ST4 Lectures on Assorted Topics in Ads3/CFT2

4 ST Lectures on Assorted Topics in AdS3/CFT2

Pinaki Banerjee˚ ˚International Centre for Theoretical Sciences Survey No. 151, Shivakote, Hesaraghatta Hobli, Bengaluru - 560 089, India. E-mail: [email protected]

Abstract: These lecture notes can broadly be divided in three parts : overview, basics and recent developments. In the ﬁrst part gauge/gravity duality is reviewed with motivations, “derivation” and the prescription. Second part deals with basic of AdS3 and CFT2 and the related thermodynamics. I also discuss Cardy formula, BTZ black hole entropy and Hawking-Page transition in this section. Finally the last part is based on some selected current activities in this area which include conformal blocks, geodesic Witten diagrams and Cardy-like formula for three point functions in AdS3/CFT2 context. These notes are based on lectures delivered at Student Talks on Trending Topics in Theory (ST4) 2018, held at the National institute for Science Education and Research, Bhubaneshwar. Contents

1 Introduction1 1.1 Why bother?1 1.2 Brief outline3

2 Lightning review of AdS/CFT4 2.1 Dualities in physics5 2.1.1 Dualities in quantum ﬁeld theory5 2.1.2 Dualities in string theory8 2.2 Strings, D-branes & Holography 10 2.2.1 String theories 10 2.2.2 D-branes 12 2.2.3 Why do we expect AdS/CFT duality? 13 2.2.4 The decoupling argument 17 2.2.5 The dictionary of parameters 24 2.2.6 Generalization to ﬁnite temperature and density 26 2.2.7 The GKPW prescription 26

3 AdS3 and its thermodynamics 31 3.1 AdS3 and its ‘excitations’ 33 3.1.1 Global AdS 34 3.1.2 Poincaré patch 36 3.1.3 Conical defect 36

3.1.4 Thermal AdS3 37 3.1.5 BTZ black hole 38 3.2 Hawking-Page transition 39 3.2.1 Crtitical Temperature 41 3.2.2 Order of the Transition 42 3.2.3 Is BTZ stable? 42 3.3 Brown-Henneaux in a nutshell 42 3.3.1 Why are boundary conditions important? 42 3.3.2 Asymptotic symmetries 44

4 CFT2 and its thermodynamics 47 4.1 ABC of CFT2 47 4.2 CFT on a torus 53 4.3 Modular invariance 56 4.4 Derivation of Cardy formula 57 4.5 BTZ from Cardy 61

– i – 5 2D blocks, 3-point functions and AdS3 61 5.1 Local observables in CFT 61 5.1.1 2-point function 61 5.1.2 3-point function 62 5.1.3 4-point function 62 5.2 Heavy-light conformal blocks at large c 64 5.2.1 Monodromy method 66 5.2.2 Geodesic Witten diagrams 69 5.2.3 Entanglement entropy : an application 70 5.3 A Cardy formula for three-point coeﬃcients 72 5.3.1 Torus 1-point function 73 5.3.2 Modular invariance 73 5.3.3 Asymptotic formula 74 5.3.4 Dual gravity interpretation 76

A Some useful tricks & results 79 A.1 Lagrange multipliers 79 A.2 ‘Solvable’ path integrals 80 A.3 Gaussian is special 80

B Maximally symmetric spaces 81 B.1 Killing vectors 81 B.2 Maximal symmetry 82

C “ 4 super Yang-Mills 83 N C.1 The Lagrangian 83 C.2 Dimensional reduction of “ 1 SYM 84 N C.3 Symmetries 85 C.4 Phases 85

1 Introduction

1.1 Why bother? AdS/CFT is very important, if not the most important ingredient of present day theoretical high energy physics research. I’ll briefly review that duality below. In case some of you are not at all familiar with it, here is the basic picture. It relates certain gravitational theories to certain field theories (without gravity). When I say relates I actually mean equates i.e. they are two completely different but equivalent descriptions.

AdS “ CFT (1.1)

– 1 – Here AdS is an abbreviation of Anti-de Sitter space which is solution to Einstein equations with negative cosmological constant and CFT is Conformal Field Theory which is a ﬁeld theory with extra symmetries known as conformal symmetries. The equation (1.1) is considered one of the most beautiful ‘equations’ of theoretical physics.

In what follows I will try to motivate why AdS3/CFT2 is particularly interesting. There will be some remarks that you probably find difficult to digest if you are not familiar with AdS/CFT or/and AdS and CFT - but please don’t panic! I shall review (though briefly) it before starting the actual content of these lectures.

Before I discuss about why AdS3/CFT2 is particularly interesting for the sake of completeness let me tell you about gauge/gravity duality in other spacetime dimensions. The duality was originally discovered by Maldacena in the context of AdS5/CFT4 where he conjectured that maximally supersymmetric Yang-Mills theory in 4 dimensions was dual to 5 type IIB string theory in AdS5 ˆS . This is the most well understood version of the duality till date. There have been a vigorous development in this context, both from the gauge theory side and the gravitational side, just after it was proposed. Then there is examples from even higher dimensional context e.g. AdS7/CFT6 where it is believed that six dimensional conformal gauge theory which lives on the world volume of stack of M5 branes is actually 4 dual to string theory in AdS7 ˆ S . This ﬁeld theory is quite “mysterious” and AdS/CFT is one possible way to understand this. In lower dimensions too there are proposed examples of similar duality - particularly popular these days is AdS2/CFT1. AdS2 space is ubiquitous d´2 in BH physics. Near horizon limit of any extremal BH is AdS2 ˆM . This was one of the major reasons string theorists were particularly interested in this duality. But the recent fuzz about it is mostly due to the SYK model1. There are diﬀerent other examples as well but let’s focus on the topic of this set of lectures i.e. AdS3/CFT2.

AdS3 3D gravity is easier! No propagating DOF and locally always AdS (for EH with • 3 negative cosmological constant). Since three-dimensional Einstein gravity has no local degrees of freedom, it is not a realistic model of the world where gravitational waves do exist2! Then why should we study such a “unrealistic” model? Because diﬀerent things get simpliﬁed in 3D and we can learn several aspects of the theory even ignoring GW3.

Calculations are often easy due to mere few number of spacetime directions. • AdS can be seen as a group manifold. This feature plays important role while • 3 studying string propagation in this background.

1See Pranjal’s lectures in ST4 2017 for more details. 2Yes, I am referring to LIGO detections [1,2]. In fact, Jahanur and Aneesh are going to talk about related aspects of gravity in de Sitter universe where we actually live in. 3One can always add matter to EH action to get dynamics i.e. contain local degrees of freedom. See TMG, NMG. Those theories are interesting by their own. We don’t discuss them in these lectures.

– 2 – Higher spin gauge ﬁelds (like gravity) also don’t have any propagating degrees of free- • dom. Little group is trivial.

HS tower can be truncated to spins s “ 2, 3,...N in AdS3.

But the most important feature is, even after all those simpliﬁcations, 3D gravity is still nontrivial enough to be interesting. It has BH solutions. BH entropy computation, Info loss, String propagation, etc.

CFT2

Conformal ﬁeld theory is QFT which enjoys “extra” symmetries including Poincare in- • variance. Poincare transformations preserve the distance or length. Roughly speaking if we relax the condition more and consider theories which remain unchanged under any angle preserving transformations (Poincare transformations will only be a subset). pd`2qpd`1q For d dimensional CFTs the symmetry group is SOpd, 2q which has 2 generators. But d “ 2 case is special, it enjoys Virasoro symmetry which has inﬁnite number of symmetry generators and therefore very constraining. One can directly compute

correlators even without writing down the Lagrangian of the theory! CFT2’s are most well studied among all their higher dimensional (or even the lower dimensional) siblings.

Infinite number of relevant operators unlike higher dimensions. So there can be arbi- • trary number of directions one can flow to get new fixed points (i.e. CFTs).

For Euclidean 2D CFT one can use well developed tools of complex analysis (partic- • ularly analyticity properties) which is extremely useful in exploring the theory.

There is modular invariance. We don’t know analogous property for higher dimen- • sional CFTs yet. It plays an important role in 2D CFTs. We will get to see its power in next lecture.

In 2D for c ă 1 there are unitary minimal models - an infinite family of discrete models • which are unitary and have finite numbers of “primary”. These can be mapped to different statistical mechanical models like Ising, Potts etc. and one can write down all the correlators of these theories without even mentioning their Lagrangians!

Hope I could convince you how AdS3 and CFT2 are special and particularly interesting (I’ll talk more about them tomorrow). So by now we all are motivated enough to start learning/discussing about AdS3/CFT2. Before that here is the plan of my lectures.

1.2 Brief outline Lecture I : Why bother? (done!) • Outline of the lectures (going on!) Lightening review of AdS/CFT

– 3 – Lecture II : AdS and its thermodynamics • 3 CFT2 and its thermodynamics

Lecture III : Cardy formula and BTZ black hole entropy • Lecture IV : Conformal blocks (cont.) • Geodesic Witten diagrams, EE computation Cardy-like formula for three point functions

Here is a list of some interesting topics in AdS3/CFT2 which I shall not cover.

D1-D5 systems and BH microstates • Higher spin theories • Strings in AdS • 3 Information loss problem • Integrability • ... •

Acknowledgments

I am thankful to Parthajit Biswas, Subramanya Hegde and Akavoor Manu for scribing part of the lecture notes. I thank all the participants of ST4 2018 for the stimulating questions, discussions, suggestions and complains which, I believe, have improved these notes a lot. We are grateful to the faculty and administration of NISER Bhubaneshwar for hospitality during the workshop ST4 2018, where these lectures were delivered.

2 Lightning review of AdS/CFT

In this section, as promised, I brieﬂy review4 AdS/CFT correspondence. There are two main motivations to study AdS/CFT [5–9].

1. It provides the only description or rather deﬁnition of non-perturbative quantum gravity, although in a particular background. The interesting bit is it describes that in terms of well understood quantities viz. observables in CFTs.

2. It works like an amazing machine which converts some classical gravity results into some useful quantities in very strongly coupled ﬁeld theories. This is the only analytic technique to study strongly coupled ﬁeld theories.

4This section is mostly taken from my PhD thesis [3] and also slightly extended version of the evening lecture [4] I gave in ST4 2017 at CMI, Chennai.

– 4 – Let me now state the modern version of Maldacena’s conjecture very crudely

The statement of AdS/CFT correspondence

String theory in asymptotically AdS space-time ” A Quantum Field Theory on its boundary

This statement is a very ‘coarse grained’ version of the original conjecture. The main aim of this lecture is to introduce you (or review, in case you are already familiar) to Maldacena’s original conjecture and the “dictionary” of the duality.

2.1 Dualities in physics Duality means equivalence between two seemingly diﬀerent theories. This is actually a very old concept in physics. In this section I discuss about dualities in quantum ﬁeld theories and also in string theory. Before I go into the details here are few typical characteristics of dualities which we will encounter in this section many times.

Characteristics of dualities Two sides (theories) of a duality are typically related by following maps. Degrees of freedom or the Lagrangian need not be same. • Global symmetries coincide. • Equation of motion ðñ Bianchi identity • Weak coupling ðñ Strong coupling • The last one typically holds but not always true. When it holds, one calls that a strong- weak duality. We will see later that AdS/CFT is a famous example of strong-weak duality.

Some useful techniques in the context of duality are provided in appendixA.

2.1.1 Dualities in quantum ﬁeld theory Here is a list of dualities from quantum ﬁeld theories and/or statistical mechanics. 1. Maxwell duality

2. Kramers-Wannier duality

3. Bosonization

4. Montonen-Olive duality

5. Seiberg-Witten duality We will elaborate on one from the list viz. the Maxwell duality and comment on other dualities only brieﬂy.

– 5 – I. Maxwell duality The oldest example of duality goes back to Maxwell. The famous equations due to Maxwell for electric ﬁeld E~ , magnetic ﬁeld B~ , charge density ρ and electric current J~ are given by

.E~ “ ρ ∇ µν ν BE~ , ðñ BµF “ J , (2.1) ˆ B “ J~ ` / ∇ Bt . .B~ “ 0 -/ ∇ µν BB~ , ðñ BµF “ 0 , (2.2) ˆ E~ “ ´ / ∇ Bt . r µν µνρσ µν where F :“ Fρσ is the Hodge-/ dual to F . The equations (2.2) are independent of sources whereas (2.1) depend on sources. We refer to (2.1) as ‘Maxwell equations’ and call (2.2) asr‘Bianchi identities’. Notice that, for ρ “ 0 and J~ “ 0, E~ Ø ´B~ is a symmetry. Actually E~ Ø ´B~ interchanges F µν Ø F µν that amounts to interchanging

rDynamical “Maxwell equations” ÐÑ Geometric Bianchi identities.

Here I choose a ‘trivial’ example to illustrate the electro-magnetic duality. Let’s consider a bunch of photons (Aµ) and they don’t interact with any sources (e.g, electrons). That’s the reason I call it a ‘trivial’ theory

´i 1 F 2 Z “ A e 4g2 µν δpB Aµq . (2.3) D µ µ ż ş The delta function ensures we are dealing with only physical of degrees of freedom. This is known as a gauge choice. Let’s perform a change of variable : Aµ Ñ Fµν i.e, our integration 5 variable will be Fµν instead of Aµ. This implies a Jacobian which is not important for the dynamics of the system and can be taken out of the integral.

´i 1 F 2 Z “ “Jacobian” ˆ F e 4g2 µν δp BνF ρσq . (2.4) D µν µνρσ ż ş We have imposed the Bianchi identities over the path integral. Now our aim is to introduce a Lagrange multiplier Cα into the path integral and integrate out the dynamical Fµν to write down a theory for the ‘fake variable’ Cα. Let’s ﬁrst introduce Cα

1 2 αβγδ ´i p F ` N Cα B F q Z « F C e 4g2 µν β γδ δpB Cµq . (2.5) D µν D α µ ż ş µ The δpBµC q is there because if one shifts Cα Ñ Cα ` Bαf in the exponent, that does nothing to the integral (extra piece vanishes due to anti-symmetry of αβγδ) and is just N 5The Jacobians are, in general, rather tricky in path integrals. They can be very important which may lead to anomaly. But for this particular example the Jacobian is innocent.

– 6 – normalization factor which is not important for this discussion. Now we integrate by parts and drop the boundary term6 assuming ‘nice’ boundary condition.

1 2 αβγδ í p F ` N B Cα F q Z « F C e 4g2 µν β γδ δpB Cµq . (2.6) D µνD α µ ż ş αβγδ Notice that due to anti-symmetry of the term BβCα has to be anti-symmetric in its indices. Therefore one can define this as the field strength for the new ‘gauge field’ Cα : Gαβ :“BαCβ ´ BβCα. The above action is a just quadratic in Fµν and therefore we can easily integrate out Fµν to obtain

2 ´ ig G2 µ Z « C e N µν δpB C q . (2.7) D α µ ż ş

This is almost the same Up1q theory but of a completely diﬀerent ‘gauge ﬁeld’ Cµ. Few remarks in order.

This is an example of self duality where a Maxwell theory goes to another Maxwell • 1 theory. But the coupling is inverted g Ñ g . This is also an example of strong-weak duality.

A and C are completely diﬀerent degrees of freedom. They are related by extremely • µ µ involved relation.

Degrees of freedom need not be the same in both sides of a duality. But in this case • a gauge ﬁeld goes to another.

II. Kramers-Wannier duality It relates the partition function of a two-dimensional square-lattice Ising model at a low temperature to that of another Ising model at a high temperature. Using this duality Kramers and Wannier [10] predicted the exact location of the critical point of 2D Ising model in 1941 before Onsager [11] could solve that model exactly in 1944.

III. Bosonization In 1+1 dimensions one can map an interacting fermionic system to a system of bosons. E.g., massive Thirring model is dual to sine-Gordan model. This is also an example of strong- weak duality. This duality was uncovered independently by particle physicists Coleman [12] and Mandelstam [13], and condensed matter physicists (Mattis, Luther and others) around 1975. Later in 1983 Witten [14] discovered its (not-at-all-straight-forward) generalization to non-abelian case.

6This step is very non-trivial. We are assuming very particular boundary conditions. By ‘nice’ I mean the ﬁeld Cα and/or its derivative dies down at the boundary. But some non-trivial boundary condition can give rise to more interesting physics - topological theories.

– 7 – Figure 1: Closed strings wrapping a compact direction in diﬀerent ‘windings’.

IV. Montonen-Olive duality This is generalization of Maxwell duality with magnetic charge and current but in “ 4 N SYM [15]. This is again a strong-weak duality and it relates ‘elementary particles’ of one side to ‘monopoles’ of the other side.

V. Seiberg-Witten duality Similar to Montonen-Olive duality but it is for IR eﬀective theory of “ 2 SUSY theory N in D “ 4 [16]. Unlike “ 4 SYM this theory not conformally invariant in general i.e. its N beta function runs – more interesting dynamics.

2.1.2 Dualities in string theory In string theory there are mainly two important dualities : T-duality and S-duality. T-duality or ‘Target space’ duality is very simple and elegant. This is also an example of duality which is not strong/weak type. Hints of this duality were noticed in [17, 18] but it was ﬁrst stated and explicitly shown to be a symmetry by Sathiapalan [19] and also independently by Nair et al [20] in 1987. We discuss this in some detail and brieﬂy mention about S-duality.

I. T-duality Einstein changed our view of space and time by marrying them. The notion of ‘space’ and ‘time’ became rather observer dependent. T-duality goes one step further to completely change our notion of space-time itself. It shows how different objects or probes perceive space-time quite differently. In that sense the notion of space-time itself is an ‘emergent’ concept. Let’s see how T-duality works in string theory. Consider a flat 1+1 dimensional space-time (higher dimensional generalization is straight forward). This is just a plane sheet of paper. Let’s compactify the spatial direction to make it an infinite cylinder with radius R (see Fig.1).

– 8 – First consider a particle (or a ﬁeld) of mass m moving on this cylinder. Its momentum

(~p) has two orthogonal components : along the circle (pθ) and along the non-compact direction (pK). But along the compact direction pθ has to obey the periodicity condition i pθp2πRq n e “ 1 i.e. pθ “ R where n P Z. Thus the total momentum

~p “ pθ eθ ` pK eK n “ e ` p e , (2.8) R θ K K where eθ, eK are the corresponding unit vectors. The energy is given by

E2 “ p2 ` m2 n2 “ pp2 ` m2q ` . (2.9) K R2 Notice that, if one takes R Ñ 0 the energy E Ñ 8, unless n “ 0. Physically this means when the compact direction is very small the particle can not ‘sense’ or probe that direction and eﬀectively ‘lives’ only in the non-compact dimension. But what happens if we replace the particle with a closed string? Unlike the particle it can wind around (ﬁg.1) the cylinder. Therefore there will be an extra contribution to the energy from these winding modes.

E2 “ p2 ` M 2 ` p“winding energy”q2 N n2 “ p2 ` ` ` p“winding energy”q2 , (2.10) K α1 R2 ˆ ˙ where α1 is the string tension and N indicates the ‘level’ of the tower of closed string states. Winding energy (Ew) of a string which wraps the cylinder w times is given by

Ew “ length of the string ˆ string tension 1 “ w ˆ p2π Rq ˆ 2πα1 w R “ . (2.11) α1 Thus the total energy becomes

N n2 w2R2 E2 “ p2 ` ` ` . (2.12) K α1 R2 α12 ˆ ˙ If we take R Ñ 0 the momentum modes along the compact direction become very ‘heavy’ as before but at the same time the winding modes become very ‘light’! On the other hand if we take R Ñ 8 momentum modes play the role of ‘light’ modes and winding modes become ‘heavy’. Clearly there is a duality at work here and to be precise if we make the following transformations α1 R Ñ R pn, wq Ñ pw, nq , (2.13)

– 9 – ? 1 the expression for energy remains unaltered. R :“ α “ ls is called the self-dual radius. Physics for R ă α1 is identical to physics with R ą α1. I just want to point out the following characteristics of T-duality. 1. It is intrinsically stringy – there is no ﬁeld theoretic analog to this. Strings perceive the spacetime quite diﬀerently compared to point particles. 2. This is not a strong-weak duality.

II. S-duality This is a strong coupling- weak coupling duality in string theory. S-duality in string theory was ﬁrst proposed Sen [21] in 1994. This duality maps one string theory with coupling gs to another string theory with coupling 1 . For example, type IIB string theory with the gs coupling constant gs is equivalent via S-duality to the same string theory with the coupling constant 1 . Similarly, type I string theory and the SO(32) heterotic string theory are dual gs to each other.

III. Gauge/string duality This duality [5] mixes the above two frameworks viz. QFTs and string theory. It was proposed by Juan Maldacena in 1997. It roughly states a particular gravitational theory in asymptotically AdS space-time is equivalent to a certain ﬁeld theory in one less number of spacetime dimensions. This statement is a very ‘coarse grained’ version of Maldacena’s original conjecture which will be reviewed in detail below.

2.2 Strings, D-branes & Holography 2.2.1 String theories General theory of relativity is arguably the most beautiful theory written down by human mind. It works remarkably accurately at the astrophysical level. Recent discovery of gravitational waves [1, 22, 23] has put it into a even firm footing. But if one tries to write down a quantum field theory of gravity it suffers from UV divergences and turns out to be non-renormalizable. String theory is the most promising candidate for quantum theory of gravity. General relativity comes out naturally from perturbative string theory. After the discovery of dualities and particularly AdS/CFT correspondence string theory seems to be promising even at the non-perturbative level. String theory avoids the UV divergences since it contains objects with finite length, namely the strings, instead of point particles. Thus the theory is described by a two dimensional worldsheet action, instead of one dimensional worldline, 1 S “ ´ dτ dσ ´detphq . (2.14) NG 2πl2 s ż This action contains a square root and hard to quantize.a That’s why people usually work with Polyakov action, 1 S “ dτ dσ p´γq1{2 γab B Xµ B Xν η , (2.15) P 4πl2 a b µν s ż which can be easily shown to be equivalent to Nambu-Goto action at classical level.

– 10 – Figure 2: Spectrum of open and closed strings.

This worldsheet theory represents d free bosons in two dimensions and γabpσ, τq is independent metric on the worldsheet. Moreover this action enjoys 1. Reparameterization invariance : tσ, τu Ñ tσ˜pσ, τq, τ˜pσ, τqu . 1 2ω 2. Weyl invariance : γab Ñ γab “ e γab . After choosing a particular gauge namely ‘conformal gauge’ the theory is conformally invariant and the worldsheet ﬁelds satisfy following free two dimensional wave equation B2 B2 ´ Xµpτ, σq “ 0 . (2.16) Bσ2 Bτ 2 ˆ ˙ So the aim is to solve this string equation of motion with appropriate boundary conditions i.e. periodic for closed strings and for open strings the boundary conditions can be Dirichlet or Neumann. It is well known that if one quantizes the theory without harming worldsheet conformal invariance and Poincaré invariance of the target space (which is spacetime itself) one has to have 26 scalars on the worldsheet which is same as having a 26 dimensional spacetime. Below I focus on the spectrum of the bosonic strings.

‚ Closed string spectrum Closed strings have two independent modes which are usually called left and right movers. Upon quantization the Fourier modes become creation and annihilation operators. It contains a tachyon and three massless fields namely Dilaton (Φ), Graviton (Gµν) and antisymmetric Kalb-Ramond field (Bµν) and then there are infinite number of massive higher spin modes.

‚ Open string spectrum The open string spectrum is given by a single copy of oscillator. This too has a tachyon and a massless Gauge ﬁeld (Aµ) and then there are inﬁnite number of massive higher spin

– 11 – modes. The bosonic string theory is not ‘physical’ for two reasons (i.) it contains tachyon in its spectrum and (ii.) it doesn’t have fermions which are fundamental constituents of matter in our universe. For these reasons one adds fermions to the theory and moreover considers supersymmetric version which is famously known as superstring theory. Again symmetries restrict the spacetime dimensionality of superstring to be 10 and there are no tachyons in its spectrum. There are ﬁve consistent superstring theories namely Type I, Type IIA, Type

IIB, Heterotic E8 ˆ E8 and Heterotic SO p32q. They have different spectra. But if one is interested in low energy effective theory one can focus only on the massless modes and forget about the infinite tower of massive higher spin modes since they are suppressed by 1 higher powers of α1 . In these notes I mainly focus on Type II theories (particularly Type IIB) as the original AdS/CFT conjecture was proposed in Type IIB theory.

Superstring theory Low energy massless ﬁelds

Type IIA Gµν,Bµν, Φ and Cµ,Cµνκ,Cµνκσρ

Type IIB Gµν,Bµν, Φ and C,Cµν,Cµνκσ

Table 1: Low energy massless ﬁelds in diﬀerent superstring theories

Notice that, Gµν,Bµν, Φ fields which come from the NS-NS sector are present in both theories. But the Ramond-Ramond (antisymmetric) fields are different for different theories - type IIA has ‘odd-form’ R-R fields where as type IIB contains only ‘even-form’ R-R fields. A d-form gauge field naturally couples to a d ´ 1 dimensional charge. E.g, a Up1q 1-form µ gauge field couples to a worldline of a charged particle as : Aµ dx or 2-form Kalb-Ramond field B naturally couples to string worldsheet as : B dxµdxν. µν µνş The natural question that comes to one’s mind is what are the corresponding charges ş for different R-R fields? The answer is D-branes!

2.2.2 D-branes D-branes [24] appear in string theory in two seemingly diﬀerent ways.

1. They are lower dimensional objects where an open string can end.

2. These are also solutions to particular supergravity e.g. type IIB SUGRA.

The first one is essentially an open string description of D-brane. If one is interested in studying D-branes from this perspective one should study the dynamics or fluctuations of open strings ending on the D-brane. The second one is effectively a closed string description which says that a D-brane as an ‘heavy’ object sources gravitons and can be described

– 12 – geometrically. In section 2.2.4 we will see how the equivalence between these two seemingly different descriptions of D-brane leads Maldacena to the AdS/CFT conjecture. In the last section we stated that D-branes source the R-R fields or equivalently they are the charges of R-R fields.

Superstring Low energy massless ﬁelds Sources coupled to ﬁelds theory

Gµν,Bµν, Φ F1 couples to Bµν

Type IIA Cµ,Cµνκ,Cµνκσρ,Cµνκσρλγ D0, D2, D4, D6

Type IIB C,Cµν,Cµνκσ,Cµνκσρλ,Cµνκσρλγδ D(-1), D1, D3, D5, D7

Table 2: D-branes appearing in IIA and IIB theories.

Note that I have added the magnetic dual ﬁelds in the table2. For example, Cµνκσρλ is dual to Cµν and thus D5 brane is magnetic dual to D1 brane. Notice that Cµνκσ is self dual ﬁeld in type IIB theory and correspondingly D3 brane is its own magnetic dual.

2.2.3 Why do we expect AdS/CFT duality? At ﬁrst sight the equivalence of gravity with gauge theory in one lower spacetime dimensions might seem very strange mainly for following reasons.

1. Two theories don’t even live in same number of spacetime dimensions.

2. One is gauge theory without gravity and other one is a gravity theory.

If we look back to few inﬂuential discoveries of theoretical physics in last few decades the duality looks more plausible.

I. Open string-closed string duality Closed string spectrum contains : {Graviton + infinite tower massive modes.} • Open string spectrum contains : {Gauge field + infinite tower of massive modes.} • If we are interested only in low energy physics, closed string has ‘gravity’ in it where as open string contains Yang-Mills ‘gauge fields’. Now let’s look at the following process in fig. 3. One can look at it in two completely different but equivalent ways namely a closed string is being exchanged between the D-branes or an open string is running in a loop between them. Roughly it means,

– 13 – (a) Open string running in a (b) Closed string is exchanged loop. between D-branes.

Figure 3: Open string-closed string duality

Closed string tree = Open string loop.

Therefore one would expect, at least in some particular sense, there should be an equivalence between gauge theory and gravity.

II. Large-N gauge theories It is established that strong nuclear force is described by QCD which is nothing but a Yang-Mills theory with gauge group SU(3). Here three indicates the number of colors. The Yang-Mills coupling undergoes dynamical transmutation and QCD doesn’t have a free parameter to play with – QCD is very difficult. What will happen if one works with infinite number of colors instead of only three? This was the question ’t Hooft asked in seventies [25]. Actually the theory simplifies7 a lot. ’t Hooft introduced a parameter N which is the number of colors and it then plays the role of a free parameter. The N Ñ 8 limit is similar to taking ~ Ñ 0 i.e, ‘classical’ limit of QCD. For this discussion we shall consider pure SUpNq YM i.e. no ‘quarks’. But adding ‘quarks’ is a very straight forward extension. The ‘gluons’ are adjoint valued elements of SUpNq and the Lagrangian

1 “ ´ F a F µνb , (2.17) L 4 g2 µνb a YM ż where is a Lorentz scalar since µ, ν indices are contracted and is singlet under SUpNq L since a, b indices are contracted. From the Lagrangian it is clear that in this theory, 2 Propagator „ gYM 1 Interaction vertices „ 2 . gYM

7This is in the same spirit in statistical mechanics. When fluctuations are important one way to handle them is to work with a lot of such fluctuating variables. 3-body problem is very difficult but a box of gas with huge number of molecules is easier to handle! Same is true with dimensionality. In lower dimensions there are lot of fluctuations. Mean field theory is easier because one works in infinite dimensions.

– 14 – I will follow the double-line notation what ’t Hooft introduced to make the counting easy – replacing each gluon propagator by a quark-antiquark pair (see ﬁg.4).

2 Figure 4: The gluon propagator „ gYM

In this notation the 3-point and 4-point functions look as follows.

1 Figure 5: 3-pt vertex „ 2 gYM

1 Figure 6: 4-pt vertex „ 2 gYM

Suppose we are interested in vacuum-to-vacuum amplitudes (see ﬁg.7). Our aim is to see how the diagrams scale with N. For that we just need to count the number of propagators and vertices. We know how they scale with the coupling gYM . On top of that whenever we have a color loop (color index is summed over) that should correspond to a factor of N, since there are total N colors.

For ﬁg. 7a : # of propagators = 3 # of vertices = 2 # of loops = 3

2 3 1 3 2 2 2 6 It scales as „ pgYM q 2 2 N “ pgYM Nq N ” λ N . pgYM q

For ﬁg. 7b : # of propagators = 6 # of vertices = 4

– 15 – (a) Planar diagram : (b) Non-planar diagram : pN 2q. pN 0q. O O Figure 7: Vacuum-to-vacuum amplitude in double-line notation

# of loops = 2

2 6 1 2 2 2 0 2 0 6 It scales as „ pgYM q 2 4 N “ pgYM Nq N ” λ N . pgYM q

8 2 I have defined a new effective coupling λ :“ gYM N and have extracted the N- dependence. If we keep λ to a fixed value as N Ñ 8, the fig. 7a contributes at pN 2q O whereas fig. 7b contributes at pN 0q. Notice that the fig. 7a can be drawn on a plane or O a sphere and is called planar diagram [25]. On the other hand fig. 7b can not be drawn on a plane – one requires a torus. This is a non-planar diagram. Therefore at large N and fixed (but small λ) one can schematically write down SU(N) YM vacuum-to-vacuum amplitude as following.

Figure 8: Large N expansion of SU(N) gauge theory

Notice that at large N one needs to consider only the planar (or sphere) diagrams but there are still inﬁnitely many such terms since it is a perturbative expansion in λ. At this point someone familiar with string perturbation theory [26, 27] can easily correlate this with perturbative string amplitude which looks as follows,

8 2 This λ is called ’t Hooft coupling since ’t Hooft introduced this quantity. Also keeping λ :“ gYM N fixed, with N Ñ 8 is known as ’t Hooft scaling limit for the same reason.

– 16 – Figure 9: Perturbative expansion of closed strings

and formally identify 1 g ô (2.18) s N α1 ô λ

Thus gauge theory at large N and string theory have similar perturbative expansions – it’s not very hard to imagine that they can be related to each other.

III. Holographic principle The holographic principle, originally proposed by ’t Hooft [28], states that the total amount of information inside a volume of space cannot be larger than the amount of information that can be encoded on its boundary. Later Susskind worked on this principle in the context of string theory [29]. We don’t really need the details. The expression for Bekenstein-Hawking entropy [30, 31] of black hole Horizon area S “ , (2.19) BH 4G will ring a bell. It scales as the area of its horizon divided by 4G. Intuitively one can argue why there should be such a principle. Suppose we have some volume of space with some matter in it and we can put more and more matter inside that volume of space, thus increasing the entropy within it. But at some point there will be so much matter inside that it will essentially collapse into a black hole! Thus we cannot increase the entropy of a volume of space indeﬁnitely; we can only increase it until it is equal to the surface area of the volume divided by 4G. We know that entropy is a measure of information. The more information the volume has, the more entropy it will get. This is the holographic principle which roughly says that the total amount of information inside a volume of space cannot be larger than the amount of information that can be encoded on the boundary of that volume. Thus in a crude way ‘volume’ is equivalent to ‘its boundary’. And therefore if a d ` 1 dimensional gravity theory is dual to a d dimensional ﬁeld theory living on its boundary, it shouldn’t be so surprising.

2.2.4 The decoupling argument It is quite clear from the above motivations that there should be some relationship between gauge theory and gravity (string theory). After the discovery of D-branes [24] in mid-

– 17 – Figure 10: Electron’s motion near proton nineties there was a rapid development in this direction [32–35]. Finally, in 1997, Maldacena proposed a duality [5] between9 “ 4 SYM and Type IIB string theory in AdS ˆ S5. N 5 His argument to reach this conjecture is famously known as the decoupling argument and I discuss this in detail in this section.

‚ Diﬀerent descriptions of same physics Before discussing about the decoupling argument let’s discuss about some simple examples of describing same physical phenomenon using two complementary point of views10. This discussion will be very useful in describing Maldacena’s decoupling limit.

QED

Let’s start with a very basic example from QED. Suppose we want to understand how an electron’s moves in presence of a proton. We can treat this problem perturbatively and sum up all possible Feynman diagrams (see fig. 10). The first diagram of fig. 10 in position space gives the standard Coulomb potential 1 V prq „ ´ r . The other diagrams are corrections to this ‘classical potential’. There will be many more diagrams as one goes to higher loops. The more number of diagrams one considers the more accurate the description will be. Effectively the extra diagrams change α e´2mer the form of the potential, V prq “ ´ r1 ` # α 3{2 ` ...s . r pmerq

There are two diﬀerent ways of describing the same phenomenon.

Picture I : The electron and the proton are in vacuum and they are interacting • via exchanging photons (see ﬁg. 10). Then sum all such Feynman diagrams.

Picture II : Another way of describing the same problem is the following. There • α is no proton but the electron is moving in a background potential V prq “ ´ r r 1 ` e´2mer # α 3{2 ` ...s (see ﬁg. 11). pmerq

9See appendixC for more details on “ 4 SYM respectively. 10In this section I heavily follow [36, 37N].

– 18 – Figure 11: Electron moving in a ‘background’ ﬁeld

String theory

What can be the analogous picture in string theory? One should replace the electron by an ‘elementary’ closed string and the ‘heavy’ proton by a heavy and extended object available in the theory – D-brane.

Figure 12: A closed string moving near a D-brane.

Again we have two diﬀerent ways of looking at this phenomenon.

Picture I : First approach would be analogous to summing over Feynman diagrams • i.e., studying the scattering of a closed string with a D-brane perturbatively. The string can split into many closed strings or can become an open string on the D-brane and then can further split into many open strings on that brane (see fig. 12). Some of the open strings can join the end points on the D-brane and leave the brane as closed strings. In the world sheet picture it is easier to keep track of the factors of couplings (similar to counting loops in QED). The number of handle indicates string splitting and number of boundary of the worldsheet signifies the interaction with the D-brane (see Fig 13). One needs to sum over all different worldsheet topologies.

Picture II : Here is another equivalent description of the same phenomenon. One • can forget about the existence of the D-brane and replace all intermediate effects (Feynman diagrams) by an effective background (see fig. 14) in which the closed string

– 19 – +

...

Figure 13: Worldsheet picture of closed string - D-brane interaction

Figure 14: A closed string moving near a D-brane.

moves. In this picture we are considering D-brane as a source of closed strings. The ‘coherent state’ of large number of closed strings eﬀectively changes the background near the brane.

Notice that ‘Picture I’ holds true only in the perturbative regime i.e, low energy action of open strings on the brane. This is described by SYM theories. Whereas in ‘Picture II’ D-brane is the source of closed strings. Since the closed strings change the background this should have some gravitational description.

– 20 – Maldacena’s Argument

In his original paper [5], Maldacena started with a stack of ND3 branes (ﬁg. 15). One can again describe the system in two alternative ways – (i) by open string dynamics or (ii) by closed string dynamics11.

Figure 15: A stack of N D3-branes

‚ Picture I Let’s see how one would describe the low energy dynamics of this system from open string perspective. The stack of N branes are described by “ 4 U(N) SYM theory plus N higher derivative terms. These higher derivative interactions come due to integrating out all massive open string modes. These are all suppressed by increasing powers of α1. Similarly away from the branes (I call it ‘bulk’) the physics should be described by 10D low energy string theory (type IIB super-string theory since D3 brane appears in IIB theory) which is known as type IIB super-gravity. Again there will be higher derivative interactions suppressed by diﬀerent powers of α1. And these two theories can interact. So schematically the action for the total system looks as following.

S “ Sbranes ` Sbulk ` Sint . (2.20)

pSYM + higher derivatives) p10D SUGRA + higher derivativesq l jh n ljhn The ‘bulk’ and the ‘branes’ interact gravitationally. Maldacena’s main aim was to turning oﬀ this interaction by tuning some coupling and to decouple the theories. Notice 1 that if we take α Ñ 0 keeping gs and N ﬁxed, it is equivalent to taking Newton’s constant

11See ﬁg. 13. If one tries to look along a D-brane one can ‘see’ the worldsheet of open string ﬂuctuating. On the other hand if one looks perpendicular to the D-brane one can ‘see’ closed string(s) being emitted (or absorbed) by the D-brane.

– 21 – Figure 16: Stack of D3-branes as a gravitational solution

? 12 1 GN Ñ 0 because GN „ gsα . But α is a dimensionful quantity, therefore it can not be taken to zero. The correct way to take the limit is to make α1 smaller compared to the energy (or inverse length) scale one is looking at i.e.,

|~x ´ ~x 1|2 α1 |~k|2 ! 1 or, " 1 . (2.21) α1 Taking such a limit amounts to turning oﬀ all the interactions and all higher derivative 1 terms since they come with positive powers of GN or α . Thus we are left with 4 D SYM and 10 D super-gravity which are not talking to each other.

Picture I : “ 4 SYM in 4 dimensions ‘ Super-gravity in 10 dimensions. N

‚ Picture II Following same chain of arguments we want to see the stack of D3 branes a gravitational solution or we should ask, what background do the stack of branes produce? This “classical” background should be described by low energy effective action of string theory which for this particular case is type IIB super-gravity (Einstein action + “other fields”). Therefore the aim is to look for a solution or a metric for this stack of N D3 branes (fig. 16).

Exploiting the symmetry of the system we start with the following ansatz

dx dx ds2 “ η µ ν ` fprq dxm dx where, r2 “ xm x . (2.22) µν fprq m m r

– 22 – To satisfy Einstein equations the unknown functions in the ansatz have to have the following form

L4 1 with 4 4 12 (2.23) fprq “ fprq “ ` 4 ,L “ gsN p πα q . c r Once we have the metric therer are two obvious ‘extreme’ limits we can look at in this picture II: (i) r Ñ 8 (ii) r Ñ 0 Far away from the branes (r Ñ 8): Far away from the branes the geometry has to be ﬂat space R9,1

2 µ m ds “ dx dxµ ` dx dxm . (2.24)

Near the branes (r Ñ 0): The metric becomes

r2 L2 ds2 “ η dx dx ` dr2 ` L2dΩ2 . (2.25) L2 µν µ ν r2 5 S5 AdS5 l jh n l jh n In GR we always talk about observables with respect to particular observers. Let’s ask the question what do we mean by ‘time’?

µ 2 2 dx dxµ “ ´dt ` d~x . (2.26)

This t is just co-ordinate time and it is ‘physical’ or ‘proper’ time only for an observer at r “ 8. Therefore the natural question arises what is the ‘time’ for an observer at arbitrary r? The proper time for an observer is related to co-ordinate time as follows. ? ∆tprop “ gtt ∆t r “ ∆t . (2.27) L On dimensional ground the ‘proper’ energy L ∆E “ ∆E. (2.28) prop r Notice that for r Ñ 0 there is an infinite red shift. So even if the near the stack of branes the energy E is arbitrarily large12 for the observer at r Ñ 8 it is finite due to the redshift. 12This is very crucial point. So let’s elaborate on it with a simple thought experiment. Suppose A and B are at r Ñ 8. A is carrying a 10100 GeV ‘lamp’. Suddenly A decides to walk towards it. Due to the red shift factor, to B the lamp energy keeps decreasing (i.e., lamp’s frequency gets smaller) as A approaches the stack. When A is very close to the branes, such that the redshift factor is 10´109 say, to B the lamp’s energy is just 1 eV . But for A it is still the 10100 GeV lamp! Therefore arbitrarily large energy near the branes is finite energy for the observer far away. The bottom line is, low energy theory for the observer at 5 infinity includes all possible high energy phenomena near the D-branes – full string theory in AdS5 ˆ S .

– 23 – Figure 17: The decoupling limit

We are interested in ‘low energy’ physics for the observer at r Ñ 8. The things near him/her are already low energy i.e, 10 D super-gravity and anything near r Ñ 0 is also low energy due to huge redshift as discussed above. Here by anything I mean full string theory 5 in AdS5 ˆ S .

5 Picture II : Full type IIB string theory in AdS5 ˆ S ‘ Super-gravity in 10 dimensions.

We need to “equate” Picture I and Picture II. Comparing these two description Mal- dacena conjectured,

“ 4 SYM in 4 dimensions ” Full type IIB string theory in AdS ˆ S5 . N 5

2.2.5 The dictionary of parameters

There are two dimensionless parameters in the gauge theory namely gYM and N. As I have 2 discussed before it is more convenient to deﬁne dimensionless ’t Hooft coupling λ ” gYM N. Thus the gauge theory has two independent dimensionless couplings gYM and λ. On the other hand the string theory in AdS ˆ S5 has one dimensionless coupling g 5 ? s 1 and two dimensionful parameters namely the string length ls “ α and the AdS radius L. Thus eﬀectively this theory also has two dimensionless parameters g and L . They are s ls related as follows.

– 24 – Gauge theory String theory

Small λ Small λ i.e, L „ ls

Perturbative SYM Highly (stringy) quantum theory

(Easy) (Hard)

Large λ Large λ i.e, L " ls

Strongly coupled theory Classical SUGRA

(Hard) (Easy)

Table 3: Diﬀerent regimes of gauge/gravity duality

2 gYM “ gs (2.29) L 4 λ ” g2 N “ YM l ˆ s ˙

Planar limit : According to the stronger version of the conjecture, the above match- ing of parameters holds true for all values of the parameters. Things get simplified if one 2 takes N Ñ 8 keeping λ “ fixed i.e, gYM Ñ 0. This is famous ’t Hooft limit. In this limit, as we have seen in the large N gauge theories, only the planar diagrams in “ 4 N SYM contribute since the other non-planar diagrams are suppressed by powers of 1{N. Analogously in the string theory side g Ñ 0 and L remains finite which means that string s ls cannot split and join (i.e, no ‘handles’ in the string world sheet). If we restrict ourselves in this planar limit where N Ñ 8 and λ “ fixed, there are two possibilities : λ can be large or small. Notice that (see table3) when one side of the duality is computationally easy the other side becomes extremely hard to handle. Thus it is not only very difficult to prove the duality but it’s even hard to check. At the same time, due to exactly the same reason, the duality is extremely powerful. One can calculate interesting quantities in strongly coupled quantum field theories by computing corresponding quantities in dual classical gravitational theory.

Symmetries of the two sides :

Conformal group SO(4, 2) ðñ Isometry group of AdS SO(4, 2) • 5

– 25 – R-symmetry of SYM SU(4) ðñ Isometry of S5 : SO(6) = SU(4) •

Q. : What about CFT2? I have mentioned in the beginning that CFT2 is special. It enjoys inﬁnite dimensional Virasoro symmetry. But if AdS/CFT is a correct duality, then it’s dual should be an AdS3 space whose isometry group is SO(2, 2) – ﬁnite dimensional with only six generators! Clear mismatch?

A. : Brown-Henneaux [38]! They showed that the asymptotic symmetry group of AdS3 is indeed inﬁnite dimensional (see section 3.3).

2.2.6 Generalization to finite temperature and density Although the AdS/CFT duality was proposed in a very particular setup it is believed to be valid for more generic systems and is usually referred to as gauge/gravity duality. This was first generalized to thermal state of CFT which is dual to a black hole in asymptotically AdS spacetime [7]. Turning on chemical potential in the field theory is equivalent to adding charge to the black hole13 (see table4).

Field theory in d-dimensions Dual gravity in pd ` 1q-dimensions

T “ 0 and µ “ 0 Pure AdSd`1

T ‰ 0 and µ “ 0 Black hole in AdSd`1 (or thermal AdSd`1)

T ‰ 0 and µ ‰ 0 Charged black hole in AdSd`1

T “ 0 and µ ‰ 0 Extremal charged black hole in AdSd`1

Table 4: Generalization of gauge/gravity duality for diﬀerent backgrounds

2.2.7 The GKPW prescription To use the above mentioned duality quantitatively one needs to have a prescription that relates the ﬁeld theory quantities to their gravity theory equivalents. Such a prescription has been given in [6,8] which state that partition function of the QFT coincides with the same of gravity theory

i i exp φ “ ZQGpφ q , (2.30) 0Oi 0 B ˆżBAdS5 ˙FCFT where φi are bulk ﬁelds in gravity theory and i are their dual boundary operators in the i O gauge theory. ZQGpφ0q is the partition function of quantum gravity with the boundary 13There are also other generalizations e.g, for rotating black holes. This is known as Kerr/CFT correspondence [39–41] which is not very well understood.

– 26 – i i conditions that φ goes to φ0 on the boundary. The conjecture becomes useful in studying strongly coupled ﬁeld theories when the gravity theory is ‘classical’. In that limit the path i integral can be approximated by saddle point. Treating φ0 as the sources of boundary ﬁeld theory one can calculate the correlators by taking functional derivative of ZQG with respect i to φ0.

CFT correlators from AdS Let us now see how to use the above prescription and calculate the boundary correlators corresponding to a free massive scalar ﬁeld in the bulk. To illustrate we shall consider the

Euclidean AdS5 in the bulk in Poincaré coordinates,

dx dxµ ` dz2 ds2 “ µ (2.31) z2 Notice z “ 0 is the boundary and z Ñ 8 is ‘center’ of AdS space. For any given value of z “ z0 the the metric is 4D Euclidean space. Thus one can think of the role of this radial 1 coordinate z is to scale the 4D boundary metric where the CFT lives. One can interpret z as the energy scale of the boundary field theory such that z Ñ 0 is the UV and z Ñ 8 is the IR of the field theory. The scalar field action in this background is given by,

5 ? µν 2 2 S “ d x gpg BµφBνφ ` m φ q ż 1 “ dzd4x z2pB φq2 ` z2pB φq2 ` m2φ2 (2.32) z5 z x ż ` ˘ Equation of motion is obtained by demanding δS “ 0 around the classical solution. We ignore the boundary term obtained by doing integration by parts, for the moment. Equation of motion thus obtained is,

1 1 m2 B B φ ` B Bµφ “ φ (2.33) z z3 z µ z3 z5 ˆ ˙ ˆ ˙ To analyze the asymptotic behavior, let us assume to begin with that φ only depends on the z coordinate. 1 z5B B φ “ m2φ (2.34) z z3 z ˆ ˙ We can take an ansatz φpzq „ z∆˜ . Equation of motion gives us,

m2 “ ∆˜ p∆˜ ´ 4q (2.35) ? Let us denote as ∆, the largest ∆˜ that solves Equation (2.35). Note that ∆ “ 2 ` 4 ` m2 ? ˜ 2 and the smaller solution, ∆´ “ 2 ´ 4 ` m “ 4 ´ ∆. The near boundary behavior of the on-shell ﬁeld is

p4´∆q ∆ φ „ φ0z ` φ1z (2.36)

– 27 – Therefore at the boundary z “ 0, φ0 corresponds to the non-normalizable part and φ1 corresponds to the normalizable part of the ﬁeld conﬁguration. Therefore φ0 acts as the source for the boundary operator and φ1 acts as the vacuum expectation value [6, 42]. µ µ Let us restore xµ dependence of φpz, x q. Near boundary behaviour of the φpz, x q is,

µ p4´∆q ∆ φpz, x q „ φ0pxqz ` φ1pxqz (2.37)

As z approaches z “ 0, we impose the boundary condition,

µ µ p4´∆q lim φpz, x q “ φ0px qz (2.38) zÑ0

µ φ0px q is the Dirichlet boundary data, upon specifying which, there is a unique solution µ µ that extends to the bulk. In this process, φ1px q is also obtained as a function of φ0px q. Consider the massless case as an example and compute the two point function of the ? µ 2 boundary operator sourced by φ0px q. For m “ 0, we have ∆ “ 2` 4 ` m “ 4. Equation of motion is 1 1 B B φ ` B Bµφ “ 0 (2.39) z z3 z µ z3 ˆ ˙ ˆ ˙ Fourier decomposing φpz, xµq along the xµ-directions,

`8 dp φpz, xµq “ φ pzqeip¨x (2.40) 2π p ż´8 we get the EOM for φppzq as follows,

1 p2 B B φ pzq ´ φ pzq “ 0 (2.41) z z3 z p z3 p ˆ ˙ 2 Let’s make a change of variables, Y ppzq “ φppzq{ppzq . The EOM reads,

d2Y ppzq dY ppzq ppzq2 ` ppzq ´ p4 ` ppzq2qY ppzq “ 0 (2.42) dppzq2 dppzq which we can recognize as the modiﬁed Bessel’s equation. The solution is,

Y ppzq “ αpI2ppzq ` βpK2ppzq (2.43)

The asymptotic behavior of I2ppzq and K2ppzq as z Ñ 0, is given by,

2 3 I2ppzq „ ppzq ` Opz q ´2 4 k2ppzq „ ppzq 1 ` Opzq ` C2ppzq ln x (2.44)

However, as z Ñ 8, we have ` ˘ epz I ppzq „ ? a pz e´pz K ppzq „ ? (2.45) a pz

– 28 – But we need a solution that is regular as z Ñ 8, therefore αp “ 0. For ∆ “ 4, we set the boundary value of φ as φ0. As the on-shell action diverges, let us put a cut-oﬀ at z “ . We will take Ñ 0 in the end.

0 φppz “ q ” φp (2.46) Therefore the solution now reads, ppzq2K ppzq 2 0 (2.47) φp “ 2 φp ppq K2ppq Let us now compute the on-shell action. The action for the scalar ﬁeld, on integration by parts is given by

? µ ? 2 S “ g φ nµ B φ ` g φp´ ` m qφ (2.48) żBM ż 2 where BM is the boundary submanifold and nµ is the normal vector on the boundary. The second term vanishes for the on-shell action due to the equations of motion. The ﬁrst term gives us φB φ S „ z (2.49) on´shell z3 We are yet to substitute the solution to the equation of motion,

4 ipx φpz, xq “ d p e φppzq (2.50) ż With this, we obtain the eﬀective action to be

1 1 0 0 1 W pφ q “ S „ dp dp δpp ` p qφ φ 1 B ln φ pzq (2.51) 0 on´shell p p z3 z p ż ˇz“ ˇ We can thus calculate the two point function ˇ ˇ 1 δW 1 4 1 d ln φppzq x ppq pp qy “ „ δpp ` p qp 3 O O δφ δφ 1 ppzq dppzq p p ˇz“ 1 ˇ „ p4 lnppzq ` ppolynomialˇ in pq ` .... (2.52) k ˇ k ÿ Second term in the last equation contains divergences which are however local. These divergent terms can be canceled by adding covariant counter terms to the action on the z “ hypersurface.Therefore we will only consider the ﬁrst term, which on Fourier transform leads to 1 x pxq px1qy „ (2.53) O O px ´ x1q8 In general14 if the operator has conformal dimension ∆ the 2-point correlator becomes,

1 x pxq px1qy „ (2.54) O∆ O∆ px ´ x1q2∆

14I considered m “ 0 to demonstrate the 2-point correlator computation. Nothing goes wrong if one considers m ‰ 0.

– 29 – One can similarly ﬁnd n-point correlators on the boundary, using the AdS/CFT dictionary. These correlators are built out of bulk to boundary and bulk to bulk propagators. It’s evident that to have such correlators we need to have interacting ﬁelds in AdS5 as follows.

1 1 n S “ d5x pBφ q2 ` m2φ2 ` λ φ φ . . . φ (2.55) 2 i 2 i i1,i2,...iq i1 i2 iq i q“3 ż ÿ ˆ ÿ ˙

E.g, if one is interested in 3-point function one should choose the interaction term to be cubic (i.e. n “ 3), and so on. The procedure for obtaining higher point functions is similar. For n-point correlator, GKPW instruct us to diﬀerentiate the on-shell bulk action with i i respect to the sources φ0 and at the end to set φ0 “ 0 i.e.

n δ Son-shell x 1 2 ... ny “ (2.56) O O O δφ1δφ2 . . . δφn ˇ i ˇφ0“0 ˇ ˇ ˇ

The above computations in the bulk geometry can be visualized in Feynmanian manner i.e. by drawing cartoons. But this time the Feynman diagrams will be drawn inside some ‘circle’ (supposed to be AdS space) and will be referred to as Witten diagrams.

[Insert images].

The above prescription is applicable to obtain Euclidean correlators. The Euclidean signature avoids some complications related to boundary conditions. However in many cases, particularly for ﬁnite temperature systems extraction of Lorentzian-signature Ad- S/CFT results directly from bulk gravity theory is inevitable. Therefore one requires to have some prescription for computing real time correlators directly from gravity. This was done by Son and Starinets [43].

Just for the sake of completeness here is the (incomplete) AdS-CFT or CFT-AdS dictionary.

– 30 – CFTd AdSd`1

Scalar operator Scalar ﬁeld φ O µ Global current J Maxwell ﬁeld Aa

µν Energy-momentum T Metric tensor gab

Fermionic operator Ψ Dirac ﬁeld ψ

Scaling dimension of operator Mass of the corresponding ﬁeld

Global symmetry Local symmetry

Temperature Hawking temperature

Phase transition Gravitational instability

Table 5: The AdS-CFT or CFT-AdS dictionary

3 AdS3 and its thermodynamics

After this review of AdS/CFT duality, I shall now focus on the AdS3 spacetime speciﬁcally and its thermodynamics. Let’s begin with Einstein’s gravity in d dimensions to set up the bigger picture. The action for Einstein’s gravity with a cosmological constant is given by

1 d ? SEH “ d x ´gpR ´ 2Λq (3.1) 16πG ż For diﬀerent choices of Λ, the solutions obey diﬀerent asymptotic structures. There are three such classes of solutions to this theory.

1. Λ “ 0 corresponds to asymptotically ﬂat solutions. Examples include Minkowski space time, Schwarzschild black hole, Kerr-Neumann black hole etc.

2. Λ ą 0 corresponds to asymptotically dS solutions. Examples include dS space time, dS Schwarzschild solution etc.

3. Λ ă 0 corresponds to asymptotically AdS solutions. Examples include AdS space time, AdS Schwarzschild solution, BTZ black hole solution etc.

In these set of lectures I shall mostly focus on the Λ ă 0 case. On varying the above action, one gets the following equation of motion. 1 R ´ Rg ` Λg “ 0 (3.2) µν 2 µν µν

– 31 – Trace of the above equation gives us, 2Λd R “ (3.3) d ´ 2 We can substitute this back to the equation of motion to obtain, 2 R “ Λg (3.4) µν d ´ 2 µν For d “ 3, this gives us

Rµν “ 2Λgµν (3.5)

But, note that, in three dimensions there is a peculiar fact that the Riemann tensor Rµνρσ can be completely determined in terms of the Ricci tensor Rµν. This is possible because in three dimensions the number of independent components is the same for the Riemann d2pd2´1q and Ricci tensor. Reimann tensor in d dimensions has 12 independent components dpd`1q where as the Ricci tensor has 2 independent components. For d “ 3, they both have 6 independent components, hence one can be expressed in terms of the other. For d dimensions, the Riemann tensor can be expressed in terms of the Ricci tensor, the metric and the Weyl tensor Cµνρσ. 2 2 R “ C ` pR g ´ R g q ´ Rg g (3.6) µνρσ µνρσ d ´ 2 µrρ σsν νrρ σsµ pd ´ 2qpd ´ 1q µrρ σsν This can, in fact, be regarded as the definition of the Weyl tensor. Note that along with the Bianchi identity and symmetry property satisfied by the Riemann tensor, the Weyl tensor µ also satisfies the tracelessness property C νµλ “ 0. Thus Weyl tensor in d dimensions has d2pd2´1q dpd`1q 12 ´ 2 independent components. In three dimensions, this gives us 0 independent components. Thus Cµνρσ “ 0 in three dimensions. Therefore the Riemann tensor can be completely expressed in terms of the Ricci tensor and the metric, without using the equation of motion. Thus, in three dimensions,

Rµνρσ “ 2pRµrρgσsν ´ Rνrρgσsµq ´ Rgµrρgσsν (3.7)

We can substitute into this equation (3.5) and its trace, obtained from the equation of motion, to get

Rµνρσ “ Λpgµρgνσ ´ gµσgρνq (3.8)

Thus, the equation of motion in three dimensions completely ﬁxes the Riemann tensor to be in the above form. The geometry is in turn completely ﬁxed by the Riemann tensor, and there are no propagating degrees of freedom. Further the form in equation(3.8) corresponds to a maximally symmetric solution. Hence, any solution to Einstein gravity in three dimensions is locally

1. Minkowski space time for Λ “ 0.

– 32 – 2. dS space time for Λ ą 0.

3. AdS space time for Λ ă 0. These maximally symmetric solutions have maximum number of killing vectors and are rather boring, as the space time looks the same at each point (homogeneous) and in every direction (isotropic). See appendixB for more details on maximally symmetric spaces. Since there are no propagating degrees of freedom, all configurations of the metric are gauge equivalent to one of the maximally symmetric spaces. But, in fact, it is not as boring as it sounds. These maximally symmetric solutions are not the only solutions to three dimensional gravity as one can obtain different solutions by global identifications. For example, we can quotient the Minkowski spacetime by a discrete symmetry group and the resulting spacetime is not a Minkowski spacetime globally. Therefore the physics of three dimensional gravity lie in global degrees of freedom. Hence, three dimensional gravity is purely topological. We will see some manifestations of this below.

3.1 AdS3 and its ‘excitations’

We can think of pure AdS3 as the solution to Einstein’s equations with maximal amount of symmetry. Other solutions to Einstein’s equations, i.e. excitations of the theory, then have less symmetry. Hence, most (asymptotically) AdS3 spacetimes can be generated by reducing the amount of symmetry in empty AdS3. This can be done by making isometry group identiﬁcations, i.e. by quotienting or orbifolding AdS3. Note that spacetimes thus obtained only diﬀer globally from vacuum AdS3.

Let’s brieﬂy review what it means to take a quotient of a spacetime. Mathematically it means the following. Suppose M is a smooth manifold and G is a discrete isometry group acting on this manifold: G ˆ M Ñ M, pg, pq Ñ gppq. The action of G on M deﬁnes an equivalence relation on M : two points p, p1 P M are equivalent if there is a g P G such that gppq “ p1. The quotient space M{G then consists of equivalence classes of points of M, called orbits. Hence, a point in the quotient space corresponds to an orbit of points in M. These quotients spaces are also called orbifolds in the literature.

One very important example is that of a torus. We can obtain a torus T 2 from the complex plane C by following identification15 : T 2 “ C{pZ ` τZq. Thus the torus is locally identical to the complex plane but globally they are quite different manifolds. Suppose a 2D cow is happily roaming around on a green grass field. But it’s halter is attached to a pole by reasonably long rope. If the grass field is a plane it will come back after a long tour to the pole without any trouble. But if the field is actually a torus sometimes it can find itself in an awkward position (see figure 18)!

16 Applying the above to AdS one can show most asymptotically AdS3 spacetimes are orbifolds of pure AdS3. Here by ‘asymptotically AdS3’ I mean that they behave similarly

15 This is going to be very important manifold when I discuss about thermodynamics of CFT2. 16I don’t go into the detail constructions. May be add an appendix collecting the them together.

– 33 – Figure 18: A 2D cow knows torus is not same as plane/sphere!

to pure AdS3 in the limit r Ñ 8, with certain falloff conditions. The falloff conditions at infinity are called ‘boundary conditions’.

3.1.1 Global AdS Consider a space Rp2,2q gridded with coordinates tx0, x1, x2, x3u and endowed with the following metric,

ds2 “ ´pdx0q2 ´ pdx1q2 ` pdx2q2 ` pdx3q2 (3.9)

p2,2q AdS3 is the hypersurface/submanifold of R given by the constraint,

´px0q2 ´ px1q2 ` px2q2 ` px3q2 “ ´L2 (3.10) where the length scale L is known as AdS radius. The constraint can be satisﬁed by following parameterizations,

t x0 “ L cosh ρ cos L ˆ ˙ t x1 “ L cosh ρ sin L ˆ ˙ x2 “ L sinh ρ cos φ (3.11) x3 “ L sinh ρ sin φ

where, t P r0, 2πLq, φ P ro, 2πq, ρ P r0, 8q.

– 34 – Induced metric on the 3D hyper surface

ds2 “ L2p´ cosh2 ρ dt2 ` sinh2 ρ ` dρ2q (3.12)

This is the metric of AdS3 in global coordinates. Now, I shall do the following change of variables : r “ L sinh ρ, to get

r2 r2 ´1 ds2 “ ´ 1 ` dt2 ` 1 ` dr2 ` r2dφ2 (3.13) L2 L2 ˆ ˙ ˆ ˙ where t P p0, 2πLq. Clearly this geometry contains closed time like curves! The conventional wisdom is to go to universal cover such that t P p´8, 8q. From this point onwards by AdS spacetime I shall always refer to the universal covering space.

Another nice way of parametrizing AdS3 is following

1 x0 ´ x2 ´x1 ` x3 SL 2 R Group Manifold (3.14) g “ 2 1 3 0 2 P p , q ø L ˜ x ` x x ` x ¸ Notice that the constraint (3.10) is imposed by, detpgq “ 1,

ñ ´px0q2 ´ px2q2 ` px3q2 ` px4q2 “ ´L2 (3.15)

Q. How to get the AdS3 metric? A. The metric on the group manifold SLp2, Rq is given by Killing-Cartan metric 1 ds2 “ tr g´1dg g´1dg (3.16) 2 The above equation manifests nice connection“` between˘ ` metric˘‰ and isometries. The metric is invariant under ‘rigid’ group action from both side.

g Ñ kL g (3.17) g Ñ g kR

Where kL, kR P SLp2, Rq and they act independently. Thus Lorentzian AdS3 isometry group is two copies17 of SLp2, Rq,

rSLp2, RqL ˆ SLp2, RqRs Z2 « SOp2, 2q. (3.18) L The generators of SO(2,2) are six linearly independent Killing vectors.

0 1 02 0 2 H “ x Bx1 ´ x Bx0 B “ x Bx2 ` x Bx0 03 0 3 21 1 2 B “ x Bx3 ` x Bx0 B “ x Bx2 ` x Bx1 (3.19) 13 3 1 23 2 3 B “ x Bx1 ` x Bx3 J “ x Bx3 ´ x Bx2

17 I mod out by Z2 because, g Ñ kL g kR and g Ñ p´kLq g p´kRq are identical. These transformations should be identiﬁed.

– 35 – Notice that in the embedding coordinates both x0 and x1 are timelike coordinates. In that embedding space H generates time translation, J 23 generates rotations and Bij which mix ‘time’ and ‘space’ coordinates generate boosts.

3.1.2 Poincaré patch The Poincaré coordinates pt, x, zq are given by the following parametrization, Lt x0 “ z L2 x1 “ z Lx x2 ` x3 “ z ´t2 ` x2 ` z2 x2 ´ x3 “ (3.20) z In the above coordinates the metric reduces to L2 ds2 “ ´dt2 ` dx2 ` dz2 (3.21) z2 where t P p´8, 8q, z ą 0 or z ă 0 pdivides` the hyperboloid˘ into two chartsq. Poincare patch only covers a part of the full space time. z Ñ 0 is the boundary which is planar. CFT2 lives on that plane in this coordinates.

3.1.3 Conical defect To obtain conical defect geometry we need to introduce the following coordinates, ir x0 “ cos pγφq γ r2 γt x1 “ ` L2 sin γ2 L d ˆ ˙ r x2 “ sin pγφq γ r2 γt x3 “ i ` L2 cos (3.22) γ2 L d ˆ ˙ The metric becomes,

r2 r2 ´1 ds2 “ ´ γ2 ` dt2 ` γ2 ` dr2 ` r2dφ2 (3.23) L2 L2 ˆ ˙ ˆ ˙ Where, t P r0, 8q, φ P r0, 2πq and γ P r1, 0s is just a parameter. γ “ 0 Ñ Global AdS and γ “ 1 Ñ Poincaré patch. One can rescale the coordinates with γ p0 ă γ ă 1q such that the 1 1 r 1 metric look more like AdS3 : t “ t γ, r “ γ , φ “ φ γ

´1 r12 r12 2 1 12 1 12 12 12 (3.24) ds “ ´ ` 2 dt ` ` 2 dr ` r dφ ˜ L ¸ ˜ L ¸

– 36 – Figure 19: Conical defect geometry : full disc (left) and ‘Pac-Man’ (right)

1 This is the metric of AdS3 but the angular coordinate lies between p0 ď φ ď 2πγq Deﬁcit angle α ” δφ1 “ 2πp1 ´ γq

This geometry can be easily visualized for 2D Euclidean space. Let’s say we take a disc and cut out a triangular piece (just like a pizza slice. See figure 19). Now if one identifies or glues the edges, one gets a cone. Needless to say the new object (i.e. the cone) doesn’t lie one the original plane. But locally it’s flat everywhere except at the origin i.e. the tip where the Ricci scalar diverges18. Conical singularity is naked singularity. But it has a nice interpretation in AdS3- caused by point particle at r “ 0 such that mass of the particle

1 ´ γ δφ1 m “ “ (3.25) 4 G 8πG

1 1 where, 0 ă m ă 4 G . For m Ñ 0 above metric reduces to global AdS3 and for m Ñ 4 G it gives Poincaré AdS3.

3.1.4 Thermal AdS3

Now we will look at the thermal properties of AdS3 solutions. The metric for it is gotten by wick rotating the time coordinate : t Ñ ´it,

r2 r2 ´1 ds2 “ 1 ` dt2 ` r2dφ2 ` 1 ` dr2. (3.26) L2 L2 ˆ ˙ ˆ ˙

This is the global AdS3 metric in Euclidean space. The symmetry group is SLp2, Cq and the topology is of a solid cylinder. To get thermal AdS3 we identify the Euclidean time coordinate. This follows from the fact that for a statistical mechanical system the partition function has a trace. Therefore the initial and final configuration should be identified. (see e.g. Zee’s QFT book). When we identify the time coordinate we get a solid torus with the

18This is easy to see if one regularizes the metric of the cone by introducing some cut-oﬀ and after computing the Ricci scalar ( ) takes that to zero. ´1 ; see e.g. [44]. Rcone Rcone „ Ñ0 δprq ˇ ˇ

– 37 – following identiﬁcations

pt, φq „ pt ` β, φ ` θq, φ „ φ ` 2π (3.27) β “ T ´1 P R; θ P R.

The θ here is the angular potential. This is EAdS3 with T ‰ 0 and θ ‰ 0. The ‘modular parameter’ is given by β 2πτ :“ θ ` i . (3.28) L 3.1.5 BTZ black hole A rotating BTZ black hole in Euclidean space is given by the following metric.

pr2 ´ r2 qpr2 ´ r2 q pr2 ´ r2 qpr2 ´ r2 q ´1 ds2 “ ` ´ dt12 ` ` ´ dr2 r2L2 r2L2 ˆ ˙ (3.29) r |r | 2 ` r2 dφ1 ` ` ´ dt1 . r2L2 ˆ ˙

Here r` and r´ are the black hole radii. This metric can be obtained by quotienting the EAdS3 solution.

pt1, φ1q „ pt1 ` β1, φ1 ` θ1q, φ1 „ φ1 ` 2π (3.30) β1 “ T 1´1 P R; θ1 P R.

Again this is also a solid torus with ‘modular parameter’,

β1 2πτ 1 :“ θ1 ` i . (3.31) L Thermodynamics of the BTZ BH Since we know the metric corresponding to the Black hole we can calculate the Hawking temperature, velocity of the horizon and the entropy from it. Here are the expressions,

κ r2 ´ r2 r T “ “ ` ´ ;Ω “ ´ H 2π 2πr L2 H r L ` ` (3.32) 2πr L L S “ ` “ 2π pML ` Jq ` 2π pML ´ Jq. BH 4G 8G 8G N c N c N Where from the second law of black hole thermodynamics

dM “ TH dSBH ` ΩH dJ. (3.33)

M is the ADM mass of the black hole, ΩH is the angular velocity, J is the angular momentum and κ is the surface gravity. We will come back to SBH formula later via a complete diﬀerent route. So try to memorize the expression!

– 38 – A vague comment

Here I make a vague comment about modular invariance in thermal AdS3 and BTZ. This can be made concrete. Both thermal AdS3 and BTZ black hole are solid torus (see [45, 46]). The boundary topology of both manifolds is torus. Then how can they diﬀer? By the “modular parameter!”

(a) Thermal AdS3 (b) BTZ black hole

Figure 20: Thermal AdS3 and BTZ are related by the transformation : t Ø φ.

All the asymptotically AdS3 spaces I discussed above can be obtained from quotienting. In particular for thermal AdS3 and BTZ black hole one can relate their modular parameters,

τ τ 1 “ ´1, 1 τ “ ´ (3.34) τ 1

The coordinate transformation that relates thermal AdS3 to BTZ interchanges t and φ (see ﬁg. 20). Notice that since for both spaces are solid torus, for BTZ black hole the t cycle that is contractible in the bulk, while the φ cycle is noncontractible. For thermal AdS3 it’s the other way round. Anyway we shall come back to the identical expression (3.34) while discussing Cardy-formula in the next section.

3.2 Hawking-Page transition Hawking and Page showed [47] that for gravity in asymptotically AdS at ﬁnite temperature there are two possible saddles19 (i.e. classical solutions) - thermal AdS and AdS black hole. For low temperature the thermal AdS dominates whereas at high temperature the black

19In the light of AdS/CFT one can think of this as follows. Given a CFT at finite temperature which ‘lives’ on the boundary of the aAdS space, there are two different ways one can ‘fill in’ the bulk : by thermal AdS or by AdS black hole. For low temperature the thermal AdS has lower free energy compared to the

AdS-BH solution. But if one keeps increasing the temperature at a particular Tc the black hole solution will be more favorable. If AdS/CFT is true shouldn’t this transition have an interpretation in the CFT as well? Yes! The corresponding CFT transition was interpreted by Witten [7] as conﬁnement-deconﬁnement transition in “ 4 SYM for AdS5/CFT4 case. N

– 39 – hole solution is dominant. Therefore at some intermediate temperature (Tc) there must be a transition from one saddle to the other. This is known as Hawking-Page transition. From the above discussion it’s quite evident how to compute the transition temperature. We just need to compute the free energies, which is nothing but the Euclidean actions, of the corresponding solutions and ﬁnd out at which temperature they are comparable.

Let’s calculate the gravitational partition function for the thermal AdS3 and BTZ. The Euclidean path integral in AdS3 gravity is given by

I rgs ´ E AdS3 “ g e ~ . (3.35) Z 2 D żBM“T For ‘small ~’ the path integral is sharply peaked around the classical minima gcl of IErgs. So, we can expand the partition function around the classical solutions which is known as saddle point approximation,

AdS3 pτ, τq “ expp´IErgs ` “loop corrections”q (3.36) Z g ÿcl where sum is over all the saddles with modular parameter τ.

Now there are two ways of computing the Euclidean action IErgs.

1. For a given metric one can readily compute the Euclidean action with proper boundary conditions. This is a standard procedure and this is how Hawking-Page and Witten computed it in their papers.

2. Alternatively, since we have learned AdS/CFT dictionary recently we can use it to

compute the IErgs. The statement in the AdS3/CFT2 context is,

pτ, τq “ pτ, τq (3.37) ZAdS3 ZCFT2

pτ, τq is nothing but the CFT partition function on a torus! Then just take a ZCFT2 logarithm to get IErgs.

CFT partition function on a torus20 is given by,

“ T rpe´βH`iθJ q, (3.38) ZCFT2 The Hamiltonian is given by H “ E “ U ´ T S. (3.39)

Using AdS3/CFT2 dictionary we get the expression for IErgs,

I « ´ln “ βM ` θJ ´ S. (3.40) E Z 20Please trust me for the time being in case you are not familiar with this expression. I’ll discuss in detail how to derive this while reviewing ‘ABC of CFT2’. If you still feel impatient please have a quick look at section 4.2.

– 40 – ‚ Thermal AdS3

For thermal AdS3 we have 1 S “ 0,J “0,M “ ´ 8GN (3.41) 6 IE “ βH M From statistical mechanics, we know the free energy in terms of partition function, 1 FT AdS3 “ ´ ln βH Z I « E (3.42) βH 1 “ M “ ´ 8GN ‚ Non-rotating BTZ For non-rotating BTZ the value of the parameters are r2 2πr J “ 0,M “ ` ,S “ ` , 8G L2 4G N N (3.43) 2 r` 2πr` 6 IE “ βH M ´ S “ βH 2 ´ 8GN L 4GN

r` The Hawking temperature is given by, TH “ 2πL2 . Therefore the free energy, 1 FBTZ “ ´ ln βH Z I « E β H (3.44) 1 “ pβH M ´ Sq βH 2 2 2 r` π l 2 “ ´ 2 “ ´ TH 8GN L 2GN 3.2.1 Crtitical Temperature To obtain the critical temperature one just needs to equate21 the two free energies as follows,

F “ F (3.45) BTZ Tc AdS3 Tc ˇ ˇ ˇ 1 ˇ T “ (3.46) c 2πL

Note that both TAdS3 and BTZ exist and valid solutions for all values of temperatures. But for T ą Tc, the more probable/stable solution is the BTZ, whereas for T ă Tc thermal AdS3 is more stable. 21This might remind the reader the way Kramers and Wannier [10] predicted the exact location of the critical point of 2D Ising model by equating high and low temperature expansion of the partition function.

– 41 – 3.2.2 Order of the Transition To see what is the order of the phase transition one has to see which derivative of the free energy is discontinuous.

dF S “ ´ dT ˇV (3.47) ˇ 2 2 ˇ π L 6 SAdS3 “ˇ 0,SBTZ “ TH GN There is a jump in the 1st derivative of free energy. Hence this is a 1st order phase transition.

3.2.3 Is BTZ stable? To check the stability let’s calculate its heat capacity,

BS C “ T ą 0, for BTZ V BT ˇV (3.48) ˇ ˇ “ 0, for Thermal AdS3 or, massless BTZ. ˇ For comparison it is useful to recall that the heat capacity for a black hole in flat spacetime is negative, hence that is unstable. But BTZ is stable. Physically a black hole in flat space Hawking radiates and therefore looses mass. Less massive black hole radiates more 1 (TH „ M ) and thus it evaporates. This shows the instability. Black hole in AdS (e.g. BTZ) on the other hand can be effectively thought of black hole in a box. The AdS potential can be thought of as ‘walls’. Therefore although the black hole radiates, the radiations are reflected back (and forth) from the ‘walls’ and eventually black hole can reach a thermal equilibrium with the ‘gas of photons’. This explains how BTZ (and other black holes in AdS) is stable.

3.3 Brown-Henneaux in a nutshell Here our approach will be more of an overall exciting highlight rather than lengthy (and possibly boring) derivation that can be found in many articles.

3.3.1 Why are boundary conditions important? Before I discuss Brown-Henneaux’s [38] analysis for 3D gravity let’s try to understand why boundary conditions play such an important role in gauge theories (e.g. gravity) in general. QFTs as Hamiltonian (Lagrangian) systems are deﬁned by,

1. Poinsson brackets (action principle),

2. boundary conditions on the ﬁelds and conjugate momenta (derivative of ﬁelds).

For a set of ﬁelds living on a given manifold we choose a radial coordinate r and M other coordinates as ~x. By boundary I mean r Ñ 8 hypersurface22. Then we specify the

22In rigorous sense if doesn’t have a boundary on needs to conformally compactify the manifold to M and then demand the fall oﬀ behaviour at the boundary of . M M

– 42 – fall-off conditions of fields and their derivatives at r Ñ 8 (keeping ~x fixed)

Φpr, ~xq “ pr#q (3.49) O B Φpr, ~xq “ pr#´1q (3.50) r O where # ă 0, that’s why it’s called fall-oﬀ behaviour!.

The fall oﬀ behaviours are important at the level of action principle. Suppose we have a theory with action SrΦs, where Φ schematically represents all possible ﬁelds of the theory. The partition function can be written as,

“ Φ eiSrΦs (3.51) Z D ż This produces all quantum mechanical transition amplitudes (if we can carry out the full path integral!). But for the time being, if we be a less ambitious and focus on the classical limit the leading contribution must come the on-shell conﬁguration. But we can take this classical limit only if there is no boundary terms in the action. Let’s see in more detail why this is the case. E.g, suppose we start with Einstein-Hilbert action with cosmological constant Λ and vary it. 1 ? 1 δS “ ddx ´gpR ´ Rg ` Λg qδg EH 16πG µν 2 µν µν µν żM 1 ? ? ` ddxB p ´ggµνδΓα ´ ´ggµαδΓλ q (3.52) 16πG α µν λµ żM boundary term l jh n The 1st integrand with no surprise gives us Einstein equations. The 2nd term which is a boundary term may or may not vanish depending on the boundary conditions on the ﬁelds and their derivatives.

If there is no boundary terms the on-shell action can be put back to the path integral to obtain the semiclassical physics. But otherwise we can’t do this i.e. semiclassical limit of the path integral is not given by on-shell ﬁeld conﬁgurations which is unacceptable! To make things consistent we need to cancel the boundary term by adding a boundary piece to the action to start with ? Srg s “ S ` dd´1x ´g pg , Bg q (3.53) µν EH L µν µν żBM The second term’s sole purpose in life is to cancel the possible boundary term coming due to variation of SEH . Few more important remarks in order.

Since we are dealing with gravitational action as an example, there is another reason • 2 to add boundary term to SEH . R and Rµν contain B g terms. One needs to add so called ‘Gibbons-Hawking-York’ [48, 49] boundary term to have a well-deﬁned action principle.

– 43 – The above discussion also shows why even if a theory that has no local bulk dynamics • can have non-trivial boundary dynamics – holography in a sense. E.g. 3D gravity has their physical degrees of freedom only at the boundary. For higher dimensional gravity theory the same holds true but they also have non-trivial bulk dynamics on top of that.

3.3.2 Asymptotic symmetries Rigid global symmetries are strikingly diﬀerent from the gauge symmetries (which are re- dundancies rather than actual symmetries). From Noether’s theorem we know that for any contentious global symmetry there is a conserved current and integrating the 0th component of that current over the spatial manifold we obtain the associated charge of that symmetry transformation. Now the immediate question we can ask what is the conserved charge associated to gauge symmetries (or rather global part of the gauge symmetries)?

Here are the steps one follows obtain the conserved charges in gauge theories (e.g. gravity) and hence the algebra generated by these charges known as the asymptotic symmetrys algebra.

1. Deﬁne an action with sensible boundary term and fall-oﬀ behaviour.

2. Find among all possible gauge transformations those preserve all the fall-oﬀ conditions – these are allowed gauge transformations. Others are forbidden gauge transformations and therefore shouldn’t be considered.

3. Associate with each allowed gauge transformation a ‘surface charge’ Q.

Trivial gauge transformation23 : Q “ 0 • Non-trivial/large gauge transformation : Q ‰ 0 •

Figure 21: Diﬀerent type of gauge transformations

Here are all three diﬀerent kind of gauge transformations (see ﬁgure 21).

23 For example, in electrodynamics inﬁnitesimal gauge transformations are of the form : δAµpxq “ Bµαpxq. We usually assume or impose the boundary condition αpx Ñ 8q Ñ 0. These correspond to trivial gauge transformations. But suppose we impose αpx Ñ 8q Ñ const. These are large gauge transformations associate to a global U(1) symmetry. The corresponding charge is the electric charge.

– 44 – Forbidden gauge transformation : Don’t obey boundary conditions ùñ Excluded. • Trivial gauge transformation : Genuine gauge transformation ùñ Redundancy. • Non-trivial gauge transformation : Maps a field configuration to physically different • one ùñ Global (part of gauge) symmetry.

Asymptotic symmetry is deﬁned as follows.

allowed guage transformations Asymptotic symmetry “ (3.54) trivial guage transformations

For gravity the allowed gauge transformations are generated by asymptotic Killing vector ﬁelds ξα. Now we know what one means by asymptotic symmetry. Let’s focus on ‘asymptotically

AdS3 spacetimes’ – we shall deﬁne bellow what mean by asymptotically AdS3. The AdS3 metric is given by

r2 r2 ´1 ds2 “ ´ 1 ` dt2 ` 1 ` dr2 ` r2dφ2 (3.55) L2 L2 ˆ ˙ ˆ ˙

Asymptotically AdS3 space-times are deﬁned as ones which behave like AdS3 in the limit r Ñ 8 with certain fall-oﬀ/boundary conditions. Following Brown-Henneaux [38] we choose.

L2 ´4 ´3 ´3 grr grφ grt r2 ` pr q pr q pr q O´3 2O O gµν “ »gφr gφφ gφtfi “ » pr q r ` p1q p1q fi (3.56) O ´3 O r2O gtr gtφ gtt pr q p1q ´ 2 ` p1q — ffi — O O L O ffi – fl – fl One might ask what makes one to chose such fall off/boundary conditions. The mantra here is – they shouldn’t be as ‘loose’ to allow (almost) all possible solutions in 3D gravity or they shouldn’t be as stringent to accommodate (almost) no solutions at all – the Buddhist middle way of doing physics! They are chosen keeping in mind the following (see [50]).

they are invariant under the AdS global isometry group • 3 they allow for the asymptotically AdS solutions of physical interest e.g. the BTZ • 3 black hole.

they yield ﬁnite charges in the canonical (i.e. Hamiltonian) formalism of GR. • One way to fulﬁll the above criteria is to start with the BTZ metric and act on it with the AdS3 isometry group in all possible ways. This procedure generates metric per- turbations which behave asymptotically as in (3.56). Interestingly, however, at the time of Brown-Henneaux (1986) [38] the BTZ black hole solution (1992) [51] was not known. But

– 45 – it turned out their boundary conditions24 accommodate even the BTZ black hole as well.

After we have chosen some set of boundary conditions ((3.56) for our case), the asymptotic symmetries are generated by the vector ﬁelds ξα which preserve the metric (i.e. the boundary conditions). In other words the ξαs transform any metric of the form (3.56) into another one of the same form and therefore ξαs are solutions of the equations

g “ p1q, g “ pr´3q, g “ p1q,... Lξ tt O Lξ tr O Lξ tφ O Meditating on these equations, one ﬁnds the most general solution may be nicely written as follows.

L3 ξt “ LpT ` ` T ´q ` pT `2 ` T ´2q ` pr´4q 2r2 O ξr “ ´rpT `1 ` T ´1q ` pr´1q O L2 ξφ “ pT ` ´ T ´q ´ pT `2 ´ T ´2q ` pr´4q (3.57) 2r2 O

` ` ` ´ ´ ´ ˘ t where T ” T px q and T ” T px q with x :“ L ˘ φ. This is clear from the above expressions that T ˘ depends on t, r, φ in very particular manner. Expanding the symmetry ˘ generators in modes T ˘ „ eimx one can ﬁnd25,

2 2 ˘ L m imr ξ˘ “ eimx B ´ B ´ B (3.58) m ˘ 2r2 ¯ 2 r ˆ ˙

˘ These ξm generate two copies of inﬁnite dimensional Witt algebra (i.e. Virasoro algebra with zero central charge).

` ` ` rξm, ξn s “ pn ´ mqξm`n ´ ´ ´ rξm, ξn s “ pn ´ mqξm`n ` ´ rξm, ξn s “ 0 (3.59)

But this is not the end of the story. To leading order in 1{r, the Poisson bracket algebra of canonical generators Hrξs is isomorphic to the Lie bracket algebra of the spacetime generators ξ (up to a possible central extension. Using this fact and suitably choosing charges (Q), Brown and Henneaux [38] showed that the asymptotic symmetry algebra of

24By the way, probably it’s the right place to stress that the boundary conditions imposed by Brown and Henneaux are not something sacred. Many people have relaxed/modiﬁed those conditions (see e.g. [52– 55]) to accommodate other interesting gravity solutions which also amounts to changing the dual boundary theory. 25Notice that we are working in leading order in r i.e. we drop subleading terms in (3.57). Recall gauge transformations that obey the boundary conditions and goes to zero at the boundary are called trivial gauge transformations. They correspond to these subleading terms and have vanishing charge (Q “ 0). To obtain the asymptotic symmetry we need to mod them out from allowed gauge transformations.

– 46 – ˘ ˘ the metric (3.56) we started with is two copies of Virasoro algebra (with Lm ” Qrξms), c rL` ,L`s “ pn ´ mqL` ` m2pm ´ 1qδ m n m`n 12 m`n,0 c¯ rL´ ,L´s “ pn ´ mqL´ ` m2pm ´ 1qδ m n m`n 12 m`n,0 ` ´ rLm,Ln s “ 0 (3.60)

The central charge in terms of gravitational parameters is given by

3L c “ c¯ “ (3.61) 2G

This is quite remarkable result. Starting with a 3D gravitational (asymptotically AdS3) theory with six symmetry generators SL(2,R) ˆ SL(2,R), Brown and Henneaux showed that the asymptotic symmetry enhances to inﬁnite dimensional Virasoro ˆ Virasoro. The asymptotic symmetry group is identical to that of CFT2. This was something more pro- found than they could perceive at that time. In a sense they encountered a baby version of AdS/CFT duality almost twelve years before Maldacena!

Ex. Fill in the gaps in the derivation of asymptotic symmetry following [38].

If there is a single thing that the reader wishes to take away from this section would be the formula for central charge (3.61). In AdS3/CFT2 context this would be the central charge for the dual CFT2. We shall always have this formula back of our mind whenever we talk about AdS3/CFT2.

4 CFT2 and its thermodynamics

This part is going to be assorted topics in CFT2. I shall introduce (or rather review) the notion of several ideas useful for this set of lectures. People comfortable with BPZ paper or, Chap 4-6 of the Yellow book [56] or, Chap 4 of Tong’s lecture notes [57] can safely ignore the section 4.1. Next I discuss CFT on torus, modular invariance and Cardy formula.

4.1 ABC of CFT2

Again before studying CFT2 we must ask why should we even study it?

1. It shows up in the title of the course – which is actually a very good reason to study

CFT2. After AdS/CFT came in CFT has been helping us in understanding (quantum) gravitational theories. Therefore it is worth studying in order to understand holography better.

2. But even before AdS/CFT era people have spent lot of their time studying CFT,

particularly CFT2 - for more practical reasons. There are phenomena e.g. statistical

– 47 – mechanical systems at critical points (or QFTs at ﬁxed points) when there is no preferred or natural scale and therefore correlators show power law behaviour. These phenomena are described by conformal ﬁeld theories. As we will see while studying CFTs, people (usually) don’t even talk about Lagrangian! There are 2D statistical mechanical models (equivalently 1D quantum spin systems) that can be completely solved26 using CFT techniques without ever referring to Lagrangian.

3. There is one more motivation to study CFT2. String world sheet is two dimensional and unitarity requires it to be a CFT. Demanding β-functions of this world sheet theory vanish one obtains Einstein equations (with few more extra ﬁelds).

Conformal symmetries : Let’s start with the deﬁnition of conformal invariance. • Transformations which locally preserve the angle between any two lines are called conformal transformations. Operationally these are coordinate transformations xµ Ñ x1µ those keep the metric invariant up to a coordinate dependent scale factor Λpxq, i.e. in ﬂat space,

Bx1ρ Bx1σ η “ Λpxqη (4.1) ρσ Bxµ Bxν µν

It should be clear from the above deﬁnition if Λpxq “ 1 then the corresponding transformations are Poincaré transformations. But here we are relaxing the conditions a bit. Therefore there should be few more transformations which will be allowed – one obvious example is local scaling i.e. x1µ “ λpxq xµ.

Considering inﬁnitesimal coordinate transformation x1µ “ xµ ` µpxq ` p2q and O substituting this into (4.1) one can obtain the The conformal Killing equations,

2 B pxq ` B pxq “ pB.pxqq η (4.2) µ ν ν µ d µν

Solving the above equations we can obtain the corresponding generators of conformal symmetry.

CFTd

For CFTd with d ě 3 here are the conformal transformations and corresponding generators.

26By solving a theory here I mean knowing all possible correlators of that theory.

– 48 – Transformations Generators

Translation Pµ “ ´iBµ

Rotation Lµν “ ipxµBν ´ xνBµq

µ Dilatation D “ ´ix Bµ

ν Special conformal transformation Kµ “ ´ip2xµx Bν ´ px.xqBµq

Table 6: Conformal transformations and their generators

dpd´1q pd`1qpd`2q Thus for CFTd with d ě 3 the number of generators is : d` 2 `1`d “ 2 . One can organize the above generators to write them as generators of SOpd, 2q. This

is also the isometry group for AdSd`1.

CFT2 The conformal Killing equation (4.2) in 2d (Euclidean space) reduces to Cauchy- Riemann conditions,

B00 “B11

B01 “ ´B10 (4.3)

A complex function whose real and imaginary parts satisfy the above equations is a holomorphic function. Let’s go to complex coordinates which will be always useful in

CFT2 context.

z “ x0 ` ix1, z¯ “ x0 ´ ix1 “ 0 ` i1, ¯ “ 0 ´ i1 1 1 B “ pB ´ iB q, B “ pB ` iB q z 2 0 1 z¯ 2 0 1

So, pzq and hence fpzq “ z ` pzq is holomorphic. This is a very strong and useful statement. It says, a holomorphic function fpzq “ z`pzq gives rise to an inﬁnitesimal two-dimensional conformal transformation z Ñ fpzq. This implies that under z Ñ fpzq, the metric tensor transforms as,

Bf Bf¯ Bf 2 ds2 “ dzdz¯ Ñ dzdz¯ “ dzdz¯ (4.4) Bz Bz¯ Bz ˇ ˇ ˇ ˇ ˇ ˇ Witt algebra : As we have observed above, for anˇ inﬁnitesimalˇ conformal transfor- • mation in two dimensions the function pzq has to be holomorphic in some open set.

– 49 – However, it is reasonable to assume that pzq in general is a meromorphic function having isolated singularities outside this open set. We therefore perform a Laurent expansion of pzq around say z “ 0. A general inﬁnitesimal conformal transformation can then be written as

8 8 n`1 n`1 pzq “ ´ anz , ¯pz¯q “ ´ a¯nz¯ (4.5) n“´8 n“´8 ÿ ÿ n`1 ¯ n`1 The corresponding generators og these transformations ln “ z Bz and ln “ z¯ Bz¯ satisfy the following commutation relations

rlm, lns “ pn ´ mqlm`n

rlm, lns “ 0 (4.6)

This is classical version of Virasoro algebra we have seen in the context of Brown- Henneaux analysis. This inﬁnite dimensional algebra is also known as Witt algebra.

Primaries, quasi-primaries, descendants : Here are some useful definitions. • Fields only depending on z, i.e. φpzq, are called holomorphic (or chiral) fields and fields φpz¯q only depending on z¯ are called anti-holomorphic (or anti-chiral) fields. If a field φpz, z¯q is said to have conformal dimensions ph, h¯q if under scalings z Ñ λz it transforms as follows,

¯ φpz, z¯q Ñ λhλ¯hφpλz, λ¯z¯q (4.7)

A ﬁeld is called a primary ﬁeld of conformal dimension ph, h¯q if under conformal transformation z Ñ fpzq it transforms as follows,

¯ φpz, z¯q Ñ f 1pzqhf¯1pz¯qhφ fpzq, f¯pz¯q (4.8) ` ˘ A ﬁeld is quasi-primary if it behaves as primary only for SL(2,R) generators i.e. only

under L´1,L0 and L1. Clearly this notion is there only for 2D CFT.

Descendants are the ﬁelds obtained by acting Virasoro generators L´ms on some primary. Given a primary there is an inﬁnite tower of such descendants. The primary with the corresponding tower of descendants is called conformal family of that primary.

Idea of OPE : The operator product expansion (OPE) is a statement about what • happens as local operators approach each other. The idea, which ﬁrst introduced by Wilson [58], is that two local operators inserted at nearby points can be closely

– 50 – approximated by a string of operators at one of these points. Let’s denote all the local operators of the CFT by , Then the OPE is Oi

pz , z¯ q pz , z¯ q “ Ck pz ´ z , z¯ ´ z¯ q pz , z¯ q ` ... (4.9) Oi 1 1 Oj 2 2 ij 1 2 1 2 Ok 2 2 k ÿ

k where Cijpz1 ´ z2, z¯1 ´ z¯2q are a set of functions which, depend only on the separation between the two operators due to translational invariance.

Central charge : One can deﬁne central charge of a CFT by taking OPE of stress • tensors as follows,

c{2 2T pz q BT pz q 2 2 (4.10) T pz1qT pz2q “ 4 ` 2 ` ` ... pz1 ´ z2q pz1 ´ z2q pz1 ´ z2q

where by “...” I denote the regular terms. The c appearing in (4.10) is the central charge of the CFT. This is a very important quantity with diﬀerent physical interpre-

tations (see [57]). For CFT2 c can be interpreted as

1 1. Degrees of freedom of the theory : c “ 1 for free boson, c “ 2 for free fermion etc. πc 2. Measure of Casimir energy : CFT2 on a cylinder E “ ´ 6 l . µ c 3. Measure of Weyl anomaly : xTµ y “ 12 R, where R is the Ricci scalar od the 2D surface. ? 4. It tells us the density of high energy states in CFT2 : Entropy SpEq „ c E. This is famous Cardy formula. I shall discuss this in detail later part of this section.

State-operator correspondence, radial quantization : Intuitively, the state- • ment of the state-operator correspondence is that given an initial state on the cylinder |ψy we can make a conformal transformation that ‘squashes’ it all to the origin on the plane. Thus it is a local perturbation on the plane, and there is a corresponding local operator p0q at the origin. One can also go the other way i.e. given some local Oψ operators on the plane one can go back to initial states on the cylinder as follows,

|ψy “ lim ψpxq|0y (4.11) xÑ0 O

This is something very particular to CFTs. For generic QFTs we don’t have such correspondence since it depends heavily on the conformal transformation - which is not a symmetry for generic QFTs. This is something which will play crucial role27 in

27Whenever I’ll say ‘state’ or particularly‘excited state’, I’ll mean some operator acts on vacuum to create that.

– 51 – rest of my lectures.

[Insert image].

In standard QFT course we mostly deal with time-ordered correlators which naturally arise in path integral formalism. Suppose we have a CFT on a cylinder. A natural way of thinking about it as a worldsheet generated by closed string propagation. But the theory enjoys conformal symmetry. We can always use a conformal transformation to map the cylinder on to a plane. Each ‘closed string’ will be mapped to a circle on the plane, while the ‘worldsheet time’ will be mapped to radial direction

t “ ´8 ñ r “ 0, and t “ 8 ñ r “ 8.

Thus time ordering on the cylinder is equivalent to radial ordering on the plane. One can quantize the theory foliating the 2D space by concentric circles, as I have mentioned above. Using the ideas of state-operator correspondence, OPE and some

contour integrals from complex analysis it can be shown that the Laurent modes Lm of the energy-momentum tensor (which are the generators of inﬁnitesimal conformal transformations) satisfy Virasoro algebra. Note that at classical level they satisfy Witt algebra as I discussed above.

Virasoro algebra, CFT Hilbert space : Virasoro algebra is central extension of • Witt algebra and is given by

c rL ,L s “ pn ´ mqL ` m2pm ´ 1qδ m n m`n 12 m`n,0 c¯ rL¯ , L¯ s “ pn ´ mqL¯ ` m2pm ´ 1qδ m n m`n 12 m`n,0 ¯ rLm, Lns “ 0 (4.12)

We have already came across this algebra in last section while discussing Brown-

Henneaux analysis in asymptotically AdS3. Once we have classified all possible operators and established a unique way to map those operators to states by above correspondence, we can construct the Hilbert space of 2D CFT – representation of Virasoro algebra. The first job is to define the vacuum which has ‘maximal symmetries’. It’s defined28 as follows,

Ln|0y “ 0, for n ě ´1 (4.13)

We can act local operators on |0y to obtain states. There will be some special operators (or ﬁelds) namely primaries and then one can act on them by the Virasoro

28 Ideally we would like to have Ln|0y “ 0, for all n. But we cannot satisfy this criterion due to the central term in the Virasoro algebra.

– 52 – generators (the ‘raising operators’) which are nothing but some derivative operators to get the descendants. By state-operator correspondence we associates states to all such operators. Number of primaries can be finite (e.g. rational CFTs) or infinite. But for each primary state there will be infinite tower of descendant states.

Null states : By deﬁnition Virasoro primaries are annihilated by L , i.e. • m

Lm|hy “ 0 for m ą 0.

A generic (descendant) state of this primary will be of the form

Lń1 Lń2 ...Lńl |hy; 1 ď n1 ď n2 ... ď nl.

If n1 ` n2 ` ... ` nl “ N, then the state is called a descendant of level N. A generic state at level N is given by,

|χy :“ cnLt´nu|hy (4.14) m ÿ where Lt´nu represents all possible combinations of ‘raising operators’ of level N. For special value of h it can happen that |χy is a primary too, i.e. Lm|hy “ 0 for m ą 0. In that case |χy is called a null state at level N. These states have following interesting properties.

1. They have zero norm : xχ|χy “ 0 2. They are orthogonal to any other state of the representation i.e. |χy ‘decouples’ from all states in the conformal family of |hy.

You may feel rather uncomfortable to have such ‘spurious states’ which are both primary and descendant; have zero norm; completely decouple from the representation – but they are our friends! They put stringent constraint29 on the correlators such that sometimes we can get lot of millage. This idea will be useful when I discuss the ‘monodromy method’ of computing the 2D Virasoro blocks in large c limit (see section 5.2).

4.2 CFT on a torus

2 The aim in this section is to write down partition function of CFT2 on a torus (T ) which is given by,

“ Tr H expp´βH ` i θJq (4.15) Z S1 „  2πi τ L ´ c 2πi τ L ´ c “ Tr e 0 24 e 0 24 . (4.16) „ ` ˘ ` ˘ 29The minimal models in 2D CFTs always have such null vectors in their spectrum (actually inﬁnitely many of them!). They help in ﬁxing higher point correlators completely.

– 53 – If you are familiar with one of these formulae (in that case I don’t even need to explain the notations here), you can happily skip this section.

Geometrically torus is something that we get from gluing two open ends of a cylinder as following cartoon.

(a) Cylinder (b) Torus

Figure 22: Torus from a cylinder

Mathematically T2 can be obtain from C by modding out it by some lattice.

T2 “ C{pZ ` τZq.

1 Im

Figure 23: Lattice that generates 2-torus

Notice that all points that diﬀer by a linear combinations of the two basic lattice vectors are identical. It’s worth noting here, to have this lattice picture we have used lot of conformal symmetries.

1. 2D diﬀeomorphisms : to straighten the coordinates.

– 54 – 2. Rotations : to ﬁx one direction point along the real axis.

3. Translation : to put a point of the lattice at the origin.

4. Global scaling : to ﬁx horizontal lattice spacing to one.

Thus the entire lattice is described by a single complex parameter τ which is known as the modular parameter. A torus is characterized by its modular parameter. Let’s try to write down the partition function on T2 using the ‘traditional wisdom’.

“ φ e´SE rφs (4.17) Z 2 D żT This formula is correct but it is useful while one deals with Lagrangians - which we have given up for good! For CFTs it’s convenient to rather work Hamiltonian of the theory because one always has Hamiltonian in CFT2 given by

H „ L0 ` L0 (4.18)

Now I try to construct the partition function in terms of H. To motivate let me start with ordinary quantum mechanics (QM) i.e. QFT in 0 ` 1 dimensions. From standard textbook QM (with periodic boundary condition along the time direction) we know that,

“ X e´SE rXs “ Tr e´βH (4.19) Z D żXp0q“Xpβq Let’s jump from QM Ñ p1`1qD QFT which roughly tells us to jump from tXu Ñ tφpx1qu. Therefore, if I assume for the time being that Reτ “ 0 i.e. τ “ ix0, then the partition function on the torus becomes

“ φ e´SE rφs “ Tr e´2πpIm τqH (4.20) Z D ż The factor of 2π is there to make contact with convention in literature (?). But this was a special case when τ “ ix0, what about the generic τ which also contains a real piece i.e. τ “ x1 ` ix0? In that case we need to glue the real part as well but should shift by 2πpReτq that can be generated by ei 2π pReτqP

“ φ e´SE rφs “ Tr e´2πpIm τqH ei 2π pReτqP (4.21) Z D ż We have written down in terms of H and P , only thing remains is to derive their expres- Z sion from Tµν.

We know how T pzq transforms when we map C to a cylinder via the conformal map : w “ eiz. c T pwq Ñ pBfq2T pfpwqq ` Spf, wq (4.22) 12

– 55 – where Schwarzian derivative is deﬁned as,

Bf B3f ´ 3 pB2fq2 Spf, wq ” 2 (4.23) pBfq2

Therefore the stress tensor on the cylinder becomes,

c T pzq “ pB wpzqq2T pwpzqq ` Spw, zq cyl z 12 c “ ´ w2T pwq ´ (4.24) 24 ˆ ˙ ´n´2 Using mode expansion on the plane, T pwq “ w Ln, I ﬁnd,

ř c T pzq “ ´ e´inz L ´ cyl n 24 „  ÿ c T pzq “ ´ e´inz L ´ (4.25) cyl n 24 „ ÿ  Once we have the Tcylpz, zq we can extract H and P as follows.

1 1 H “ dx1T “ ´ dpRezq T pzq ` T pzq 2π 00 2π cyl cyl ż ż „  c c “ L ´ ` L ´ (4.26) 0 24 0 24 ˆ ˙ ˆ ˙ 1 1 P “ dx1T “ ´ dpRezq T pzq ´ T pzq 2π 01 2π cyl cyl ż ż „  c c “ L ´ ´ L ´ (4.27) 0 24 0 24 ˆ ˙ ˆ ˙ Just to make things clear H and P are time and space translation on the cylinder but they are actually dilatation and rotation on the complex plane, respectively. So ﬁnally we have the CFT partition function on T2

c c 2πi τ L0´ 2πi τ L0´ 2 pτ, τq “ Tr e 24 e 24 (4.28) ZT „ ` ˘ ` ˘ as advertised in the beginning of this section.

4.3 Modular invariance

We have deﬁned our theory on torus in terms of the bases of lattice on the complex plane :“1” and “τ”.

– 56 – Re

τ `1 τ

1 Im

Figure 24: t1, τu and t1, τ `1u give the same torus because the shaded regions are identical.

It is easy to see that all lattice are not independent i.e. there are diﬀerent lattices which give the same torus. E.g. t1, τu and t1, τ ` 1u give the same torus which is quite evident from ﬁg. 24. This notion can be generalized further and it can be shown that there two such transformations namely : τ Ñ τ ` 1 T 1 : τ Ñ ´ (4.29) S τ that generate the full symmetry group - also known as the modular group. The most general modular transformation has the form aτ ` b τ Ñ ; a, b, c, d P Z, ad ´ bc “ 1. (4.30) cτ ` d

This modular group is isomorphic to SLp2, Zq{Z2 » PSLp2, Zq.

Ex. Check that, p q3 “ 2 “ 1. ST S The bottom-line here is partition function on a torus has to obey modular invariance - just like functions on sphere need to be periodic. We will see while deriving Cardy formula this is actually a very powerful constraint. We will just use the transformation for that S purpose.

4.4 Derivation of Cardy formula The Cardy formula gives us the asymptotic density of states (hence the entropy) in a 2D CFT determined only by a few features namely the central charge c and the ground state 30 conformal weight ∆0 and independent of any details of the dynamics . Since E. Verlinde

30The advantage of the Cardy formula is that while it let us count states, it does not require detailed knowledge of the states being counted. Actually the above statement which is the strong point of Cardy

– 57 – [59] extended this formula to arbitrary d-dimensional CFTs it is also called Cardy-Verlinde31 formula. The Cardy-Verlinde formula was later shown by Kutasov and Larsen [61] to be invalid for weakly interacting CFTs (violates Bekenstein bound on the entropy). In fact, since the entropy of higher dimensional (meaning d ą 2) CFTs is dependent on exactly marginal couplings, it is believed that a Cardy formula for the entropy is not achievable when d ą 2. The Cardy formula is of great importance for the AdS/CFT correspondence, since it matches precisely with the horizon entropy of BTZ black holes, as shown by Strominger [62]. Carlip computed the logarithmic correction to this formula [63]. We will see modular invariance which we have already discussed in sec4 plays a crucial role in the derivation of the Cardy formula. We start with a 2D CFT with central charge c with Virasoro algebra c rL ,L s “ pn ´ mq ` npn2 ´ 1qδ n m 12 n`m,0 c rL , L s “ pn ´ mq ` npn2 ´ 1qδ n m 12 n`m,0 rLn, Lms “ 0 (4.31)

The torus partition function with c “ c is given by,

2πi τ L ´ c 2πi τ L ´ c pτ, τq “ Tr e 0 24 e 0 24 (4.32) Z „ ` ˘ ` ˘ Let’s focus on the τ part; τ I can restore later.

2πi τ E´ c pτq “ ρpEqe 24 Z Eě0 ` ˘ ÿ8 2πi τ E´ c “ dE ρpEq e 24 (4.33) ż0 ` ˘ Here I have just taken the trace over an energy eigenstate |Ey which has energy eigenvalue E. L0 |Ey “ E |Ey ρpEq denotes the density of states with energy eigenvalue E. This is the quantity that we will be interested in. In microcanonical ensemble entropy is deﬁned as follows.

Smicro “ log ρpEq (4.34)

Notice that for T “ 0 (lowest energy) there is only one state i.e. the vacuum. So, ρpE0q “ 1 and therefore Smicro “ 0. For large temperature (T " 1) there are many states with a given large energy E, the entropy Smicro ‰ 0. formula, is also its main weakness, although we can count microstates without a full quantum theory of gravity we have no clue about the actual states we are counting. Nevertheless these results suggest an interesting effective description of a black hole entropy. 31It should rather be called Nahm-Cardy-Verlinde formula. Actually Nahm found this formula in 1974 (see eq. (47) of [60]) with the log corrections. The paper has got only 5 citations and it’s funny that it’s quite difficult to find ‘Cardy formula’ in Cardy’s original paper we all refer to! I thank Shouvik Datta for drawing my attention to Nahm’s paper [60].

– 58 – In what follows we will be interested in entropy for E " 1 and high temperature. We iβ choose our modular parameter purely imaginary, τ “ 2πl . Therefore T Ñ 8 ô τ Ñ 0. As I have discussed in the last section the partition function has to be modular invariant i.e. 1 pτq “ ´ (4.35) Z Z τ ˆ ˙

Bellow we use this fact to derive entropy formula for CFT2 at T Ñ 8. This goes by the name of Cardy formula.

we want to extract ρpEq from and for that we need to take an inverse Laplace Z transform

´2πiτ E´ c ρpEq “ dτ pτq e 24 (4.36) Z ¿C ` ˘ Again we are interested in T Ñ 8 i.e. τ Ñ 0 limit. This limit the integral is dominated by pτq . But how to obtain pτ Ñ 0q? We only know pτq in terms of ρpEq. But our Z Z Z ˇτÑ0 ˇ ultimateˇ aim is to ﬁnd ρpEq! We are in a loop here. ˇ Now we see how modular invariance rescues us from/ this crisis. We can use modular invariance (actually just the transformation) of pτq to relate the T Ñ 8 behaviour to S Z T Ñ 0 behaviour.

pτq “ p´1{τq Z Z 2πi 1 c “ e τ 24 p´1{τq (4.37) Z ´2πi 1 L where p´1{τq ” Tr e τ 0 . Z r As we are interested in τ Ñ 0, the quantity p´1{τq is dominated by the lowest eigen- Z value ofrL0 (let’s take it to be E “ 0. Since we are not interested in logarithmic corrections to the entropy it doesn’t matter) and since the CFTr vacuum is unique i.e. p´1{τq Ñ 1 as Z τ Ñ 0. Thus, r 2πi 1 c pτq “ e τ 24 as, τ Ñ 0. (4.38) Z Now we are through thanks to modular invariance. We have the expression for pτq, all Z we need to do is to substitute this into the expression for ρpEq to obtain

2πi 1 c ´2πiτ E´ c ρpEq « dτ e τ 24 e 24 ¿C ` ˘ ´2πi τ E´ c ´ 1 c “ dτ e 24 τ 24 (4.39) ¿C “ ` ˘ (‰

– 59 – ´2πifpτ˚q Here we need to do a saddle point approximation i.e. ρpEq „ e where τ˚ is the value for τ at the stationary point of fpτq.

c τ˚ “ i (4.40) 24 E ´ c c 24 ` ˘ 32 c Notice that we are in τ Ñ 0 regime i.e. the saddle point is valid only when E " 24 .

ρpEq « e´2πifpτ˚q ? 2π c pE´ c q “ e 6 24 (4.41)

Ex. Derive the above result using saddle point approximation.

We can do identical analysis with τ part (and take cL ‰ cR) and take logarithm to obtain

c c c c S “ 2π L pE ´ L q ` 2π R pE ´ R q (4.42) Cardy 6 24 6 24 c c

Remarkable fact about the entropy formula is it depends only on the central charge c and the energy eigenvalues of the CFT2.

Cardy formula in canonical ensemble We can write down this formula in canonical ensemble. L BS 1 “ 2π BNL TL L BS 1 “ (4.43) 2π BNR TR where,

1 NL ´ cL{24 1 NR ´ cR{24 TL “ ,TR “ (4.44) Ld cL{24 Ld cR{24

The entropy formula is given by,

πL S “ c T ` c T (4.45) Cardy 6 L L R R ˆ ˙ valid for TR,TL " 1.

32In the next section we compute Bekenstein-Hawking entropy for BTZ black hole using this result. Interestingly there we will be in the semiclassical limit i.e. c Ñ 8 which is clearly diﬀerent regime than what we are in. But still the entropy formulas match! I comment on this again in the next chapter.

– 60 – 4.5 BTZ from Cardy Now we will see how Strominger used Cardy formula to match the Bekenstein-Hawking entropy for BTZ black hole. In his paper [62]33 Strominger combined results from some older woks and magically reproduced the BTZ entropy. Here is what he did.

3L 1. Brown-Henneaux (1986) [38]: c “ 2G

cL cR 2. Banados (1994) [64]: ML “ L0 ´ 24 ` L0 ´ 24 ; J “ L0 ´ L0.

cL cL cR cR 3. Nahm(’74)-Cardy(’86) [60, 65]: S “ 2π 6 pE ´ 24 q ` 2π 6 pE ´ 24 q Strominger took the above know results anda substituted #1 anda #2 into #3 (with cL “ cR “ c) to obtain the BTZ black hole entropy

L L 2πr` SBH “ 2π pML ` Jq ` 2π pML ´ Jq “ (4.46) c4G c4G 4G

This is identical to the formula we got from BTZ metric in previous section.

5 2D blocks, 3-point functions and AdS3

5.1 Local observables in CFT Local observable of CFT are the n-point correlators of local primary operators. All other correlators of descendants can be computed taking derivatives once the former ones are known. As I discussed earlier that conformal symmetries are extremely constraining – sometimes ﬁx correlators up to some constants. Let’s see how it works for some lower point correlators. For this subsection our discussion will be applicable to arbitrary spacetime dimensions. Next subsection onwards we will restrict ourselves to 2D.

5.1.1 2-point function Using translational symmetry we can ﬁx one of the points at the origin without loss of generality and the rotational invariance dictates that the form of the 2-point correlator should be a function of |x|.

x ˆ pxq ˆ p0qy “ fp|x|q (5.1) O1 O2 Scaling (dilation) symmetry requires, fpλxq “ λ´∆fpxq, which constraints

ˆ ˆ c12 x 1pxq 2p0qy “ O O |x|∆1`∆2

Ex. Check that, special conformal transformation (Kµ) (which can be implemented as combination of inversion (I) and spacetime translation (Pµ)) demands

33 # of citations This paper is amazingly simple! Probably, it has ‘simplicity’ “ max, in the ﬁeld of hep-th. ,

– 61 – dimp ˆ q “ dimp ˆ q O1 O2 6 ∆1 “ ∆2 “ ∆ psayq.

Therefore ﬁnally we have (after normalization),

δ x ˆ pxq ˆ p0qy “ 12 (5.2) O1 O2 |x|2∆

5.1.2 3-point function Similar manner we can ﬁx 3-point correlators. Using translational and rotational invariance,

c ˆ ˆ ˆ 123 (5.3) x 1px1q 2px2q 3px3qy “ 2a 2b 2c O O O |x12| |x13| |x23|

∆1`∆2`∆3 Dilatation invariance ﬁxes : a ` b ` c “ 2 . Special conformal invariance ﬁxes,

2a “ ∆1 ` ∆2 ´ ∆3

2b “ ∆1 ` ∆3 ´ ∆2

2c “ ∆2 ` ∆3 ´ ∆1

Ex. Check the above results.

5.1.3 4-point function If the reader is seeing the above results for the ﬁrst time, (s)he might be tempted to ﬁx the 4-point function using similar steps. I encourage him/her to try. It will be clear that these objects are more interesting. One can construct functions of coordinates which are conformally invariant and are known as conformal cross ratios. In arbitrary dimensions there exist two independent cross ratios

|x |2|x |2 : 12 14 u “ 2 2 |x13| |x24| |x |2|x |2 : 14 23 (5.4) v “ 2 2 |x13| |x24| Thus 4-point correlator can be ﬁxed up to a function of u and v and formally can be expressed as,

4 ∆ ´∆ ´∆ x ˆ px q ˆ px q ˆ px q ˆ px qy “ fpu, vq x 3 i j (5.5) O1 1 O2 2 O3 3 O4 4 ij iăj ź

– 62 – 4 where ∆ “ ∆i. i“1 ÿ Notice that 4-point correlators contain dynamical information about the CFT. But there are quantities called conformal blocks which contain all the information ﬁxed by conformal symmetries. These quantities are purely kinametical. They are the central character of this part of these notes. Let’s discuss what I actually mean by conformal blocks. For that we just need to calculate 4-point function using OPE. Pictorially,

x ˆ px q ˆ px q ˆ px q ˆ px qy “ c c (5.6) O1 1 O2 2 O3 3 O4 4 12∆ 34∆ ∆ ÿ

Let’s try to understand what this picture actually means.

ˆ ˆ ˆ ˆ c12∆ c34∆1 ˆ ˆ x 1px1q 2px2q 3px3q 4px4qy “ 1 xp 2px2q ` ...qp 4px4q ` ...qy O O O O |x |∆1`∆2´∆ |x |∆3`∆4´∆ O O ∆,∆1 12 34 ÿ c12∆ c34∆ ˆ ˆ “ x 2px2q 4px4qy ` ... |x |∆1`∆2´∆ |x |∆3`∆4´∆ O O ∆ 12 34 ÿ ˆ ˙ c c 1 “ 12∆ 34∆ ` “desc. contributions” |x |∆1`∆2´∆ |x |∆3`∆4´∆ |x |2∆ ∆ 12 34 24 ÿ ˆ ˙ ” c c px q (5.7) 12∆ 34∆W∆,l i ∆ ÿ px q are called conformal partial waves (CPW) and are completely ﬁxed by con- W∆,l i formal invariance. They get contribution from entire conformal family of primaries with scaling dimension ∆ and spin l. Then what are the conformal blocks? They are very similar to CPW – just diﬀer by a coordinate dependent scale factor.

∆3´∆4 ∆1´∆2 1 x2 2 x2 2 px q ” 14 24 pu, vq (5.8) ∆,l i ∆1`∆2 ∆3`∆4 2 2 ∆,l W 2 2 2 2 x x G px12q px34q ˆ 13 ˙ ˆ 14 ˙

pu, vq are known as conformal blocks. G∆,l

So far our discussion true for arbitrary spacetime dimensions. In fact these conformal blocks in D ą 2 are extensively used in the study of conformal bootstrap34. But we are interested in 2D CFTs. As we have seen several times by now CFT2 is special. Here also I can make stronger statements – CPWs or equivalently the conformal blocks factorize into holomorphic and anti-holomorphic pieces. It’s also very common in case of 2D CFT to ﬁx

34See ST4 2017 lecture notes by Parijat Dey and Apratim Kaviraj for more details on conformal bootstrap.

– 63 – three out of the four points to 0, 1 and 8 using Möbius transformation. Thus the functional form of the 4-point correlator simpliﬁes to

x ˆ p8, 8q ˆ p1, 1q ˆ pz, z¯q ˆ p0, 0qy “ c c ph , ∆, zq ¯ph , ∆¯ , z¯q (5.9) O1 O2 O3 O4 12∆ 34∆ F i F i ∆ ÿ In what follows I focus on ph , ∆, zq. I call them Virasoro conformal blocks. Unlike higher F i dimensional blocks we still don’t know closed form expressions for these blocks in general. We know their forms in diﬀerent limiting situations.

5.2 Heavy-light conformal blocks at large c Here I discuss about one particular limit namely ‘heavy-light’ limit where the form of the Virasoro blocks is known. I also describe some physical applications where one can use them. I shall be very brief (and skip almost all computational details!) in this section and refer to relevant papers for enthusiastic readers. For this part of the notes I shall mainly follow [66]. Virasoro conformal blocks are known is so called ‘heavy-light’ limit. In this limit the central charge c of the CFT is very large i.e. c Ñ 8 and both the heavy (OH ) and the light operators (OL) in the theory scale as the central charge (c) but they obey the following relations h h H „ p1q, L ! 1. (5.10) c O c Due to Zamolodchikov [67, 68] it is known that Virasoro conformal blocks exponentiates when c Ñ 8 as follows c h pz q « exp f i , z . (5.11) F i 6 c i ˆ ˆ ˙˙ This is motivated by Liouville theory but so far there is no honest derivation of this “classical limit”. Assuming this holds and using a technique called “monodromy method” one can compute the conformal blocks, pz , z¯ q. This limit is particularly interesting because in F i i this c Ñ 8 limit (perhaps with some more conditions) we can expect to have a dual classical gravitational description of the CFT. This can be seen from the Brown-Henneaux formula 3l (3.61) for the central charge : c “ 2G Ñ 8 means l " G and one can think the dual gravity theory as “classical”. Conformal blocks also play an important role in the context of holography. There has also been very strong evidence that conformal blocks are intimately related to geodesics in AdS [69–74]. One of the important objects in this context, for a 2d CFT, is the correlator of two heavy operators with two light operators [69–77]. As c Ñ 8, the ratio of the conformal dimension with the central charge remains ﬁxed for heavy operators, whilst that of light operators is much smaller than unity. One can think of these heavy operators being responsible for creating an excited state after a global quench [78, 79]. On the gravity side, this excited state corresponds to the conical defect background [79]. It has been shown that the conformal block of this correlator is precisely reproduced from holography from

– 64 – an appropriate worldline configuration in this bulk geometry [69, 71]. Moreover, the correlation functions in this excited state mimic thermal behaviour if the conformal dimension of the heavy operator exciting the state is greater than c{24 [76, 77]. This is an example of a pure microstate (with a sufficiently high energy eigenvalue) behaving effectively like a mixed state being a part of the thermal ensemble.

Here I discuss about conformal blocks of two heavy operators and arbitrary number pmq of light operators. We work in the heavy-light approximation and utilize the monodromy method to derive the (m ` 2)-point conformal block. We expand the correlation function in a basis which involves pairwise fusion of the light operators . In the strict heavy-light limit and at large central charge, I show that, for this class of OPE channels :

The conformal block having an even number of light operators and two heavy opera- • tors factorizes into a product of 4-point conformal blocks of two heavy and two light operators.

The conformal block having an odd number of light operators and two heavy operators • factorizes into a product of 4-point conformal blocks of two heavy and two light operators and a 3-point function involving one light and two heavy operators.

These results for conformal blocks from the CFT are reproduced from the bulk by considering suitable generalizations of the worldline configurations considered in [71]. The choice of OPE channels in the CFT are in one-to-one correspondence to geodesic configurations in the bulk. This implies that these higher-point conformal blocks can be fully recast in terms of bulk quantities. This outcome nicely fits within the notion of emergence of locality from a conformal field theory [80] and serves as an explicit demonstration of the same not only for higher-point correlation functions but also for non-vacuum states. As an application we will see the results can be used to compute entanglement entropy of heavy states with multiple intervals. The light operators will correspond to twist operators, with conformal dimension c{24 pn ´ 1{nq, (which implement the replica trick) in the limit n Ñ 1, where n is the replica index [81]. One can then utilize the higher-point conformal block to obtain the entanglement entropy of an arbitrary number of disjoint intervals.

Here I shall brieﬂy sketch how to go about computing the HLL . . . LLH conformal blocks. We are interested in the following correlator

m`1 x pz , z¯ q pz , z¯ q pz , z¯ qy. (5.12) OH 1 1 OL i i OH m`2 m`2 i“2 ź In terms of cross-ratios x “ pzm`2´ziqpz2´z1q , the above quantity becomes, i pzm`2´z2qpzi´z1q

m`1 p8q p1q px q p0q . (5.13) OH OL OL i OH « i“3 ﬀ A ź E

– 65 – We work in c Ñ 8 limit for which c pz , h , h˜ q “ exp ´ f pz , , ˜ q . (5.14) Fppq i i i 6 ppq i i i ” ı We shall also work in the heavy-light limit

6h 6h “ H „ p1q , “ L ! 1 (5.15) H c O L c And this particular OPE channel (see ﬁg. 25) where light operators fuse together.

1 x x x x x xm xm+1 L L L L L L L L

˜p ˜p ˜p ˜p

0 ∞ H ˜Q ˜R H

Figure 25: Particular choice of OPE or monodromy contour : light operators fuse in pairs.

5.2.1 Monodromy method Recall a typical conformal block looks like

pz , h , h˜ q :“ x pz q pz q|αy xα| pz q|βy ¨ ¨ ¨ xζ| pz q pz qy (5.16) Fp i i i O1 1 O2 2 O3 3 Op´1 p´1 Op p Let’s insert an additional operator, ψˆpzq whose dimension doesn’t scale with c. So it cannot change the exponential behaviour but can merely modify the block by some multiplicative factor as follows,

Ψpz, z q :“ x pz q pz q|αy xα|ψˆpzq pz q|βy ... xζ| pz q pz qy i O1 1 O2 2 O3 3 Op´1 p´1 Op p c « ψpz, z q exp ´ f pz , , ˜ q (5.17) i 6 ppq i i i ” ı The operator ψˆpzq is arbitrary. Let’s choose that ψˆpzq obeys the null-state condition at level 2. This requirement imposes strong constraint on |Ψy which has to satisfy35 the following diﬀerential equation,

3 1 9 L ´ L2 |Ψy “ 0, with, h cÑ8“ ´ ´ . (5.18) ´2 2p2h ` 1q ´1 ψ 2 2c „ ψ  The diﬀerential operator representation of the Virasoro generators gives an ODE

p d2ψpzq c ` T pzqψpzq “ 0, with, T pzq “ i ` i . (5.19) dz2 pz ´ z q2 z ´ z i“1 i i ÿ „  35In section 4.1 I told you null-states are our friends! See Chap 7 and 8 of [56] for more details. Why did we choose null-state at level 2? Level p null-state would give rise to a p-th order diﬀerential equation. The lowest p that we can have is 2.

– 66 – Here, i “ 6hi{c and ci are the accessory parameters

Bfppqpzi, i, ep˜ iq Bci Bcj ci “ ´ satisfying “ (5.20) Bzi Bzj Bzi

Now the strategy should be as follows.

1. Solve for the ci, by using the monodromy properties of the solution ψpzq around the singularities of T pzq.

2. Integrate ci to get fppqpzi, i, ˜iq.

3. Finally exponentiate f pz , , ˜ q to obtain pz , h , h˜ q. ppq i i i Fp i i i The above steps involve technicalities. Interested reader can see [66]. Essentially, monodromy around a contour γk contains info about the resultant operator which arises upon fusing the operators within γk. The slogan is,

Choice of monodromy contour ô Choice of OPE channel

We choose the contours such that each of them contains a pair of light operators within.

This is equivalent to looking at the OPE channel in which light operators fuse in pairs. Obviously this choice in no way is unique but is geared towards entanglement entropy calculations.

1 x3 x4 x5 L L L L γ γ1 2

= 1 x3 x4 x5 ˜p ˜p

0 H H ∞

Figure 26: OPE channel/monodromy contour for 6-point function.

1 x3 x4 L L γ1 γ2 L = 1 x3 x4 ˜p

0 H H ∞

Figure 27: OPE channel/monodromy contour for 5-point function.

The monodromy conditions for all the contours form a coupled system of equations for the accessory parameters. Performing the exercise for 5- and 6-point blocks provides

– 67 – suﬃcient intuition to guess the solutions. The accessory parameters can now be used to obtain the conformal block ˜ Bfppqpzi, i, iq ˜ c ci “ ´ ppqpzi, hi, hiq “ exp ´ fppqpzi, i, ˜iq (5.21) Bzi F 6 ” ı Even-point conformal blocks

The pm ` 2q-point block factorizes into a product of m{2 4-point conformal blocks

c ptx u; , ; ˜ q “ exp ´ f px , x ; , ; ˜ q Fpm`2q i L H p 6 p4q p q L H p ΩiÞÑtpźp,qqu ” ı “ px , x ; , ; ˜ q. (5.22) Fp4q p q L H p ΩiÞÑtpźp,qqu

Ωi : Indicates the OPE channels / monodromy contours.

Odd-point conformal blocks

The pm`2q-point block factorizes into a product of pm´1q{2 4-point conformal blocks and a 3-point function

´L c pm`2qptxiu; L, H ; ˜pq “ xs exp ´ fp4qpxp, xq; L, H ; ˜pq F A 6 Ωi ÞÑtpźp,qqu ” ı ´L “ xs p4qpxp, xq; L, H ; ˜pq. (5.23) A F Ωi ÞÑtpźp,qqu

xα ´ xα xα{2 ` xα{2 where, f px , x ; , ; q “ p1 ´ αq log x x ` 2 log i j ` 2˜ log 4α j i p4q i j L H p L i j α p α{2 α{2 ˆ ˙ « xj ´ xi ﬀ

with α “ 1 ´ 24hH {c. a

Caveats The above results are not true in general. They hold true provided the following require- ments are satisﬁed.

The central charge is large. • The operators obey the ‘heavy-light’ limit. • The speciﬁc OPE channels are chosen. •

– 68 – 5.2.2 Geodesic Witten diagrams

In this section I shall try to provide a dual 3D classical gravity picture of the above CFT2 computation. The heavy excited state is dual to the conical defect geometry

α2 1 ds2 “ ´dt2 ` dρ2 ` sin2ρ dφ2 , with α “ 1 ´ 24h {c. (5.24) cos2ρ α2 H ˆ ˙ a

π The metric is written in global coordinates : ρ “ 2 is the boundary where the CFT ‘lives’ and α is the deﬁcit angle which is related to the dimension of the heavy operator. The conformal blocks can be reproduced by considering length of suitable worldline conﬁgurations in this bulk geometry.

˜p

wi L

Figure 28: Bulk picture of the 3-point function (left) and 4-point conformal block (right) in CFT. Two of the operators are heavy which deform the background geometry (from the vacuum AdS to the conical defect) and the other light operators are described by geodesics of massive probe particles.

Here I brieﬂy show one sample geodesic computation for 4-point block from the bulk.

x p8q px q px q p0qy OH OL i OL j OH

Figure 29: Computing 4-point conformal block from the geodesic lengths.

The action for the dual light ﬁeld (actually they too scale as c and therefore can be

– 69 – thought of as massive particles) can be approximated as following worldline action

S “ LlL ` ˜plp (5.25)

π But the boundary (ρ “ 2 )of the conical defect geometry is a cylinder. Therefore to compare iwi results with CFT computation we need to map everything from cylinder to plane : xi “ e iwj and xj “ e . Thus we get the 4-point block,

c ´hL ´hL ´ Spw ,w q px , x q “ x x ˆ e 6 i j (5.26) Fp4q i j i j ˇwi,j “´i log xi,j ˇ ˇ This matches our CFT result for 4-point HLLH block. ˇ Once I have shown how to obtain how to reproduce the 4-point block by adding up geodesic lengths, its straightforward to generalize this bulk picture for arbitrary higher point functions as follows.

w w

L 1 x x x x x L L L L L L ˜p

w 0 L L  ˜p ˜p ˜p ˜p ˜p

0 H H ∞ w w

Figure 30: Even point function. 8-point function in this particular case.

w

1 x x x x L L L L L

L ˜p ˜p w ˜p ˜p 0 0 L L H H ∞ w w

Figure 31: Odd point function. 7-point function in this particular case.

5.2.3 Entanglement entropy : an application I have motivated before that entanglement entropy of multiple disjoint intervals for excited states can be computed using above results. Here is how one goes about calculating it. Suppose we have N disjoint intervals as in ﬁg. 32.

A A AN 1 x3 x4 x5 x2N x2N+1

Figure 32: N disjoint intervals.

– 70 – The standard way to compute EE is via computing Rényi entropy

pnq 1 n S “ log trA pρAq (5.27) A 1 ´ n and then taking n Ñ 1. Eﬀectively one needs to compute the following correlator of twist– anti-twist operators [81] (for n Ñ 1)

Gnpxi, x¯iq “ xΨ|σp1qσ¯px3qσpx4qσ¯px5qσpx6qσ¯px7q . . . σpx2N qσ¯px2N`1q|Ψy 2N “ x0| Ψp8q σp1qσ¯px3q σpxiqσ¯pxi`1q Ψp0q |0y (5.28) i“4,6,¨¨¨ ź Dimensions of the twist and anti-twist operators [81] are given by

c 1 h “ h “ n ´ σ σ¯ 24 n ˆ ˙ In the limit n Ñ 1,

σ, σ¯ : Light operators, Ψ : Heavy operator.

Thus in this limit we use the result we have derived for two heavy and 2N light operators to obtain the following formula for EE

α α pnq c pxp ´ xq q SA “ lim SA “ min log α´1 . (5.29) nÑ1 3 i # ˜ αpxpxqq 2 + ΩiÞÑtpÿp,qqu

where, α “ 1 ´ 24hH {c. a

Figure 33: Bulk picture of EE for three disjoint intervals in excited states.

– 71 – Therefore depending on the values of the cross-ratios xi the relevant OPE channel is chosen in the CFT and analogously the geodesic conﬁguration of minimal length is the one that reproduces the corresponding entanglement entropy of the heavy excited state.

5.3 A Cardy formula for three-point coeﬃcients

In section4 I introduced 2D CFTs and stressed that CFTs are (almost) completely speciﬁed by so called conformal data t∆i, cijku. In the last chapter I derived Cardy formula using modular invariance of the CFT2. It captures the information about ∆i which appears in the partition function. There can be diﬀerent generalisations to this formula. I focus on one36 of them here. Let us start by asking the following questions.

Q. What is the simple most object in CFT that contains the info about c ? • ijk A. The thermal 1-point function — x y . O β Q. Can there be a Cardy-like formula for these objects? • A. Yes!

Q. We matched S to S . Can there be a dual bulk interpretation of this • Cardy BTZ formula? A. Yes! (actually the average version of it.)

Therefore here we will be interested in x y . O β x y “ Tr e´βH “ xi| |iy e´βEi (5.30) O β O O i ÿ ciiO l jh n where is a primary and x y transforms in a known way under modular transformations O O β (which is equivalently temperature inversion β Ñ 4π2{β). Broadly in this chapter here are the two goals we are going to achieve using the above modular transformation.

1. Modular transformation will determine behaviour of ciiO (when Ei are large) in terms of three point coeﬃcients of low dimensional operators. We are going to derive a

universal formula for the average value of three point coeﬃcient ciiO as a function of Ei.

2. There will be a nice holographic interpretation of this result. xi| |iy matches with O one point function in BH background. Our bulk computation will match only average

value of ciiO which is not unexpected. It just says that black hole geometry emerges only when we coarse grain over all heavy microstates |iy.

36I shall heavily follow Kraus-Maloney paper [82].

– 72 – 5.3.1 Torus 1-point function We start with a primary operator with scaling dimension pH, Hq inserted on a torus with O modular parameter τ.

x y “ Tr qL0ć{24 qL0ć{24 where, q ” ei2πτ O τ O “ “xi| |iy q∆ić{24 q∆i´‰c{24 (5.31) O i ÿ

In the last line I have just expanded the trace in a bases of states with dimensions p∆i, ∆iq. From the translational invariance of torus it shouldn’t matter where I insert the operator on the torus i.e. operator can only be a function of τ. O O Also the states |iy are eigenstates of pL , L q. Therefore xi| |iy is a constant. This is 0 0 O actually the three point coeﬃcient for correlator x y on sphere i.e. OiOOi

xi| |iy “ x p8, 8q p1, 1q p0, 0qy 2 . O Oi O Oi S 5.3.2 Modular invariance We have already seen in the last chapter that partition function on a torus remain invariant under modular transformation aτ ` b τ Ñ γτ “ ; a, b, c, d P Z, ad ´ bc “ 1. (5.32) cτ ` d

A primary operator transforms with modular weight pH, Hq. Under z Ñ z where z O cτ`d is the location of primary on the torus (elliptic variable) the one point function transforms as,

x y “ pcτ ` dqH pcτ ` dqH x y (5.33) O γτ O τ Check : What we are doing here is just generalization to the analysis of last chapter. There- fore in each step we can replace “ 1 and get back similar analysis as Cardy formula. E.g. O for the last step we get, 1 “ 1 which is same as saying partition function remains invariant under modular transformation.

As before we are only interested in the -transformation i.e. τ Ñ ´1{τ, S x y “ pτqH pτqH x y . (5.34) O ´1{τ O τ The interesting fact about this formula is it relates T Ñ 0 behaviour to T Ñ 8 behaviour. We have already seen the power of this in last chapter at the level of partition function.

For : It mapped the behaviour of asymptotic states of the theory ô dimension of Z ground state on the cylinder.

For x y : It will map the asymptotics of xi| |iy when |iy is heavy ô three point O O coeﬃcient of light operators.

– 73 – 5.3.3 Asymptotic formula β As before let’s take τ “ i 2π .

∆ić{24 ∆ić{24 x yi β “ xi| |iy q q O 2π O i ÿ “ xi| |iy e´βp∆ić{24q e´βp∆ić{24q (5.35) O i ÿ

Now let’s ask, for low temperature i.e. β Ñ 8 which state(s) will dominate x yi β ? Clearly O 2π it will be dominated by the lightest operator (χˆ) which has non-vanishing three-point coef- ﬁcient with i.e. xχ| |χy ‰ 0. We assume there is only one37 such operator χˆ. Therefore O O we can approximate,

´βpEχ´c{12q x yi β « xχ| |χy e , where Eχ “ ∆χ ` ∆χ. (5.36) O 2π O All other terms are exponentially suppressed at low temperature. We want the high temperature (β Ñ 0) behaviour of x y . By now we all know modular invariance (rather O β covariance in this case) can relate low to high temperature behaviour. Therefore from (5.34) we can write

x y “ pτq´H pτq´H x y . (5.37) O τ O ´1{τ

H`H β 4π2 H´H ´ β pEχć{12q x yi β “ piq xχ| |χy e O 2π 2π O ˆ ˙ EO 2 β ´ 4π pE ć{12q “ piqs xχ| |χy e β χ (5.38) 2π O ˆ ˙ where I have defined, spin s “ H ´ H and total ‘energy’ of the operator , EO “ H ` H. O From the definition of x yi β , we can write O 2π

∆ić{24 ∆ić{24 x yi β “ xi| |iy q q O 2π O i ÿ “ xi| |iy e´βpEić{12q O i ÿ c “ dE T pEq exp ´ E ´ β (5.39) O 12 ż ˆ ˆ ˙ ˙ where TOpEq which quantifies total contributions coming from operators of dimension E is defined as,

TOpEq ” xi| |iy δpE ´ E q. (5.40) O i i ÿ 37This is an assumption. There is no guarantee that there won’t be two or more states with lowest a a dimension p∆χa , a “ 1, 2,...q which have xχ | |χ y ‰ 0. We make this assumption just for the sake of O simplicity.

– 74 – Notice that at high temperature i.e. β Ñ 0, eq. (5.39) is dominated by states with E " 1. Comparing (5.38) and (5.39) we can extract

EO 2 s β c c 4π TOpEq « piq xχ| |χy dβ exp E ´ β ´ E ´ (5.41) O 2π 12 χ 12 β ¿ ˆ ˙ „ˆ ˙ ˆ ˙  At large E the above integral is dominated by following saddle point

c 12 ´ Eχ EO β˚ « 2π c ` c ` ... (5.42) d E ´ 12 2pE ´ 12 q

c We focus only on light χ operators i.e. Eχ ă 12 such that the saddle (β˚) is real.

? c EO{2´3{4 c c TOpEq « 2π O xχ| |χy E ´ exp 4π ´ E β ´ E ´ ` ... N O 12 12 χ 12 ˆ ˙ „ dˆ ˙ ˆ ˙  (5.43) where

EO{2`1{4 s c O ” i ´ E (5.44) N 12 χ ˆ ˙ which is independent of E. Notice that (5.43) is a particular ‘smeared’ version of (5.40) i.e. instead of δpE ´ Eχq we have a spreading. We are almost done. But I have already mentioned in the beginning of this section that we are more interested in the typical (i.e. average) values of the three point function coeﬃcients xE| |Ey, rather than the total three point function TOpEq. We can deﬁne it in O a standard manner since we already know the density of states ρpEq at high energy from Cardy formula,

T pEq xE| |Ey ” O (5.45) O ρpEq where

? c ´3{4 c ρpEq « 2π E ´ exp 4π ´ E ` ... as E Ñ 8. (5.46) 12 12 ˆ ˙ „ dˆ ˙  Finally we obtain the desired form average light-heavy-heavy three point coeﬃcient

c EO{2 πc 12E 12E xE| |Ey « N xχ| |χy E ´ exp ´ 1 ´ 1 ´ χ ´ 1 O O 12 3 c c ˆ ˙ „ ˆ c ˙c 

(5.47) This equation (5.47) is the main result of this section - this is as the same footing as the Cardy formula of last chapter. There we computed entropy of BTZ black hole using that

– 75 – formula. Here we can ask similar question - Is there some bulk gravitational interpretation of this formula? Below we discuss about it.

Before we discuss about the dual gravity interpretation of the result, let us try to see what are the important diﬀerences between this result with Cardy formula.

1. Three point functions are not positive deﬁnite unlike Cardy density formula where ρpEq is sum of positive terms. Here we can see from (5.47) as E Ñ 8, the average value xE| |Ey Ñ 0 (exponentially) but individually each term in the sum can be O large and of opposite signs.

2. In this derivation I made no assumption on the central charge c. So in principle this should hold true even for minimal models for which c ă 1. Of course, as I have 38 discussed before, to have a semiclassical gravity dual in AdS3 one takes c " 1 limit for which the expression simpliﬁes

12E EO{2 12E xE| |Ey « N xχ| |χy ´ 1 exp ´ 2πE ´ 1 (5.48) O O c χ c ˆ ˙ „ c  r where, EO ! c, Eχ ! c and c Ñ 8.

5.3.4 Dual gravity interpretation A typical ﬁnite-c CFT is not expected to have a dual semi-classical gravity. One might therefore expect that one must make certain assumptions about the CFT in order to match a bulk derivation – such as large c or a sparseness constraint on the light spectrum or three- point coeﬃcients. We shall see that’s not necessary. Although (5.47) and (5.48) above were derived assuming the following

E " c • Existence of χˆ such that xχ| |χy ‰ 0 • O E ă c , • χ 12 39 they can nevertheless be reproduced using semi-classical AdS3 gravity .

AdS3 setup In what follows I try to reproduce (5.47) and (5.48) using bulk geodesic diagrams in 3D gravity (very similar to what I have done in sec. 5.2.2). The two formulae are in two diﬀerent regime of parameters.

38We will see this is not necessary to have a gravitational description. Even for ﬁnite c one can reproduce the result (5.47) from bulk geodesic prescription! 39This is in the same spirit of reproducing Bekenstein-Hawking formula for BTZ black hole using Cardy formula (see section 4.5). The former is in semi-classical regime and valid in c Ñ 8 limit. Whereas the latter holds true for E " c.

– 76 – c I. EO,Eχ ! 12 ; c Ñ 8 : In this case ˆ and χˆ are light scalar primary operators in CFT . Thus one can think of O 2 their dual as perturbative scalar ﬁelds φO and φχ in the 3D bulk.

p ˆ, χˆq ðñ pφO, φ q O χ 2 The bulk theory will contain an interaction term φ φO with coupling „ xχ| |χy. χ O Now I wish to compute xE| |Ey in a typical state with energy |Ey " c . Since |Ey O 12 is a high energy state it should be described40 by BTZ black hole background – this is a reasonable argue using eigenstate thermalization hypothesis (ETH). |Ey ðñ ‘BTZ black hole’ The BTZ metric is given by dr2 2 2 2 2 2 2 (5.49) ds “ ´pr ´ r`q dt ` 2 2 ` r dφ r ´ r`

12E where, r` “ c ´ 1 is the horizon radius, φ „ φ ` 2π and I have taken AdS radius L “ 1 for simplicity.b From BTZ thermodynamics the area (rather the ‘circumference’) of the horizon,

AH “ 2πr` " 1. (5.50)

So it’s a large black hole. Let’s consider φO, φχ are very massive such that,

EO « mO " 1,Eχ « mχ " 1; but, mO, mχ ! c (5.51)

Figure 34: Thermal 1-point function from bulk geodesic conﬁgurations in BTZ.

40A vigilant reader would complain here. Indeed individual microstate |Ey will not necessarily have a geometric interpretation. The BTZ metric emerges upon coarse graining over suitable family of microstates. Here I really consider xE| |Ey over all states with energy E. O

– 77 – In this approximation one can think of the bulk ﬁelds as ‘heavy particles’ and therefore the bulk 2-point functions „ emL, where m is the mass of the particle and L is the length of the geodesic segment.

For thermal 1-point function, as shown in ﬁgure 34, the amplitude of that diagram is proportional to xχ| |χy. Total contribution coming from the geodesics can be obtained O computing the lengths of the ‘circular’ (φχ) and the ‘linear’ (φO) geodesics.

´mχAH ´mχ2πr` F rom φχ loop : e “ e (5.52)

´mOLO F rom φO line : e (5.53)

LO will have UV divergence coming from the boundary (r Ñ 8). Therefore I need to regularize it by introducing a cutoﬀ R,

R dr ´1 R LO “ “ cosh « log R ´ log r` (5.54) r 2 2 r` ż ` r ´ r` ˆ ˙ b 41 reg We regularize LO to get LO « ´ log r`. Thus the contribution from the φO line is given by,

reg ´mOLO O e « r` (5.55)

Thus the average thermal 1-point function,

EO 12E 2 12E xE| |Ey « N˜Oxχ| |χy ´ 1 exp ´2πE ´ 1 (5.56) O O c χ c ˆ ˙ « c ﬀ

This is identical to (5.48).

c I. Eχ ă 12 : The formula we reproduced above using geodesics in BTZ is the c Ñ 8 approximation of c (5.47). In deriving (5.47) we just assumed Eχ ă 12 . But if Eχ « c, in the dual gravity picture it cannot be thought as a perturbative ﬁeld – but it should backreact to make AdS3 a conical defect geometry,

dr2 2 2 2 2 2 2 (5.57) ds “ ´pr ´ r`q dt ` 2 2 ` r dφ r ´ r` where, , φ „ φ ` 2π ´ ∆φ. The deﬁcit angle ∆φ is related to the mass of ‘χ-particle’,

c ∆φ m “ (5.58) χ 6 2π

41By regularize here I mean blindly dropping the divergent piece log R as R Ñ 8. Nothing fancy.

– 78 – The ADM mass Eχ and mχ are related by gravitational redshift factor

c 12E m “ 1 ´ 1 ´ χ (5.59) χ 6 c ˆ c ˙ Now if we just insert this expression for mχ in the previous set up (i.e. in (5.52)) and proceed we get back (5.47),

c EO{2 πc 12E 12E xE| |Ey « N xχ| |χy E ´ exp ´ 1 ´ 1 ´ χ ´ 1 . O O 12 3 c c ˆ ˙ „ ˆ c ˙c 

(5.60)

Whatever we have seen so far can be generalized to CFT2 with an additional global up1q symmetry. See [83] for details.

‹ Note : The above bulk pictures are meant to be intuitive not deﬁnitive. There are some subtitles in these pictures which Kraus and Maloney [82] have addressed by more careful Witten diagram derivation of (5.48). I have skipped that part due to lack of time.

A Some useful tricks & results

Here I list some very useful techniques and results in the context of dualities in section 2.1.1. These are widely used in quantum ﬁeld theory literature.

A.1 Lagrange multipliers In classical physics to impose constraint on a system one introduces Lagrange multiplier. Let’s describe a particular dynamical system to illustrate how to deal with Lagrange multiplier. Consider a free scalar field φ that can take value only on a unit sphere in field space i.e, φ satisfies the constraint φ2 “ 1. The Lagrangian which describes this free scalar field on a sphere is given by 1 “ B φ Bµφ ´ Λpφ2 ´ 1q . (A.1) L 2 µ

We have introduced Lagrange multiplier Λ which is a non-dynamical ﬁeld since its conjugate momentum vanishes. The EOM of Λ is B L “ 0 , BΛ φ2 ´ 1 “ 0 , φ2 “ 1 . (A.2)

This shows that our Lagrangian correctly incorporates the constraint on the ﬁeld and the constraint equation comes from the EOM of the Lagrange multiplier.

– 79 – So far this was all classical. To do the same thing in quantum theory one just needs to introduce same Λ into the path integral. A free un-constrained scalar ﬁeld in d dimensions is represented by the following partition function

“ φ ei Srφs , (A.3) Z D ż d`1 1 µ where Srφs “ d x r 2 Bµφ B φs . To impose the constraint one has to introduce a ‘fake’ variable Λ into the partition ş function

d`1 2 “ φ Λ ei Srφs´ i d x Λ pφ ´1q . (A.4) Z D D ż ş

A.2 ‘Solvable’ path integrals Path integrals are usually tricky objects. Only a very special subclass of them namely those with quadratic action can be exactly solved.

ˆ “ φ e´ φ θ φ , (A.5) Z D ż ş where θˆ is the kinetic energy operator or inverse propagator e.g, θˆ “ p ´ m2q for free massive scalar ﬁeld. And for this particular case 2

ˆ “ φ e´ φ θ φ “ det θˆ . (A.6) Z D ż ş

The result det θˆ is a formal expression. It can be realized by diagonalizing θˆ and then multiplying all its eigenvalues.

A.3 Gaussian is special In the previous section I discussed that we can exactly compute only those path integrals whose integrands are Gaussian in the ﬁeld variables. Gaussian has one more special property namely Fourier transform of a Gaussian is also a Gaussian

2 2 2 1 i ωt ´ t σ ´ σ ω dω e e 2σ2 “ e 2 . (A.7) 2π 2π ż c

The interesting fact to notice here is the ‘width’ or the standard deviation of the Gaussian 1 gets inverted i.e, σ Ñ σ after the Fourier transformation. Thus if the Fourier transformation is performed at the level of path integrals, σ may be interpreted as a ‘coupling’ for some theory. Then roughly speaking this transformation effectively maps a ‘weakly coupled’ theory of ‘field’ t with coupling constant σ to a ‘strongly coupled’ one of ‘field’ ω with 1 coupling σ . (For a more concrete example see Maxwell duality in section 2.1.1.) .

– 80 – B Maximally symmetric spaces

Q. Why is “spherical cow” so (in)famous as a frequent example in theoretical physics?

A. It’s maximally symmetric!

Let’s ask a practical question – given a metric of a sphere

ds2 “ dθ2 ` sin2θdφ2, how can we know it’s maximally symmetric?

φ Ñ φ ` const. is an obvious symmetry. But what about θ direction?

To answer the above question let’s start from the basics. Metric being a rank-2 covariant tensor transforms as follows, Bxµ Bxν g1 px1q “ g pxq (B.1) ρσ µν Bx1ρ Bx1σ 1 1 1 1 Statement of isometry under coordinate transformation x Ñ x is gρσpx q “ gρσpx q i.e. Bxµ Bxν g px1q “ g pxq (B.2) ρσ µν Bx1ρ Bx1σ (B.1) and (B.2) look almost identical but they are not! Track the ‘primes’! Equation (B.2) are not easy to solve.

B.1 Killing vectors Let’s introduce idea similar to Lie algebras. Suppose two points in space are separated inﬁnitesimally,

x1µ “ xµ ` ζµpxq (B.3)

Expanding (B.2) in linear order,

α α µ µ ν ν gρσpx ` ζ pxqq “ gµνpxq δρ ´ Bρζ pxq δσ ´ Bσζ pxq λ ν µ ùñ gρσpxq ` ζ Bλgρσpxq “ gρσpxq` ´ pgρνBσζ ` g˘`µσBρζ q ˘ (B.4) Therefore the above condition reduces to famous Killing equations,

g :“ ζλB g pxq ` g B ζν ` g B ζµ “ 0 (B.5) Lζ µν λ ρσ ρν σ µσ ρ where ζµ are the Killing vector ﬁelds. The above equations g “ 0 tells us Lie derivative Lζ µν of metric along direction of a Killing vector ζ vanishes.

– 81 – µ There can be more than one Killing vector, say ζpaq. Linear combination of Killing µ vectors will also be a Killing vector : ca ζpaq. a µ µ To any vector A we can associateÿ a diﬀerential operator A Bµ. Similarly for Killing µ vectors we can associate diﬀerential operator : ζpaq :“ ζpaqBµ. Given a set of Killing vectors, one can study the symmetry (or rather the isometry) algebra generated by commuting the

ζpaq. We have developed the formalism, now let’s work out some examples.

3D Euclidean (E3)

x y x The Killing equations are : Bx ζ “ 0, Bx ζ ` By ζ “ 0 and so on. One can easily solve these to obtain,

ζ1 “ p1, 0, 0q, ζ2 “ p0, 1, 0q, ζ3 “ p0, 0, 1q (B.6)

ζ4 “ py, ´x, 0q, ζ5 “ p0, z, ´yq, ζ6 “ p´z, 0, xq (B.7)

Clearly ζ1,2,3 generate translations in x, y and z directions and ζ4,5,6 generate rotations along x ´ y, y ´ z and z ´ x planes respectively.

2D sphere (S2) 2 Here the metric is non-trivial : gθθ “ 1 and gφφ “ sin θ. The Killing equations are as follows,

θ Bθ ζ “ 0, (B.8) 2 φ θ 2 2 sin θBφ ζ ` ζ Bθ sin θ “ 0 (B.9) θ 2 φ Bφζ ` sin θBθ ζ “ 0 (B.10)

One can solve for ζµ and construct the diﬀerential Killing operators,

ζ1 “ sin φBθ ` cot θ cos φBφ (B.11)

ζ2 “ cos φBθ ´ cot θ sin φBφ (B.12)

ζ3 “Bφ (B.13)

Clearly ζ3 is the Killing vector we predicted in the beginning of this appendix. But the symmetries involving θ-direction are more involved. At any given point on the sphere, one can translate in two directions and rotate around one axis ‘without changing anything!’ Hence there are three Killing vectors.

B.2 Maximal symmetry Q. How many Killing vectors can a space have?

DpD´1q A. D-dimensional Riemannian manifold can at most have : D translations and 2 DpD`1q rotations i.e. 2 symmetries. It’s evident that both of our examples saturate the limit

– 82 – which is also geometrically intuitive. These spaces are examples of maximally symmetric spaces. Now the claim is with these many Killing vectors one can uniquely ﬁx the Riemann curvature tensor Rµνρλ. As I have stressed before maximally symmetric spaces are boring – every directions look identical. Then what could be Rµνρλ? It can be written only in terms of the metric gµν (using the symmetries of Riemann tensor) as follows,

Rµνρσ “ Kpgµρgνσ ´ gµσgνρq (B.14)

Contracting µ and ρ one gets the Ricci tensor

Rµν “ KpD ´ 1q gµν (B.15) And from there one gets the Ricci scalar,

R “ KDpD ´ 1q “ const. (B.16)

So we conclude that a maximally symmetric space has constant curvature i.e. each point D D of the space the curvature is same. Here are few examples of such space : E ,S , AdSD, dSD.

C “ 4 super Yang-Mills N Here I brieﬂy describe diﬀerent features of “ 4 super Yang-Mills theory. For more details N see for example, [84, 85].

C.1 The Lagrangian “ 4 supersymmetric Yang-Mills theory in four dimensions was written down for the ﬁrst N time in [86, 87] in 1976 by applying the method of dimensional reduction to “ 1 super N Yang-Mills in ten dimensions. For “ 4 supersymmetric gauge theory in four dimensions N the gauge multiplet is the only possible multiplet (unlike “ 1 or “ 2 theories) and a i a N N is given by pAµ, Ψα, Φ q where Aµ is a vector ﬁeld, Ψα with pa “ 1,... 4q are Weyl spinors and Φi with pi “ 1 ... 6q are real scalars. And the Lagrangian is given by

1 θ “ tr ´ F F µν ` I F F˜µν ´ i Ψ¯ σ¯µ D Ψ ´ D Φi DµΦi L 2g2 µν 8π2 µν a µ a µ # a i ÿ ÿ g2 ` g CabΨ rΦi, Ψ s ` g C¯ Ψ¯ a rΦi, Ψ¯ bs ` rΦi, Φjs2 , i a b iab 2 a,b,i a,b,i i,j + ÿ ÿ ÿ

(C.1)

– 83 – where g is the coupling constant, θI is the so-called instanton angle, Fµν is the usual field ˜ strength of the gauge field, Dµ is the usual gauge-covariant derivative, F is the Hodge dual ab of F , and Ci are the structure constants of R-symmetry SUp4qR. The trace is over the gauge indices to make the action gauge invariant. The full action is invariant under the “ 4 supersymmetry transformations. N C.2 Dimensional reduction of “ 1 SYM N The Lagrangian of “ 4 SYM in D “ 4 may look complicated but it arises naturally if N one dimensionally reduce ten dimensional “ 1 SYM to four dimensions and this is how N it was first obtained [86, 87]. Let’s start with that 10D theory

1 S “ d10x tr ´ F F MN ` i ΨΓM D Ψ , (C.2) 10 2g2 MN M ż ˆ ˙ where M,N “ 0, 1,..., 9 ; µ, ν “ 0, 1, 2, 3 and Ψ is a 16 component real (Majorana-Weyl) spinor. It has 8 degrees of freedom (DOF). The gauge ﬁeld in 10 D also has 8 degrees of freedom, as it should be the case due to supersymmetry. ΓM are the ten dimensional gamma matrices. Here the trace actually means

9 N N N N M tr ΨD{ Ψ ” pΨAqab ΓAB pBM ΨB ` i rAM , ΨBsqba , (C.3) M“0 b“1 a“1 B“1 A“1 ÿ ÿ ÿ ÿ ÿ where M is a Lorentz index, A, B are number of spinors and a, b represent color indices. Dimensionally reducing this theory from ten to four dimensions on a torus (T6) involve the following steps.

Nothing depends on co-ordinates x4, x5, . . . , x9. Therefore B ,..., B must vanish. • x4 x9 The gauge ﬁeld should be decomposed as A “ pA , Φ q and the gamma matrices as • M µ i ΓM “ pγµ, γiq . If we follow the above ‘rules’ 0 0 ¨¨* ¨¨* FMN “BµAν ´ BνAµ ` i rAµ,Aµs ` ¨BiΦj ´ ¨BjΦi 0 ¨¨* ` i rΦi, Φjs ` BµΦi ´ ¨BiΦµ ` irAµ, Φis ,

“ Fµν ` irΦi, Φjs ` DµΦi , MN µν µ ´ FMN F “ ´ FµνF ´ DµΦiD Φi ` rΦi, ΦjsrΦi, Φjs , (C.4)

0 ¯ ¯ µ ¯ i ¨¨* iΨD{ Ψ “ iΨγ tBµΨ ` rAµ, Ψsu ` i Ψγ ¨BiΨ ` rΦi, Ψs " * ¯ µ ¯ i “ iΨγ tBµΨ ` rAµ, Ψsu ` i Ψγ rΦi, Ψs . (C.5) Adding RHS of (C.4) and (C.5) and keeping track of the coupling constants one obtains the “ 4 SYM action in 4D (C.1). Notice that the γis which are in the compactiﬁed T6 N play the tole of Yukawa couplings in this theory.

– 84 – C.3 Symmetries The above Lagrangian is “ 4 Poincaré supersymmetry invariant by construction. From N dimensional analysis

i rAµs “ rΦ s “ 1, rΨas “ 3{2 , 6 rgs “ rθI s “ 0, we can see that the theory is also classically scale invariant theory since all fields are massless and there is no dimensionful parameter. It is well known that relativistic theories which posses both Poincaré symmetry and scale invariance are actually invariant under enhanced symmetry group called the conformal symmetry. For four spacetime dimensions the group is SO(2,4) „ SU(2,2). Furthermore, the “ 4 Poincaré supersymmetry and N conformal invariance combines themselves to an even more larger symmetry group known as superconformal symmetry which is given by the supergroup SU(2,2|4). Remarkably the theory remains scale invariant even after quantization. Its β-function vanishes to all order in perturbation and it is believed to vanish even non-perturbatively. Thus the superconformal group SU(2, 2|4) is a symmetry even at the quantum mechanical level. This UV finiteness makes “ 4 SYM a very special quantum field theory. N In addition to superconformal symmetry as described above this theory enjoys a discrete global symmetry known as S-duality or Montonen-Olive duality (see section 2.1.1). This symmetry can be described better by combining the coupling constant (g) and instanton angle (θI ) into a single complex coupling as follows θ 4πi τ ” I ` . 2π g2 The quantized theory is invariant under τ Ñ τ `1. Montonen-Olive conjectured the theory to be invariant under a full SL(2,Z) symmetry group which is realized as following.

aτ ` b τ Ñ . cτ ` d where ad ´ bc “ 1 and a, b, c, d P Z. Note that when θI “ 0 this duality transformation 1 relates g Ñ g which is equivalent to exchanging weak coupling and strong coupling.

C.4 Phases To study the phases of the theory one needs to analyze the potential energy term of “ 4 N SYM

g2 6 V pΦq “ ´ tr rΦi, Φjs2 . (C.6) 2 i,j“1 ÿ This is a sum of positive terms. The ground state obtained when

rΦi, Φjs “ 0 .

This criterion can be satisﬁed by following two diﬀerent ways.

– 85 – xΦiy “ 0 for all i “ 1, ..., 6. • The gauge algebra and the superconformal symmetry SU(2, 2|4) are unbroken. This is known as superconformal phase.

xΦiy ‰ 0 for at least one i. • Superconformal symmetry is spontaneously broken since the non-zero vacuum expectation value of xΦiy sets a scale. This is known as Coulomb phase.

***

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