t2)

ξ±k(t2 − t±k)(

1 +O(t2 − t±k))

=

∮t±k

dt12πi

∂Xa(t1)

ξ±k(t1 − t±k)(

1 +O(t1 − t±k)) 1

ξ±k∂Xb(t±k)

=

∮t±k

dt12πi

1

ξ2±k

[1 + η1(t1 − t±k) + η2(t1 − t±k)2 +O

((t1 − t±k)3

)]t1 − t±k

∂Xa(t1) ∂Xb(t±k)

=1

ξ2±k

(∂Xa ∂Xb(t±k)−

δab

4(t±k)2(t±k − 1)2

), (4.87)

where the arrow in the second line denotes that we passed from the base to the cover

using the map which locally looks like (4.68), and η1 and η2 are coefficients obtained

from expanding the map. The form of this operator may again be expected because

: ∂X∂X : needs to be regulated, and this regulation causes this object to not transform as

a tensor. However, the transformation properties of this object for finite transformations

can be easily determined, and one recognizes the the Schwarzian derivative f(t), t =

−3(t− 1)−2 t−2/2 of the map (4.65) appearing in (4.87). The same relation is obtained

for the anti-holomorphic part of the operator. The operator has the two-point function

〈φst(0, 0)φst(1, 1)〉 = 28, which is used to normalize the four-point function. Using the

above equation (4.87), the deformation (4.38), and taking into account the bare twist


contribution, the four-point function normalized by the two-point functions is evaluated

|b|−32

⟨φtst(t±1, t±1) λOtd(1, 1) λOtd(0, 0) φtst(t±2, t±2)

⟩= (4.88)

|b|4

216λ2 |a1|−4 |a2|−4 |a1 − b|−4 |a2 − b|−4 ×

5 + 4(t±1 − 1)2(t±2)2

(t±1 − t±2)2+ 4

(t±1)2(t±2 − 1)2

(t±1 − t±2)2+ 8

(t±1)2(t±1 − 1)2(t±2)2(t±2 − 1)2

(t±1 − t±2)4

×

5 + 4(t±1 − 1)2(t±2)2

(t±1 − t±2)2+ 4

(t±1)2(t±2 − 1)2

(t±1 − t±2)2+ 8

(t±1)2(t±1 − 1)2(t±2)2(t±2 − 1)2

(t±1 − t±2)4

.

4.4.2 Image sums

We finally sum over the images of the insertion points of the two non-twist operators and

compute the complete normalized four-point function. To make the notation compact,

we write down the result in terms of the cross ratio, R = a1(a2 − b)/(a2(a1 − b)). We

obtain

⟨ 2∑κ=1

φst (κ)(a1, a1) λOd(b, b) λOd(0, 0)2∑

κ′=1

φst (κ′)(a2, a2)⟩

=

|b|4

212λ2 |a1|−4 |a2|−4 |a1 − b|−4 |a2 − b|−4

25

4+ 5

((R + 1)2

(R− 1)2+

(R + 1)2

(R− 1)2

)+16

R12 (R + 1)

(R− 1)2

R12 (R + 1)

(R− 1)2+ 4

(R + 1)2

(R− 1)2

(R + 1)2

(R− 1)2

+10

(R (R2 + 6R + 1)

(R− 1)4+R (R2 + 6R + 1)

(R− 1)4

)+64

(R

12 (R + 1)

(R− 1)2

R32 (R + 1)

(R− 1)4+R

32 (R + 1)

(R− 1)4

R12 (R + 1)

(R− 1)2

)+8

((R + 1)2

(R− 1)2

R (R2 + 6 R + 1)

(R− 1)4+R (R2 + 6R + 1)

(R− 1)4

(R + 1)2

(R− 1)2

)+256

R32 (R + 1)

(R− 1)4

R32 (R + 1)

(R− 1)4+ 16

R (R2 + 6R + 1)

(R− 1)4

R (R2 + 6 R + 1)

(R− 1)4

. (4.89)

Note that this is the correct form for the 4-point function for weights (2,2), (1,1), (1,1),

(2,2).


4.4.3 Coincidence limit and operator mixing

We investigate the operator mixing by taking the coincidence limit (a1, a1) → (0, 0)

and (a2, a2) → (b, b). Similar to the case of the dilaton in section (4.3.4), there are

singularities which correspond to mixing with quasi-primary operators with half-integer

conformal weights. All the descendants of these fields also have half-integer weights. This

type of mixing contributes to the wave function renormalization (4.57). It does not affect

the anomalous dimension of the string state which has conformal weight (2, 2). Under the

above coincidence limit, the singular part of the four-point function which corresponds

to mixing with integer weight operators is of the form

2−12

81

4

1

a21 (a2 − b)2 a2

1 (a2 − b)2 |b|4+

81

2

(a1 b− (a2 − b) b+ a1 b− (a2 − b) b

)a2

1 (a2 − b)2 a21 (a2 − b)2 |b|6

+27

a21 (a2 − b)2 a2

1 (a2 − b)2 |b|8(− 5 a1 (a2 − b) b2 − 5 a1 (a2 − b) b2

+ 3 a1 a1 |b|2 − 3 a1 (a2 − b) |b|2 − 3 (a2 − b) a1 |b|2 + 3 (a2 − b) (a2 − b) |b|2)

+54

a21 (a2 − b)2 a2

1 (a2 − b)2 |b|10

(5 a1 (a2 − b) (a2 − b) b b2 − 5 a1 (a2 − b) a1 b b

2

− 5 a1 a1 (a2 − b) b2 b+ 5 (a2 − b) a1 (a2 − b) b2 b)

+900

a1 (a2 − b) a1 (a2 − b) |b|8+ · · ·

, (4.90)

where “ · · · ” corresponds to subleading singularities. We can now move on to subtracting

the relevant conformal family.

4.4.4 Conformal family subtraction and mixing coefficients

Let us consider the leading singular term in the above expansion with the coefficient 81/4.

According to (4.80), this term corresponds to operator mixing with a quasi-primary, or a

linear combination of quasi-primaries, with conformal weight (1, 1). Charge conservation

requires this operator to be a singlet under the R-symmetry SU(2)L × SU(2)R and the


internal symmetry SU(2)1 × SU(2)2. We show that the weight (1,1) operator is indeed

the deformation operator (4.38). To see this, we evaluate the three-point function

limy,y→∞

y2hd y2hd⟨ 2∑κ=1

φst (κ)(a1, a1) λOd(0, 0) λOd(y, y)⟩

=CiAA

ahst+hd−hd1 ahst+hd−hd

1

(4.91)

where the indices i and A in CiAA correspond to φst and the deformation, respectively.

(hst, hst) and (hd, hd) are the conformal weights of φst and the deformation, respectively.

The above three-point function, normalized by two-point functions, is

CiAA

ahst+hd−hd1 ahst+hd−hd

1

=18

28

1

|a1|4, (4.92)

which then gives CiAA = 9 × 2−7. Since we have normalized the operators by their

two-point functions, the coefficient of the three-point function and the coefficient of the

most singular term in the OPE (4.80) are equal: CiAA = CA 0,0iA . Denoting the complete

deformation operator as OA, we find the operator algebra

2∑κ=1

φst (κ)(a1, a1)OA(0, 0) =∑k,k

CA k,kiA a−2+K

1 a−2+K1 O

k,kA (0, 0), (4.93)

where the deformation operator is the ancestor of this conformal family. The operator

algebra for the other two operators∑2

κ=1 φst (κ)(a2, a2) andOA(b, b) is obtained in a similar

way. Inserting the two OPEs in the four-point function (4.89) we obtain the two-point

function

⟨( ∑k′,k′

C′A k′,k′iA (a2 − b)−2+K′ (a2 − b)−2+K′O

k′,k′A

)(b, b)×

(∑k,k

CA k,kiA a−2+K

1 a−2+K1 O

k,kA

)(0, 0)

⟩. (4.94)


Let us first consider the most singular terms of the two OPEs which corresponds to the

ancestor field with k, k = k′, k′ = 0, 0. The above two-point function then reads

⟨(C′A 0,0iA

(a2 − b)2 (a2 − b)2O0,0A

)(b, b)

(CA 0,0iA

a21 a

21

O0,0A

)(0, 0)

⟩=

1

212

81

4

1

|a1|4 |a2 − b|4 |b|4. (4.95)

This gives the leading singular term that we found in the expansion (4.90) with exactly the

same coefficient. Therefore, the deformation operator accounts for the leading singularity

of the four-point function.

We will next have to compute the contribution of the descendant operators in (4.94)

and subtract them from the subleading singular terms in the expansion (4.90). We

will only need to find the contribution of the descendants O1,0A , O

0,1A , and O

1,1A ,

since higher descendants have conformal weights larger that (2, 2) and play no role in

determining the anomalous dimension of the string state. Computation of the structure

constants of the descendant fields is explained in [233]. In general, in the operator algebra

of two quasi-primary fields O1 and O2 with conformal weights h1 and h2,

O1(z, z)O2(0, 0) =∑p

∑k,k

Cp k,k12 zhp−h1−h2+K zhp−h1−h2+KOk,kp (0, 0),

the structure constants of the descendants Cp k,k12 are determined by the structure con-

stant of the ancestor field through the relation

Cp k,k12 = C

p 0,012 β

p k12 β

p k12 , (4.96)

where βk12 and β

k12 are coefficients which depend only on the central charge and the


conformal weights. For the descendant states at level 1 the coefficients are given by

βp112 =

hp + h1 − h2

2hp, β

p 112 =

hp + h1 − h2

2 hp. (4.97)

In our case, hp = 1, h1 = 2, and h2 = 1 and we have:

βA1iA = 1, β

A 1iA = 1. (4.98)

The structure constants then read

CA 1,1iA = C

A 1,0iA = C

A 0,1iA = C

A 0,0iA =

9

27. (4.99)

We can now calculate the contribution of the descendant fieldsOk,kA (0, 0) andO

k′,k′A (b, b)

in (4.94). As mentioned earlier, for each OPE, we are interested in the three level-1 de-

scendants (O1,0A , O

0,1A , and O

1,1A ). Therefore there are nine terms to evaluate. For

example, for k, k = 1, 0 and k′, k′ = 0, 0, we have K = 1, K = K ′ = K ′ = 0,

and obtain

⟨(C′A 0,0iA

(a2 − b)2 (a1 − b)2O0,0A

)(b, b)

(CA 1,0iA

a1 a21

O1,0A

)(0, 0)

⟩=

1

212

81

4

1

a1 (a2 − b)2

1

a21 (a2 − b)2

L−1 〈O0,0A (b, b) O0,0A (0, 0)〉 =

1

212

81

4

1

a1 (a2 − b)2

1

a21 (a2 − b)2

∂z〈O0,0A (b, b) O0,0A (z, z)〉

∣∣∣z→0

=

1

212

81

2

a1 b

a21 (a2 − b)2 a2

1 (a2 − b)2 |b|6. (4.100)

We perform similar computations for the eight remaining terms and subtract them from

the corresponding singularities in the expansion (4.90). The remaining singular terms

are


2−12

27

a21 (a2 − b)2 a2

1 (a2 − b)2 |b|8(− 1

2a1 (a2 − b) b2 − 1

2a1 (a2 − b) b2

)+

27

a21 (a2 − b)2 a2

1 (a2 − b)2 |b|10

(a1 (a2 − b) (a2 − b) b b2 − a1 (a2 − b) a1 b b

2

− a1 a1 (a2 − b) b2 b+ (a2 − b) a1 (a2 − b) b2 b)

+171

a1 (a2 − b) a1 (a2 − b) |b|8+ · · ·

. (4.101)

The first line of the above equation has two terms in it

− 27

2

1

a1 (a2 − b) a21 (a2 − b)2 b4 b2

, (4.102)

−27

2

1

a21 (a2 − b)2 a1 (a2 − b) b2 b4

. (4.103)

The first term (4.102) shows that there is operator mixing with a quasi-primary or a

linear combination of quasi-primaries with conformal weight (2, 1). These fields are the

ancestors of their conformal families. The second term (4.103) signals mixing with quasi-

primaries with conformal weight (1, 2). We can use the conformal algebra of the theory

and construct quasi-primary operators with the required conformal weights which con-

tribute to these mixings. In our case, however, we do not need to identify explicitly all

the (2, 1) or (1, 2) operators which mix with φst. We can indeed evaluate the structure

constants of all the descendant fields using (4.96). Thus the only information needed

are the coefficients βp k12 and β

p k12 . We are only interested in the level-1 descendants

which have conformal weights (2, 2). The values of βp 112 and β

p 112 are therefore the same

as above (4.98): βp 112 = β

p 112 = 1. Following similar computations as in the previous

case we compute the contributions of the descendant fields and subtract them from the


corresponding singular terms in (4.101). The remaining singularity is of the form

2−12 9

a1 (a2 − b) a1 (a2 − b) |b|8+ · · · . (4.104)

This shows that there is a quasi-primary or a linear combination of quasi-primary oper-

ators which have conformal dimension (2, 2) and mix with our candidate operator φst at

the first order in perturbation theory. Since these quasi-primaries have the same weight

as φst, they contribute to the anomalous dimension of the candidate operator at the first

order. Operators with different weights which mix with φst will contribute to the wave

function renormalization.

We can see the above results in an alternative way, using a trick that is available

for this case. In the coincidence limit we considered above, we set a1, a1 → 0 and

a2, a2 → b, b. We can equally well send the deformation operator OA to the vicinity of

φst. Let us assume that the two deformations are at positions z = z1 and z = z2 in

the base space. We take the coincidence limit (z1, z1) → (a1, a1), (z2, z2) → (a2, a2) and

then set z1 = 0, z2 = b. Under this coincidence limit, the singular part of the four-point

correlation function (4.89) is of the form

2−12

81

4

1

a21 (a2 − b)2 a2

1 (a2 − b)2 |a1 − a2|4

− 27

2

(a1 (a2 − b) (a1 − a2)2 + a1 (a2 − b) (a1 − a2)2

)a2

1 (a2 − b)2 a21 (a2 − b)2 |a1 − a2|8

+9

a1 (a2 − b) a1 (a2 − b) |a1 − a2|8+ · · ·

. (4.105)

The first line shows operator mixing with a (1, 1) quasi-primary. As was shown in (4.95),

this operator is the deformation operator, which accounts for the leading singularity. We

note that in the present coincidence limit we have h1 = 1 and h2 = 2 in the opera-

tor algebra (4.80). Therefore, the coefficients βp 112 and β

p 112 in (4.97) for the level-1


descendants of the deformation operator are now

βA1iA = 0, β

A 1iA = 0. (4.106)

Hence, the structure constants of these descendants vanish and only the ancestor of the

family contributes to the expansion. The second line of equation (4.105) then tells us

that there are quasi-primaries with weight (2, 1) and (1, 2) which contribute to operator

mixing. The coefficient of this singularity agrees with that obtained in (4.101). Again,

the structure constants of the descendants of these operators vanish and there will be no

contribution from these descendants to the expansion. Finally, the last line in (4.105)

implies that there is at least one (2, 2) quasi-primary operator which mixes with φst at the

first order. The coefficient agrees with (4.104). While this shortcut is available for this

computation, this is not always the case. The earlier procedure is generically applicable.

In order to evaluate the anomalous dimension to the first order we need to determine

all (2, 2) quasi-primary operators that contribute to the operator mixing in (4.104). We

then need to determine all the (2, 2) quasi-primaries that mix with these operators. We

have to continue this search until we find all such (2, 2) operators. We will then be able

to diagonalize the matrix of the structure constants and find the anomalous dimension

of our candidate string state.

4.5 Summary and outlook

In this chapter and the previous chapter, we began an investigation into how the anoma-

lous dimensions of low-lying string states in the D1-D5 SCFT are lifted as we perturb

away from the orbifold point where string calculations are easiest to do. We found evi-

dence of operator mixing at first order, which means that as we increase the deformation

parameter the anomalous dimensions of some of the string states will head downwards

while others will head upwards.


Our method starts with evaluating four-point functions involving the operator of

interest, the deformation operator, and their Hermitian conjugates. The two deformation

operator insertions are of course required because we want to perturb away from the

orbifold point (towards the gravity limit). Using four-point functions may seem a tad

roundabout, but it is actually more efficient than starting with three-point functions.

The reason is that we can use factorization channels of four-point functions to identify

which intermediate quasi-primary operators participate in mixing. This cuts down on

the number of independent three-point functions we need to calculate. Once we know the

conformal weight of such an intermediate quasi-primary, we can subtract its conformal

family from the leading-order singular limit of the four-point function. Such a family

contains an ancestor quasi-primary operator and all its descendants under the Virasoro

algebra. After subtraction, some leading-order singularities in coincidence limits will

typically remain. We will therefore continue iterating the procedure until we exhaust all

the intermediate conformal families which mix with our original operator. Diagonalizing

the resulting matrix of structure constants will eventually yield the anomalous dimensions

that we wanted in the first place, at leading-order in perturbation theory.

In general, solving for string state anomalous dimensions is an extremely hard prob-

lem. We need the iteration procedure outlined above to truncate, in order to be able

to find the anomalous dimensions for the string states we are after. Diagonalizing an

infinite-dimensional matrix of structure constants, after all, would be practically impos-

sible. The key observation is that for low-lying string states the number of intermediate

quasi-primaries which can mix with them should be finite. This is why we have fo-

cused on a low-lying string state which is about as simple as possible (but no simpler):

∂X∂X∂X∂X.

In order to be able to compute the four-point and three-point functions we needed, we

had to further develop Lunin-Mathur symmetric orbifold technology. In particular, we

needed to know how lifting to the covering space worked, including all the details of twist


and non-twist operators, fractional modings, ramification points, images, bosonization of

fermions, and so forth. We illustrated our developments of Lunin-Mathur symmetric

orbifold technology in Chapter 3 with a simple example and with a more complicated

one involving excitations, fermions, and currents. In section 4.2.2 of this work, we also

found a suitable representation of cocycles transforming correctly under SU(2)L×SU(2)R

R-symmetry and the internal SU(2)1×SU(2)2, which involved features that we had not

seen elsewhere.

We have several items on our remaining to-do list. Of highest priority is to enumerate

all the operators that mix with ∂X∂X∂X∂X – and the operators that they mix with

in turn. Once we have that complete list of operators, we can subtract their conformal

families, iterating our procedure until all of the mixing coefficients are nailed down.

This will permit us to diagonalize the matrix of structure constants and find the desired

anomalous dimensions. We are also investigating other choices of low-lying string states

coming from both the non-twist and the twist sector. Our other immediate goal is

to connect with the work [163] of Gava and Narain, who also investigated anomalous

dimensions of particular low-lying string states, in the context of proving pp-wave/CFT2

duality. The string states they considered are right-chiral; the left sector has excitations

with fractional modes of conserved currents. They analyzed the anomalous dimension

for a class of states by calculating three-point functions of the CFT. They found that

it is proportional to (k/n)2, where n is the twist order and k gives the fractional mode

number. In order to compute the anomalous dimension of more general string states

with excitations on both the left and right sides, it is necessary to compute full four-

point functions of the CFT. We plan to investigate this directly by using the methods of

this chapter.

Chapter 5

Conclusions

In this dissertation we have investigated the AdS/CFT correspondence in two differ-

ent contexts. In the first part of the thesis we presented research on the holographic

AdS/CMT correspondence and studied thermodynamic properties of a class of strongly

coupled gauge field theories with non-relativistic symmetry. In the second part of the

thesis we presented research on the AdS3/CFT2 correspondence in the context of the

D1-D5 brane system in type IIB string theory. We focused on the microscopic CFT per-

spective of the fuzzball programme and investigated the difference between string states

and supergravity states.

In the first part of the thesis, Chapter 2, we constructed a particular class of candidate

holographic gravity duals to quantum field theories with Lifshitz scaling symmetry in gen-

eral d-dimensions. In the setup considered in our studies we provided a UV completion of

the Lifshitz black branes by embedding these geometries in an asymptotically AdS space.

Our constructed Lifshitz models may be relevant to describe quantum critical behaviour

of some condensed matter systems including high temperature superconductors.

In the context of the phenomenologically motivated holographic models, it is impor-

tant to investigate how the holographic gravity duals can be properly embedded in micro-

scopic string theory/M-theory. It has been shown in [234, 235, 236] that Lifshitz space-

130

Chapter 5. Conclusions 131

times contain singularities. Lifshitz dual models embedded in ten or eleven-dimensional

supergravity also contain singularities and quantum corrections become important at the

deep interior of the spacetime. Therefore, our Lifshitz model needs to be completed by

considering string corrections in the interior region of the Lifshitz spacetime.

The second part of the thesis comprises Chapters 3 and 4. In this part we con-

sidered the AdS3/CFT2 correspondence in the context of the fuzzball programme and

investigated the deformation of the CFT description of the D1-D5 system away from

the orbifold point. We used the generalization of the Lunin-Mathur covering space tech-

nique and conformal perturbation theory and developed a programme to evaluate the

anomalous dimensions of string states in the D1-D5 CFT.

In Chapter 3 we generalized the Lunin-Mathur covering space technique to the non-

twist sector and showed how to evaluate correlation functions which involve both twist

and non-twist sector operators. We also showed how to compute correlation functions

containing twist sector operators which are excited by operator modes orthogonal to the

twist directions. The procedure is achieved by determining appropriate maps from the

base space to the covering surface and summing over all the images of the twist insertions

in the cover. We worked out two examples and computed correlation functions of SN -

invariant operators in the (1 + 1)-dimensional bosonic symmetric orbifold CFT and in

the (1+1)-dimensional N = (4, 4) orbifold CFT of the D1-D5 system.

In Chapter 4 we performed conformal perturbation theory to the first order and de-

formed the symmetric product orbifold CFT of the D1-D5 system away from the orbifold

point towards the point in the moduli space which corresponds to black hole physics.

This was done by acting with an exactly marginal deformation operator belonging to the

twist-2 sector of the CFT. We studied lifting of a low-lying string state of the CFT under

the deformation. The string state that we considered belongs to the non-twist sector of

the CFT.

We used the generalized Lunin-Mathur method developed in Chapter 3 and computed


four-point functions with two insertions of the string operator of interest and two inser-

tions of the marginal deformation operator. We then took the coincidence limit as one of

the deformation operators approaches a string operator and the other string operator co-

incides with the second deformation operator. We analyzed the factorization channels of

the four-point function and showed how to account for the contributions from conformal

families of descendant operators and subtract them from the leading coincidence limit.

We found that at first-order in the deformation operator the low-lying string operator

of interest mixes with quasi-primary operators which have the same conformal weight as

that of the string operator. This yields the lifting of the anomalous dimension of the

string state. We computed the precise coefficient of mixing at first-order in perturbation

theory.

The orbifold CFT technology that we have developed enables us to examine quanti-

tatively the difference between string states and supergravity states. Enigmatic phases of

the D1-D5 CFT at the orbifold point were recently discovered [237, 238] which have larger

entropies than the previously known phases. The dual bulk configurations corresponding

to these CFT ensembles were studied and it was found that they are entropically dom-

inant over the BMPV black hole [15]. The interesting observation is that the entropy

of the CFT configuration is larger than the entropy of the bulk configuration. Since the

CFT and bulk phases correspond to different points in the moduli space, this observation

implies that some CFT states lift as one moves away from the orbifold point towards the

point with black hole description. It will be interesting to explore further lifting of the

CFT states of these novel phases in the context of the conformal deformation theory.

The non-renormalization theorem for general three-point correlation functions of half-

chiral primary states of the D1-D5 symmetric product CFT was recently proved [179].

This generalization relies on the assumption that the half-chiral states do not get lifted

under the deformation away from the orbifold point. It will be interesting to explore this

further by applying the deformation theory to the symmetric product CFT and investi-


gating the possibility of lifting of the half-chiral states and its potential contributions to

the three-point functions.

It is of great interest to investigate physical processes of the infall of high energy

quanta into a black hole within the context of the microscopic CFT perspective of the

fuzzball programme. The infalling quanta result in left and right excitations of the CFT

states. To study the evolution of this system one then applies the deformation operator

and deforms the CFT away from the orbifold point toward the point with a gravity dual

in the moduli space and investigate how the energy of the initial excited quanta is spread

over a larger number of quanta as the system evolves.

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