Ads/CFT, Black Holes, and Fuzzballs by Ida G. Zadeh a Thesis Submitted

Home , Extremal black hole, Fuzzball (string theory), Gravitational collapse, Holographic principle

AdS/CFT, Black Holes, And Fuzzballs

Ida G. Zadeh

A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Physics University of Toronto

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 programme, 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

N of the form M /SN , with M 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 excitations 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, subtracting 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

N 3 Twist-nontwist correlators in M /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 (4 Gd+2 s) versus ln(LT ) for ﬁxedµ ˆ = 1...... 45 2.2 The plot of ln (c(z, d)) as a function of ln(z) for ﬁxed value ofµ ˆ = 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:

1 ls R− 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 functions. 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

7 8 2 16π G10 = (2π) ls gs . (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 Poincar´esymmetry

of the theory. In the presence of D-branes, Poincar´einvariance 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- Chapter 1. Introduction 3

branes is related to gs and ls through the relation

1 τD = . (1.2) p p p+1 (2π) ls gs

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 Chapter 1. Introduction 4 black hole entropy

Ad SBH = , (1.3) 4 Gd

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 Chapter 1. Introduction 5 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, where M4 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

p Smicro = 4π Nw Np, (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

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

2 gsN5 1, and gs Nm 1. This 3-charge BPS black hole has a macroscopic horizon area. The Bekenstein-Hawking entropy of the black hole is given by

p SBH = 2π N1N5Nm. (1.5) Chapter 1. Introduction 6

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 Chapter 1. Introduction 7 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 Chapter 1. Introduction 8 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, temperature, 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 description 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

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

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

the D3-branes which is the solution of the form

3 6 1 X 1 X 2 − 2 2 2 2 2 ds = H3 (r) − dt + dxµ + H3 (r) dxi , (1.6) µ=1 i=1 2Φ 2 e = gs , (1.7)

1 1 C4 = −gs− H3− (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

P9 2 1/2 perpendicular to the brane, and r = ( i=4 xi ) . Here Φ is the dilaton, C4 is the R-R

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

4 r3 4 2 H (r) = 1 + , r = 4π g Nα0 . (1.9) 3 r 3 s

r We now deﬁne the new coordinate u ≡ α0 and take the decoupling limit proposed by Maldacena [30] r α0 → 0, u = = ﬁxed. (1.10) α0 Chapter 1. Introduction 9

The harmonic function in the decoupling limit reads

4π g N → s H3(r) 2 4 , (1.11) α0 u

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

2 3 2 ! 2 u 2 X 2 p du p 2 ds = α0 √ − dt + dx + 4π gsN + 4π gsN dΩ , (1.12) 4π g N µ u2 5 s µ=1

5 where dΩ5 is the unit ﬁve-sphere metric. This metric has the geometry AdS5 × S . The radius of the sphere and the AdS radius are equal and are given by

1 1 R = (4π gsN) 4 α0 2 . (1.13)

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

5 decouples from the asymptotically ﬂat region (r r3). It is the AdS5 × S 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

5 the AdS/CFT correspondence: type IIB superstring theory on AdS5 ×S is equivalent to (3+1)-dimensional N = 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 Chapter 1. Introduction 10

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

4 limit. M-theory on AdS7×S was conjectured to be dual to (5+1)-dimensional N = (2, 0) conformal ﬁeld theory which describes the low-energy worldvolume theory of a stack of N

3 4 coincident M5 branes. Another example is the type IIB string theory on AdS3 ×S ×M , 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 ×

7 S /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 Chapter 1. Introduction 11

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

Poincar´esymmetry 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 condensed matter theory (CMT) [50]−[61]. Poincar´esymmetry is not preserved in these non-relativistic systems. The AdS/CMT correspondence has been intensively used to investigate properties of strongly coupled condensed matter systems. Moreover, the holographic principle suggests that these systems may exhibit stringy behaviour in some regimes.

According to the holographic dictionary, black hole geometries with horizons are associated 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 holographic AdS/CMT, charged black holes describe field theories at finite temperature and chemical potential [62, 63, 64]. Chapter 1. Introduction 12

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

z t → λ t, xi → λxi, z ≥ 1. (1.14)

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

dr2 ds2 = L2 r2zdt2 + r2dxidxjδ + , i ∈ {1, 2...d}. (1.15) ij r2

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

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 Chapter 1. Introduction 13 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 translational 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 quantum 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 Chapter 1. Introduction 14 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. Chapter 1. Introduction 15

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 between 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

γ a† b† e |0ia|0ib, (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 Chapter 1. Introduction 16

the system is very small: S(p) < , where is the norm of the change to the initial state and 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 , (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 ). 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 Chapter 1. Introduction 17 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 configurations 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 ingredients. 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. Chapter 1. Introduction 18

Figure 1.1: The traditional geometry of a black hole is a coarse graining over an exponentially large number of microstates. These microstates have the same asymptotic properties as that of the black hole but diﬀer from each other in a region of the size of the 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 black hole 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 Chapter 1. Introduction 19

Sbh tunnels into the fuzzball microstates. Fuzzball density of states is e , 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. Chapter 1. Introduction 20

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

4 1 1 constructed by compactifying N5 D5 branes on T ×S and N1 D1 branes on S . 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 Chapter 1. Introduction 21

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 possess 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

4 1 Consider D1-D5 system constructed by wrapping N5 D5 branes on M × S 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+1)-dimensional N = (4, 4) conformal ﬁeld theory that lives on S . The AdS3/CFT2

3 4 correspondence asserts that superstring theory on AdS3 × S × T 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 Chapter 1. Introduction 22

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

4 N1 N5 target space (M ) /SN1 N5 . This special point in the moduli space is referred to as

4 the orbifold point, i.e., the target space of the sigma model is N = N1 N5 copies of M 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 Chapter 1. Introduction 23 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

5 chiral operators of the supergravity limit of type IIB string theory on AdS5 × S 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

3 4 evaluating correlation functions of the superstring theory on AdS3 ×S ×M 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 orbifold CFT were investigated in [176] and the apparent existing disagreement was resolved. Chapter 1. Introduction 24

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 correlator. 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 Chapter 1. Introduction 25

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 correlation functions of the chiral primary operators and found agreement with the previous results mentioned above. Chapter 1. Introduction 26

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 operators 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 Chapter 1. Introduction 27 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 properties of the microscopic degrees of freedom of black holes and to investigate the microscopic 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 understand 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. Chapter 1. Introduction 28

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

N CFT near the orbifold point” [183] and “Twist-nontwist correlators in M /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 performed 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- Chapter 1. Introduction 29 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 perturbation 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 directions 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 condensed matter systems. Quantum critical systems typically exhibit a scaling symmetry

z t → λ t, xi → λxi, z ≥ 1. (2.1)

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

dr2 ds2 = L2 −r2zdt2 + r2dxidxjδ + , i ∈ {1, 2...d}. (2.2) ij r2

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]

Z 2 1 4 √ 1 2 c 2 S0 = d x −g R − 2Λ − G − A (2.3) 16πG4 4 2

30 Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 31

(we call this model S0) and [66]

1 Z √ S = d4x −g R − 2Λ − 2(∇φ)2 − e2αφG2 , (2.4) 16πG4

(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 S0 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 S0, 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]

1 Z √ S = dd+2x −g R − 2Λ − 2(∇φ)2 − e2αφG2 . (2.5) 16πGd+2

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- Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 32 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

T d/z 4G s = c(z, d)(Lµ)d (2.6) d+2 µ 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

d E = (T s + µn) (2.7) d + 1 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, Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 33

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

1 Z √ S = dd+2x −g R − 2Λ − 2(∇φ)2 − e2αφG2 , (2.8) 16πGd+2

and reduce on the following Ansatz

2 2A(r) 2 2B(r) i j 2C(r) 2 ds = −e dt + e dx dx δij + e dr ,

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 d 1 Z Y Z S = 2dt dxi drL1D (2.10) 16πGd +2 i=1

1We keep track of normalization for later use. Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 34

where the one dimensional Lagrangian is given by

A+dB C d(d − 1) A+dB C 2 A+dB C+2G+2αφ 2 L = de − ∂A∂B + e − (∂B) + e− − (∂G) 1D 2 A+dB C 2 A+dB+C −e − (∂φ) − e Λ, (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 −

G G δ1, C, φ, G + dδ1), 2) e → e + 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 Q 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

A+dB C d(d − 1) A+dB C 2 A+dB C+2G+2αφ 2 de − ∂A∂B + e − (∂B) + e− − (∂G) 2 A+dB C 2 A+dB+C −e − (∂φ) + e Λ = 0, (2.12)

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

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

A+dB C+G+2αφ e− − ∂G = Q. (2.15) Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 35

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

1 ﬁnd the combinations D0 = α D0 +d P0,P0 = 2 P0 more convenient, and so the equations become

A+dB C d(d − 1) A+dB C 2 A+dB C+2G+2αφ 2 de − ∂A∂B + e − (∂B) + e− − (∂G) 2 A+dB C 2 A+dB+C −e − (∂φ) + e Λ = 0, (2.16)

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

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

A+dB C+G+2αφ e− − ∂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

r ! r d+1 A(r) = ln Lr 1 − h , r B(r) = ln(Lr),   L C(r) = ln  q  , rh d+1 r 1 − r

φ(r) = φb, A = gbdt. (2.20)

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

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

the solution has Tˆ = 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

d d d d d s = rh/(4Gd+2), or in thermodynamic terms s = (4π) T L / (d + 1) 4Gd+2 . This shows

d that the number of degrees of freedom in the ﬁeld theory is linear in L /(4Gd+2). Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 36

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

r ! r z+d A(r) = ln L a rz 1 − h ,, L L r B(r) = ln(Lr),   LL C(r) = ln  q  ,, rh z+d r 1 − r 2d 2αφ(r) = ln r− Φ , d z  d z+d rh +  (z − 1)L aLr 1 − r G(r) = ln   . (2.21) 2Q

The constants are given by

(z + d)(z + d − 1) L2 = L2 , L d(d + 1) d 12 2 Q/L − (z + d − 1) Φ = , (2.22) d(d + 1)(z − 1)

p aL is arbitrary (and may be removed by time rescaling), and α = 2d/(z − 1). The conserved quantities are

p z+d d P0 = 0,D0 = d/ [2(z − 1)] rh aL L 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 rd s = h . (2.24) 4 Gd+2

ˆ z Finally, we see that T = rhaL(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. Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 37

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), L C(r) = ln + C (r),G(r) = ln(L) + G (r). (2.25) r 1 1

We further deﬁne dimensionless conserved quantities

d 1 d d Q = L − Q,ˆ D0 = L Dˆ 0,P0 = L Pˆ0 = 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

2αφ e− :

2αφ 1 2d+2 A1+C1 A1 C1 e− = dr α(d + 1) e − e − 2r2Qˆ2αeA1+C1 ! d+1 d+1 G1 C1 A1 G1 2 −2r Dˆ 0 − 4dr e αQˆ + 2αe − (αQeˆ ) . (2.27)

Using this, the other diﬀerential equations may be written as

C1 G1 e Dˆ 0 + 2dαQeˆ ∂eA1 − = 0, (2.28) dαrd+2 2C1 2A1 C1 2 2 2G1 A1 d+1 G1 A1 d+1 e − 2αe α Qˆ e − 2de r αQeˆ − Dˆ 0e r ∂eC1 − = 0, (2.29) dαr2d+3 2d+2 G1 αdr A1+C1 A1 C1 ∂e − (d + 1) e − e − (2.30) 2αQrˆ d+2 ! d+1 G1 C1 A1 2 2 2G1 −2r Dˆ 0 + 2dαQeˆ + 2αe − α Qˆ e = 0. Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 38

These functions may be expanded around a regular horizon as

1 3 A1 e = a0 (r − rh) 2 + a1(r − rh) 2 + ··· ,

c0 1 C1 2 e = 1 + c1(r − rh) ··· , (r − rh) 2 G1 2 e = a0 g0(r − rh) + g1(r − rh) + ··· . (2.31)

One may read that the dilaton goes to a constant. With Pˆ0 = 0, we ﬁnd that the equations of motion require

ˆ 2 2 4 2 2 2 2 2c0D0 α d(d + 1) c0 − 2drh(α − 2)(d + 1)c0 + rh (α d − 4 − 6d) a0 = 2+d , a1 = 3 , αdrh 8rh 2 2 4 2 2 2 2 c0 (3α d(d + 1) c0 − 2drh(2 + 3α )(d + 1)c0 + rh (3α d + 4 + 6d)) c0 = c0, c1 = 3 , 8rh 2 d d((d + 1)c0 − rh)rh g0 = , (2.32) 2c0Qˆ 2 d d 2 2 3 6 d 1 2 2 4 g1 = rh− α (d + 1) c0 − rh− α (d + 1) c0 8Qcˆ 0rh ! d 2 2 d+1 2 −rh(α + 2)(d + 1)c0 + rh α + 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

2 ˆ d rha0 D0 rh T = = d , s = . (2.33) 4πLc0 αdrh2πL 4Gd+2

Note that, in the above expressions, c0, a0, and Dˆ 0 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 √ 2 p p satisfies c0 = α + 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. Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 39

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:

A+dB C d(d − 1) A+dB C 2 A+dB C+2G+2αφ 2 de − ∂A∂B + e − (∂B) + e− − (∂G) 2 A+dB C 2 A+dB+C −e − (∂φ) + e Λ = 0, (2.34)

A+dB C d (2∂φ + α (∂A − ∂B)) e − = D0, (2.35)

A+dB C G e − ∂φ + αQe = P0, (2.36)

A+dB C+G+2αφ e− − ∂G = Q. (2.37)

We use the expansion:

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

B(r) = ln(Lr), L C(r) = ln + C (r), (2.38) r 1

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)

D + 2dαQg − 2dP Q2 − 0 b 0 A1(r) = d d+1 + 2α 2(d 1) 2d , (2.39) d(d + 1)αL r d(d − 1)Φb L − r D + 2dαQg − 2dP 2Q2 0 b 0 − C1(r) = d d+1 2α 2(d 1) 2d , (2.40) d(d + 1)αL r (d − 1)(d + 1)Φb L − r Q − G1(r) = 2α d 2 d 1 , (2.41) (d − 1)gbΦb L − r − αQg − P αQ2 b 0 − φ1(r) = d d+1 2α 2(d 1) 2d . (2.42) (d + 1)L r 2d(d − 1)Φb L − r Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 40

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

1 E = − N dK −d K − N µp rˆν , (2.43) 8π t 0 t µν

µ d where Nt is the lapse function, Nt is the shift vector, K is the extrinsic curvature of the

d d-dimensional spatial boundary slice inside the constant-t slice, K0 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, andr ˆ is the spatial unit vector normal to the constant-r surface. The reference background we take is pure AdS in the

Poincar´epatch.

A µ C For our metric, Nt = e , Nt = 0,r ˆ = e , and the energy density is

− 1 A d 2B (d 2)B C Cref E = lim e ∂r e e − e− − e− . (2.44) 8π 2 r →∞

Cref L For the pure AdS reference, we have e = r . The total energy density then reads

2d rd+1 E = lim C1 (2.45) r →∞ 16πGd+2L

where we have dropped terms that go to zero in the large r limit. Therefore, we read the

energy density to be 1 2 (D + 2dαQg ) E 0 b = d+1 . (2.46) 16πGd+2 (d + 1) αL

Now we need to normalize the charge density Q. For this, we follow the discussion of Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 41

[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

Vd µgeom S 2 = qgeom, (2.47) G 2 T

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 S 2 G and ﬁnd

1 2Vd µgeom 2 S = d+1 Q, (2.48) G 16πGd+2 L T

and so we identify the charge Q as

16πG Ld+1 Q = d+2 q . (2.49) 4 geom

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

d E = (T s + µn) , (2.50) d + 1

2 2 where we have changed to ﬁeld theory quantities µgeom = µL , qgeom = n/L . 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

r r rh p rh < c < (2 + α2) (2.51) d + 1 0 d + 1 Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 42 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,

A G φ given a c0, we take these asymptotic values of the ﬁelds e , e , e 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 Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 43

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,

eG1 eαφ µˆ = . (2.52) eA1 r= ∞ 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

αφ αφ invariant qgeomµgeom = e |r= qgeom e− |r= µgeom . The second expression is made ∞ ∞ αφ out of global symmetry invariants (e.g e |r= qgeom ). These, therefore, can inﬂuence ∞ Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 44

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 Tˆ for ﬁxedµ ˆ = 1 and various dimensions.

As seen in Figure 2.1, the entropy density is smooth and monotonic in Tˆ for a wide

range of z, and there are no possible discontinuous phase transitions associated with

going from Tˆ µˆ to Tˆ µˆ. 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 (Tˆ µˆ), and the slope

approaches d in the AdS regime (Tˆ µˆ).

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 Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 45

(a) d = 3. (b) d = 5.

Figure 2.1: The plots of ln (4Gd+2s) versus ln(LT ) for fixedµ ˆ = 1. Figures (2.1a), (2.1b), (2.1c) and (2.1d) correspond to d = 3, 5, 7, and 9, respectively. The different curves in each 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 z p via α = 2d/(z − 1). Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 46 the Lifshitz-like theories. We note first that the entropy density of the system in the

Lifshitz-like limit is of the form

T d/z 4G s = c(z, d)(Lµ)d , (2.53) d+2 µ where c(z, d) is a function of the dimension and the critical exponent. We have chosen

µˆ = 1 in the plots shown in Figure 2.1, and so the above expression reduces to 4Gd+2s = c(z, d) Tˆd/z. Therefore, if we ﬁt the ln(Tˆ) ln(ˆµ) 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 Tˆ µˆ 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

  n1/`     ~k = n /` . (2.54)  2   .  . Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 47

Figure 2.2: The plot of ln (c(z, d)) as a function of ln(z) for fixed value ofµ ˆ = 1. The different 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 behave similarly. First of all, notice that the curves’ intercept at ln(z) = 0 is given by the value ln (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 monotonic in d. Given these two facts, it may be no surprise that the curves ln(c(z, d)) for fixed d are generically non-monotonic for small values of z and that, in fact, they cross each other.

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

√ √ 2 z 1 z 2 z 1 ω = ( k ) µ − δ = ( n ) µ δ (2.55) ~n (`µ)z 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 Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 48

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

βωn βωn the ground state is e− F 0 e− . 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 chemical 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

∂ X βωn E = − F,F = F e− . (2.56) ∂β n

Calculating F , and approximating the sum as an integral, we ﬁnd

Z dn δ z d βµ (`µ)z n F = Ωd 1 n F e− − n d Z Ωd 1 (`µ) d/z dnˆ nˆ = − nˆ F e− (2.57) z (δβµ)d/z nˆ

and deﬁne, therefore, Z d d/z dnˆ nˆ Fˆ = nˆ F e− . (2.58) z nˆ Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 49

Finally, we read that the energy is

d d (`µ) 1+d/z d E = Ωd 1 T Fˆ . (2.59) − z2 (δµ)d/z z

d d Note that the constant ` = Vd/(2π) . Hence, we may take derivatives of the above by T at constant volume in a straightforward way. Therefore

Z 1 ∂E S = dT V T ∂T V d d + z (`µ) d d/z = Ωd 1 Fˆ T . (2.60) − z2 (δµ)d/z z

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

d E = T s (2.61) d + z 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. Chapter 2. Lifshitz black brane thermodynamics in higher dimensions 50

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 ∼

d L /(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

S 4G s = 4G d+2 d+2 (2π`)d d/z d/z Ωd 1 d + z d d T = δ(z, d)− κ(z, d) − (Lµ) Fˆ (2.62) (2π)d z2 z µ

where, on the right hand side, the function κ(z, d) measures the number of species exactly:

d NS = κ(z, d)L /(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

d/z d/z Ωd 1 d + z d d T 4G s = δ(z, d)− κ(z, d) − (Lµ) Γ , (2.63) d+2 (2π)d z2 z µ

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

d 2 d 2 1 factor Γ z (d + z)/z . In the large z limit Γ z (d + z)/z → d and 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 N M /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)

N supersymmetric sigma model with target space M /SN [153, 155]. Here, M is either K3

4 N or T , and SN is the symmetric group. The target space M 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 N Chapter 3. Twist-nontwist correlators in M /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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 53

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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 54

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 rôleof twist fields. Twist fields change the boundary conditions that the fundamental fields must satisfy. In particular, they consider the effects of the N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 55

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:

p r n(N − n)! X n(N − n)! X σn = √ σg(12 n)g−1 = σc. (3.2) − ··· N! n(N n)! N! g c C[(12 n)] ∈ ···

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

hσn1 σn2 σn3 · · · i. (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.

hσ(1,2,3 n1) σ(2,3,4, n2+1) σ(8,9,10 n3+7) · · · i. (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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 56

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

ni z − z0 = b(t − t0) + ··· . (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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 57

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 hσ(12)σ(23)σ(321)i. 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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 58

non-twist ∞

σ(23) Φ1 σ(321)

∞ σ(12) Φ3

Φ2

∞

(a) base space (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 in figure 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 Φ on 3 patches. We further show the three images of a generic point (where we will later include 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

1 1 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 59

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

φ is scaled back to a reference metric g = e gc. The change to the measure by this rescaling of the metric is given by the exponential of the Liouville action [227]

c Z √ S = d2t −g [∂ φ∂ φgµν + 2R(g )φ] . (3.6) L 96π c µ ν c c

This factor, coming from an anomaly, simply multiplies the correlators in the covering

space,

Y SL Y h Oiibase = e h Oˆiicover (3.7) i i 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 Oˆi 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

1 2 1 n dz n 1 2 z = at + · · · → ψ(t) = ψ(z) = ant − + ··· ψ(z) (3.8) dt

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, N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 60

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

1 a bare twist σ(12) and we wish to excite it with a mode of a bosonic X operator, α3, 1 , − 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˜20 = ∂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. N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 61 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 ofσ ˜20 along with itself and σ3, there will be terms of the form

h(∂X3σ(12))(∂X1σ(23))(σ(321))i or h(∂X4σ(12))(∂X4σ(23))(σ(321))i. (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

1 dz − × ∂X(t ) (3.11) dt o,1 t=to,1

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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 62

of 1 dz − × ∂X(t ) (3.12) dt x,3 t=tx,3

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

2 2 2 3 2 2 ∂ X(z) → (∂z/∂t)− ∂ X(t) − (∂z/∂t)− (∂ z/∂t )∂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

N 1 X O0 = √ ∂Xκ (3.14) N κ=1 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 63 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

s N 1 X 1 X O0 = √ ∂Xκ + √ ∂Xκ N κ=1 N κ=s+1

= O0, + O0, (3.15) k ⊥

where O0, is the ﬁrst s terms (copies along the twists), and O0, is the other terms (copies k ⊥ 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 k possible excitations along the ﬁrst s copies of the CFT, generically of the form

Y A = hO0, (z0) σn` (z`)i (3.16) k `

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. k N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 64

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

1 Ps diagram 3.1b. Next, we note that the operator √N κ=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 k 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

1 dz − × ∂X(t ). (3.17) dt 0,i t=t0,i

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 ··· P describes an non-SN -invariant piece of an SN -invariant operators S i Oi1 iq , which N ( k) ··· 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

1 operator ∂X1∂X2 (here 1, 2 are copy indices) we would put an image of (∂z/∂t)− ∂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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 65 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 hΣ2Σ20 Σ3i, 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

n points in the cover, the map is locally z − z0 = b(t − t0) + ··· . 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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 66 map to use is

2 z − a1 = (a1 − a2)t (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

2 z − a2 = (a1 − a2)(2t + 1)(t − 1) (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

3 t = ∞, z = ∞, again with the correct behavior z = 2(a1 − a2)t + ··· .

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

t2([1 + 2ω]t − 3ω)(a − a )(a − b)(ω − 1)2 − − 1 2 1 z a1 = 2 2 3 . (3.20) t ([1 + 2ω]t − 3ω)(a1 − a2)(ω − 1) + (t − ω) (a2 − b)

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

a a (t + 1)3 − 1 2 z = 2 3 . (3.21) 4t (t − 3)(a1 − a2) − (t + 1) a2 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 67

which one can also write as

4t2(t − 3)(a − a )a − − 1 2 1 z a1 = 2 3 (3.22) 4t (t − 3)(a1 − a2) − (t + 1) a2 (t − 1)2(5t + 1)(a − a )a − − 1 2 2 z a2 = 2 3 (3.23) 4t (t − 3)(a1 − a2) − (t + 1) a2

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]

2 1 |C , , | = (3.24) 2 2 3 1 c 5 c 3 12 2 9

and so for twist operators located at ﬁnite points, we have

2 |C2,2,3| hσ(1,2)(a1) σ(2,3)(a2) σ(3,2,1)(b)i = 2c 2c 2c (3.25) |a1 − a2| 72 |a1 − b| 9 |a2 − b| 9

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˜20 already discussed:

σ˜20 = ∂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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 68

out the operator as N N X X σ˜20 = ∂Xkσ(ij). (3.27) i,j=1 k=1 i

N N 1 X X σ20 = ∂Xkσ(ij). (3.28) q N (N − 2) i,j=1 k=1 2 i

The 3-point function we wish to consider is

hσ20 (a1)σ20 (a2)σ3(b)i (3.29)

where σ3 is just a bare twist three operator, i.e.

1 X σ3 = σ(i,j,k) (3.30) q N 2 3 3-cycles

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

N N 2 factor of 2 . Then, there are 1− ways to assign the last index k. Thus, we ﬁnd that

hσ20 (a1)σ20 (a2)σ3(0)i = (3.31) p * N N + (N − 3)!3! X X 2 √ √ ∂Xi(a1)σ(12)(a1) ∂Xj(a2)σ(23)(a2) σ(321)(0) 2 N! i=1 j=1 i=1,2 j=2,3 6 6 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 69

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

hσ20 (a1)σ20 (a2)σ3(0)i = (3.32) p (N − 3)!3! 2 √ √ ∂X3(a1)σ(12)(a1) ∂X1(a2)σ(23)(a2) σ(321)(0) 2 N! !

+(N − 3) σ(12)(a1)σ(23)(a2)σ(321)(0) h∂X4(a1)∂X4(a2)i (3.33)

The second term above clearly has the correct form of a 3-point function. It simply

2 gives an additional factor of 1/(a1 − a2) , which is what should happen: the holomorphic

weight h of the twist-two operators have both increased by 1. This aﬀects the 1/(a1 −

h1+h2 h3 h1 h2+h3 a2) − terms, but not the terms of the form 1/(a1 − 0) − , 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)   *" 1 # 1 + ∂z − ∂z − 1 → ∂X(3)  ∂X −  (3.35) ∂t ∂t 5 t=3 t= 1 − 5

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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 70

trivial, and gives

* + ∂z ∂z 1 16a 16a 1 − 2 1 ∂X(3) ∂X = 2 ∂t t=3 ∂t t= 1 5 9a1(a1 − a2) 225a2(a1 − a2) (3 − (−1/5)) − 5 1 = 2 (3.36) 81(a1 − a2)

This extra dressing, just like the orthogonal piece, changes the 3-point function to be of the proper form. Combining everything, we ﬁnd

p 2 1 (N − 3)!3! |C2,2,3| 81 + N − 3 hσ0 (a1)σ0 (a2)σ3(0)i = 2 √ √ c 2c 2c c 2c 2c . (3.37) 2 2 +2 2 N! 72 9 9 72 9 9 (a1 − a2) a1 a2 (¯a1 − a¯2) a¯1 a¯2

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

* N N N N + 1 X X X X hσ0 (1)σ0 (0)i = ∂Xk(1)σ(ij)(1) ∂Xk0 (0)σ(i0j0)(0) 2 2 (N − 2)N 2 i,j=1 k=1 i,j=1 k0=1 i

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. N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 71

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

D E σ0 σ200 σ200 (3.39)

where σ0 is a non-twist insertion, and σ200 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 tn z = a (3.41) tn − (t − 1)n

where the location of the twist operators is now z = 0 and z = a. It is easy to check that

(t − 1)n z − a = a (3.42) tn − (t − 1)n 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 Xi, and four real fermions in both the left and right moving

sectors ψj, ψek.

The presence of fermions complicates the lift to the covering surface when the twist N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 72

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 φei(¯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 φe. 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, N 1 X O = √ A · ∂φi (3.44) N i=1

which is clearly invariant under the permutation group SN , and is also properly normalized 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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 73

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

1 if φ/2 if φ/2 OA(12)(z) → √ :(A · ∂φ)e− · : + : (B · ∂φ)e · : (3.45) 2b5/8 1 if φ/2 if φ/2 OA(12)(z)† → √ :(A∗ · ∂φ)e · : + : (B∗ · ∂φ)e− · : . (3.46) 2b5/8

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 OA† (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

a a z = 0 for convenience) ∂φ1 and ∂φ2 for a = 1, 2, 3, 4 do not have well-deﬁned periodicity

a a conditions. Instead, a more natural combination is ∂φ1 + exp (2πim/2)∂φ2. 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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 74 naturally define modes of these operators as

I (12) dz 1 1 m/2 2πim/2 A∗ · α m/2 ≡ z − − A∗ · ∂φ1(z) + e ∂φ2(z) (3.48) − 2πi 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

I (12) dz 1 1 m/2 2πim/2 [A∗ · α m/2,OA,(12)(z)†] = z − − A∗ · ∂φ1(z) + e ∂φ2(z) OA,(12)(0)† − 2πi I 1 1 dt dz − m/2 → (z(t))− A∗ · ∂φ(t) (3.49) 2πi dt 1 if/2 φ if/2 φ ×√ :(A∗ · ∂φ)e · (0) : + : (B∗ · ∂φ)e− · (0) : . 2b5/8

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

m/2 above operator is normal ordered. We simply need to expand z(t)− 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

(12) [A∗ · α 2/2,OA,† (12)] (3.50) − 1 5/8 b− − 2 b1 if/2 φ if/2 φ → √ : A∗ · ∂ φ − ∂φ (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · : 2 b where we have expanded the map to second order as

2 3 4 z = bt + b1t + b2t + ··· (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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 75

to apply I dz 2 1+` L = z − T (z). (3.52) ` 2πi

To apply this in the covering space, recall that the stress tensor does not transform

tensorially, but rather transforms as

2 dz − c T (z) → T (t) − {z(t), t} , (3.53) dt 12

with {z(t), t} denoting the Schwarzian derivative. In the base space, the stress tensor is

just 1 T (z) = − : ∂φa∂φa :, (3.54) 2

and c = 6 in the cover. Thus, to check if our operator is a quasiprimary, we lift to the

covering surface and compute

h (12) i L`, [A∗ · α 2/2 ,OA,† (12)] − 1 5/8 I 1 b− − dt dz − 1+` 1 a b 1 → √ (z(t)) − : ∂φ ∂φ (t): δab − {z(t), t} 2 2πi dt 2 2 2 b1 if φ/2 if φ/2 × : A∗ · ∂ φ − ∂φ (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · (0) : . (3.55) b 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

1 dz − 1 3b 8b b − 9b2 = 1 − 1 t − 2 1 t2 + ··· , (3.56) dt 2bt 2b 4b2 and

(1 + `)b (1 + `)(2b b + b2`) (z(t))1+` = b1+`t2+2` 1 + 1 t + 2 1 t2 + ··· . (3.57) b 2b2 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 76

Therefore, the leading order behavior of these terms put together is t1+2`. Putting this

together, we expand out

h (12) i L`, [A∗ · α 2/2,OA,† (12)] − b 1 5/8+` I dt 3b (1 + `)b → − −√ t1+2` 1 − 1 t + ··· 1 + 1 t + ··· 2 2 2πi 2b b 1 3 × − : ∂φa∂φb(t): δ + + ··· 2 ab 4t2 2 b1 if/2 φ if/2 φ × : A∗ · ∂ φ − ∂φ (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · (0) : . (3.58) b

We can expand the OPE as

1 a b 2 b1 if/2 φ if/2 φ − δ : ∂φ ∂φ (t):: A∗ · ∂ φ − ∂φ (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · : 2 ab b 2 if/2 φ if/2 φ if/2 φ = : A∗∂φe · (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · : t3 2 2 1 (f/2) 2 (f/2) b1 + : A∗ · 3 + ∂ φ − 2 + ∂φ t2 2 2 b h if/2 φ if/2 φi × (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · : " # 1 2 b1 if/2 φ if/2 φ + ∂ A∗ · ∂ φ − ∂φ (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · + ··· (3.59) t b

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

2+2` term in the integral is t− . 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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 77

h (12) i L0, [A∗ · α 2/2,OA,† (12)] − 1 5/8 b− − b1 if φ/2 if φ/2 if φ/2 → − √ : A∗∂φe · (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · : (3.60) 2 2 b 1 5/8 2 2 b− − (f/2) 2 (f/2) b1 + √ : A∗ · 3 + + 3/4 ∂ φ − 2 + + 3/4 ∂φ 2 2 2 2 b h if φ/2 if φ/2i × (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · :

h (f/2)2 i 3 + + 3/4 1 5/8 2 b− − 2 b1 = √ : A∗ · ∂ φ − ∂φ 2 2 b h if φ/2 if φ/2i × (A∗ · ∂φ)e · + (B∗ · ∂φ)e− · :

(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 1 √ · ··· ··· (12) ··· N A (∂φ1(a) + ∂φ2(a) + ) OA,(12)(b)+ [α m/2,OA,(12)(0)†] + − N 2 (3.62)

where ··· denotes all of the other terms that lead to permutation invariant operators. The

qN extra factors of 1/ 2 normalize the sum of twist two operators OA,(i,j) with respect N to N. Above, there are 2 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

N 2 terms, all of which give the same contribution, canceling the normalization factors coming from the twist two operators. We ﬁnd

1 h (12) i √ A · (∂φ1(a) + ∂φ2(a)) OA,(12)(b) α m/2,OA,(12)(0)† . (3.63) N − N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 78

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

t2 t2 z(t) = b = b . (3.64) t2 − (t − 1)2 2t − 1

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

t2 a = b ± (3.66) 2t − 1 ± to ﬁnd p a ± a(a − b) t = . (3.67) ± b

Further, because the operator ∂φ1(a) transforms under conformal mapping, we will need to compute

∂z 2bt (t − 1) 2a(a − b) ± ± = 2 = (3.68) ∂t t t (2t − 1) bt (t − 1) = ± ± ± ± 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 ±

* + h (12) i A · ∂φ1(a) + ∂φ2(a) OA,(12)(b) α 2/2,OA,(12)(0)† . (3.69) −

We lift this computation to the covering surface, needing two images of the non-twist N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 79

operator, which we then add together. This gives

D h (12) iE A · (∂φ1(a) + ∂φ2(a)) OA,(12)(b) α 2/2,OA,(12)(0)† − * ∂z ∂z → A · (t+)∂φ1(t+) + (t )∂φ1(t ) ∂t ∂t − − 1 if φ/2 if φ/2 ×√ (A · ∂φ)e− · + (B · ∂φ)e · (1) (3.70) 2b5/8 1 5/8 + (−b)− − 2 b1 if φ/2 if φ/2 × √ A∗ · ∂ φ − ∂φ A∗ · ∂φ e · + B∗ · ∂φ e− · (0) . 2 b

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

13/8 (−1)− X bt (t − 1) = ± ± (3.71) b1+5/4 2a(a − b) ± " if φ/2 2 if φ/2 × A · ∂φ(t )[A · ∂φe− · (1)][A∗ · (∂ φ − 2∂φ)A∗ · ∂φe · (0)] ± # if φ/2 2 if φ/2 + A · ∂φ(t )[B · ∂φe · (1)][A∗ · (∂ φ − 2∂φ)B∗ · ∂φe− · (0)] . ±

We see that the ﬁrst expectation value has two types of contractions to remove ∂φa, while the second has only one. These become

13/8 (−1)− X bt (t − 1) = ± ± (3.72) b1+5/4 2a(a − b) ± " −2 −1 −1 if φ/2 if φ/2 × (A∗ · A) − 2 (A∗ · A) he− · (1)e · (0)i t3 t2 (1)2 ± ± −1 −2 −1 if φ/2 if φ/2 +(A∗ · A) (A∗ · A) − 2 he− · (1)e · (0)i t2 13 12 ± # −2 −1 −1 if φ/2 if φ/2 +(A∗ · A) − 2 (B∗ · B) he · (1)e− · (0)i . t3 t2 12 ± ±

Note that the second line above gives no contribution. Further, there are several combi- N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 80 nations of the form t (t − 1) (t − 1) ± ± = ± . (3.73) t2 t ± ± Explicitly summing these contributions using (3.67), we ﬁnd

(t+ − 1) (t − 1) + − = 0. (3.74) t+ t −

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

13/8 4(−1)− t+ − 1 t − 1 − = 5/4 2 + 2 . (3.75) b a(a − b) t+ t −

Evaluating the sum, after plugging in t+ and t , we ﬁnd that the 3-point function (3.69) − lifts to the remarkably simple result

(12) A · (∂φ1(a) + ∂φ2(a)) OA,(12)(b) α 2/2,OA,(12)(0)† − −8(−1) 13/8 → − . (3.76) b5/4a2

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

3/4 dress the above computation with a factor of b− . Our total 3 point function becomes

h (12) i A · ∂φ1(a) + ∂φ2(a) OA,(12)(b) α 2/2,OA,(12)(0)† − −8(−1) 13/8 ∝ − (3.77) b2a2 up to the normalization factors from [171] which we have not included, and normalizations N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 81

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

1 5/8 5/8 phase ambiguity in (−1)− − because there would be a factor of (−¯b)− : 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:

C BC C BC A(a)B(b)C(0) = A = A (3.78) hA+hB hC hA+hC hB hA+hB hC 2 2 (a − b) − a − b − a b

where h = 1, h = 1, h = 2. Note that this behavior comes about only after we summed A B C 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) N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 82

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 N Chapter 3. Twist-nontwist correlators in M /SN orbifold CFTs 83 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 supergravity 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 superstrata [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

1 4 N is (T ) /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 factorization 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 perturbation 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. Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point86 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 corresponds 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 ψ˜1, ψ˜2, ψ˜3, ψ˜4.

i The bosonic ﬁelds X can be written as doublets of SU(2)1 and SU(2)2 [168]

1 XAA˙ = √ Xi (σi)AA˙ , (4.1) 2 where σ1, σ2, and σ3 are the Pauli spin matrices and σ4 = i I, with I being the identity 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

αA˙ complex fermions which transform as doublets of the SU(2)L and SU(2)2: ψ . The index α corresponds to the SU(2)L doublet. The reality condition imposes the following Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point87 constraint on the complex fermions

αA˙ βB˙ (ψ )†(z) = −αβ A˙B˙ ψ (z). (4.2)

Thus we count four complex fermions ψαA˙ , with two linearly independent reality conditions. 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

α˙ A˙ which transform as doublets of the SU(2)R and SU(2)2: ψ˜ , where the indexα ˙ corresponds to the SU(2)R doublet. Again, the reality condition

˜α˙ A˙ ˜β˙B˙ (ψ )†(¯z) = −α˙ β˙ A˙B˙ ψ (¯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

αA a energy tensor T , the four supercurrents G , and the SU(2)L R-symmetry current J which are given by

1 AA˙ BB˙ 1 αA˙ βB˙ T = ˙ ˙ ∂X ∂X + ˙ ˙ ψ ∂ψ , 4 AB AB 2 αβ AB √ αA αA˙ BA˙ G = 2A˙B˙ ψ ∂X ,

a 1 αA˙ a β γB˙ J = ˙ ˙ ψ (σ∗ ) ψ . (4.4) 4 AB γ

The generators of the superconformal algebra in the right-moving sector are T˜, G˜αA˙ , and

J˜a and are given by similar expressions.

The OPE of bosonic and fermionic ﬁelds are

A˙B˙ AB AA˙ BB˙ ∼ ∂X (z1) ∂X (z2) 2 , (4.5) (z1 − z2) αβ A˙B˙ αA˙ βB˙ ψ (z1) ψ (z2) ∼ − . (4.6) z1 − z2 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point88

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

i i i i i X(1) → X(2) → X(3) · · · → X(n) → X(1), (4.7)

i i i i i ψ(1) → ψ(2) → ψ(3) → · · · → ψ(n) → ψ(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

a consider, for example, the R-current J acting on the twist operator σn. The fractional mode operator is deﬁned

I n dz X m m a a n 2πi (k 1) n J m (z) σn(0) ≡ J(k)(z) z− e− − σn(0), (4.9) − n 2πi z=0 k=1

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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point89 because the integrand is periodic as one circles around the origin in the contour integral.

a The operator J 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 bosonizing 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 anticommutation 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

2 ﬁelds, φ˜5, φ˜6 . These ﬁelds have the corresponding momenta p5, p6, p˜5, p˜6 which satisfy the algebra

[φi, pj] = iδij, [φ˜i, p˜j] = iδij, (4.10)

where any of the pi andp ˜i 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 distinction between the index being up or down. Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point90

Therefore, it is natural to consider operators of the form

i (k5φ5+k6φ6+k˜5φ˜5+k˜6φ˜6) Ck e , (4.11)

where ki and k˜j are real numbers and Ck is the cocycle operator which explicitly depends on ki and k˜j. We deﬁne the cocycle Ck to be of the form

  p5     p6 iπck ≡ ˜ ˜   Ck = e , ck k5 k6 k5 k6 M   , (4.12)   p˜5   p˜6 where M is a 4 × 4 matrix. Individual fermions are special operators for which one of the ki or k˜i 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

  1 1 − 1 1  2 2 2 2    − 1 − 1 1 − 1   2 2 2 2  M =   . (4.13)  1 − 1 − 1 − 1   2 2 2 2    1 1 1 1 − 2 2 2 2

1 1 i (φ5+φ6) i (φ˜5+φ˜6) We choose this so that operators of the form e± 2 or e± 2 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- Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point91

tor, ψ˜α˙ A˙ (¯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

ik φ ik φ i( k) φ (Cke · )† = e− · (Ck)† = e − · C k. (4.14) −

Using the matrix M deﬁned above, we ﬁnd

+1˙ iπc iφ6 +2˙ iπc iφ5 ψ = e e− , ψ = e e ,

2˙ iφ6 iπc 1˙ iφ5 iπc ψ− = −e e− , ψ− = e− e− , (4.15) where

iπ iπc (p5+p6 p˜5+˜p6) e ≡ e 2 − . (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

+˙ 1˙ iπc˜ iφ˜6 +˙ 2˙ iπc˜ iφ˜5 ψ˜ = e e− , ψ˜ = e e ,

˙ 2˙ iφ˜6 iπc˜ ˙ 1˙ iφ˜5 iπc˜ ψ˜− = −e e− , ψ˜− = e− e− , (4.17) where

iπ iπc˜ (p5 p6 p˜5 p˜6) e ≡ e 2 − − − . (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

1 OPEs. The singular part is proportional to (z1 − z2)− , 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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point92

a singular OPE, passing the cocyle operators across the exponentiated scalars produces

the minus sign. For instance, we consider the OPE

π π +1˙ +2˙ i (p5+p6 p˜5+˜p6) iφ6 i (p5+p6 p˜5+˜p6) iφ5 ψ (z) ψ (w) = e 2 − e− (z) e 2 − e (w)

π π π [ iφ6,i p6] i (p5+p6 p˜5+˜p6) i (p5+p6 p˜5+˜p6) iφ6 iφ5 = e − 2 e 2 − e 2 − e− (z) e (w)

π π π π [ iφ6,i p6] [i p5,iφ5] i (p5+p6 p˜5+˜p6) iφ5 i (p5+p6 p˜5+˜p6) iφ6 = e − 2 e 2 e 2 − e (w) e 2 − e− (z)

= −ψ+2˙ (w) ψ+1˙ (z), (4.19) where we have used the Baker-Campbell-Hausdorﬀ formula and the commutation relations (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+1˙ ψ+2˙

1˙ 2˙ and J − = −ψ− ψ− . 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

1˙ 1˙ the OPEs for ψ± as an example. For ψ± we have

π π π + 1˙ i (p5+p6 p˜5+˜p6) iφ6 i (p5+p6 p˜5+˜p6) iφ5 iφ5 i (p5+p6 p˜5+˜p6) J (z)ψ− (w) = e 2 − e− e 2 − e (z) e− (w) e− 2 − i π (p +p p˜ +˜p ) iφ +1˙ e 2 5 6 5 6 e 6 (w) ψ (w) ∼ − − = , (4.20) z − w z − w

and

π π π +1˙ iφ5 i (p5+p6 p˜5+˜p6) iφ6 i (p5+p6 p˜5+˜p6) i (p5+p6 p˜5+˜p6) iφ6 J −(z)ψ (w) = e− e− 2 − e e− 2 − (z) e 2 − e− (w) iφ i π (p +p p˜ +˜p ) 1˙ e 5 (w) e 2 5 6 5 6 ψ (w) ∼ − − − = − . (4.21) z − w z − w Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point93

2˙ Similar considerations hold for ψ± . We thus obtain

+ A˙ +A˙ +A˙ A˙ [J0 , ψ− ] = ψ , [J0−, ψ ] = ψ− . (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

+ +1˙ 1˙ +2˙ 2˙ have J = ψ ψ− , J − = −ψ ψ− . Using the bosonized ﬁelds (4.15) and (4.17) and performing similar computations as in (4.20) and (4.21), we ﬁnd

+ α2˙ α1˙ α1˙ α2˙ [J0 , ψ ] = ψ , [J0−, ψ ] = ψ , (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 covering surface. For spin fields we have ki = ±1/2, k˜i = ±1/2. Let us consider the holomorphic sector first. There are four spin fields in this sector Sα and SA˙ . Following Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point94

the procedure described above, we ﬁnd that

i i + iπc (φ5 φ6) iπc (φ5 φ6) S ≡ e e 2 − , S− ≡ e− e− 2 − , (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

+ + + + [J0 , S−] = S , [J0−, S ] = S−, (S )† = S−. (4.25)

α The assigned cocycles make the two spin ﬁelds S transform as a doublet of SU(2)L and

a singlet of SU(2)2.

For spin ﬁelds SA˙ we ﬁnd

π i π i 1˙ i (φ5+φ6) 2˙ i (φ5+φ6) S ≡ e 2 e− 2 , S ≡ e− 2 e 2 . (4.26)

We note that the structure of the cocycles we deﬁned in (4.12) and (4.13) is such that

i (φ5+φ6) the cocycles associated with exponentials of the form e± 2 are just 1. The rescaling of the spin ﬁelds with the above phases guarantees that they satisfy the SU(2)2 algebra

+ 2˙ 1˙ 2˙ 2˙ 1˙ 2˙ [J0 , S ] = S , [J0−, S ] = S , (S )† = S . (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

π i ˜ ˜ π i ˜ ˜ +˙ i (p5 p6 p˜5 p˜6) (φ5 φ6) ˙ i (p5 p6 p˜5 p˜6) (φ5 φ6) S˜ ≡ e 2 − − − e 2 − , S˜− ≡ e− 2 − − − e− 2 − , (4.28)

π i ˜ ˜ π i ˜ ˜ 1˙ i (φ5+φ6) 2˙ i (φ5+φ6) S˜ ≡ e− 2 e− 2 , S˜ ≡ e 2 e 2 . (4.29)

Finally, we note the the fermionic zero modes act on the spin fields and map them Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point95 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

˙ π ˙ ˙ π ˙ 1 + i 2 1 2 + i 2 2 ψ0− S (0) = e− S (0), ψ0− S (0) = e− S (0),

+1˙ 1˙ +2˙ 2˙ ψ0 S−(0) = −S (0), ψ0 S−(0) = −S (0),

˙ ˙ π ˙ ˙ +2 1 +i 2 + 2 1 ψ0 S (0) = e S (0), ψ0− S (0) = S−(0),

˙ ˙ π ˙ ˙ +1 2 i 2 + 1 2 ψ0 S (0) = e− S (0), ψ0− S (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 deformation 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.

αA The deformation operators are constructed by applying modes of supercurrent G 1/2 − ˜αB˙ ββ˙ and G 1/2 to the super-chiral and anti-chiral primaries of the twist-2 sector, σ2 , 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:

αA ˜αB˙ ββ˙ Od ∝ AB αβ α˙ β˙ G 1 G 1 σ2 . (4.31) − 2 − 2 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point96

ββ˙ We now split the above twist operator into left-moving and right-moving parts σ2 (z, z¯) =

β β˙ A + +A σ2 (z)σ2 (z). It is shown in [185] (Appendix B) that G− 1 σ2 ∝ G 1 σ2−. Therefore, − 2 − 2 A + G− 1 σ2 is a singlet of SU(2)L by itself. The same reasoning holds for the right-moving − 2 sector. This could also be seen by the discussion in [226] (§5.1) that a single application

A + of supercharges G− 1 on a super-chiral primary σn constructs Virasoro primaries which − 2 + are annihilated by J0 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

A ˜ ˙ B ++˙ Od = AB G− 1 G− 1 σ2 . (4.32) − 2 − 2

The perturbation that we add to deform the symmetric product CFT away from the

orbifold point is Z 2 Sint = λ d z Od(z, z¯) + a.c., (4.33)

where a.c. refers to the anti-chiral ﬁelds acted on by the Hermitian conjugate supercurrent

modes.

√ αA αA˙ BA˙ The supercurrents are G = 2 ψ ∂X A˙B˙ , where the summation over an omit- ted target space copy index is implicit. Inserting this in (4.32) we obtain

√ h 1˙ 21˙ 2˙ 11˙ ˙ 1˙ 22˙ ˙ 2˙ 12˙ Od = 2 (ψ− ∂X − ψ− ∂X ) 1 (ψ˜− ∂X¯ − ψ˜− ∂X¯ ) 1 − 2 − 2 i 1˙ 22˙ 2˙ 12˙ ˜ ˙ 1˙ ¯ 21˙ ˜ ˙ 2˙ ¯ 11˙ ++˙ −(ψ− ∂X − ψ− ∂X ) 1 (ψ− ∂X − ψ− ∂X ) 1 σ2 . (4.34) − 2 − 2

The modes of the supercurrents in the left-moving sector are given by

I αA dz αA h+m 1 G = G z − , (4.35) m 2πi

˜αA˙ where h = 3/2. The same procedure holds for the right-moving supercurrents Gm . Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point97

˙ In the twist-2 sector, the insertion of super (anti-)chiral primary σ2±± operators in ˙ the base space corresponds to the insertion of spin ﬁelds S±± on the covering surface

˙ [172]. The operators σ2±± 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

1 b− 8 S± where b is speciﬁed by the map from the base space to the cover in the vicinity

2 2 of the insertion point of the spin field: (z − z0) ≈ b (t − t0) (1 + b1t + O(t )). 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’êtrefor subsection 4.2.2.

We will now construct Od. Using (4.35) we have:

I I A ˙ B ˙ dz dz¯ 3 1 3 1 A ˙ B ˙ ˜ ++ 2 2 1 2 2 1 ˜ ++ G− 1 G− 1 σ2 (0, 0) = z − − z¯ − − G− (z) G− (¯z) σ2 (0, 0) − 2 − 2 0 2πi 0 2πi 1 3 1 3 I 2 I ¯ 2 dt dz − dt dz¯ − A ˙ B 1 ++˙ → G− (t) G˜− (t¯) S (0) ¯ 1 1 0 2πi dt 0 2πi dt b 8 ¯b 8 I dt I dt¯ 1 1 = 1 1 × 2πi 2πi 2 ¯ 2 0 0 3 b1 2 ¯¯ 3 b1 ¯ ¯2 2bt(1 + 2 b t + O(t )) 2bt(1 + 2 ¯b t + O(t )) A ˜ ˙ B 1 ++˙ × G− (t) G− (t¯) 1 S (0) |b| 4 ¯ − 3 b1 2 − 3 b1 ¯2 2 I dt I dt¯ 1 4 b t + O(t ) 1 4 ¯b t + O(t ) ≈ 5 1 1 × 2|b| 4 0 2πi 0 2πi t 2 t¯2 1˙ 2˙A 2˙ 1˙A ˙ 1˙ 2˙B ˙ 2˙ 1˙B ++˙ × ψ− ∂X − ψ− ∂X (t) ψ˜− ∂X¯ − ψ˜− ∂X¯ (t¯) S (0, 0),(4.36)

where the arrow in the second line implies that we passed from the base to the covering

2 2 sphere using the map which is locally of the form z ≈ b t (1 + b1t + O(t )). Here and in what follows, we simplify notation by suppressing the contribution from the Liouville Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point98 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

I I ¯ A ˙ B ++˙ 1 dt dt 1 1 G− 1 G˜− σ (0, 0) = × 2 5 ¯ − 2 |b| 4 0 2πi 0 2πi t t i 2˙A 2˙B iφ5(t) iφ˜5(t¯) ( φ6+φ5 φ˜6+φ˜5)(0,0) : ∂X ∂X¯ e− − e 2 − − :

i 2˙A 1˙B iφ5(t)+iφ˜6(t¯) ( φ6+φ5 φ˜6+φ˜5)(0,0) + : ∂X ∂X¯ e− e 2 − − :

i 1˙A 2˙B +iφ6(t) iφ˜5(t¯) ( φ6+φ5 φ˜6+φ˜5)(0,0) + : ∂X ∂X¯ e − e 2 − − :

i 1˙A 1˙B +iφ6(t)+iφ˜6(t¯) ( φ6+φ5 φ˜6+φ˜5)(0,0) + : ∂X ∂X¯ e e 2 − − :

1 ˙ ˙ i ˜ ˜ ˙ ˙ i ˜ ˜ 2A ¯ 2B 2 ( φ6 φ5 φ6 φ5) 2A ¯ 1B 2 ( φ6 φ5+φ6+φ5) = 5 : ∂X ∂X e − − − − : + : ∂X ∂X e − − : |b| 4 i i 1˙A 2˙B (+φ6+φ5 φ˜6 φ˜5) 1˙A 1˙B (+φ6+φ5+φ˜6+φ˜5) + : ∂X ∂X¯ e 2 − − : + : ∂X ∂X¯ e 2 : (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

1 i ˜ ˜ ˙ ˙ ˙ ˙ 2 ( φ6 φ5 φ6 φ5) 21 ¯ 22 22 ¯ 21 Od (0, 0) = 5 : e − − − − ∂X ∂X − ∂X ∂X : |b| 4 i ( φ6 φ5+φ˜6+φ˜5) 21˙ 12˙ 22˙ 11˙ + : e 2 − − ∂X ∂X¯ − ∂X ∂X¯ :

i (+φ6+φ5 φ˜6 φ˜5) 11˙ 22˙ 12˙ 21˙ + : e 2 − − ∂X ∂X¯ − ∂X ∂X¯ : i (+φ +φ +φ˜ +φ˜ ) 11˙ 12˙ 12˙ 11˙ + : e 2 6 5 6 5 ∂X ∂X¯ − ∂X ∂X¯ : (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 σ2−−. The Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point99

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

˙ ˜˙ ++˙ σ2−−(z0, z¯0) = J0−(z) J0−(¯z) σ2 (z0, z¯0). (4.39)

The anti-chiral part of the deformation operator is of the form

+A ˜+˙ B ˙ +A ˜+˙ B ˜˙ ++˙ G 1 (z0) G 1 (¯z0) σ2−− = G 1 (z0) G 1 (¯z0) J0−(z) J0−(¯z) σ2 (z0, z¯0). (4.40) − 2 − 2 − 2 − 2

˙ +A +˙ A A ˙ A One can then pass J − and J˜− to the left of G and G˜ which gives G− and G˜−

++˙ acting on σ2 (z0, z¯0). Therefore the two chiral and anti-chiral terms are identical and

the full deformation operator is of the form λ Od, 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 αβ and α˙ β˙ 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 summation 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

ij i j δ hφ (z1, z¯1) φ (z2, z¯2)i0 = , (4.41) hi h˜i z12 z¯12 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point100

where the subscript “0” refers to the unperturbed CFT, hi (h˜i) 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:

Z X 2 δS = λA d zOd,A(z, z¯), (4.42) A

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

ij ij i j δ δ hφ (z1, z¯1) φ (z2, z¯2)iλ = = „ « A 2h (λ ) 2h˜ (λ ) ∂h˜ (λ ) i A i A “ ∂hi(λA) ” h˜ λ i A z z¯ 2 hi+λA 2 i+ A ∂λ 12 12 ∂λA A z12 z¯12 „ ˜ « ∂hi(λA) ∂hi(λA) 2λA ln(z12)+ ln(¯z12) − ∂λA ∂λA i j = e hφ (z1, z¯1) φ (z2, z¯2)i0

˜ !! ∂hi(λA) ∂hi(λA) i j ≈ 1 − 2λA ln(z12) + ln(¯z12) hφ (z1, z¯1)φ (z2, z¯2)i0, (4.43) ∂λA ∂λA

∂hi(λA) ∂h˜i(λA) where hi(λA) = hi+λA and h˜i(λA) = h˜i+λA to the ﬁrst order in perturbation ∂λA ∂λA Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point101 theory. We then have

∂ i j hφ (z1, z¯1) φ (z2, z¯2)iλA = ∂λA ˜ ! ∂hi(λA) ∂hi(λA) i j −2 ln(z12) − 2 ln(¯z12) hφ (z1, z2) φ (z2, z¯2)i0.(4.44) ∂λA ∂λA

If the derivatives of the conformal weights (hi(λA), h˜i(λA)) vanish to the ﬁrst order, then one moves to the second order in perturbation theory

2 ∂ i j 2 hφ (z1, z¯1) φ (z2, z¯2)iλA ∂λA 2 2 ! ∂ hi(λA) ∂ h˜i(λA) = −2 ln(z ) − 2 ln(¯z ) hφ (z , z¯ ) φ†(z , z¯ )i , (4.45) ∂λ2 12 ∂λ2 12 i 1 1 j 2 2 0 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

R 2 R Sfree+λA d z d,A(z,z¯) i j i j d[X, ψ] e− O φ (z1, z¯1) φ (z2, z¯2) hφ (z1, z¯1) φ (z2, z¯2)iλA = R 2 R Sfree+λA d z d,A(z,z¯) d[X, ψ] e− O

R Sfree R 2 2 i j d[X, ψ] e− 1 + λA d zOd,A(z, z¯) + O(λA) φ (z1, z¯1) φ (z2, z¯2) = , (4.46) R Sfree R 2 2 d[X, ψ] e− 1 + λA d zOd,A(z, z¯) + O(λA) where we expanded the perturbative terms to the ﬁrst order in perturbation theory in the second line. The above equation reads

i j hφ (z1, z¯1) φ (z2, z¯2)iλA i j R 2 i j 2 hφ (z1, z¯1) φ (z2, z¯2)i0 + λA d zhφ (z1, z¯1) Od,A(z, z¯) φ (z2, z¯2)i + O(λA) = R 2 2 1 + λA d zhOd,A(z, z¯)i + O(λA) Z i j 2 i j 2 = hφ (z1, z¯1) φ (z2, z¯2)i0 + λA d zhφ (z1, z¯1) Od,A(z, z¯) φ (z2, z¯2)i + O(λA)(4.47), Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point102

where hOd,A(z, z¯)i = 0. We therefore obtain

Z ∂ i j 2 i j hφ (z1, z¯1) φ (z2, z¯2)iλA = d zhφ (z1, z¯1) Od,A(z, z¯) φ (z2, z¯2)i. (4.48) ∂λA

One then needs to evaluate the above three-point function

Z Z 2 i j 2 d zhφ (z1, z¯1) Od,A(z, z¯) φ (z2, z¯2)i = CiAj d z × (4.49) 1 × , hi+1 hj hj +1 hi hi+hj 1 h˜i+1 h˜j h˜j +1 h˜i h˜i+h˜j 1 (z1 − z) − (z − z2) − (z12) − (¯z1 − z¯) − (¯z − z¯2) − (¯z12) −

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| > and

|z − z2| > . As discussed in [230, 231], one can make the SL(2,C) transformation

z z y(z) = 12 , (4.50) z12 − z which brings the above integral to the form

y = 1 iπ(h h +1 (h˜ h˜ +1)) Z | | e j i j j dydy¯ − − − (4.51) hj hi+1 h˜j h˜i+1 y = 2 y − y¯ − | | |z12| 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 ofz ¯ij, 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 = h˜j − h˜i + 1

hj − h˜j = hi − h˜i. (4.52) Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point103

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, h˜i = h˜j (hence, i = j,

i j or explicitly φ = φ given our assumption), and 2) hi =6 hj, h˜i =6 h˜j (i =6 j) [232]. For hi = hj the integral reads

Z 2 i i d zhφ (z1, z¯1) Od,A(z, z¯) φ †(z2, z¯2)i = Z C d2z iAi = 1 1 2hi 1 1 1 2h˜i 1 (z1 − z) (z − z2) (z12) − (¯z1 − z¯) (¯z − z¯2) (¯z12) − Z 2 1 2 |z12| CiAi d z . (4.53) h h˜ 2 2 2 i 2 i |z1 − z| |z − z2| z12 z¯12

The SL(2,C) transformation (4.50) simpliﬁes the integral in the third line of the above equation

Z 2 Z 2 CiAi 2 |z12| CiAi d y ˜ d z = ˜ (4.54) 2hi 2hi 2 2 2hi 2hi 2 2 z z1 >, z z2 > |z1 − z| |z − z2| < y < z / |y| z12 z¯12 | − | | − | z12 z¯12 | | | 12 2 i i = 2πCiAi ln(z12) + 2πCiAi ln(¯z12) − 2πCiAi ln( ) hφ (z1, z¯1) φ †(z2, z¯2)i0.

The -dependent term which diverges in the limit → 0 has to be absorbed in the

renormalization of φi.

For hi =6 hj the integral (4.49) reads:

Z Z 2 i j 2 d zhφ (z1, z¯1) Od,A(z, z¯) φ (z2, z¯2)i = CiAj d z × (4.55) 1 × . hi+1 hj hj +1 hi hi+hj 1 h˜i+1 h˜j h˜j +1 h˜i h˜i+h˜j 1 (z1 − z) − (z − z2) − (z12) − (¯z1 − z¯) − (¯z − z¯2) − (¯z12) − Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point104

Performing the same SL(2,C) transformation (4.50) we obtain:

Z 2(hi hj ) Z 2 2 i j |z12| − d y d zhφ (z1, z¯1) Od,A(z, z¯) φ (z2, z¯2)i = δs ,s CiAj h +h h˜ +h˜ i j 2(hi hj +1) i j i j < x < z2 / |y| − z12 z¯12 | | | 12 ! dj di di dj − 1 − 1 = −2π δsi,sj CiAj ˜ + ˜ , (4.56) dj − di 2hj 2hj di − dj 2hi 2hi z12 z¯12 z12 z¯12

i where where si = hi − h˜i is the spin and di ≡ hi + h˜i is the scaling dimension of φ . In the limit → 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

dj di X − φi → φi + λ π ln(2) C φi + λ 2π δ C φj. (4.57) iAi si,sj d − d iAj j j i

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:

∂h ∂h˜ i = −πC , i = −πC . (4.58) ∂λ iAi ∂λ iAi

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 lnz ¯12, determine the anomalous 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, h˜i = h˜j or hi =6 hj, h˜i =6 h˜j. Therefore, this does not aﬀect the sector where hi =6 hj because this requirement means that the ﬁelds Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point105

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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point106

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 |φii with non-degenerate conformal weight (hi, h¯i) is given by (4.58)

∂h ∂h˜ i = −πC , i = −πC , ∂λ iAi ∂λ iAi

i i where CiAi are the structure constants corresponding to the three-point functions hφ Od φ †i. 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-

i j tween φ and other quasi-primary operators φ with the same conformal weight (hi, h˜i). 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

i i φ (z1, z¯1) Od(z2, z¯2) Od(z3, z¯3) φ †(z4, z¯4) . (4.59)

We take the coincidence limit as z1 → z2, z3 → z4 (or equivalently z1 → z3, z2 → z4) and Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point107

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

k this leading singularity. The sum over CiA 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 the descendants of quasi-primaries under the Virasoro algebra. As shown in section 4.2.3, it turns out that one needs consider only contributions from quasi-primary (i.e., non-derivative) operators. Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point108

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 PN i ¯ i marginal operator κ=1 ∂x(κ)∂xi (κ), where x , 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 N X i ¯ X AA˙ ¯ BB˙ ∂x(κ)∂xi (κ) = −AB A˙B˙ ∂X(κ) ∂X(κ) . (4.60) κ=1 κ=1

The dilaton operator is self-conjugate. The four-point function which we would like to compute is of the form:

* N X AA˙ ¯ BB˙ −AB A˙B˙ ∂X(κ) ∂X(κ) (z1, z¯1) λ Od(z2, z¯2) × κ=1 N + X A˙ 0A0 ¯ B˙ 0B0 λ Od(z3, z¯3) −A0B0 A˙ 0B˙ 0 ∂X(κ0) ∂X(κ0) (z4, z¯4) . (4.61) κ0=1

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, Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point109

b ≡ z2 − z3, and a2 ≡ z4 − z3, we obtain

* N X AA˙ ¯ BB˙ ¯ −AB A˙B˙ ∂X(κ) ∂X(κ) (a1, a¯1) λ Od(b, b) × κ=1 N + X A˙ 0A0 ¯ B˙ 0B0 λ Od(0, 0) −A0B0 A˙ 0B˙ 0 ∂X(κ0) ∂X(κ0) (a2, a¯2) . (4.62) κ0=1

0 The deformation operator contains a twist-2 operator σ2 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) ≡ AA˙ ¯ BB˙ −AB A˙B˙ ∂X(κ) ∂X(κ) . The four-point function (4.62) is rewritten as

N N D X ¯ X E φdil (κ)(a1, a¯1) λOd(b, b) λOd(0, 0) φdil (κ0)(a2, a¯2) = κ=1 κ0=1 2 2 D X ¯ X E φdil (κ)(a1, a¯1) λOd(b, b) λOd(0, 0) φdil (κ0)(a2, a¯2) + κ=1 κ0=1 2 N D X ¯ X E φdil (κ)(a1, a¯1) λOd(b, b) λOd(0, 0) φdil (κ0)(a2, a¯2) + κ=1 κ0=3 N 2 D X ¯ X E φdil (κ)(a1, a¯1) λOd(b, b) λOd(0, 0) φdil (κ0)(a2, a¯2) + κ=3 κ0=1 N N D X ¯ X E φdil (κ)(a1, a¯1) λOd(b, b) λOd(0, 0) φdil (κ0)(a2, a¯2) . (4.63) κ=3 κ0=3

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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point110

correlation function in the last line factorizes and we obtain

N N D X ¯ X E φdil (κ)(a1, a¯1) λOd(b, b) λOd(0, 0) φdil (κ0)(a2, a¯2) = κ=1 κ0=1 2 2 D X ¯ X E φdil (κ)(a1, a¯1) λOd(b, b) λOd(0, 0) φdil (κ0)(a2, a¯2) + κ=1 κ0=1 N N D X X ED ¯ E φdil (κ)(a1, a¯1) φdil (κ0)(a2, a¯2) λOd(b, b) λOd(0, 0) . (4.64) κ=3 κ0=3

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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point111

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

t2 z = b . (4.65) 2t − 1

In the vicinity of the two insertion points we have:

2 3 4 z → b, t → 1, (z − b) ≈ b1(t − 1) + b10 (t − 1) + O((t − 1) ),

2 3 4 z → 0, t → 0, z ≈ b0 t + b0 t + O(t ), (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 ±

1 p t k = ak ± ak(ak − b) . (4.67) ± b

In the vicinity of a generic point the map is of the form:

2 3 (z − ak) ≈ ξ k (t − t k) + ξ0 k (t − t k) + O((t − t k) ), (4.68) ± ± ± ± ± Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point112 where dz 2 ak (ak − b) ξ k = = . (4.69) ± dt t t b t k (t k − 1) = ±k ± ±

There are two diﬀerent contributions to the four-point function (4.64): a bare-twist

0 part and a mode-insertion part. The bare-twist part contains the two insertions of σ2 twists, and is computed using the Liouville action. Using the Lunin-Mathur method, the normalized two-point function is

0 ¯ 0 hσ2(b, b) σ2(0, 0)i 4 h 0 − σ2 0 0 = |b| , (4.70) hσ2(1, 1) σ2(0, 0)i

where h 0 = 3/8. σ2

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

I I AA˙ BB˙ dz dz¯ 1 1 1 1 1 1 AA˙ ¯ BB˙ α 1 (κ) α˜ 1 (κ) = (z − ak) − − (¯z − a¯k) − − ∂X(κ) (z) ∂X(κ) (¯z) − − ak 2πi a¯k 2πi I 1 1 I ¯ 1 1 dt dz − 1 dt dz¯ − 1 AA˙ BB˙ → (z − a )− (¯z − a¯ )− ∂X (t) ∂X¯ (t¯) k ¯ k t±k 2πi dt t¯±k 2πi dt I dt ∂XAA˙ (t) I dt¯ ∂X¯ BB˙ (t¯) = t±k 2πi t¯±k 2πi ξ k(t − t k) 1 + O(t − t k) ξ¯ k(t¯− t¯ k) 1 + O(t¯− t¯ k) ± ± ± ± ± ± 1 AA˙ ¯ BB˙ ¯ = 2 ∂X (t k) ∂X (t k), (4.71) |ξ k| ± ± ± 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. Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point113

The dilaton operator is then given by

−1 t ¯ ≡ AA˙ ¯ BB˙ ¯ φdil(t k, t k) 2 A˙B˙ AB ∂X (t k) ∂X (t k), (4.72) ± ± |ξ k| ± ± ± 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

X X t ¯ t t t ¯ φdil(t 1, t 1) λOd(1, 1) λOd(0, 0) φdil(t 2, t 2) , (4.73) ± ± ± ± t±1 t±2

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

3 t t t t 2 ¯ ¯ |b|− φdil(t 1, t 1) λOd(1, 1) λOd(0, 0) φdil(t 2, t 2) = ± ± ± ±

2 ( 2 2 2 2 λ 2 2 2 2 |t 1| |t 1 − 1| |t 2| | t 2 − 1| | |− | |− | − |− | − |− ± ± ± ± a1 a2 a1 b a2 b 8 4 4 |t 1 − t 2| ± ± 2 2 2 2 (t 1)(t 2 − 1) |t 1 − 1| |t 2| (t 1 − 1) (t 2) |t 1| |t 2 − 1| ± ± ± ± ± ± ± ± +2 2 + 2 2 (t¯ 1)(t¯ 2 − 1) (t 1 − t 2) (t¯ 1 − 1) (t¯ 2)(t 1 − t 2) ± ± ± ± ± ± ± ± 2 2 2 2 (t¯ 1)(t¯ 2 − 1) |t 1 − 1| |t 2| (t¯ 1 − 1) (t¯ 2) |t 1| |t 2 − 1| ± ± ± ± ± ± ± ± +2 2 + 2 2 (t 1)(t 2 − 1) (t¯ 1 − t¯ 2) (t 1 − 1) (t 2)(t¯ 1 − t¯ 2) ± ± ± ± ± ± ± ± 2 2 2 2 |t 1| |t 2 − 1| |t 1 − 1| |t 2| ± ± ± ± +2 2 2 + 2 2 2 |t 1 − 1| |t 2| |t 1| |t 2 − 1| ± ± ± ± ) (t 1)(t 2 − 1) (t¯ 1 − 1) (t¯ 2) (t 1 − 1) (t 2)(t¯ 1)(t¯ 2 − 1) − ± ± ± ± − ± ± ± ± . (4.74) (t 1 − 1) (t 2)(t¯ 1)(t¯ 2 − 1) (t 1)(t 2 − 1) (t¯ 1 − 1) (t¯ 2) ± ± ± ± ± ± ± ±

We normalize the four point function by two-point functions as in Chapter 3:

D E t ¯ t t t ¯ φdil(t 1, t 1) λOd(1, 1) λOd(0, 0) φdil(t 2, t 2) ± ± ± ± . (4.75) t t D t t E φdil(0, 0) φdil(1, 1) Od(0, 0) Od(1, 1) Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point114

(The normalization of the bare twist contribution has been accounted for in (4.70)). The two-point functions are found to be

D t t E Od(0, 0) Od(1, 1) = 8, (4.76)

t t φdil(0, 0) φdil(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

D E X X t ¯ t t t ¯ φdil(t 1, t 1) λOd(1, 1) λOd(0, 0) φdil(t 2, t 2) = ± ± ± ± t±1 t±2 t ¯ t ¯ t φdil(t+1, t+1) + φdil(t 1, t 1) λOd(1, 1) × − − E t t ¯ t ¯ λOd(0, 0) φdil(t+2, t+2) + φdil(t 2, t 2) . (4.78) − − Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point115

Inserting the four-point function evaluated in the previous section (4.74), the complete normalized four-point function reads

2 2 D X ¯ X E φdil (κ)(a1, a¯1) λOd(b, b) λOd(0, 0) φdil (κ0)(a2, a¯2) = κ=1 κ0=1 λ2 2 4 4 + 2 |a1 − a2| |b| ¯ 1 1 a1 (a2 b) 1 1 a¯1 (¯a2 b) 2 2 1 + − ¯ 2 2 1 + ¯− 1 (a1 − b) a2 (a1 b) a2 (¯a1 − b) a¯2 (¯a1 b)a ¯2 +2− − − + 1 1 2 2 1 1 2 ¯2 2 2 (a1 − a2) b 2 ¯ 2 (¯a1 − a¯2) b a1 (a2 − b) a¯1 (¯a2 − b) ¯ a1 (a2 b) 1 1 a¯1 (¯a2 b) 1 + − ¯ 2 2 1 + ¯− 3 (a1 b) a2 (¯a1 − b) a¯2 (¯a1 b)a ¯2 +2− − − + 3 1 1 3 1 1 2 ¯2 2 2 2 2 2 ¯ 2 (¯a1 − a¯2) b a1 (a1 − b) a2 (a2 − b) a¯1 (¯a2 − b) ¯ 1 1 a1 (a2 b) a¯1 (¯a2 b) 2 2 1 + − 1 + ¯− 3 (a1 − b) a2 (a1 b) a2 (¯a1 b)a ¯2 +2− − − + 1 1 2 2 3 1 1 3 2 2 (a1 − a2) b 2 ¯ 2 2 ¯ 2 a1 (a2 − b) a¯1 (¯a1 − b) a¯2 (¯a2 − b) 1 3 3 1 1 3 3 1 3 − 2 2 − 2 2 − 2 ¯ 2 − 2 ¯ 2 +2− a1 (a1 − b)− a2 (a2 − b)− a¯1 (¯a1 − b)− a¯2 (¯a2 − b)− +

3 1 1 3 3 1 1 3 3 − 2 2 − 2 2 − 2 ¯ 2 − 2 ¯ 2 +2− a1 (a1 − b)− a2 (a2 − b)− a¯1 (¯a1 − b)− a¯2 (¯a2 − b)− +

1 3 3 1 3 1 1 3 4 − 2 2 − 2 2 − 2 ¯ 2 − 2 ¯ 2 −2− a1 (a1 − b)− a2 (a2 − b)− a¯1 (¯a1 − b)− a¯2 (¯a2 − b)− +

3 1 1 3 1 3 3 1 4 − 2 2 − 2 2 − 2 ¯ 2 − 2 ¯ 2 −2− a1 (a1 − b)− a2 (a2 − b)− a¯1 (¯a1 − b)− a¯2 (¯a2 − b)− . (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, a¯1) → (0, 0) and (a2, a¯2) → (b, ¯b), and consider the leading singularity. The

OPE of two quasi-primary operators O1 and O2 has the form [233]

˜ X X p k,k hp h1 h2+K h˜p h˜1 h˜2+K˜ k,k˜ O1(z, z¯) O2(0, 0) = C12{ }z − − z¯ − − Op{ }(0, 0), (4.80) p k,k˜ { } Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point116

where Op is a quasi primary, {k, k˜} denotes a collection of indices ki and k˜i which corre- P ˜ P ˜ 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-

3/2 3/2 morphic part of the four-point function scales as a1− (a2 − b)− . 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 singularities 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

a b ¯ c ¯ d ∂Xm (κ1) ∂Xn (κ2) ∂Xk (κ3) ∂Xl (κ4) |0iR, (4.81)

where κi corresponds to the copy of the target space and |0iR 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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point117

is given by

N N X X a b ¯ c ¯ d φst (κ) |0iR ≡ δab δcd ∂X 1 (κ) ∂X 1 (κ) ∂X 1 (κ) ∂X 1 (κ) |0iR. (4.82) − − − − κ=1 κ=1

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 N N X X Rh0| φst (κ) λOd(z, z¯) λOd(z0, z¯0) φst (κ0)|0iR. (4.83) κ=1 κ0=1 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 spectral 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

αm φ(z) → z− 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

++ ++ σ2 with R-charge (1/2, 1/2) under SU(2)L ×SU(2)R. The holomorphic part of σ2 thus

+ α/2 + transforms as σ2 (z) → z σ2 (z) under the spectral ﬂow. The supercurrents transform Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point118

A α/2 A as G− (z) → z− G− (z). We then have

I I dz0 dz0 α α A + A + 2 A 2 + G− 1 (z0) σ2 (z) = G− (z0) σ2 (z) → z0− G− (z0) z σ2 (z) − 2 z 2πi z 2πi I dz0 α 1 A + = 1 − (z0 − z)z− + ··· G− (z0) σ2 (z) z 2πi 2 A + α 1 A + − 0 − − − 0 ··· = G 1 (z ) σ2 (z) z G+ 1 (z ) σ2 (z) + , (4.85) − 2 2 2 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

N N X X NSh0| φst (κ) λOd(z, z¯) λOd(z0, z¯0) φst (κ0)|0iNS. (4.86) κ=1 κ0=1

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 denote 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 P2 κ=1 φst (κ0). We will next explicitly compute φst on the covering surface and then evaluate 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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point119

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)

3/2 and is given by |b|− . To evaluate the mode insertion contribution to the four-point function, we lift the operators to the covering sphere.

a ¯ b ¯ The operator φst = ∂X 1 (κ) ∂Xa, 1 (κ) ∂X 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

I I a b dz1 1 1 1 a dz2 1 1 1 b ∂X 1 (κ) ∂X 1 (κ) = (z1 − ak) − − ∂X(κ)(z1) (z2 − ak) − − ∂X(κ)(z2) − − ak 2πi ak 2πi I 1 1 I 1 1 dt1 dz1 − 1 a dt2 dz2 − 1 b → (z1 − ak)− ∂X (t1) (z2 − ak)− ∂X (t2) t±k 2πi dt1 t±k 2πi dt2 I dt ∂Xa(t ) I dt ∂Xb(t ) = 1 1 2 2 2πi 2πi t±k ξ k(t1 − t k) 1 + O(t1 − t k) t±k ξ k(t2 − t k) 1 + O(t2 − t k) ± ± ± ± ± ± I a dt1 ∂X (t1) 1 b = ∂X (t k) t 2πi ξ k ± ±k ξ k(t1 − t k) 1 + O(t1 − t k) ± ± ± ± h 2 3i I 1 + η1(t1 − t k) + η2(t1 − t k) + O (t1 − t k) dt1 1 ± ± ± a b = ∂X (t1) ∂X (t k) 2 ± t±k 2πi ξ k t1 − t k ± ± 1 δab a b − = 2 ∂X ∂X (t k) 2 2 , (4.87) ξ k ± 4(t k) (t k − 1) ± ± ± 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} =

2 2 −3(t − 1)− t− /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

8 hφst(0, 0) φst(1, 1)i = 2 , 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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point120 contribution, the four-point function normalized by the two-point functions is evaluated

3 D t t t t E 2 ¯ ¯ |b|− φst(t 1, t 1) λOd(1, 1) λOd(0, 0) φst(t 2, t 2) = (4.88) ± ± ± ± 4 |b| 2 4 4 4 4 λ |a |− |a |− |a − b|− |a − b|− × 216 1 2 1 2 ( 2 2 2 2 2 2 2 2 ) (t 1 − 1) (t 2) (t 1) (t 2 − 1) (t 1) (t 1 − 1) (t 2) (t 2 − 1) ± ± ± ± ± ± ± ± × 5 + 4 2 + 4 2 + 8 4 (t 1 − t 2) (t 1 − t 2) (t 1 − t 2) ± ± ± ± ± ± ( 2 2 2 2 2 2 2 2 ) (t¯ 1 − 1) (t¯ 2) (t¯ 1) (t¯ 2 − 1) (t¯ 1) (t¯ 1 − 1) (t¯ 2) (t¯ 2 − 1) ± ± ± ± ± ± ± ± 5 + 4 2 + 4 2 + 8 4 . (t¯ 1 − t¯ 2) (t¯ 1 − t¯ 2) (t¯ 1 − t¯ 2) ± ± ± ± ± ±

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 2 D X ¯ X E φst (κ)(a1, a¯1) λOd(b, b) λOd(0, 0) φst (κ0)(a2, a¯2) = κ=1 κ0=1 ( 4 2 ¯ 2 |b| 2 4 4 4 4 25 (R + 1) (R + 1) λ |a |− |a |− |a − b|− |a − b|− + 5 + 212 1 2 1 2 4 (R − 1)2 (R¯ − 1)2

1 1 2 2 R 2 (R + 1) R¯ 2 (R¯ + 1) (R + 1) (R¯ + 1) +16 + 4 (R − 1)2 (R¯ − 1)2 (R − 1)2 (R¯ − 1)2 R (R2 + 6R + 1) R¯ (R¯2 + 6R¯ + 1) +10 + (R − 1)4 (R¯ − 1)4 1 3 3 1 R 2 (R + 1) R¯ 2 (R¯ + 1) R 2 (R + 1) R¯ 2 (R¯ + 1) +64 + (R − 1)2 (R¯ − 1)4 (R − 1)4 (R¯ − 1)2 (R + 1)2 R¯ (R¯2 + 6 R¯ + 1) R (R2 + 6 R + 1) (R¯ + 1)2 +8 + (R − 1)2 (R¯ − 1)4 (R − 1)4 (R¯ − 1)2 3 3 2 2 ) R 2 (R + 1) R¯ 2 (R¯ + 1) R (R + 6 R + 1) R¯ (R¯ + 6 R¯ + 1) +256 + 16 . (4.89) (R − 1)4 (R¯ − 1)4 (R − 1)4 (R¯ − 1)4

Note that this is the correct form for the 4-point function for weights (2,2), (1,1), (1,1),

(2,2). Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point121

4.4.3 Coincidence limit and operator mixing

We investigate the operator mixing by taking the coincidence limit (a1, a¯1) → (0, 0)

and (a2, a¯2) → (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

( a1 ¯b − (a2 − b) ¯b +a ¯1 b − (¯a2 − ¯b) b 12 81 1 81 2− + 2 2 2 ¯ 2 4 2 2 2 ¯ 2 6 4 a1 (a2 − b) a¯1 (¯a2 − b) |b| 2 a1 (a2 − b) a¯1 (¯a2 − b) |b| 27 + − 5 a (a − b) ¯b2 − 5a ¯ (¯a − ¯b) b2 2 2 2 ¯ 2 8 1 2 1 2 a1 (a2 − b) a¯1 (¯a2 − b) |b| 2 2 2 2 + 3 a1 a¯1 |b| − 3 a1 (¯a2 − ¯b) |b| − 3 (a2 − b)a ¯1 |b| + 3 (a2 − b) (¯a2 − ¯b) |b| 54 + 5 a (a − b) (¯a − ¯b) b ¯b2 − 5 a (a − b)a ¯ b¯b2 2 2 2 ¯ 2 10 1 2 2 1 2 1 a1 (a2 − b) a¯1 (¯a2 − b) |b| 2 2 − 5 a1 a¯1 (¯a2 − ¯b) b ¯b + 5 (a2 − b)a ¯1 (¯a2 − ¯b) b ¯b ) 900 + + ··· , (4.90) 8 a1 (a2 − b)a ¯1 (¯a2 − ¯b) |b|

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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point122

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

D 2 E 2hd 2h˜d X CiAA lim y y¯ φst (κ)(a1, a¯1) λOd(0, 0) λOd(y, y¯) = (4.91) y,y¯ hst+hd hd h˜st+h˜d h˜d →∞ κ=1 a1 − a¯1 − where the indices i and A in CiAA correspond to φst and the deformation, respectively.

(hst, h˜st) and (hd, h˜d) are the conformal weights of φst and the deformation, respectively. The above three-point function, normalized by two-point functions, is

C 18 1 iAA = , (4.92) h h h h˜ h˜ h˜ 8 4 st+ d d st+ d d 2 |a1| a1 − a¯1 −

7 which then gives CiAA = 9 × 2− . Since we have normalized the operators by their two-point functions, the coeﬃcient of the three-point function and the coeﬃcient of the

A 0,0 most singular term in the OPE (4.80) are equal: CiAA = CiA{ }. Denoting the complete deformation operator as OA, we ﬁnd the operator algebra

2 X X A k,k˜ 2+K 2+K˜ k,k˜ φst (κ)(a1, a¯1) OA(0, 0) = CiA{ }a1− a¯1− OA{ }(0, 0), (4.93) κ=1 k,k˜ { } where the deformation operator is the ancestor of this conformal family. The operator P2 ¯ algebra for the other two operators κ=1 φst (κ)(a2, a¯2) and OA(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

* X A k0,k˜0 2+K0 ¯ 2+K˜ 0 k0,k˜0 ¯ CiA0 { }(a2 − b)− (¯a2 − b)− OA{ } (b, b) × k0,k˜0 { } + X A k,k˜ 2+K 2+K˜ k,k˜ CiA{ }a−1 a¯1− OA{ } (0, 0) . (4.94) k,k˜ { } Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point123

Let us ﬁrst consider the most singular terms of the two OPEs which corresponds to the

ancestor ﬁeld with {k, k˜} = {k0, k˜0} = {0, 0}. The above two-point function then reads

* A 0,0 A 0,0 + CiA0 { } 0,0 CiA{ } 0,0 O{ } (b, ¯b) O{ } (0, 0) 2 ¯ 2 A 2 2 A (a2 − b) (¯a2 − b) a1 a¯1 1 81 1 = 12 4 4 4 . (4.95) 2 4 |a1| |a2 − b| |b|

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

1,0 0,1 1,1 will only need to find the contribution of the descendants OA{ }, OA{ }, and OA{ }, 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,

˜ X X p k,k hp h1 h2+K h˜p h˜1 h˜2+K˜ k,k˜ O1(z, z¯) O2(0, 0) = C12{ }z − − z¯ − − Op{ }(0, 0), p k,k˜ { }

p k,k˜ the structure constants of the descendants C12{ } are determined by the structure constant of the ancestor ﬁeld through the relation

p k,k˜ p 0,0 p k ˜p k˜ C12{ } = C12{ } β12{ } β12{ }, (4.96)

k ˜ k˜ where β12{ } and β12{ } are coeﬃcients which depend only on the central charge and the Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point124

conformal weights. For the descendant states at level 1 the coeﬃcients are given by

˜ ˜ ˜ p 1 hp + h1 − h2 ˜p 1 hp + h1 − h2 β12{ } = , β12{ } = . (4.97) 2 hp 2 h˜p

In our case, hp = 1, h1 = 2, and h2 = 1 and we have:

A 1 ˜A 1 βiA{ } = 1, βiA{ } = 1. (4.98)

The structure constants then read

A 1,1 A 1,0 A 0,1 A 0,0 9 C { } = C { } = C { } = C { } = . (4.99) iA iA iA iA 27

k,k˜ k0,k˜0 ¯ We can now calculate the contribution of the descendant ﬁelds OA{ }(0, 0) and OA{ }(b, b) in (4.94). As mentioned earlier, for each OPE, we are interested in the three level-1 de-

1,0 0,1 1,1 scendants (OA{ },OA{ }, and OA{ }). Therefore there are nine terms to evaluate. For example, for {k, k˜} = {1, 0} and {k0, k˜0} = {0, 0}, we have K = 1, K˜ = K0 = K˜ 0 = 0, and obtain

* A 0,0 A 1,0 + CiA0 { } 0,0 CiA{ } 1,0 O{ } (b, ¯b) O{ } (0, 0) = 2 ¯ 2 A 2 A (a2 − b) (¯a1 − b) a1 a¯1 1 81 1 1 0,0 ¯ 0,0 L 1 hOA{ }(b, b) OA{ }(0, 0)i = 12 2 2 ¯ 2 − 2 4 a1 (a2 − b) a¯1 (¯a2 − b)

1 81 1 1 0,0 ¯ 0,0 ∂zhOA{ }(b, b) OA{ }(z, z¯)i = 212 4 a (a − b)2 a¯2 (¯a − ¯b)2 z 0 1 2 1 2 → 1 81 a ¯b 1 . (4.100) 12 2 2 2 ¯ 2 6 2 2 a1 (a2 − b) a¯1 (¯a2 − b) |b|

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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point125

( 12 27 1 2 1 2 2− − a (a − b)¯b − a¯ (¯a − ¯b) b 2 2 2 ¯ 2 8 1 2 1 2 a1 (a2 − b) a¯1 (¯a2 − b) |b| 2 2 27 + a (a − b) (¯a − ¯b) b¯b2 − a (a − b)a ¯ b¯b2 2 2 2 ¯ 2 10 1 2 2 1 2 1 a1 (a2 − b) a¯1 (¯a2 − b) |b| 2 2 − a1 a¯1 (¯a2 − ¯b) b ¯b + (a2 − b)a ¯1 (¯a2 − ¯b) b ¯b ) 171 + + ··· . (4.101) 8 a1 (a2 − b)a ¯1 (¯a2 − ¯b) |b|

The ﬁrst line of the above equation has two terms in it

27 1 − , (4.102) 2 ¯ 2 4 ¯2 2 a1 (a2 − b)a ¯1 (¯a2 − b) b b 27 1 − . (4.103) 2 2 ¯ 2 ¯4 2 a1 (a2 − b) a¯1 (¯a2 − b) b b

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 contribute 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 p k ˜p k˜ are the coefficients β12{ } and β12{ }. We are only interested in the level-1 descendants p 1 ˜p 1˜ which have conformal weights (2, 2). The values of β12{ } and β12{ } are therefore the same p 1 ˜p 1˜ as above (4.98): β12{ } = β12{ } = 1. Following similar computations as in the previous case we compute the contributions of the descendant fields and subtract them from the Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point126

corresponding singular terms in (4.101). The remaining singularity is of the form

12 9 2− + ··· . (4.104) 8 a1 (a2 − b)a ¯1 (¯a2 − ¯b) |b|

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, a¯1 → 0 and

a2, a¯2 → 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, z¯1) → (a1, a¯1), (z2, z¯2) → (a2, a¯2) 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

( 12 81 1 2− 2 2 2 ¯ 2 4 4 a1 (a2 − b) a¯1 (¯a2 − b) |a1 − a2| 2 ¯ 2 27 a1 (a2 − b) (¯a1 − a¯2) +a ¯1 (¯a2 − b)(a1 − a2) − 2 2 2 ¯ 2 8 2 a1 (a2 − b) a¯1 (¯a2 − b) |a1 − a2| ) 9 + + ··· . (4.105) 8 a1 (a2 − b)a ¯1 (¯a2 − ¯b) |a1 − a2|

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- p 1 ˜p 1˜ tor algebra (4.80). Therefore, the coeﬃcients β12{ } and β12{ } in (4.97) for the level-1 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point127 descendants of the deformation operator are now

A 1 ˜A 1 βiA{ } = 0, βiA{ } = 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 anomalous 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 evidence 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. Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point128

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 problem. 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 impossible. 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 focused 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 Chapter 4. String states mixing in the D1-D5 CFT near the orbifold point129 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 perspective 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 general 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 important to investigate how the holographic gravity duals can be properly embedded in microscopic 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 considered 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 technique 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 deformed 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 Chapter 5. Conclusions 132 four-point functions with two insertions of the string operator of interest and two insertions 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 quantitatively 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- Chapter 5. Conclusions 133 gating the possibility of lifting of the half-chiral states and its potential contributions to the three-point functions.

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