Lectures on AdS/CFT from the Bottom Up

Jared Kaplan

Department of Physics and Astronomy, Johns Hopkins University

Abstract

AdS/CFT from the perspective of Effective Field Theory and the Conformal Bootstrap.

Contents

1 Introduction 21.1 Why is Quantum Gravity Different? . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 AdS/CFT – the Big Picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Quantum Field Theory – Two Philosophies . . . . . . . . . . . . . . . . . . . . . . . 41.4 Brief Notes on the History of Holography . . . . . . . . . . . . . . . . . . . . . . . . 10

2 Anti-deSitter Spacetime 112.1 Euclidean Version and the Poincare Patch . . . . . . . . . . . . . . . . . . . . . . . 132.2 Gauss’s Law in AdS and Long Range Interactions . . . . . . . . . . . . . . . . . . . 152.3 A First Look at the Boundary of AdS . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 A Free Particle in AdS 193.1 Classical Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193.2 Single Particle Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Free Fields in AdS 294.1 Classical and Quantum Fields, Anywhere . . . . . . . . . . . . . . . . . . . . . . . . 294.2 Back to AdS3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.3 General Approaches to the Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . 334.4 Implementing Holography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.5 Correlators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

5 Generalized Free Theories and AdS/CT 395.1 The Operator/State Correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . 405.2 Conformal Transformations of Operators . . . . . . . . . . . . . . . . . . . . . . . . 425.3 Primaries and Descendants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.4 The Operator Product Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485.5 Correlators and Conformal Symmetry Invariants . . . . . . . . . . . . . . . . . . . . 505.6 Connection to Infinite N . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

6 General CFT Axioms from an AdS Viewpoint 536.1 What is a C(F)T? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546.2 Radial Quantization and Local Operators – The General Story . . . . . . . . . . . . 566.3 General CFT Correlators and the Projective Null Cone . . . . . . . . . . . . . . . . 596.4 Deriving the OPE from the CFT, and from an AdS QFT . . . . . . . . . . . . . . . 616.5 The Conformal Partial Wave Expansion and the Bootstrap . . . . . . . . . . . . . . 63

7 Adding AdS Interactions 667.1 Anomalous Dimensions from Basic Quantum Mechanics . . . . . . . . . . . . . . . . 667.2 Perturbative Interactions and λφ4 Theory . . . . . . . . . . . . . . . . . . . . . . . 687.3 Conformal Partial Waves and λφ4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

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7.4 Physical Discussion of AdS/CFT with Interactions . . . . . . . . . . . . . . . . . . 757.5 A Weinbergian Proof of Conformal Symmetry . . . . . . . . . . . . . . . . . . . . . 767.6 Effective Conformal Theory and the Breakdown of AdS EFT . . . . . . . . . . . . . 78

8 On the Existence of Local Currents Jµ and Tµν 798.1 An SU(2) Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 808.2 A Very Special Current – Tµν . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 818.3 Separating Q and Jµ – Spontaneous Breaking and Gauging . . . . . . . . . . . . . . 83

9 Universal Long-Range Forces in AdS and CFT Currents 859.1 CFT Currents and Higher Spin Fields in AdS . . . . . . . . . . . . . . . . . . . . . 859.2 CFT Ward Identities and Universal OPE Coefficients . . . . . . . . . . . . . . . . . 889.3 Ward Identites from AdS and Universality of Long-Range Forces . . . . . . . . . . . 919.4 Cluster Decomposition and Long-Range Forces from the Bootstrap . . . . . . . . . 95

10 AdS Black Holes and Thermality 10010.1 The Hagedorn Temperature and Negative Specific Heat . . . . . . . . . . . . . . . . 10110.2 Thermodynamic Details from AdS Geometry . . . . . . . . . . . . . . . . . . . . . . 103

11 Final Topics 10411.1 Various Dictionaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10411.2 Breaking of Conformal Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10711.3 String Theory on AdS5 × S5 and the N = 4 Theory . . . . . . . . . . . . . . . . . . 10811.4 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

1 Introduction

1.1 Why is Quantum Gravity Different?

Fundamental physics has been largely driven by the reductionist program – look at nature withbetter and better magnifying glasses and then build theories based on what you see. In a quantummechanical universe, due to the uncertainty relation, we cannot refine our microscope without usingcorrespondingly larger momenta and energies. Hence the proliferation of larger and larger colliders.

Sometimes when you go to higher energies (shorter distances), physics changes radically at somecritical energy scale. Theoretical particle physicists get up in the morning to find and understandthe mechanisms behind new, frontier energy scales. For example, in Fermi’s theory of beta decaythere was the dimensionful coupling

GF ∼1

(293 GeV)2∼(7× 10−17 m

)2(1.1)

and Fermi’s theory was seen to break down at energies somewhere below 293 GeV. Fermi’s theoryhas been replaced by the electroweak theory, where physics does change drastically before this

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energy scale – one starts to produce new particles, the W and Z bosons. Even more dramatically,when physicists first probed the QCD scale, around a GeV (or 2× 10−14 m), we found a plethora ofnew strongly interacting particles; the resulting confusion led to the first S-Matrix program, StringTheory, and one of the first uses of Effective Field Theory in particle physics. But we’ve sincelearned to describe both the QCD and the Weak scale, and much else, using local quantum fieldtheories, and they no longer remain a mystery.

Gravity appears to differ qualitatively from these examples. The point is that if you were to tryto resolve distances of order the Planck length `pl ∼ 10−35 meters, you would need energies of orderthe Planck mass, ~/`pl, at which point you would start to make black holes. We are fairly certain ofthis because the universally attactive nature of gravity permits gedanken experiments in which wecould make black holes without passing through a regime of physics we don’t understand. Pumpingup the energy further just results in larger and larger black holes, and the naive reductionist programcomes to an end. So it seems that there’s more to understanding quantum gravity than simplyfinding a theory of the “stuff” that’s smaller than the Planck length – in fact there is no well-definednotion of smaller than `pl.

In hindsight, we have had many hints for how to proceed, and most of the best hints were decadesold even back in 1997, when AdS/CFT was discovered. The classic hint comes from Black Holethermodynamics, in particular the statement that Black Holes have an entropy

SBH =A

4`2pl

(1.2)

proportional to their surface area, not their volume. Since you can throw anything into a blackhole, and entropies must increase, this BH entropy formula should be a fundamental feature of theuniverse, and not just a property of black holes. The largest amount of information you can store inany region in spacetime will be proportional to its surface area. This is radical, and differs fromgeneric non-gravitational systems, e.g. gases of particles.

So spacetime has to die – at short distances it stops making sense, and it doesn’t seem to storeinformation in its bulk, but only on its boundary. What can we do with this idea? Well, sincethe 60s it has been known that gravitational energy is not well-defined locally, but it can be madewell-defined at, and measured from, infinity. But energy is nothing other than the Hamiltonian.So in gravitational theories, the Hamiltonian should be taken to live at infinity. Perhaps there’sa whole new description of gravitational physics, with not just a Hamiltonain but all sorts ofinfinity-localized degrees of freedom... a precisely well-defined theory, from which bulk gravityemerges as an approximation?

1.2 AdS/CFT – the Big Picture

AdS/CFT is many things to many people.For our purposes, AdS/CFT is the observation that any complete theory of quantum gravity

in an asymptotically AdS spacetime defines a CFT. For now you can just view AdS as ‘gravity ina box’. By ‘quantum gravity’ we simply mean any theory that is well-approximated by generalrelativity in AdS coupled to matter, i.e. scalars, fermions, and gauge fields. AdS/CFT says that the

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Hilbert spaces are identical

HCFT = HAdS−QG (1.3)

and all (physical, or ‘global’) symmetries can be matched between the two theories. In particular,the spacetime symmetries or isometries of AdSd+1 form the group SO(2, d) and these act on theCFTd as the conformal group1, containing the Poincare group as a subgroup. All states in boththeories come in representations of this group. We will explain why quantum field theories in AdSproduce CFTs, and we will (hopefully) explain which CFTs have perturbative AdS effective fieldtheory duals.

An incomplete theory of quantum gravity in AdS, such as a gravitational effective field theory(e.g. the standard model of particle physics including general relativity), defines an approximate oreffective CFT. Finding a complete theory of quantum gravity, a ‘UV completion’ for any given EFT,amounts to finding an exact CFT that suitably approximates the effective CFT. Among other things,this means that we can make exact statements about quantum gravity by studying CFTs. Generaltheorems about CFTs can be re-interpreted as theorems about all possible theories of quantumgravity in AdS.

When placed in AdS, quantum field theories without gravity define what we’ll refer to asConformal Theories (CTs). These are ‘non-local’ theories that have CFT-like operators withcorrelators that are symmetric under the global conformal group and obey the operator productexpansion (OPE), but that do not have a local stress-energy tensor Tµν . By thinking about simpleexamples of physics in AdS, we can understand old, familiar results in a new, holographic language.

A famous and revolutionary fact is that (large) black holes in AdS are just CFT states. Thismeans that AdS quantum gravity should be a unitary quantum mechanical theory. Roughly speaking,the temperature and entropy of AdS black holes correspond with the temperature of the CFT andthe number of CFT states excited at that temperature, respectively. Another famous and importantthought experiment explains how one obtains QCD-like confinement from strings that dip into AdS.

The purpose of these lectures is to explain these statements in detail. In the next two subsectionswe will make some philosophical and historical comments; hopefully they will provide some perspectiveon holography and QFT, but don’t feel discouraged if you do not understand them. They can beskipped on a first pass, especially for more practically or technically minded reader.

1.3 Quantum Field Theory – Two Philosophies

Why is Quantum Field Theory the way it is? Does it follow inevitably from a small set of morefundamental principles? These questions have been answered by two2 very different, profoundlycompatible philosophies, which I will refer to as Wilsonian and Weinbergian.

1In d = 2 there are many more symmetries (although no more isometries), and these form the full 2-d conformalgroup, with Virasoro algebra, as originally shown by Brown and Henneaux.

2There’s a third and oldest viewpoint which is also important, and plays a primary role in most textbooks. Itlacks the air of inevitability, although it’s very practical: one just takes a classical theory and ‘canonically quantizes’it. This was how QED was discovered – we already knew electrodynamics, and then we figured out how to quantizeit. This viewpoint is certainly worth understanding; Shankar’s text gives a quick explanation of classical → quantum.

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AdS CFT

∆t

ρ

exp

∆t

R

Figure 1: This figure shows how the AdS cylinder in global coordinates corresponds to the CFT inradial quantization. The time translation operator in the bulk of AdS is the Dilatation operator inthe CFT, so energies in AdS correspond to dimensions in the CFT. We make this mapping veryexplicit in section 4.3.2.

The Wilsonian philosophy is based on the idea of zooming out. Two different physical systemsthat look quite different at short distances may behave the same way at long distances, becausemost of the short distance details become irrelevant. In particular, we can think of our theories asan expansion in `short/L, where `short is some microscopic distance scale and L is the length scalerelevant to our experiment. We study the space of renormalizable quantum field theories becausethis is roughly equivalent to the space of universality classes of physical systems that one can obtainby ‘zooming out’ to long distances. Here are some famous examples

• The Ising Model is a model of spins on a lattice with nearest-neighbor interactions. We canzoom out by ‘integrating out’ half of the spins on the lattice, leaving a new effective theory forthe remainder. However, at long distances the model is described by the QFT with action

S =

∫ddx

1

2

((∂φ)2 − λφ4

)(1.4)

The details of the lattice structure become ‘irrelevant’ at long distances.

• Metals are composed of some lattice of various nuclei along with relatively free-floating electrons,

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but they have a universal phase given by a Fermi liquid of their electrons. Note that the Fermitemperature, which sets the lattice spacing for the atoms, is around 10, 000 K whereas weare most interested in metals at ∼ 300K and below. At these energies metals are very welldescribed by the effective QFT for the Fermi liquid theory. See [1] for a beautiful discussion ofthis theory and the Wilsonian philosophy. Research continues to understand the effective QFTthat describes so-called strange metals associated with high temperature superconductivity.

• Quantum Hall fluids seem to be describable in terms of a single Chern-Simons gauge field; onecan show that this is basically an inevitable consequence of the symmetries of theory (includingbroken parity), the presence of a conserved current, and the absence of massless particles.

• The Standard Model and Gravity. There are enormous hierarchies in nature, in particularfrom the Planck scale to the weak scale.

• Within the SM, we also have more limited (and often more useful!) effective descriptions ofQED, beta decay, the pions and nucleons, and heavy quarks. Actually, general relativity plus‘matter’ is another example of an effective description, where the details of the massive matterare unimportant at macroscopic distances (e.g. when we study the motion of the planets, it’sirrelevant what they are made of).

So if there is a large hierarchy between short and long distances, then the long-distance physics willoften be described by a relatively simple and universal QFT.

Some consequences of this viewpoint include:

• There may be a true physical cutoff on short distances (large energies and momenta), and itshould be taken seriously. The UV theory may not be a QFT. Effective Field Theories with afinite cutoff make sense and may or may not have a short-distance = UV completion.

• UV and IR physics may be extremely different, and in particular a vast number of distinct UVtheories may look the same in the IR (for example all metals are described by the same theoryat long distances). This means that a knowledge of long-distance physics does not tell us allthat much about short-distance physics – TeV scale physics may tell us very little about theuniverse’s fundamental constituents.

• Symmetries can have crucially important and useful consequences, depending on whether theyare preserved, broken, emergent, or anomalous. The spacetime symmetry structure is essentialwhen determining what the theory can be – high-energy physics is largely distinguished fromcondensed matter because of Poincare symmetry.

• QFT is a good approach for describing both particle physics and statistical physics systems,because in both cases we are interested in (relatively) long-distance or macroscopic properties.

For a classic review of the Wilsonian picture of QFT see Polchinski [1]. A nice perspective betweenhigh-energy and condensed-matter physics is provided by Cardy’s book Scaling and Renormalization[2]. Slava Rychkov’s notes give a CFT-oriented discussion of some of these ideas.

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A natural question: what if we have a theory that does not change when we zoom out? Thiswould be a scale invariant theory. In the case of high energy physics, where we have Poincaresymmetry, scale invariant theories are basically always conformally invariant QFTs, which are calledConformal Field Theories (CFTs). It’s easy to think of one example – a theory of free masslessparticles has this property. One reason why asymptotic freedom in QCD is interesting is that itmeans that QCD can be viewed as the theory you get by starting with quarks that are arbitrarilyclose to being free (they have an infinitesimal interaction strength or coupling), and then zoomingout. This makes it possible to rigorously define QCD as a mathematical theory.

It seems that all QFTs can be viewed as points along an Renormalization Flow (or RG flow, thisis the name we give to the zooming process) from a ‘UV’ CFT to another ‘IR’ CFT. Renormalizationflows occur when we deform the UV CFT, breaking its conformal symmetry. QCD was an exampleof this – we took a free theory of quarks and ‘deformed it’ at high energies by adding a smallinteraction. This leads to a last implication of the Wilsonian viewpoint as applied to relativisticQFTs:

• Well-defined QFTs can be viewed as either CFTs or as RG flows between CFTs. We canremove the UV cutoff from a QFT (send it to infinite energy or zero length) if it can beinterpreted as an RG flow from the vicinity of a CFT fixed point.

So studying the space of CFTs basically amounts to studying the space of all well-defined QFTs.And as we will see, the space of CFTs also includes all well-defined theories of quantum gravity thatwe currently understand!

The Weinbergian philosophy [3] finds Quantum Field Theory to be the only way to obtain aLorentz Invariant, Quantum Mechanical (Unitary), and Local (satisfying Cluster Decomposition)theory for the scattering of particles. Formally, a “theory for scattering” is encapsulated by theS-Matrix

Sαβ = 〈αin|S|βout〉 (1.5)

which gives the amplitude for any “in-state” of asymptotically well-separated particles in the distantpast to evolve into any “out-state” of similarly well-separated particles in the future. Some aspectsof this viewpoint:

• Particles, the atomic states of the theory, are defined as irreducible representations of thePoincare group. By definition, an electron is still an electron even if it’s moving, or if I rotatearound it! This sets up the Hilbert space of incoming and outgoing multi-particle states. Thefact that energies and momenta of distant particles add suggests that we can use harmonicoscillators ap to describe each momentum p, because the harmonic oscillator has evenly spacedenergy levels.

• The introduction of creation and annihilation operators for particles is further motivated bythe Cluster Decomposition Principle3. This principle says that very distant processes don’taffect each other; it is the weakest form of locality, and seems necessary to talk sensibly about

3In AdS/CFT we can prove that this principle holds in AdS directly from CFT axioms.

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well-separated particles. Cluster decomposition will be satisfied if and only if the Hamiltoniancan be written as

H =∑

m,n

∫d3qid

3kiδ

(∑

i

qi

)hmn(qi, ki)a

†(q1) · · · a†(qm)a(k1) · · · a(kn) (1.6)

where the function hmn must be a non-singular function of the momenta.

• We want to obain a Poincare covariant S-Matrix. The S operator defining the S-Matrix canbe written as

S = T exp

(−i∫ ∞

−∞dtV (t)

)(1.7)

Note that this involves some choice of t, which isn’t very covariant-looking. However, if theinteraction V (t) is constructed from a local Hamiltonian density H(x) as

V (t) =

∫d3~xH(t, ~x) (1.8)

where the Lorentz-scalar H(x) satisfies a causality condition

[H(x),H(y)] = 0 for (x− y)2 spacelike (1.9)

then we will obtain a Lorentz-invariant S-Matrix. How does this come about? The point isthat the interactions change the definition of the Poincare symmetries, so these symmetries donot act on interacting particles the same way they act on free particles. To preserve the fullPoincare algebra with interactions, we need this causality condition.

• Constructing such an H(x) satisfying the causality condition essentially requires the assemblyof local fields φ(x) with nice Lorentz transformation properties, because the creation andannhiliation operators themselves do not have nice Lorentz transformation properties. Theφ(x) are constructed from the creation and annihilation operators for each species of particle,and then H(x) is taken to be a polynomial in these fields.

• Symmetries constrain the asymptotic states and the S-Matrix. Gauge redundancies must beintroduced to describe massless particles with a manifestly local and Poincare invariant theory.

• One can prove (see chapter 13 of [3]) that only massless particles of spin ≤ 2 can couple in away that produces long range interactions, and that massless spin 1 particles must couple toconserved currents Jµ, while massless spin 2 particles must couple to Tµν . This obviously goesa long way towards explaining the spectrum of particles and forces that we have encountered.

In theories that include gravity, the Weinbergian philosophy accords perfectly with the idea ofHolography : that we should view dynamical spacetime as an approximate description of a morefundamental theory in fewer dimensions, which ‘lives at infinity’. Holography was apparently not a

8

motivation for Weinberg himself, and his construction can proceed with or without gravity. But thephilosophy makes the most sense when we include gravity, in which case the S-Matrix is the onlywell-defined observable in flat spacetime.

One of the eventual takeaway points of these lectures is that Weinberg’s derivation of flat spacequantum field theory from desired properties of the S-Matrix can be repeated in the case of AdS/CFT,with S-Matrix → CFT correlators. Quantum field theory in AdS can be derived in an analogousway from properties of correlation functions

〈O1(x1)O2(x2)O3(x3)O4(x4)〉 (1.10)

of local CFT operators Oi. Specifically:

• Particle-like objects in AdS arise as irreducible representations of the conformal group.

• The crossing symmetry of CFT correlators, which gives rise to ‘the Bootstrap Equation’, canbe used to prove the Cluster Decomposition Principle in AdS, guaranteeing long-range locality.AdS cluster decomposition is a consequence of unitary quantum mechanics in CFT.

• If the correlators of low-dimension operators in the CFT are approximately Gaussian (deter-mined entirely by 2-pt correlators), then the AdS/CFT spectrum is approximated by a Fockspace4 of particles in AdS at low energies.

• Crossing symmetric, unitary, interacting (non-Gaussian) CFT correlators built perturbativelyon a CFT Fock space will be derivable from an AdS Effective Field Theory description. Thecutoff of this effective field theory in AdS can be related to properties of the CFT spectrum.Symmetries play a similarly important role for AdS/CFT.

The key point to notice is that the argument for flat space QFT from the S-Matrix is directlymirrored by the argument for AdS QFT from CFT correlators. The concepts of Poincare Symmetry,Unitarity, and Cluster Decomposition have been replaced with Conformal Symmetry, Unitarity, andCrossing Symmetry. As we will eventually see, transitioning from flat space to AdS brings additionalcomplications and challenges, but also certain benefits.

The relation between the two approaches is no coincidence – in fact, flat space S-Matrices can bederived from a limit of CFT correlation functions, where the dual AdS length scale R→∞. Fromthis point of view, AdS space simply serves as an infrared regulator for flat space.

The converse of both Weinbergian arguments is simpler. It is comparatively straightforward (i.e.it’s the subject of all standard textbooks) to see that local flat space QFT produces a Poincarecovariant S-Matrix. It is also relatively easy to see why AdS QFTs produces good CFT correlationfunctions. Also, since we obviously have an interest in UV completing quantum gravity, it’s interestingto understand why gravity in AdS must be a CFT. So this is the plan of attack for most of theselectures – we will start by studying physics in AdS, introducing CFT ideas wherever useful, and

4The Hilbert space formed from any number of non-interacting particles.

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show how field theories in AdS naturally produce CFTs5. Only at the end may we return to see howAdS EFT actually follows from simple assumptions about the CFT.

1.4 Brief Notes on the History of Holography

Some anachronistic and ideosyncratic comments about the history of Holography:

• In any theory where energy can be measured by a boundary integral, one must have holography,because ‘energy’ is just the Hamiltonian. The fact that this is the case in general relativity hasbeen known for a long time, dating at least to the 1962 work of Arnowitt, Deser, & Misner [4],see for example the classic paper of Regge and Teitelboim [5] for a thorough treatment. BriceDeWitt and others were aware [6] in the 60s that the only exact observables are associatedwith the boundary, but they didn’t suggest that a different (local!) theory (e.g. a CFT) couldlive there and compute them.

• Most famously, holography can and usually is motivated via Black Hole thermodynamics,which dates to Bekenstein and Hawking in the 70s. The most important point is that BH’shave an entropy proportional to their area, not their volume. The fact that one can throwanything into a BH, combined with the 2nd law of thermodynamics, means that informationin a gravitational universe must follow an area law.

• Work studying the asymptotic symmetries of GR proceeded in the 70s and 80s, leading to thefamous Brown and Henneaux result [7] that the asymptotic symmetry group in AdS3 is theinfinite dimensional Virasoro algebra = the conformal algebra in 2 dimensions. Note that thiswork is quite non-trivial because one has to understand the symmetries of spacetime whileallowing spacetime to fluctuate. In a certain sense the full Virasoro symmetry of gravitationalAdS3 is always spontaneously broken to the global SL(2) conformal group.

• The Strominger-Vafa results on BH entropy, which used a CFT and some Brown and Henneaux-esque ideas, were also a precursor to AdS/CFT, and are now understood in these terms.

• ’t Hooft suggested the idea of taking gravity as a boundary theory seriously, and Susskindsubsequently named the idea holography. ’t Hooft was unfortunate enough to make a specificand incorrect suggestion for the boundary theory... both suggested the idea that the boundarytheory has ‘pixels’, although I’m not sure if the intention was to have a local boundary theory.

• Anomaly inflow, and the basic point of Witten that certain anomalies can be viewed as beinggiven by d+ 1 dimensional integrals, was also suggestive, although this result only relies onthe topology and not the specific geometry of the bulk.

5Actually, only UV complete theories in AdS produce exact CFTs. Effective field theories in AdS only produceapproximate CFTs, which break down when one studies operators of large scaling dimension. If one side of the dualityis incomplete, the other must be as well.

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• Polyakov had some ideas about strings propagating in an extra dimension being dual toconfinement (see his talk ‘String Theory and Quark Confinement’ from 1997), and (anecdotally)these ideas inspired Maldacena’s discovery.

• Amusingly, Dirac wrote a paper in 1936 on singleton representations in AdS3 (today wewould call this a scalar boson in AdS with a particular negative mass squared, so that itcorresponds to a free field in the CFT satisfying ∂2φ = 0); the paper was titled “A Remarkablerespresentation of the 3+2 de Sitter group”.

2 Anti-deSitter Spacetime

So what is Anti-deSitter (AdS) spacetime?AdSd+1 is a maximally symmetric spacetime with negative curvature6. It is a solution to Einstein’s

equations with a negative cosmological constant. A particularly useful coordinate system for it,often referred to in the literature simply as ‘global coordinates’, is given by

ds2 =1

cos2(ρR

)(dt2 − dρ2 − sin2

( ρR

)dΩ2

d−1

)(2.1)

In what follows we will set the AdS length scale R = 1. However, it’s important to note that wecannot have an AdS spacetime without choosing some particular distance (and curvature) scale.Lengths and energies in AdS can and usually will be measured in these units.

Here the radial coordinate ρ ∈ [0, π2), while t ∈ (−∞,∞), and the angular coordinates Ω cover a

(d− 1)-dimensional sphere. For example, if d = 3 then we can write

dΩ2 = dθ2 + sin2 θ dφ2 (2.2)

to cover the familiar S2. Note that although ρ only runs over a finite range, the spatial distancefrom any ρ < π/2 out towards π/2 diverges, so AdSd+1 is not compact. In global coordinates we canpicture AdS as the interior of a cylinder, as in figure 2. Note that in these coordinates there is anobvious time translation symmetry, and also an obvious SO(d) symmetry of rotations on the sphere.

By maximally symmetric, we mean that AdSd+1 has the maximal number of spacetime symmetries,namely 1

2(d+ 1)(d+ 2). This is the same number as we have in d+ 1 dimensional flat spacetime,

where we have d + 1 translations, d boosts, and 12d(d − 1) rotations. The easiest way to see the

symmetries of AdS is to embed it as the solution of

XAXA ≡ X2

0 +X2d+1 −

d∑

i=1

X2i = R2 (2.3)

6The use of d+ 1 dimensions is conventional when studying AdS/CFT, because the dual CFT is taken to have dspacetime dimensions.

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AdS

t

Figure 2: This figure shows AdS in global coordinates. The center is at ρ = 0, while spatial infinityis approached as ρ→ π/2. The global time coordinate t runs from −∞ to ∞.

Note again the appearance of the AdS length R. Using the abstruse equation cos2 t+ sin2 t = 1 andthe related equation sec2 ρ = tan2 ρ+ 1 we can map the global coordinates into the XA via7

X0 = Rcos t

cos ρ(2.4)

Xd+1 = Rsin t

cos ρ(2.5)

Xi = R tan ρ Ωi (2.6)

The advantage of the XA as a presentation of AdS is that all of the symmetries are just the naiverotations and boosts of the XA. In particular, we have 1

2d(d − 1) rotations among the Xi with

1 ≤ i ≤ d, we have one rotation between the two timelike directions X0 and Xd+1, and then we have2d boosts that mix X0 and Xd+1 with the Xi. All of these transformations can be represented by

LAB = XA ∂

∂XB−XB ∂

∂XA(2.7)

which generate the group SO(2, d) of linear transformations of the XA leaving equation (2.3) invariant.The group SO(2, d) is both the group of isometries of AdSd+1 and also the conformal group in d

7To get all of AdS we need to unwrap the XA to their universal cover, so that t and t+2πR are no longer identified.

12

dimensions; so we will often refer to it as the conformal group in what follows. Let’s take a look atthe ‘rotation’ in the timeline directions X0 and Xd+1. Note that

∂

∂t=∂X0

∂t

∂

∂X0+∂Xd+1

∂t

∂

∂Xd+1= L0

d+1 (2.8)

so in other words, L0,d+1 is the generator of time translations in AdS. So it is the Hamiltonian! The

purely space-like generators Lij just generate the rotations of Ωd. These are just the isometries ofthe sphere Sd−1, and they form the group SO(d).

2.1 Euclidean Version and the Poincare Patch

In many cases it will be useful and important to study the Euclidean version of AdS and theEuclidean conformal group, which is SO(1, d+ 1). The embedding space becomes

X20 −

d+1∑

i

X2i = R2 (2.9)

In this case the dt2 term in equation (2.1) flips sign, and we have cos t→ cosh t and sin t→ sinh t inequation (2.4). This leads to the coordinate identifications in equation (2.10).

There’s another coordinate system for AdS, called the Poincare patch (PP). It doesn’t cover theentire spacetime (it can be extended, although this isn’t immediately obvious), but it’s actuallyused in the literatue as or more often than the global coordinates (it’s the coordinate system touse when studying RS models, holographic QCD, and basically any theory with broken conformalsymmetry... it’s mostly used in Sundrum’s notes [8]). The reason for its importance is that it makesthe d-dimensional Poincare subgroup of the conformal group manifest.

The relationship between Euclidean embedding, global, and Poincare patch coordinates is

X0 = Rcosh τ

cos ρ=

1

2

(z2 + ~r2 +R2

z

)(2.10)

Xd+1 = Rsinh τ

cos ρ=

1

2

(z2 + ~r2 −R2

z

)

Xi = R tan ρΩi =R

zri

where ~r is a spatial d-vector, and we have written the global coordinate time as τ . Aspects of thisrelationship are pictured in figure 3. Note that z runs from 0 to ∞; this is necessary so that X0

has a fixed sign, which is itself required by equation (2.9). In the PP coordinates, dilatations aregenerated by

D = L0,d+1 = z∂z + ri∂ri (2.11)

This operator acts on ~r by stretching the space, as expected. The fact that this is truly L0,d+1 canbe checked by noting that (z∂z + ri∂ri)Xi = 0, and that

z∂z + ri∂ri = Xd+1∂X0 −X0∂Xd+1 (2.12)

13

Euclidean AdS

z = z0

Figure 3: This figure depicts Euclidean AdS. The constant global Euclidean time surfaces are justballs at fixed height on the cylinder. In constrast, the constant z surfaces begin at ~r = 0 at somefixed τ with ρ = 0, but as ~r →∞ we simultaneously have ρ(r)→ π/2 with τ →∞, as depicted onthe figure. Note that in Euclidean coordinates, both the Poincare patch and the global coordinatescover all of AdS; in contrast, the PP only covers a portion of AdS in the Lorentzian case.

This is easily seen by writing the differential operator D = z∂z + ri∂ri as D(XA)∂XA .The relationship in the Lorentzian case is basically given by analytic continuation. The geometry

is depicted in figure 4. It’s most natural to switch the labels of Xd and Xd+1 to satisfy equation(2.3). This gives

X0 = Rcos τ

cos ρ=z

2

(1 +

R2 + ~x2 − t2z2

)(2.13)

Xd+1 = Rsin τ

cos ρ=R

zt

Xi<d = R tan ρΩi =R

zxi

Xd = R tan ρΩd =z

2

(1− R2 − ~x2 + t2

z2

)

14

where ~x is a spatial d− 1 vector, and we have written the global time as τ to avoid confusion. Wecan solve for (t, z, xi) in terms of the global coordinates (τ, ρ, Ωi) as

t = Rsin τ

cos τ − Ωd sin ρ(2.14)

z = Rcos ρ

cos τ − Ωd sin ρ

~xi = RΩi sin ρ

cos τ − Ωd sin ρ

It should be clear from this that even when t→ ±∞ we only cover a finite range in τ , because thislimit is only achieved by causing the denominators on the RHS to vanish. In these coordinates weobtain the metric

ds2 =1

z2

(dt2 − dz2 −

d−1∑

i=1

dx2i

)(2.15)

where the z coordinate only covers the range 0 < z <∞. A very commonly used alternative versionre-writes dy = dz/z so that z = ey. Randall Sundrum models are probably most commonly studiedwith these variables.

Generally speaking, the global coordinates make an SO(2)× SO(d) sub-group of the SO(2, d)conformal group obvious, while the Poincare patch makes the d-dimensional Poincare group anddilatations (somewhat) manifest.

2.2 Gauss’s Law in AdS and Long Range Interactions

Now let’s consider Gauss’s law in AdS, in order to understand how electromagnetic and gravitationalfields behave at long distances.

For this purpose it’s natural to transform to a coordinate κ with dκ = dρcos ρ

, so that coshκ = sec ρand sinhκ = tan ρ. Note that since AdS is maximally symmetric, the point κ = 0 is equivalent toall other points. In these coordinates the metric takes the form

ds2 = cosh2(κ)dt2 − dκ2 − sinh2(κ) dΩ2d−1 (2.16)

Now consider a constant time surface, such as the surface t = 0, and let’s imagine we have a chargesitting at κ = 0. Gauss’s law says that the total amount of flux through any sphere around thecharge at κ = 0 will be a constant. Note that since gκκ = 1 a point with coordinate κ∗ is literally adistance κ∗ from the point κ = 0. The surface area of a sphere of (geometrical) radius κ∗ is just[sinh(κ∗)]

d−1 times the area of an Sd−1 with radius one. This means that the potential due to apoint charge at κ = 0 will fall off with κ as

V (κ) =Q

[sinh(κ)]d−2(2.17)

15

Lorentzian AdS

t = 0

t = 1

t = 1

Figure 4: This figure emphasizes the region corresponding to the Poincare patch, with constant PPtimes labeled. The PP does not cover all of Lorentzian AdS (in contrast to the Euclidean case).

In the limit that κ 1, this is just the usual κ2−d potential appropriate to an (approximately) flatd+ 1 dimensional spacetime. But for κ 1 this becomes

V (κ) ≈ e−(d−2)κ (2.18)

We can derive these results more systematically by studying the equation of motion for a sphericallysymmetric, static electric potential.8 This follows from studying the action for the electromagneticfield in AdS, namely

S =

∫dd+1X

√−g1

4FµνF

µν (2.19)

The equation of motion we want can be most easily derived by writing ~A = 0 and A0 = V , giving

∂κ(gttgκκ

√−g∂κV)

= δ(κ) (2.20)

8It’s worth noting that this differs from the potential for a massless scalar field. In fact, the potential due to ascalar will only agree with this electric potential if the scalar has a slightly negative mass m2 = −2(d− 2) in AdSunits, and then it will only agree for κ 1. Scalars with slightly negative masses can be stable in AdS.

16

or

∂2κV + ((d− 1) cothκ− tanhκ) ∂κV = δ(κ) (2.21)

which has the solution given in equation (2.17).These observations have broader implications. For one thing, they imply that IR divergences are

regulated by the AdS geometry. But at a more basic level, there are implications for the relationshipbetween angular momentum and radius of orbit. Since a sphere of radius κ will have an area oforder e(d−1)κ, to cover such a sphere with AdS scale resolution requires spherical harmonics up to` ∼ e(d−1)κ as well. This also means that objects in nearly circular orbits with angular momentum `are only a characteristic distance κ ∼ R log ` from AdS.

2.3 A First Look at the Boundary of AdS

2.3.1 A Penrose Diagram Aside – Flat Spacetime

The purpose of a Penrose diagram (or ‘causal diagram’ or ‘conformal diagram’) is to encapsulatethe causal structure of a spacetime in a neat little picture. To make a Penrose diagram, we (vastly)distort the geometry in order to map the entire spacetime into a finite region, while maintaining thenotions of timelike, spacelike, and null separations between points. Penrose diagrams are also usefulfor understanding the structure of (or ‘boundary at’) infinity because they turn infinity into a finitecodimension one spacetime with a causal structure inherited from the bulk.

How can we distort the geometry without affecting causality? An easy way is to multiply theentire metric by a Weyl factor f(x), sending gµν(x)→ f(x)gµν(x).

Let’s review how this works in flat spacetime. If we write the flat space metric as

ds2 = dt2 − dr2 − r2dΩ2 (2.22)

then we can introduce coordinates 2 tanU = t+ r and 2 tanV = t− r so that U and V run from−π/2 to π/2. In these coordinates the metric takes the form

ds2 =4dUdV

cos2 U cos2 V+ (tanU − tanV )2dΩ2 (2.23)

We can perform a Weyl transformation and multiply by cos2 U cos2 V . Now using the new coordinatesT = U + V and R = U − V and a trig identity, we find

ds2 = dT 2 − dR2 + sin2(R)dΩ2 (2.24)

with T and R only running over a finite diamond. This is the derivation of the usual diamondshaped Penrose diagram for flat spacetime.

The boundary (or ‘conformal infinity’) is a null diamond. There are special points where time-likeand space-like geodesics terminate, and then the null surfaces where all light rays approach infinity.Notice that the boundary is not a nice spacetime; for example, it does not inherit a Lorentzianmetric with timelike directions. This explains why it is non-trivial to find a holographic descriptionof flat spacetime – the holographic theory cannot be a conventional QFT, since it would have to liveon a bizarre spacetime.

17

2.3.2 The Penrose Diagram and Boundary of Global AdS

When we study AdS/CFT we will constantly refer to the boundary of AdS. The factor of 1/ cos2(ρ)multiplies the entire metric in equation (2.1), so we can determine the AdS Penrose diagram bysimply dropping it (or more precisely, by multiplying the AdS metric by the Weyl factor cos2(ρ)).This gives a new metric

ds2 = dt2 − dρ2 − sin2(ρ)dΩ2 (2.25)

Since 0 ≤ ρ < π/2, we can now draw a spatially finite Penrose diagram. Time still runs from −∞to ∞, and we cannot alter this fact while maintaining the feature that light rays move at 45 degreeangles in the t-ρ plane.

In terms of the global coordinates that cover the entirety of the AdS manifold, the boundary ofAdS is the cylinder R× Sd−1 that we obtain by taking ρ→ π

2, its limiting value at spatial infinity.

We can simply use the coordinates t and Ω to parameterize this boundary cylinder. Note that unlikein flat spacetime, the boundary of AdS is a nice spacetime with a metric that inherits the signatureof the AdS metric.

2.3.3 What About the Poincare Patch?

Recall that in the Poincare Patch coordinates the AdS metric is

ds2 =1

z2

(dt2 − dz2 −

d−1∑

i=1

x2i

)(2.26)

where the z coordinate only covers the range 0 < z <∞.Clearly if we multiply by the Weyl factor z2 we simply have the metric for a flat d+1 dimensional

spacetime, so the Penrose diagram must look the same as that for half of Minkowski spacetime. Inother words, it will look like a half-diamond.

What does this mean for the boundary of the Poincare patch? Well, the surface z = 0 must bepart of the boundary, since the Penrose diagram abruptly ends there. From the identification inequations (2.13) we see that z → 0 corresponds with ρ→ π/2, as expected. We also see that z →∞corresponds to ρ→ π/2, but only at the unique point on the boundary cylinder where Ωi = 0 andτ = 0 (in global coordinates). If we take some combinations of ~x and t to infinity we can reach theend of the Poincare patch, but not the boundary of the spacetime, because we are free to extend thespacetime past this ‘infinity’, which can actually be reached in a finite proper time.

But the real punchline is that by taking z → 0, we find a different (but only partial) boundary forthe AdS Poincare patch. This boundary is parameterized by (t, ~x), and it just looks like it inheritsthe geometry of a d-dimensional Minkowksi spacetime.

18

2.3.4 Cosmology and DeSitter Space

Let us make a couple of comments about cosmology, which naturally transpires (in inflation) indeSitter spacetime. dSd+1 is the hyperboloid

X20 −

d+1∑

i=1

X2i = −R2 (2.27)

Setting R = 1 (in other words, the Hubble constant is 1) and applying sinh2 t− cosh2 t = −1 we canparameterize dS as

ds2 = dt2 − cosh2(t)dΩ2 (2.28)

Now using tan(η/2) = tanh(t/2) we can re-write this metric as

ds2 =1

cos2 η

(dη2 − dΩ2

)(2.29)

Note the similarity to AdS in global coordinates. Now we have a metric that has an obvious Penrosediagram. Note that spatially there is no boundary, and the spacetime only ends in time, at η = ±π/2.This is the origin of the usual statement that deSitter space only has a boundary in the infinitepast and infinite future. The absence of any notion of time on the boundary is one (mild) reason toworry about attempts at holography in deSitter spacetime... the deeper problems are that dS isunstable to bubble nucleation, that classically there will be a spacelike singularity in the past (dueto the singularity theorems), and that dS behaves like a thermal system, so it is ‘asymptotically hot’– fields at infinity never cease their fluctuations.

3 A Free Particle in AdS

3.1 Classical Equations of Motion

Now let’s study the simplest possible example of AdS physics, a free scalar particle in AdS3. Themost straightforward way to work out its behavior is to write down a Lagrangian for the particle,and then study the classical equations of motion and their canonical quantization. The action for afree particle of mass m in any spacetime can be written as

S = m

∫dτ = m

∫dt

√gµν(X(t))

dXµ(t)

dt

dXν(t)

dt(3.1)

where τ is the proper time, and in the second line Xµ(t) is the spacetime position of the particle attime t. This action being the correct one for a free particle is equivalent to the statement that freeparticles move on geodesics, since this action computes the geodesic path length in units of 1/m.

It’s worth noting an alternative description (see e.g. [9]). Introducing the lagrange multiplier α,we can write the action as

S =

∫dt

(1

2αgµν(X(t))

dXµ(t)

dt

dXν(t)

dt+α

2m2

)(3.2)

19

AdS CFT

Figure 5: This figure shows an object at rest in AdS and the CFT dual, which is a primary state.

The α equation of motion is trivially solved, and substituting the solution back in gives our firstaction. But this second version is useful; for example to study massless particles we can just setm2 = 0 from the beginning.

In the simple special case of AdS3, we can take Xµ(t) = (t, ρ(t), θ(t)), so we obtain the action

S = m

∫dt

√1− ρ2 − θ2 sin2 ρ(t)

cos ρ(t)(3.3)

If we are interested in the classical physics of our free particle, we could proceed to derive theEuler-Lagrange equations for this action, but this gets messy9.

Instead, let’s take a smarter approach. We can view AdS as the surface XAXA = R2, as defined

in equation (2.3). So why not view a particle in AdS as a particle in 2 + d spacetime dimensionsconstrained to live on this surface? If we write the action for the coordinates XA(τ), where τ is

9One can still easily check that circular orbits (those with ρ = 0) exist iff θ = 1.

20

simply an arbitrary wordline parameter, simply becomes10

S[XA, λ] =

∫dτ[XAX

A + λ(R2 −XAXA)]

(3.4)

where λ is a Lagrange multiplier that we would also integrate over in the path integral. Recall thatthe Lagrangian formalism was originally invented for exactly this purpose – to describe the motionof objects constrained to live on some specific surface.

The equations of motion are

XA = −λXA (3.5)

along with the equation of motion from λ, the constraint equation

XAXA −R2 = 0 (3.6)

The solution to the first set of equation depends on our choice for the variable λ, which is effectivelyunconstrained. By rescaling the unphysical worldline coordinate τ , we can multiply λ in equation(3.5) by any positive quantity (since the derivative is quadratic). This means that we effectivelyhave three choices: λ = 1, 0, or −1.

These three choices for λ correspond to timelike, null, or spacelike trajectories (geodesics) inAdS. Let us focus on the case λ = 1 for now, which corresponds to the motion of a massive particlein AdS. Then, setting τ = t from now on, we have the solutions

XA = vcA cos t+ vsA sin t (3.7)

Then the other constraint equation becomes

R2 = vcAvcA cos2 t+ vsAv

sA sin2 t+ vcAvsA sin(2t) (3.8)

This immediately leads to the constraints vcAvsA = 0 and vcAv

cA = vsAvsA = R2. This leaves us with

2(d+ 2)− 3 = 2d+ 1 solutions for the motion of a particle in a d+ 1 dimensional spacetime. This isone too many solutions; however we can eliminate one by trading the parametric t coordinate forone of the XA. This gives the correct number of solutions.

The most trivial solution is

X0 = R cos t and Xd+1 = R sin t (3.9)

with all the Xi = 0. This represents a massive particle at rest at ρ = 0. A more interesting solutioncan be easily found by taking

X0 =R cos t

cos ρ∗(3.10)

Xd+1 = R sin t

X1 = R tan ρ∗ cos t

10If we choose intrinsic AdS coordinates for XA, the constraints are automatically satisfied. Then this actionreduces to

∫ds dsdτ =

∫ds for the appropriate choice of paramerization, dτ = ds, yielding the correct action.

21

In this case the particle is oscillating back and forth between ρ = ±ρ∗ in the 1 direction. Anothercanonical example is

X0 =R cos t

cos ρ∗(3.11)

Xd+1 =R sin t

cos ρ∗X1 = R cos t tan ρ∗

X2 = R sin t tan ρ∗

This represents a particle at fixed ρ = ρ∗ making a circular orbit in the 1-2 plane. Other morecomplicated solutions involving various elliptical orbits can be obtained via some simple algebra.The spacelike and null geodesics of AdS can also be obtained.

One might wonder why we have solutions corresponding to various orbits, given that AdS is amaximally symmetric spacetime. The reason is that by choosing a specific time coordinate t wehave chosen some particular center for AdS, and according to this t-evolution objects move in orbits.Any given orbit might be transformed to a particle at rest via a conformal transformation.

A very non-trivial feature of these solutions is that they all have the same frequency with respectto the AdS time t. An immediate consequence of this fact is that when we quantize AdS free particle,we must have energy levels that are integer spaced. In other words, we must find that all energylevels satisfy

E = ∆ +m (3.12)

for some ∆ and for integers m = 0, 1, 2, · · · . This follows from both the WKB approximation, andfrom a consideration of the behavior of the oscillatory behavior of superpositions of wavefunctions.Let us now turn to the quantum theory and confirm this prediction.

3.2 Single Particle Quantum Mechanics

Now we would like to study our single particle in AdS quantum mechanically. Note that there aretwo physical scales in the problem – the AdS length scale R and the mass of the particle m, fromwhich we can form the dimensionless number mR on which the wavefunction can depend. We didn’tsee this classically because a particle of arbitrary mass can move on any timelike geodesic.

We can straightforwardly derive the AdS Schrodinger equation via canonical quantization ofthe action for a free particle (note that there isn’t any problem describing free relativistic particlesusing first-quantized quantum mechanics). Let’s illustrate this for AdS2 for simplicity. The action inequation (3.1) in the case of AdS2 is

S =

∫dt

m

cos ρ

√1− ρ2 (3.13)

The canonical momentum conjugate to ρ is

Pρ =∂L

∂ρ= − mρ

cos ρ√

1− ρ2(3.14)

22

AdS CFT

@µ1···@µ`

@2n O|0i n`(t, ,)

Figure 6: This figure shows an object moving in AdS and the CFT dual, a descendant state.

Using this we can construct the Hamiltonian, which can be written as

H = Pρρ− L =

√m2

cos2 ρ+ Pρ

2 (3.15)

Quantum mechanically, we must impose the canonical commutation relation [ρ, Pρ] = i, which isconveniently implemented by Pρ = −i∂ρ in the ρ-basis of states. The Schrodinger equation thensays that i∂t = H. Applied to the wavefunction for a particle this would say

id

dtΨ(t, ρ) =

(√m2

cos2 ρ− ∂2

ρ

)Ψ(t, ρ) (3.16)

It’s unclear how to solve this equation, but things become easier if we study instead the square ofthe time evolution operator, −∂2

t = H2, which produces an equation that must also be satisfiedwhen applied to Ψ. So we would want to solve

−∂2t Ψ(t, ρ) =

(m2

cos2 ρ− ∂2

ρ

)Ψ(t, ρ) (3.17)

23

to describe a single quantum mechanical particle in AdS2. In other words, we have derived arelativistic version of the single-particle Schrodinger equation for a free particle in AdS2 from itsclassical action. We will see later on that this is also compatible with the Klein-Gordon equation inAdS that we obtain from relativistic field theory.

How do we solve this system? We could just type the differential equation into mathematica,and find a bewildering answer involving hypergeometric functions. But to actually understand thephysics, we need to use symmetry. In AdS2 we actually have 3 symmetry generators, corresponding(in the embedding space) to

L02 = ∂t (3.18)

and also L01 and L1

2. Let’s look at one of the latter transformations. In the embedding space withX2

0 +X22 −X2

1 = R2 we have

X0 = Rcos t

cos ρ(3.19)

X2 = Rsin t

cos ρ(3.20)

X1 = R tan ρ (3.21)

The operator L01 acts as

X0 → X0 + εX1 and X1 → X1 + εX0 (3.22)

We can translate this into a transformation of ρ and t by looking for functions ft and fρ, witht→ t+ εft and ρ→ ρ+ εfρ such that

cos(t+ εft)

cos(ρ+ εfρ)≈ cos t

cos ρ+ ε tan ρ (3.23)

tan(ρ+ εfρ) ≈ tan ρ+ εcos t

cos ρ(3.24)

The solution to these equations is

ft = − sin t sin ρ (3.25)

fρ = cos t cos ρ (3.26)

This tells us that the symmetry generator

L01 = − sin t sin ρ∂t + cos t cos ρ∂ρ (3.27)

Similarly we find that

L12 = cos t sin ρ∂t + sin t cos ρ∂ρ (3.28)

24

These generators should have the SO(2, 1) commutation relations; in fact we find

[L02, L

01] = −L1

2, [L02, L

12] = L0

1, [L01, L

12] = L0

2 (3.29)

This means that if we define D = L02, P = 1

2(L0

1 + iL12), and K = 1

2(L0

1 − iL12) then we find

[D,P ] = iP, [D,K] = −iK, [K,P ] = iD (3.30)

This is great – it means that we can choose to label our states as eigenstates of D, so that

D|ψ〉 = ∆|ψ〉 (3.31)

and with respect to this eigenvalue, P acts as a raising operator while K acts as a lowering operator.In particular, the ground state satisfies

K|ψ0〉 = 0 (3.32)

and all of the other states can be built by applying the operator P to this state. Note that, veryexplicitly, we have

K =1

2e−it (−i sin ρ∂t + cos ρ∂ρ) (3.33)

for this operator acting in the (t, ρ) basis of states.So the ground state of our system must obey the equation

−i sin ρ∂tψ0(t, ρ) + cos ρ∂ρψ0(t, ρ) = 0 (3.34)

Taking ψ0 = ei∆tχ(ρ) with ∆ the eigenvalue of D, we see that

∆χ = − cot ρ(∂ρχ) (3.35)

with the solution

ψ(t, ρ) = ei∆t cos∆ ρ (3.36)

In fact, this is the correct answer. Now every other state in the system can be obtained by actingwith the operator

P =1

2eit (i sin ρ∂t + cos ρ∂ρ) (3.37)

which is the raising operator for the system. This explains the fact that all AdS orbits have thesame period. Note that flipping t→ −t just exchanges P and K. It turns out that the nth energylevel has a wavefunction that can be written in terms of hypergeometric functions, but we will waitto introduce those until we talk about general AdSd+1.

Now let’s move on and write down the corresponding Schrodinger equation for AdSd+1, which wewill then solve. As with e.g. a hydrogen atom, we would like to use the symmetries of the problem to

25

our advantage. To be maximally slick we would use the full SO(2, d) group of conformal symmetriesimmediately, with generators from equation (2.7). We will get to that point soon, but first let us seewhat we can do with spherical symmetry, which is just the SO(d) subgroup of the conformal group.

When we have spherical symmetry, we can immediately separate variables into a radial wavefunc-tion and an angular wavefunction. The latter will just take the form of a d-dimensional sphericalharmonic Y`J , which are the eigenfunctions of the Laplacian on the sphere Sd−1 with eigenvalue`(` + d − 2). We can then write a Schrodinger equation for the the ρ-dependent part of thewavefunction ψ`(ρ), which has energy ω2. It takes the form

−ψ′′(ρ) +1− d

cos ρ sin ρψ′(ρ) +

(`(`+ d− 2)

sin2 ρ+

m2

cos2 ρ

)ψ(ρ) = ω2ψ(ρ) (3.38)

We can simplify this equation by defining

ψ(ρ) = χ(ρ) sin(ρ)[cot(ρ)]d2 (3.39)

to give the equation for χ(ρ)

−χ′′(ρ) +1

4

((2`− 3 + d)(2`− 1 + d)

sin2 ρ+

4m2 + d2 − 1

cos2 ρ

)χ(ρ) = ω2χ(ρ) (3.40)

We can interpret this equation using intuition from non-relativistic quantum mechanics. The termin parentheses represents an effective potential with an angular momentum barrier and a potentialthat diverges as ρ→ π/2.

It’s not especially obvious how to solve this equation exactly. However, it turns out that if wechoose the ` = 0 mode for simplicity, and take m2 = ∆(∆− d) and ω = ∆, there is a very simplesolution for the ground state

ψ(ρ) = cos∆(ρ) (3.41)

Quantization and the determination of ∆ follows as usual from boundary conditions – the wavefunc-tion must vanish as ρ→ π/2 and it must have ψ′(0) = 0 to avoid cusps, since ρ is a radial coordinate.Physically, we see that ψ0(ρ) has its largest support at small ρ and falls to zero as ρ→ π/2. For∆ ∼ 1 the particle has been confined to a region of order one size R, the AdS length scale. When∆ 1 we have

ψ(ρ) ≈ e−∆ ρ2

2 when ∆ 1 (3.42)

so we see that the particle is an approximately Gaussian state so that it is confined to a region ofsize R/

√∆ due to the AdS curvature.

Actually, there is a slick way of deriving the relation between bulk mass and ∆. We alreadyderived the form of the ground state using the lowering condition Kµψ0 = 0. Once we know its form,we can simply plug it back into the Schrodinger equation, noting that ω = ∆. A quick computationshows that this equation can then only be satisfied with m2 = ∆(∆− d).

26

The general solutions to the Schrodinger equation can also be obtained (we’ll write them outlater), but they do not look especially enlightening. Instead of proceeding along these lines, let usinstead use the full power of the conformal symmetry group.

As we mentioned above, the fact that all classical solutions oscillate with integer period impliesthat the energy levels are E = ∆+n for integers n. Let us understand this fact quantum mechanically.The conformal algebra SO(d, 2) can be re-written in a conventional form as

[Mµν , Pρ] = i(ηµρPν − ηνρPµ), [Mµν , Kρ] = i(ηµρKν − ηνρKµ), [Mµν , D] = 0

[Pµ, Kν ] = −2(ηµνD + iMµν), [D,Pµ] = Pµ, [D,Kµ] = −Kµ (3.43)

where we have the index µ = 1, 2, ..., d. Referring to the generators LAB from equation (2.7),the ‘dilatation operator’ D ≡ −iL0,d+1, while the ‘momentum generator’ Pµ ≡ iLd+1,µ + L0,µ and‘special conformal generators’ Kµ ≡ iLd+1,µ −L0,µ . The rotation generators Mµν are just the SO(d)generators −iLµν .

Recall that D = −iL0,d+1 = −i ∂∂t

. In other words, the dilatation operator D is actually theHamiltonian for AdS physics, since it is the generator of time translations. For now you can justthink of it as the AdS Hamiltonian; soon we’ll learn what role it plays in conformal field theoriesand thereby understand its name. So what does equation (6.9) mean physically? The first line justindicates that Pµ and Kµ transform as vectors under rotations, while the dilatation operator Dis a scalar. Since D commutes with Mµν , we can simultaneously diagonalize D and the angularmomentum generators. So we can label states by their energy (D eigenvalue) and their angularmomentum.

The second line of equation (6.9) gives us even more crucial information. It says that withrespect to the Dilatation operator D, Pµ acts as a raising operator while Kµ acts as a loweringoperator. This should remind you of the creation and annihilation operators for the harmonicoscillator. It immediately explains the fact that the energy levels of one-particle states are integerspaced (since energies are just the D eigenvalues). Furthermore, since D is the Hamiltonian, theremust be one-particle state with minimum eigenvalue ∆. This state cannot be lowered, so it must beannihilated by all d of the Kµ generators. This gives us an equation

Kµ|ψ0〉 = 0 (3.44)

where ψ0 is called (again using CFT terminology) a primary state. From the point of view of AdSquantum mechanics, ψ0 is simply the lowest energy state for our particle, and it has energy ∆. Oncewe have the primary/ground state ψ0, we can construct general states of the form11

|ψn,`〉 =(P 2µ

)nPµ1Pµ2 · · ·Pµ` |ψ0〉 (3.45)

These states have energy En,` = ∆ + 2n+ ` and angular momentum `. These form a basis for allone-particle states in AdS.

We can compute ψn,` as wavefunctions in AdS by writing out Kµ explicitly as a differentialoperator. This is a straightforward exercise using the coordinate transformations from equation (2.4),

11This equation is rather schematic for ` > 0, because we need to arrange the indices to form definite irreduciblerepresentations of SO(d), but this is the basic idea.

27

the definitions of LAB from equation (2.7), and the fact that Kµ ≡ iLd+1,µ − L0,µ as we discussedabove. For example, in AdS3

K± = ie−it±iθ(

sin ρ∂t + i cos ρ∂ρ ∓1

sin ρ∂θ

)(3.46)

P± = ieit±iθ(

sin ρ∂t − i cos ρ∂ρ ±1

sin ρ∂θ

)(3.47)

where K± = K1 ± iK2 and similarly P± = P1 ± iP2. The ground state ψ0(t, ρ) will be independentof θ so it must satisfy

(sin ρ∂t + i cos ρ∂ρ)ψ0(t, ρ) = 0 (3.48)

and this means that it takes the form

ψ0(t, ρ) = ei∆t cos∆(ρ) (3.49)

as expected. We saw above that m2 = ∆(∆ − d). All of the higher states can be constructed byacting on this state with the differential operators P±. The state and these methods also generalizeto higher d. In complete generality, the wavefunctions ψn,` in AdSd+1 can be written in terms of ahypergeometric function as

ψn,`J(t, ρ,Ω) =1

N∆n`

e−iEn,`tY`J(Ω)

[sin` ρ cos∆ ρ 2F1

(−n,∆ + `+ n, `+

d

2, sin2 ρ

)](3.50)

where N∆n` is a normalization factor and En,` = ∆+2n+`. The J quantum number just encapsulatesother angular momentum quantum numbers, e.g. ‘m’ for AdS4 with Y`m spherical harmonics. In thecase of Euclidean AdS, we have t→ it.

A Comment on Multiparticle Quantum Mechanics in AdS

Nothing stops us from studying simple examples of multi-particle quantum mechanics in AdS. Forexample, consider a hydrogen atom. In the usual description in flat space, its wavefunction can bespecified in terms of the position of the proton and the position of the electron. Usually it makesmore sense to break these up into a center of mass degree of freedom (which is somewhat trivial) anda coordinate representing the relative position between the electron and proton. Then we computethe wavefunctions, energy levels, etc.

We can repeat this same process in AdS, and we could even go on to compute the wavefunctionsand binding energies due to the Coulomb attraction. According to what we’ll soon learn aboutAdS/CFT, we could then interpret the results in terms of the scaling dimensions and correlationfunctions of operators in a putative CFT. We won’t take this approach here (although you can readabout a very similar approach in [10]), but it’s worth emphasizing that even the most elementaryresults from quantum mechanics can be applied in the AdS/CFT context. Via AdS/CFT we canreinterpret many familiar ideas holographically.

28

4 Free Fields in AdS

In the previous section we discussed AdS geometry and the physics of a classical or quantum freeparticle in AdS. Now let us see how to describe any number of quantum free particles in AdS.

The key feature of states constructed from many free particles (Fock space states) is that theirquantum numbers are just the sum of the quantum numbers of the individual particles. In particular,their energy (for us the D eigenvalue) is just the sum of the energies of the individual particles. Thisis why the Hilbert space of free quantum field theory is built using harmonic oscillator creation andannihilation operators – the harmonic oscillator has evenly spaced energy levels, so any two statescan be combined by just adding their energies.

For this reason, to describe any number of free particles in AdS we simply introduce the creationand annihilation operators an`J and a†n`J with the usual commutator

[an`J , a†n`J ] = 1 (4.1)

A basis for many particle states can be constructed as usual; a generic k-particle state is

a†n1`1J1a†n2`2J2

· · · a†nk`kJk |0〉 (4.2)

where |0〉 is the vacuum, which has no particles. In terms of these operators, the AdS HamiltonianD is simply

D =∑

n,`J

(∆ + 2n+ `)a†n`Jan`J (4.3)

because it sums the energy of all the particle, and a†n`Jan`J is the number operator that counts thenumber of particles with quantum numbers n, `, J . The other conformal generators can also beexpressed in terms of the creation and annihilation operators. So we have an infinite collection ofharmonic oscillators in AdS.

It’s straightforward to see that this is just what we get from quantizing a free scalar field in AdS,with action

S =

∫

AdS

dd+1x√−g

(1

2(∇Aφ)2 − 1

2m2φ2

)(4.4)

Before we explain why, let’s detour to review everything you ever wanted to know about free fieldsbut were afraid to ask.

4.1 Classical and Quantum Fields, Anywhere

Here we will review, at a rather formal level, how one goes about solving for the time evolutionof a classical field. Then we will explain how this relates to quantization. These ideas are worthunderstanding for reasons that go beyond AdS – for example, they are an elementary ingredient inHawking’s derivation of black hole evaporation.

29

Let us study a scalar field theory in a completely general spacetime, with free action

S =

∫ddxdt

√−g(

1

2gµν∇µφ∇νφ−

m2

2φ2

)(4.5)

This action implies the usual equations of motion

gµν∇µ∇νφ+m2φ = 0 (4.6)

Let us imagine that we solve this equation with some general solutions fn(t, x) labeled by an index

n that may be discrete or continuous. For example in flat spacetime we usually take12 n = ~k and

fk(t, x) = eiωtei~k·~x with ω2 = ~k2 +m2. In AdS we find solutions labeled by n, `, J .

Even if we are only solving the theory at a classical level, we would still like to be able to write ageneral solution as (we will get to quantization below)

φ(t, x) =∑

n

(cnfn(t, x) + c†nf

†n(t, x)

)(4.7)

given some initial data consisting of φ0(0, x) and φ0(0, x) on the t = 0 surface. The initial datajust corresponds to the initial position and velocity of the φ field. We need to know both initialconditions to solve for φ(t, x) because its equation of motion is second order.

We know that φ(t, x) can be expressed as a sum over the modes fn(t, x). But given some initialcondition, how do we solve for the cn coefficients? We need an inner product defined on spacelikesurfaces at fixed time. A natural place to start is with the equations of motion – one often obtainsinner products by noting that some integral

∫fdg = −

∫dfg, and if the operator “d” can be

diagonalized, then this integral must vanish unless f and g have the same eigenvalue. Therefore forany two functions ψ1 and ψ2 on spacetime13 we can define an integral

∫

Ω

ddxdt√−g

[ψ†1(gµν∇µ∇ν −m2

)ψ2 −

[(gµν∇µ∇ν −m2

)ψ†1

]ψ2

](4.8)

over some region Ω. Let’s imagine that Ω spans all of space, but ends at some finite times ti and tfon Cauchy surfaces Σi and Σf . We can re-write this integral as a boundary integral

∫

Σf−Σi

ddx√−gg00

[ψ†1∇tψ2 −

(∇tψ

†1

)ψ2

](4.9)

where we assumed that the t direction is orthogonal to Σ, for simplicity, and we took t pointingforward in time for both surfaces. If both ψ1 and ψ2 obey the equations of motion, then our originalintegral must vanish, so we see that

〈ψ1, ψ2〉 ≡ i

∫

Σf

ddx√−gg00

[ψ†1∇tψ2 − ψ2∇tψ

†1

]= i

∫

Σi

ddx√−gg00

[ψ†1∇tψ2 − ψ2∇tψ

†1

](4.10)

12Actually, the t dependence should technically be forced to be either cos(ωt) or sin(ωt) so that the solutions arereal, assuming we are studying a real scalar field. Then we would express the full solution as a sum over sines andcosines, instead of complex exponentials with complex coefficients. The real case is simpler, but the complex case ismore useful and standard. It’s also noteworthy that if the t direction points along a Killing vector, then we have atime translation symmetry, and so the time dependence will always just be e±iωt.

13We assume they vanish with some sufficient rapidity at spatial infinity.

30

We have defined a conserved inner product. In particular, if we normalize 〈fn, fm〉 = δmn and〈f †n, fm〉 = 0 at some time then this normalization will be time independent. This gives the usualnormalization 1√

2k0eik·x for modes in flat spacetime (noting that in that case, since k is continuous,

we get a delta function in place of δmn).This means that if we write φ(t, x) as in equation (4.7) then the cn will be time independent, as

desired. More importantly, it means that we can compute the cn from

cn = 〈φ(0, x), fn〉 = i

∫

Σf

ddx√−gg00

[φ†0(x)fn(x)− fn(x)φ†0(x)

](4.11)

using our inner product along with both the φ(0, x) and φ(0, x) initial conditions. So we see that ourinner product automatically knows how initial conditions work in classical theories. At the classicallevel, it seems that if we had a higher order differential operator for a kinetic term, then we wouldend up with an inner product involving higher time derivatives, in keeping with expectations for theinitial value problem.

Now consider the canonical momentum

π(x) =δL

δφ=√−g(x)g00(x)φ(x) (4.12)

In either the classical or the quantum theory we need to satisfy the canonical commutation relations(or Poisson brackets)

[π(x), φ(y)] = −iδd(x− y) (4.13)

and as usual, when dealing with the quantum theory we can write14

φ(t, x) =∑

n

1

Nn

[fn(t, x)an + f †n(t, x)a†n

](4.14)

in terms of anniliation and creation operators an and a†n, and a normalization factor Nn that wehave added in case we need it. The canonical commutation relations are satisfied if

√−g(x)g00(x)

∑

n

1

N2n

[fn(0, x)f †n(0, y)− f †n(0, x)fn(0, y)

]= −iδd(x− y) (4.15)

However, this can simply be viewed as an expression for the coefficients cn when we write thefunction δd(x − y) (viewed as a function of y, with x serving as a fixed paramter) in the fn(0, y)basis. This means we can use the inner product to write

cn =

∫ddy√−g(y)g00(y)δd(x− y)fn(0, y)

=√−g(x)g00(x)fn(0, x) (4.16)

in this very special case. This determines that in fact all the normalization constants Nn = 1 oncewe work in terms of modes fn normalized with our inner product as 〈fn, fm〉 = δmn .

14As with the previous footnote, we could also re-group this into the coefficient of an + a†n and an − a†n to avoidcomplex valued functions.

31

4.2 Back to AdS3

Let us return and consider the AdS3 example for concreteness. Then the action is

S =

∫dtdρdθ

sin ρ

cos2 ρ

1

2

(φ2 − (∂ρφ)2 − 1

sin2 ρ(∂θφ)2 − m2

cos2 ρφ2

)(4.17)

The canonical momentum conjugate to φ is

Pφ =δL

δφ=

sin ρ

cos2 ρφ (4.18)

So we will want to impose the canonical commutations relations

[φ(t,X), Pφ(t, Y )] = i(2π)2δ2(X − Y ) (4.19)

We will do this by first solving the equations of motion for φ, and then writing φ in terms of a sumover harmonic oscillator modes. As we discussed in generality above, the canonical commutationrelation then gives a normalization condition for the modes.

The equations of motion are

φ− cos2 ρ

sin ρ∂ρ(sin ρ cos−2 ρ∂ρφ

)− 1

sin2 ρ∂2θφ+

m2

cos2 ρφ = 0 (4.20)

Taking φ = eiωt+i`θψ(ρ), we have

−ψ′′(ρ)− 1

cos ρ sin ρψ′(ρ) +

(`2

sin2 ρ+

m2

cos2 ρ

)ψ(ρ) = ω2ψ(ρ) (4.21)

which is just the equation for a one-particle wavefunction in d = 2 that we obtained previously. Thesolutions were given in equation (3.50).

The quantized field which obeys canonical commutation relations is

φ(t, ρ,Ω) =∑

n,`

ψn`J(t, ρ,Ω)an` + ψ∗nlJ(t, ρ,Ω)a†n`, (4.22)

with the wavefunctions ψn` that were defined in equation (3.50). To find the normalizations of theψn` we impose

〈ψn`, ψn′`′〉 = δnn′δ``′ =

∫

AdS

ddx√−gg00(ψn`∂tψn′`′ − ∂tψn`ψn′`′) (4.23)

which translates in AdS3 to

1 =1

(N∆n`)2

∫dρdθ sin ρ 2(∆ + 2n+ `)ψ†n`(ρ, θ)ψn`(ρ, θ) (4.24)

32

Note that it is easy to see that modes with different ` are orthogonal, as the θ integral then vanishes.The orthogonality of modes with different n can be verified directly. One finds that the normalizationsof the ψn`J are

N∆n` = (−1)n

√n!Γ2(`+ d

2)Γ(∆ + n− d−2

2)

Γ(n+ `+ d2)Γ(∆ + n+ `)

. (4.25)

by a direct computation. This can be checked for small n and `.A crucial condition obeyed by φ(x) is that, as in flat spacetime, [φ(x), φ(y)] = 0 whenever x and

y are spacelike separated in AdS. This means that if we construct local interactions V[φ(x)] thenwe will also have [V(x),V(y)] = 0 outside the lightcone. Weinberg [3] showed (see section 3.5 of hisbook) that this condition was necessary for QFT to produce a sensible Poincare invariant S-Matrix.In section 2.4 of [10] this reasoning was generalized to AdS and the conformal algebra.

We saw in section 3.2 that a single free particle in AdS transforms under an irreduciblerepresentation of the conformal group. The particle’s ground state is a primary state of theconformal algebra, because it is annihilated by all the ‘lowering operators’, the special conformalgenerators Kµ. All the other states can be built with Pµ, the momentum generator or ‘raisingoperator’ with respect to D. So what are the analagous statements for the quantum field φ(x)?

4.3 General Approaches to the Boundary

4.3.1 Lessons from the Poincare vs Global Approach

So it seems we have two different ways to approach the boundary – in global coordinates, we naturallyobtain a cylinder parameterized by (τ, Ω) when we take ρ→ π/2 at the same rate for all (τ, Ω). Incontrast, in the Poincare patch coordinates we find a Minkowksi space parameterized by (t, ~x) whenwe take z → 0 at the same rate for all (t, ~x). Actually, there are infinitely more possibilities.

The idea is that in the global coordinate metric of equation (2.1), we can approach different pointson the boundary, parameterized by (t, Ωi), at different rates. This effectively allows us to performfurther Weyl transformations as we take ρ→ π/2 in order to alter the geometry of the boundary.

Specifically, we can choose some function f(t, Ωi

)in the neighborhood ρ = π/2− εf(t, Ωi) with

small ε, and then take ε→ 0. This results in an effective metric on the boundary set by

ds2 =[f(t, Ωi

)]−2 (dt2 − dΩ2

i

)(4.26)

which is related to the metric on the cylinder by a Weyl transformation (again, this just means themetric is muliplied by an overall function of the coordinates). So by taking the limit as we approachthe boundary in different ways, we can obtain a CFT on any conformally flat manifold, a.k.a. anymanifold related to flat spacetime by a Weyl transformation. Note that a cylinder is conformallyflat since we can take f(t, Ωi) = e−t and switch to the coordinate r = et to find a flat space metric(at least in Euclidean signature).

33

4.3.2 The Euclidean Version

We parameterized Euclidean AdSd+1 using global or Poincare coordinates as

X0 = Rcosh τ

cos ρ=

1

2

(z2 + ~r2 +R2

z

)(4.27)

Xd+1 = Rsinh τ

cos ρ=

1

2

(z2 + ~r2 −R2

z

)

Xi = R tan ρΩi =R

zri

where ~r = (~x, t). Note that this means that ~ri = Ωieτ sin ρ and z = eτ cos ρ. A nice thing about this

version is that now as z → 0, we have ρ→ π/2 along with z → 0, with a mapping

eτ/2Ωi√π2− ρ =

ri√z

(4.28)

This naturally produces a flat spacetime metric ds2 = d~r2 for the boundary.We can also see this directly in global coordinates without the mapping – the Euclidean AdS

immediately gives us radial quantization. In particular, with the metric

ds2 =1

cos2 ρ

(dτ 2 + sin2 ρdΩ2

)(4.29)

we can simply approach the boundary as

ρ =π

2− εe−τ (4.30)

with ε→ 0. Then we find the effective boundary metric

ds2∂AdS → e2τ (dτ 2 + dΩ2)→ dr2 + r2dΩ2 (4.31)

with r = eτ . We see that r = 0 maps to t = −∞, the infinite past in Euclidean AdS, while r =∞corresponds to the infinite future.

4.3.3 Getting a DeSitter Boundary Geometry

Finally, note that the deSitter metric from equation (2.29) is also of the general form that we canobtain. If we choose

ρ =π

2− ε cos(t) (4.32)

Then we find

1

cos2 ρ(ε; t)(dt2 − sin2 ρdΩ2)→ 1

cos2 t(dt2 − sin2 ρdΩ2) (4.33)

This is precisely the metric for global deSitter space, where in equation (2.29) we called t the variableη. In particular, now we only have −π/2 < t < π/2. This scaling makes it possible to obtain a CFTliving in a deSitter spacetime background!

34

4.3.4 Lorentzian Poincare Patch Version

Let’s go back and make this explicit in the case of the Poincare patch. The boundary of the Poincarepatch is obtained by taking z → 0 with fixed t, ~x, so let us translate this into the global coordinatesρ, τ, Ω. Using the relations of equation (2.13), we see that if we take the coordinate ρ towards π/2at a rate dictated by

ρ(z; τ, Ωi) =π

2− z

R2(cos θ − cos τ) as z → 0 (4.34)

where θ is the complement of the angle Ωd. This means that in the limit as we approach the boundarywe obtain the metric

ds2 =1

(cos θ − cos τ)2(dτ 2 − dθ2 − sin2 θdΩ2

d−1) (4.35)

This is familiar from our discussion of the Penrose diagram for flat spacetime – if we first define2U = −θ + τ and 2V = θ + τ then we obtain a new metric

ds2 =1

sin2 U sin2 V

(dUdV − sin2(U + V )dΩ2

d−1

)(4.36)

Now we can define cotu = U and cot v = V so that the metric becomes simply

ds2 = dudv − (u+ v)2

4dΩ2

d−1 (4.37)

Finally, we can write this in terms of 2t = u− v and 2r = u+ v and see that it is in fact the flatspacetime metric.

4.3.5 Embedding Space Coordinates and the Projective Null Cone

It is natural to translate all of these statements into the embedding space coordinates, wherethe full conformal symmetry is manifest. For this purpose we can use equation (2.4). The limitρ = π

2− εf(τ,Ω) with ε→ 0 sends the XA to infinity (defined by equation (2.10)) while fixing the

null projective cone coordinates

PA ≡ εXA (4.38)

while ε→ 0. These satisfy

PAPA = 0 and PA ∼= λPA (4.39)

where the second statement means that all PA are equivalent up to an overall positive rescaling.The conformal group SO(2, d) acts on the PA in the obvious way inherited from the XA. Also, therescaling by λ effectively just rescales ε, and therefore rescales the boundary metric by an overallfactor, so it is a global dilatation. This is a section of the projective null cone, discussed at length ine.g. chapter 2 of Slava’s notes.

35

In the Euclidean case where the conformal group is SO(1, d+ 1), the PA can be identified withflat space, or any conformally flat manifold. For example, using our choice

ρ =π

2− εe−τ (4.40)

with ε→ 0 means that

PA = (P+, P−, Pµ) = (e2τ , 1,Ωieτ ) (4.41)

where P± = P0 ± Pd+1. If we take ri = eτΩi, where ri is a coordinate in Euclidean flat spacetime,then we obtain the section of the null projective cone corresponding to flat space.

4.4 Implementing Holography

The basic idea of holography is that physics15 in d+ 1 dimensions can be reproduced by a ‘hologramat infinity’, in d or fewer dimensions. It’s always a good idea to take ideas maximally seriously, makethem as concrete and specific as possible, and see where you are led. Now that we have constructeda quantum field φ(x) depending on the AdS coordinate x, we can do just that – we can study φ(x)in the limit that x approaches infinity and see what we find. Why is this a reasonable guess? Ifthere exists a holographic dual to effective field theory in AdS, then it must reproduce at least someof the familiar observables in AdS. Correlation functions of operators φ(x) as x→∞ would be areasonable starting point, since they can be easily computed in the AdS QFT, but are also definedpurely in terms of data at infinity. The S-Matrix is a flat space analog of these observables. However,let us begin with an even broader approach by studying φ as an operator with x→∞.

For this purpose let us switch to the Euclidean version of AdS, where the conformal group isSO(1, d + 1). We saw in section 2.3 that the boundary of AdS (aka ‘infinity’) is just the limitρ→ π/2 in global coordinates. Being a bit more careful, we saw in section 4.3.2 that we need totake this limit as

ρ(ε) =π

2− εe−t (4.42)

with ε→ 0 in a coordinate independent way in order to recover a Euclidean flat spacetime metricon the boundary.

Now we are ready to consider φ(t, ρ,Ω) in the limit ρ→ π/2. As we can see from the explicitwavefunctions in equation (3.50), in this limit ψn`J → 0 as cos∆ ρ. Physically, this follows becausefinite energy particles in AdS have vanishing probability of making it to the boundary, due to theAdS curvature. However, we can define a new quantum operator O(t,Ω), ‘dual to φ’ via16

O(t,Ω) ≡ limε→0

φ(t, ρ(ε),Ω)

ε∆(4.43)

15As we have said before, this actually only makes perfect sense for theories with gravity. However, since gravitationaltheories also involve other degrees of freedom, and we can talk about a decoupling limit where gravity shuts off, itmakes sense to also see how these ideas are reflected in non-gravitational theories.

16Note that we are not rescaling φ with cos ρ, but only with ε. This is necessary so that the conformal (AdSisometry) transformation properties of φ are not altered by the rescaling, and so to obtain the conformal correlators.

36

AdS CFT

∆t

ρ

exp

∆t

R

Figure 7: This figure shows how the AdS cylinder in global coordinates corresponds to the CFT inradial quantization. The time translation operator in the bulk of AdS is the Dilatation operator inthe CFT, so energies in AdS correspond to dimensions in the CFT. We made this mapping veryexplicit in section 4.3.2.

which is finite on the boundary of AdS. Computing this limit for the wavefunction ψn`J we note that

sin(ρ) → 1 (4.44)

cos(ρ) → εe−t∆ (4.45)

which means that since

ψn`J ∝ e−En,`tY`J(Ω)

[sin` ρ cos∆ ρ 2F1

(−n,∆ + `+ n, `+

d

2, sin2 ρ

)](4.46)

we have

ψn`J →1

N∆n`

e−(2∆+2n+`)tY`J(Ω) 2F1

(−n,∆ + `+ n, `+

d

2, 1

)(4.47)

ψ†n`J →1

N∆n`

e(2n+`)tY †`J(Ω) 2F1

(−n,∆ + `+ n, `+

d

2, 1

)(4.48)

37

Note that the factors of cos∆(ρ) are responsible for a shift in the t-dependence. Also, a useful factabout hypergeometric functions is that

2F1

(−n,∆ + `+ n, `+

d

2, 1

)=

Γ(`+ d2)Γ(d

2−∆)

Γ(n+ `+ d2)Γ(d

2− n−∆)

(4.49)

This means that the operator O inherits a normalization

1

NOn`∝ (−1)nΓ(`+ d

2)Γ(d

2−∆)

Γ(n+ `+ d2)Γ(d

2− n−∆)

(n!Γ2(`+ d

2)Γ(∆ + n− d−2

2)

Γ(n+ `+ d2)Γ(∆ + n+ `)

)−1/2

(4.50)

=

√Γ(∆ + n+ l)Γ

(∆ + n− d−2

2

)Γ(d2

)

n!Γ(∆)Γ(∆− d−2

2

)Γ(d2

+ n+ l) (4.51)

where in the second line we multiplied by a constant factor of√

Γ(∆ + d−22

)/Γ(∆) so that O will

have a 2-pt correlator with a simple normalization. Now we can write

O(t,Ω) =∑

n,`

1

NOn`

(e−(2∆+2n+`)tY`J(Ω)an` + e(2n+`)tY †`J(Ω)a†n`

)(4.52)

Let us now see why this operator has the correct correlators in flat spacetime.

4.5 Correlators

Now let’s look at the correlation function of our new operator O. In the next section we will studyits conformal transformation properties, the operator state correspondence, and the OPE.

4.5.1 Computation from AdS

In the limit of large distances (which is all we need because the AdS points are approaching infinity),the 2-pt correlator of φ is

〈φ(X)φ(Y )〉 ≈ Γ(∆)

Γ(∆ + 1− d

2

)e−∆σ(X,Y ) (4.53)

where σ(X, Y ) is the geodesic distance between X and Y in AdS. One can compute this 2-pointcorrelation function in many ways, e.g. by directly by summing over modes created by the oscilatorsan`J . It is also guaranteed to take this form at large distances (up to a normalization) because ofthe AdS isometries combined with our knowledge of the behavior of solutions to the Klein-Gordonequation due to a point source.

Geodesic distances in the ρ direction are just given by an integral∫

sec(ρ)dρ. In the ε→ 0 limitthis is just log(ρ − π/2) = t − log ε. Without loss of generality we can take X and Y to be onopposite sides of the Euclidean AdS cylinder at some fixed time t. This gives a 2-pt correlator for O

〈O(x1)O(x2)〉 =Γ(∆)

Γ(∆ + 1− d

2

) exp [−∆(2t− 2 log ε)]

ε2∆=

Γ(∆)

Γ(∆ + 1− d

2

) 1

e2∆t(4.54)

38

Restated in terms of the boundary points x1 = Ωet and x2 = −Ωet and dropping the normalizationfactor for simplicity, this is

〈O(x1)O(x2)〉 =1

(x1 − x2)2∆=

1

(2P1 · P2)∆(4.55)

where we have also stated the result in terms of the projective null cone coordinates. Although thiscan be computed directly, the result actually follows entirely from conformal symmetry. The onlyconformal invariant that can be made from P1 and P2 is P1 · P2, and then the power of ∆ followsfrom the D transformation properties that O inherits from φ.

Note that since we were studying a free theory in AdS, all correlators of the φ field can be reducedto Wick contractions leading to products of 2-pt (Gaussian) correlators. So in fact we already knowall correlation functions of O(x) as well, for the same reason.

4.5.2 Computation from the CFT Operator

As we will soon see, O inherits conformal transformation properties from the AdS isometrytransformations of φ. In particular, it certainly inherits translations, so WLOG we can compute

〈O(~r)O(0)〉 (4.56)

with, as usual ~r = etΩ. However, note that

O(0)|0〉 =

(∑

n,`

1

NOn`e−(2∆+2n+`)tY`J(Ω)an` + e(2n+`)tY †`J(Ω)a†n`

)|0〉

= a†0,0|0〉 (4.57)

This follows because all the annihilation operators destroy the vacuum, while the other creationoperators are multiplied by positive powers of ~r = 0. Now evaluating the correlator is extremelyeasy, because the only term that survives is proportional to a0,0a

†0,0, and we simply find

〈O(~r)O(0)〉 =1

r2∆(4.58)

which is exactly what we expected. Notice that it was crucial that we scaled ρ appropriately toobtain O from φ – if we had simply taken ρ→ π/2 uniformly then O would have been proportionalto an additional e−t∆; this would have given the correct correlators on a cylinder, but not in flatspacetime!

5 Generalized Free Theories and AdS/CT

In this section we will show how our Generalized Free Theory, defined by the holographic limit ofthe AdS bulk field φ, satisfies and in many cases physically illustrates the general abstract propertiesexpected for a local operator in a conformal theory. We refrain from using the words ”CFT” becausethe conformal theory we are discussing lacks a stress-energy tensor.

39

AdS CFT

O|0i

O|0i

| O(t)i

| O(t)i

Figure 8: This figure portrays radial quantization in the CFT, while also providing the correspondingpicture in AdS. The insertion of an operator at the origin in the CFT defines a specific AdS/CFTstate; interpreted in AdS this sets up an initial state in the infinite past that evolves to ψO(t) ata finite time. Note that et ∼ |~r|, so the global coordinate time is the logarithm of the radius of acircle about the origin in the CFT.

5.1 The Operator/State Correspondence

First of all, what are these so-called operators?Clearly we have the object O(~r) that we defined previously in terms of the AdS field φ. To study

the question in general, let us specialize for convenience to the case of AdS3. The reason is that wecan significantly simplify the notation in that case by writing

~r = (x, y) (5.1)

in terms of

z = x+ iy and z = x− iy (5.2)

Since we are studying the boundary of Eucliean AdS3 as the 2-dimensional Euclidean plane, theangular coordinate Ω→ θ just parameterizes a single circle. In terms of z and z we have

r2 = zz and e2iθ =z

z(5.3)

40

and the Y` = ei`θ. Therefore we can write

O(z, z) =∑

n,`

1

NOn,`

[(zz)−(∆+n+

|`|2 )(zz

) `2an` + (zz)n+

|`|2

( zz

) `2

a†n`

](5.4)

When we work in terms of z and z, it’s simpler to define the operator in terms of integers h and hwith

O(z, z) =∞∑

h,h=0

1

NOh,h

[(zz)−∆z−hz−hah,h + zhzha†

h,h

](5.5)

where 2n+ |`| = h+ h and ` = h− h, so that n = min(h, h).Above we saw that when we act O(0) on the vacuum, we obtain the state

O(0)|0〉 = a†0,0|0〉 (5.6)

Now let us consider what we get from Pµ acting on O. We can simply write the momentum operatorin terms of a basis as Pz = ∂z and Pz = ∂z. Thus we find

(PzO)(0) =1

NOh=1,h=0

a†h=1,h=0

|0〉 (5.7)

Similarly, we can act with any number of Pz and Pz on O. So we have a whole host of operators ofthe form

((∂z)

h(∂z)hO)∣∣∣

z,z=0(5.8)

These create all of the states a†h,h

! Crucially, these are just the one-particle states in AdS3 labeledby n and `. So we see that all one-particle states can be created by some particular CT operator!This is the beginning of the operator/state correspondence; it’s not some abstract idea but a veryconcrete point that can be visualized directly in AdS.

What about multipartcle states? We would like to define an operator like

O2(0) (5.9)

and we would expect it to create 2 AdS particles, both in their ground state. But if we evaluate thisdirectly, there appears to be a singularity. For example, if we look at

O(z, z)O(w, w) =

∞∑

h,h=0

1

NOh,h

[(zz)−∆z−hz−hah,h + zhzha†

h,h

]

×

∞∑

h,h=0

1

NOh,h

[(ww)−∆w−hw−hah,h + whwha†

h,h

] (5.10)

41

then we have singularities from the limit of ω → 0 in the second operator, as well as singularitiesthat arise when the annhilation operators in O(z, z) hit the creation operators in O(w, w). Thus wecan define O2(0) as the normal ordered object with these singularities removed, so that we get

(O2)

(0) ≡(

1

NO0,0

)2

a†0,0a†0,0 (5.11)

Simillarly, we can define

(O3)

(0) ≡(

1

NO0,0

)3

a†0,0a†0,0a

†0,0 (5.12)

and also the operator

(OPzO) (0) ≡(

1

NO0,0NO

1,0

)a†0,0a

†1,0 (5.13)

We can proceed to define operators with an aribtrarily large string of Pz, Pz, and Os. Again, thecrucial point is that by acting with general linear combinations of such operators on the vacuum |0〉,we can obtain every single state in the theory – in particular, there is a one-to-one correspondencebetween AdS/CFT states in the Hilbert space and local operators acting at the origin, z = 0. This isthe operator/state correspondence for the CT defined by a free field in AdS. From the CT point ofview, this system is called a Generalized Free Theory, a Gaussian CFT, a CFT at infinite N , or amean field theory. I will usually stick with the first term.

In closing, we should note that throughout this section we have used radial quantizatoin withoutdrawing much attention to it. This is the natural quantization scheme in mapping from AdS inglobal coordinates to a CFT on a plane.

5.2 Conformal Transformations of Operators

We obtained the Conformal Theory operator O(~r) from a limit of φ(t, ρ,Ω), a free field in AdS withan action symmetric under the AdS isometries aka conformal transformations. This means that Onaturally inherits conformal transformation properties from AdS. The intelligent way to approachthis question would be to use the embedding space coordinates. However, we can also be somewhatless clever and see how it works using the global coordinate system.

We already computed the Dilatation operator D as the AdS Hamiltonian, it is

D =∑

n,`J

(∆ + 2n+ `)a†n`Jan`J (5.14)

and by definition, since in AdS there was no explicit time dependence, we have that

[D,φ(t, ρ,Ω)] =d

dtφ (5.15)

42

Let’s compute its action on O in two different ways. Note that if we had defined O by simply takingρ→ π/2 in a t-independent way, then we would find that O obeys the same relation as φ. However,in order to obtain a CT in flat Euclidean space, we took ρ = π/2 − εe−t, producing an explicitt-dependence.

In particular, an infinitesimal shift t→ t+ a leads to

[D,O(t,Ω)] = limε→0

(∂

∂t

φ(t, π/2− εe−t,Ω)

ε∆

)

≈ ∂

∂tO(t,Ω)− lim

ε→0

∂ρ

∂t∂ρφ(t, ρ,Ω)

ε∆

≈(

∆ +∂

∂t

)O(t,Ω) (5.16)

with ρ = π/2− εe−t as usual. This can be written in ~r = etΩ coordinates as

[D,O(~r)] =

(∆ + r

∂

∂r

)O(~r) (5.17)

This is the correct conformal transformation rule for a CFT operator of dimension ∆. The lessonis that time derivatives on O act both explicitly (via partial derivative) and with a shift of ∆ inorder to cancel the t dependence that arises through ρ = π/2− εf(t,Ω). We can also compute this adifferent way, using our explicit formula for O and D in terms of oscillators in AdS3/CFT2. We find

∑

h,h

(∆ + h+ h)a†h,hah,h,

∞∑

h,h=0

1

NOh,h

[(zz)−∆z−hz−hah,h + zhzha†

h,h

]

=∞∑

h,h=0

1

NOh,h

[−(∆ + h+ h)(zz)−∆z−hz−hah,h + (∆ + h+ h)zhzha†

h,h

]

= (∆ + z∂z + z∂z)O(z, z) (5.18)

as promised, so the two methods agree.In AdS3 we looked at the momentum operator P±. Converting to the Euclidean case, it is

P± = e−t±iθ(

sin ρ∂t + cos ρ∂ρ ±i

sin ρ∂θ

)(5.19)

We cannot immediately drop cos ρ∂ρ because it scales as a constant as ρ→ π/2. However, this termjust produces −∆ when acting on φ, and this cancels with the shift of ∆ from ∂t that we observedwhen we examined the dilatation operator. So as we approach the boundary, we simply find that itacts on O as

P± → e−t±iθ (−∂t ± i∂θ)

=1√zz

(zz

)± 12

[−(z∂z + z∂z)± (z∂z − z∂z)] (5.20)

43

This just translates into the statement that P− = −2Pz and P+ = −2Pz. So the Pµ isometry of AdSjust turns into the translation operator, as we would expect.

In AdS3 there is only one angular momentum generator, proportional to ∂θ. This does not changewhen we take it to the boundary, and so we just find

L = ∂θ =1

2(z∂z − z∂z) (5.21)

in the 2-dimensional plane.Finally, we can consider the special conformal generators, which take the form

K± = et±iθ(

sin ρ∂t − cos ρ∂ρ ∓1

sin ρ∂θ

)(5.22)

in AdS3. Here the contribution of cos ρ∂ρ does not cancel, instead it contributes additively with theshift in ∂t. Otherwise this operator behaves similarly to the translation operator, and we find

K± → et±iθ (2∆− ∂t ∓ i∂θ)

=√zz(zz

)± 12

[2∆− (z∂z + z∂z)∓ (z∂z − z∂z)] (5.23)

This leads to

K+ = −2z∆− 2z2∂z (5.24)

which is the correct result, up to an overall constant. Note that, as expected, the holomorphic andanti-holomorphic parts of the algebra can be decoupled from each other. We have derive the actionof all of the conformal generators on our CT operator O(z, z).

To summarize, we have found that the 2d global conformal algebra SO(1, 3) acts on the operator

[D,O(x)] =

(∆ + r

∂

∂r

)O(z, z) (5.25)

[Pµ,O(x)] = ∂µO(z, z) (5.26)

[Kz,O(x)] =(2z∆ + 2z2∂z

)O(z, z) (5.27)

[Kz,O(x)] =(2z∆ + 2z2∂z

)O(z, z) (5.28)

[L,O(x)] =1

2(z∂z − z∂z)O(z, z) (5.29)

Note that we derived these relations from AdS. When we study general CFTs in the next section,these relations will take on a fundamental character – in fact, one way of defining a CT is as atheory of operators transforming in exactly this way. But we have understood these transformationproperties as an inevitable consequence of AdS physics.

We can also write these symmetry generators in terms of ah,h, just as we did for D. There is analgorithm to do this – we just need to find the conformal generators in terms of the AdS field φ, and

44

then use the expansion of that field in creation and annhilation operators. We can also try to getthe answer by guessing. For example, to obtain P I find we need to take

Pz ≡∑

h,h

−√

min(h, h)(∆ + min(h, h)− 1)a†h,hah−1,h

Pz ≡∑

h,h

−√

min(h, h)(∆ + min(h, h)− 1)a†h,hah,h−1 (5.30)

Note that (crucially!) these operators also satisfy the commutation relations [D,Pµ] = Pµ and[Pz, Pz] = 0. It’s fairly trivial to find an operator Pz that acts on O(z, z) as ∂z, the propertythat actually makes this a Conformal Theory is that the operators D,Pµ, Kµ, and Mµν that actappropriately on O(x) also satisfy the commutation relations of the conformal algebra.

Finally, note that in Lorentzian AdS we had the relation P †µ = Kµ. This has a simple meaning inAdS – it just says that since complex conjugation sends it→ −it, it exchanges the past and future.This is what we are used to in Quantum Mechanics; for example when considering scattering wehave

(|in〉)† = 〈out| (5.31)

as usual. However, we we go from AdS → CFT, we went to Euclidean space and took r2 = zz = e2t.Thus in the CFT, t→ −t means

r → 1

r(5.32)

or in other words, complex conjugation corresponds to performing a spacetime inversion about theorigin. The states built at the origin transform like

O(0)|0〉 → limr→∞

r2∆〈0|O(r) (5.33)

under this transformation. More generaly, for a primary operator O we have

[O(~r)]† = r−2∆O∗(~r

r2

)(5.34)

where the ∗ means that we take a → a†. The reason for the factor of r can be seen explicitly inequation (4.52). There we see that the an`J annihilation operators are accompanied by this factor,whereas the creation operators are not, so we need to multiply so that the operator is Hermitian, orin other words so that [O(r)]† = O(r).

5.3 Primaries and Descendants

In our earlier discussions we noted that states in a theory with conformal symmetry can be classifiedas primary or descendant. Primary states have the property that

Kµ|ψ0〉 = 0 (5.35)

45

so they are annihilated by all the special conformal (or lowering) operators. One example of aprimary states is a†0,0|0〉; this corresponds to a free particle in its quantum mechanical ground statein AdS3. States formed by acting on this state with Pµ are descendants.

But there are many many more primaries. One example is

Ok(0)|0〉 =(a†0,0

)k|0〉 (5.36)

but this is just the beginning. This can be interpreted as a condensate of k particles in AdS3, all intheir ground state.

Let us pause to recall why this is a primary operator. In AdS, note that the CoM wavefunction is

ψk(tCoM , ρCoM) = 〈φk(tCoM , ρCoM)Ok(0)〉 (5.37)

= e−k∆tCoM cosk∆ ρCoM (5.38)

So we see it has the correct form for a primary wavefunction with dimension = k∆. In particular, itwill be annihilated by all the SCT Kµ in AdS. In the CFT, the fact that is primary is tested byasking if

KµOk(0)|0〉 = 0 (5.39)

We can compute this by commuting Kµ through so that it hits the vacuum, where we obtain 0. Soin other words, this state will be a primary if

[Kµ,Ok(0)

]= 0 (5.40)

This is the condition for an operator to be a primary. A primary operator is just an operator thatcreates a primary state when it acts at the origin.

We can also understand the converse statement, namely that if O(0)|0〉 is a primary state, thatwe can justifiably assume [Kµ,O(0)] = 0, because this commutator has vanishing correlators withother operators. The argument is very simple.

A more interesting and non-trivial example is the state

[∂zO∂zO −O∂z∂zO] |0〉 (5.41)

where the operators O are normal ordered. One can check that this is always a primary state byusing the conformal commutation relations. Note that

∂zO = [Pz,O] (5.42)

and so this state takes the form

([Pz,O][Pz,O]−O[Pz, [Pz,O]) |0〉 (5.43)

To check that it is a primary we need to show

Kz/z ([Pz,O][Pz,O]−O[Pz, [Pz,O]) |0〉 = 0 (5.44)

46

and this will be true iff

[Kz, ([Pz,O][Pz,O]−O[Pz, [Pz,O]) (0)] = 0 (5.45)

and similarly for z → z, since the conformal generators act on commutators via the Lie algebracommutator. Note that the operator we wish to show is primary is evaluated at the origin. Wecould also compute this directly in AdS using the known wavefunctions.

Carrying out the computations using [A, [B,C]] + [B, [C,A]] + [C, [A,B]] = 0, we find that

[Kµ, [Pz,O][Pz,O]] = 2∆O (δzµ[Pz,O] + δzµ[Pz,O]) (5.46)

while a similar computation gives

[Kµ,O[Pz, [Pz,O]] = 2∆O (δzµ[Pz,O] + δzµ[Pz,O]) (5.47)

so the linear combination we wrote down is in fact a primary operator. Note that it was crucialfor these computations that we are dealing with a generalized free (or quadratic) theory. Thisassumption was used whenever we assumed that

limx→0

[A,O(x)O(0)] = [A,O(0)]O(0) +O(0)[A,O(0)] (5.48)

an identity we used implicitly above. We also implicitly normal ordered the operators to subtractoff the identity contribution.

This state corresponds to 2 particles in AdS3 in an excited state with ` = 0 and with theircenter of mass degree of freedom in its ground state. However, this state has a non-trivial s-wavemomentum, so neither particle is expected to be in its ground state. Because the center of massdegree of freedom in AdS3 is at rest, this state is primary. Acting on this state with Pµ will generatedescendant 2-particle states.

We already established a one-to-one operator/state correspondence. So we see that not only states,but also operators can be classified as either primaries or descendants. Usually this classificationis stated in terms of the conformal transformation properties of operators, but the operator/statecorrespondence and the AdS picture give a more physical understanding.

Primary operators transform according to the rules similar to those we derived above for O(z, z);we will give the general story below. Descendants do not transform as simply because they take theform

Odesc ∼ [Pµ1 , [Pµ2 , [· · · , [Pµx ,Oprim(x)] · · · ] (5.49)

and so the conformal generators must be commuted through to evaluate their commutation relationswith descendant operators. In general CFTs primary operators are often defined as the operatorswith the ‘good’ transformation properties.

In a unitary quantum mechanical theory, all states must have positive normalization. However,since different states are connected by raising and lowering using P and K, the norm of one state

47

can be computed in terms of the norm of another, and this puts restrictions on the theory. Unitaritybounds follow from evaluating the matrix element (for a scalar state/operator)

〈∆|KαKαPµP

µ|∆〉 (5.50)

and demanding that it is positive definite, since it must be the norm of a physical state. Note thatthis follows because P †µ = Kµ; this relation must hold so that e.g.

(Pµ|ψ〉)† = 〈ψ|Kµ (5.51)

and raising and lowering work correctly. Let us derive the unitarity relation. First we consider

∆|KαPµ|∆〉 = ∆|P µKα + 2ηαµD + 2Mαµ|∆〉 (5.52)

= 2∆〈∆|∆〉 (5.53)

and from this we learn something – namely that ∆ ≥ 0 is required in order to have a positive (orzero) norm descendant. Note that in fact ∆ = 0 is allowed – this is the identity operator 1, whichclearly does not have any descandents at all, since [Pµ, 1] = ∂µ1 = 0. This is allowed because it is inthe trivial representation of the conformal group.

We will now see that if ∆ 6= 0 then we must have a stricter bound. Let us go on and evaluatethe norm of a 2nd level descendant. This is

〈∆|KαKαPµP

µ|∆〉 = 〈∆|KαPµKαP µ + 2Kα(ηµαD +Mµα)P µ|∆〉 (5.54)

= 〈∆|2KαP µ(ηµαD +Mµα) + 2K · P (∆ + 1 + (1− d))|∆〉 (5.55)

= 2(2∆ + 2− d)〈∆|K · P |∆〉 (5.56)

by assumption the matrix element in the last line must be positive, and we assume that 〈∆|K ·P |∆〉is positive since if it was not then the theory already would not have been unitary at the first level.This means that we must have

∆ ≥ d− 2

2(5.57)

as promised in order to have a unitary representation. For operators/states with spin ` a similaranalysis shows that

∆ ≥ d− 2 + ` (5.58)

for unitarity.

5.4 The Operator Product Expansion

Finally, we are ready to discuss one of the other key attributes of a conformal theory – the OperatorProduct Expansion (OPE) as applied to generalized free theories. The idea behind the OPE inAdS/CFT is pictured in figure 9.

48

AdS CFT

| 12iO1O2

|0i

| 12i O1O2

|0i

Figure 9: This figure shows how the OPE can be derived in the CFT, and how the derivation ismirrored in AdS. The final point left off the diagram is that the state ψ12 can be created by a localoperator acting at the origin in the CFT, corresponding to an operator in the infinite past in globalcoordinate AdS. This follows from the operator/state correspondence of radial quantization.

Let us examine the general operator quantity

O(z, z)O(w, w) =

∞∑

h,h=0

1

NOh,h

[(zz)−∆z−hz−hah,h + zhzha†

h,h

]

×

∞∑

h,h=0

1

NOh,h

[(ww)−∆w−hw−hah,h + whwha†

h,h

] (5.59)

We cannot say much about this quantity that is useful, in general. However, if it is the case thatbetween these two operators the CFT is in its vacuum state, then we can translate w → 0 forsimplicity and note that this operator product creates the linear combination of states

O(z, z)O(0)|0〉 =

∞∑

h,h=0

1

NOh,h

[(zz)−∆z−hz−hah,h + zhzha†

h,h

] a†0,0|0〉 (5.60)

49

That’s quite a bit simpler!The AdS picture is the following: assume that the theory is in its vacuum state in the infinite

past. Then if we insert two operators before some time t = log(zz) then we can expand the resultingstate in local operators inserted at the origin, corresponding to t = −∞ in AdS. Then all matrixelements with operators appearing at t > log(zz) can be computed using the terms in this OPE.

Now we can ask what linear combination of local operators, acting at the origin, creates thisstate? We already saw that every CFT state is associated with an operator, so can write the answerto this question in terms of a sum over primary and descendant operators. Since all descendants arejust ∂z or ∂z acting on a primary, we must have that

O(z, z)O(0) =∑

prim n,`

(Cn,`(z, z, ∂z, ∂z)On,`) (0) (5.61)

as an operator statement, for some coefficient function Cn`. The On,` are just the primary operatorsmade from two of the fundamental O operators. This is the OPE. The coefficients Cn,` can becomputed once and for all, as they are determined by conformal symmetry up to a constant (whichcan also be determined for generalized free theories).

Let us go back and look at the generalized free theory result more explicitly. We can write

O(z, z)O(0)|0〉 =

∞∑

h,h=0

1

NOh,h

[(zz)−∆z−hz−hah,h + zhzha†

h,h

] a†0,0|0〉 (5.62)

=

1

(zz)∆+

∞∑

h,h=0

1

NOh,h

zhzha†h,ha†0,0

|0〉

=

[1

(zz)∆+

1

NO0,0

O2(0) +z

NO0,1

O∂zO(0) +z

NO1,0

O∂zO(0) + · · ·]|0〉

so we can explicitly see various operators appearing. Note also the appearance of the identityoperator = 1, from the Wick contraction.

In fact, the OPE has a much more precise structure determined by conformal symmetry. Thepoint is that once we write down the OPE as a sum over all of the various operators that can appearon the RHS, including both primaries and descendants, we can then act with conformal symmetrygenerators on both sides to find relations among the coefficients. In fact, the only free coefficientsare those in front of the primary operators! We will return to explain this fact in detail when wediscuss general CFTs.

5.5 Correlators and Conformal Symmetry Invariants

Recall that in d Euclidean dimensions, a free boson has the 2-pt function

〈φ(x)φ(0)〉 =1

xd−2(5.63)

50

All of the other correlators are just obtained from this one by Wick contraction. Another way todiscover generalized free theories is to define (really just make up!) a theory that generalizes this ina simple and conformally invariant way. It will be a theory whose entire Hilbert space is generatedby the action of a local operator O(x) that has purely ‘Gaussian’ correlators – in other words, allcorrelation functions of O(x) will be determined by its 2-pt function via Wick contraction. To bespecific, the 2-pt function has the form

〈O(x)O(0)〉 =1

x2∆(5.64)

where ∆ ≥ d2− 1 is a number that we are free to specify. The 3-pt correlator of O vanishes, as do

all correlators with an odd number of Os. The 4-pt correlator is

〈O(x1)O(x2)O(x3)O(x4)〉 =1

x2∆12

1

x2∆34

+1

x2∆13

1

x2∆24

+1

x2∆14

1

x2∆23

(5.65)

Working out any n-pt correlator is just an exercise in combinatorics. These correlators show whythis is a ‘generalized free theory’ – in a free theory we would have ∆ = d

2− 1, but here we are letting

the exponent be a general parameter.So how do we see the conformal symmetry of these correlators? Obviously these correlators are

invariant under translations and rotations. To study the other transformations, we can use theconformal algebra directly, recalling that [D,O] = (∆ + r∂r)O so that we have

0 = 〈DO(P1)O(P2)〉 (5.66)

= 〈[D,O(P1)]O(P2)〉+ 〈O(P1)[D,O(P2)]〉 (5.67)

This means that the 2-pt correlator must obey

(r1 · ∂r1 + r2 · ∂r2) 〈O(~r1)O(~r2)〉 = −2∆〈O(~r1)O(~r2)〉 (5.68)

which is exactly what we find. We will study the constraints of conformal symmetry on correlatorsin more detail in the next section in order to fully constrain the correlators.

Conformally invariant correlators can only be a function of the conformal cross ratios

uijkl ≡(Pi · Pj)(Pk · Pl)(Pi · Pk)(Pj · Pl)

(5.69)

that are invariant under scalings of the individual Pi. In particular, when we study the 4-pt correlator,we only have two cross ratios, conventially called u and v

u =(P1 · P2)(P3 · P4)

(P1 · P3)(P2 · P4)and v =

(P1 · P4)(P2 · P3)

(P1 · P3)(P2 · P4)(5.70)

Note that in terms of these variables, our generalized free theory correlator can be written as

〈O(x1)O(x2)O(x3)O(x4)〉GFT =1

(P1 · P3)∆(P2 · P4)∆

(u−∆ + 1 + v−∆

)(5.71)

51

The factor out front just encodes the necessary kinematical data associated with the scaling dimensionof O, while the terms in parentheses contain all of the dynamical information about the theory. In ageneral theory this correlator would take the form

〈O(x1)O(x2)O(x3)O(x4)〉 =1

(P1 · P3)∆(P2 · P4)∆A(u, v) (5.72)

5.6 Connection to Infinite N

Many physicists would refer to a Generalized Free Theory as ‘a CFT at infinite N ’. There are awhole host of examples. For instance, we might have

• A large N gauge theory with gauge field Aµab, as well as matter in the adjoint such as ψaband Φab fields, or vector-like matter ψa, perhaps with flavor quantum numbers. The classicAdS/CFT example is the maximally supersymmetric N = 4 SYM Theory.

• A vector-like theory such as the O(N) model, with scalar fields φa and an O(N) symmetry.

• Any other theory with a large N Lie Group symmetry forcing the observables to take asymmetric form, such as φaφ

a or Tr[(F ab

µν)k].

Let us understand why large N relates to AdS/CFT. Consider a free theory of an SO(N) matrixvalued field φab where these indices a, b run from a = 1, 2, · · · , N . The action will simply be

S =

∫ddx

1

2

∑

a,b

∂µφab∂µφab

=

∫ddx

1

2Tr [∂µφ∂

µφ] (5.73)

Thus the φab fields have extremely simple correlation functions, namely

〈φab(x)φcd(y)〉 =δacδbd + δadδbc

(x− y)d−2(5.74)

or it can be written as

〈φA(x)φB(y)〉 =δAB

(x− y)d−2(5.75)

where A indexes the generators TAab of SO(N), since φ is in the adjoint representation. None of thesedetails matter much for our purposes here. All of the n-pt correlators of φ are just built on thesevia Wick contraction, since this is just a free theory with N2 free fields.

The interesting point is that now we can consider more general operators made out of φ. Forexample, consider

O2(x) =1

N√

2Tr[φ2(x)

]=

1

N√

2

∑

ab

φab(x)φba(x) (5.76)

52

What is the 2-pt correlator of O2? It’s easy to compute, it’s simply

〈O(x)O(y)〉 =1

(x− y)2(d−2)(5.77)

This is the form of a generalized free field with dimension ∆ = (d − 2). If we were to considerOk = Tr

[φk]

then we could get any ∆ = k(d− 2)/2.But now let’s consider a 4-pt correlator

〈O2(x1)O2(x2)O2(x3)O2(x4)〉 = =1

x2∆12

1

x2∆34

+1

x2∆13

1

x2∆24

+1

x2∆14

1

x2∆23

+1

N(· · · ) (5.78)

In the limit that N → ∞, the 4-pt correlator of O2, and in fact of all properly normalized Ok,will be given entirely by evaluating pairs of 2-pt functions. Thus in the large N limit, this theoryreproduces what we wanted – a generalized free theory with many choices for ∆, the dimensionof the operator. It’s that last point that differentiates perturbation theory in 1/N from the morefamiliar examples of perturbation theory in a small coupling. For further reading, Witten’s ‘Baryonsand the Large N Expansion’ [11] is a great intro to the 1/N expansion.

An interesting point to note here is that the quantum mechanical expansion of φab is in termsof free field oscillator modes, which are not much like the AdS oscillator modes of a generalizedfree field. So the relationship between an operator like tr[φk] and O isn’t straightforward quantummechanically.

6 General CFT Axioms from an AdS Viewpoint

There are a bunch of concepts about CFTs that we will discuss/review. We will do it in a somewhatunusual way, because we are being led by AdS and holography. Here is a list of the main ideas:

• Conformal transformations consist of the Poincare group of transformations, plus scaletransformations and special conformal transformations. The extra symmetries can be mostquickly derived by demanding Poincare invariance plus an additional symmetry under inversionsabout a point, where xi → xi/x2. They are best understood as coordinate transformationsthat do not leave the metric invariant, but rescale it by an overall spacetime-dependent factor.

• CFTs can be quantized in the standard way, along flat spacelike surfaces that evolve with time.However, an alternative radial quantization in Euclidean space is often more useful, and willplay a large role in what follows; in this quantization we start at a point and evolve outwardson expanding spheres. This idea is pictured in figure 7.

• Radial quantization associates the entire Hilbert space of CFT states with an arbitrarily smallregion about a point. Associating states on arbitrarily small circles with operators at the centerof these circles leads to the operator-state correspondence. This can be made very explicitwhen there is a path integral description of the CFT, but the existence of a path integral isnot necessary.

53

• Using conformal symmetry we can classify states according to irreducible representations ofthe conformal group, SO(1, d+ 1) in Euclidean space, and therefore describe every state as alinear combination of primaries and their descendants; the latter are obtained from primariesby acting with conformal generators.

• The conformal symmetry allows us to move these operators around, so that from O(0) at theorigin we obtain O(x) at any point x. The correlators of any 2 or 3 CFT operators are entirelyfixed by symmetry, up to a finite set of constants.

• As usual in a quantum mechanical theory, we can multiply any two operators to obtain someother (new) operator. So what about operators at different points in space? The operator-statecorrespondence applied to a product O1(x1)O2(x2) leads us to the Operator Product Expansion(OPE), which has a finite radius of convergence in any CFT. This can be made very explicitwhen there is a path integral description of the theory (see e.g. Weinberg Volume 2), but theexistence of a path integral is not necessary.

• Conserved currents are special and extremely important. Spin-1 currents Jµ satisfying ∂µJµ = 0generate global symmetries, while the spin-2 conserved current Tµν is even more special (ifthese exist). Conventionally a theory is only defined to be a CFT if a conserved Tµν exists; ifthere is no conserved spin 2 current the lore holds that the theory is ‘non-local’.17

6.1 What is a C(F)T?

Although later on we will take a more abstract point of view, let’s review the standard approach toCFTs. Conventionally, conformal isometries are defined as a change of coordinates or diffeomorphismxµ → x′µ(x) such that the metric

dx2 → dx′2 = Ω2(x)dx2 (6.1)

where Ω(x) is an arbitrary function of the coordinates. This choice is interesting and special becauseit preserves angles, but not distances. This means that in the Lorentzian case, it always preservesthe causal structure of the spacetime.

Thinking infinitesimally, we have x′µ = xµ + vµ(x), where we will treat vµ as small. The newmetric is

dx′αdx′α = dxµdxν (ηµν + ∂µvν + ∂νvµ + · · · ) (6.2)

and so we must have ∂µvν + ∂νvµ = ω(x)ηµν , where Ω = 1 + ω/2. This immediately gives theequation

∂µvν + ∂νvµ −2

d∂αv

αηµν = 0 (6.3)

17in quotes because this phrase is vague... different physicists use it to mean very different things... as with ‘entropy’

54

A well-known fact is that in 2-dimensions there are infinitely many solutions, whereas in d > 2dimensions there are only a finite number of solutions, the transformations that form the conformalgroup SO(2, d).

Note that there is a significant difference between a Weyl transformation gµν(x)→ W (x)gµν(x),where we are changing the physical metric, and a conformal isometry, which is a change of coordinates(diffeomorphism) xµ → x′µ(x) such that in the new coordinates the metric takes the form of the oldmetric multiplied by some function Ω2(x).

To make this all clearer, let’s consider why a free massless scalar field theory is invariant under aconformal isometry. We have φ(x) → φ(x + v) ≈ φ(x) + vµ(x)∂µφ(x). Then the free Lagrangiantransforms as

δL = ∂µφ(x)∂µ(vα(x)∂αφ(x))

= ∂µvα

(∂µφ∂αφ− 1

2ηµα(∂φ)2

)= ∂µvαT

µα (6.4)

When we take vµ = λxµ, corresponding to a dilatation, or the more complicated vµ = bµx2− 2xµb · x

for a special conformal transformation, we find that δL ∝ T µµ , the trace of the energy momentumtensor, as expected.

You might notice that Tµν for the scalar is only traceless in d = 2 spacetime dimensions; thisis because we have automatically (and in general incorrectly) assigned it a scaling dimension of0 under dilatations. This can be fixed by adding a local improvement term to Tµν , namely somemultiple of the conserved total derivative (∂µ∂ν − ηµν∂2)φ2. The point is that the scale and specialconformal transformations can involve a combination of Tµν and other local conserved currents; see[12] for more details. For example, for the case of dilatations as long as we can define

Sµ = xνTµν − Vµ (6.5)

with T µµ = ∂µVµ then the current Sµ is a scale current, which generates dilatations as we desire. Inthe case of a free theory we have Vµ ∝ ∂µφ

2.This can be understood more directly. When we computed Tµν above we assumed that φ(x)→

φ(x+ v) ≈ φ(x) + vµ(x)∂µφ(x), but this is not correct for e.g. dilatations for any choice of vµ. Weknow that φ will be a primary operator of dimension ∆ = d

2− 1, and so it should transform as

δφ(x)→ (∆ + x · ∂x)φ(x). But taking vµ = εxµ does not lead to this transformation rule. To obtaina scale current we therefore want to take δφ→ ε(x)(∆ + x · ∂x)φ. This leads to a scale current

Sµ = xν(∂µφ∂αφ− 1

2ηµα(∂φ)2

)−∆φ∂µφ (6.6)

which accords with the form we found above by the inclusion of an explicit improvement terminto Tµν . The effect of the improvement term is simply to shift the scaling dimension of φ withoutaltering the conservation of the stress-energy tensor.

It’s worth noting that in the theories we are used to studying, e.g. λφ4 theory, gauge theories,etc., the fact that conformal symmetry is broken quantum mechanically means that T µµ ∝ β(λ),

55

where β is the usual beta function for the running coupling. This means that these classicallyconformal theories have an anomaly.

We will take a rather different and more abstract point of view that follows more naturally fromthinking about AdS. Conformal Theories (CTs) are defined as quantum mechanical theories where

• All states fall into representations of the d-dimensional conformal algebra:

[Mµν , Pρ] = i(ηµρPν − ηνρPµ), [Mµν , Kρ] = i(ηµρKν − ηνρKµ), [Mµν , D] = 0

[Pµ, Kν ] = −2(ηµνD + iMµν), [D,Pµ] = Pµ, [D,Kµ] = −Kµ (6.7)

• There are operators Oi(x) called primaries that transform covariantly. We already saw whatthis means for Generalized Free Theories in the previous section. As we will see again in thenext section, there is a one-to-one correspondence between all the operators at any point xand all of the states in the Hilbert space.

• We can multiply operators O1(x)O2(y). If the CFT is in its vacuum state between theseoperators, then the product can be re-written as a convergent sum over all the operators in thetheory at any point between x and y. This is called the Operator Product Expansion (OPE),to be derived in general below. The coefficients of all descendant are determined by conformalsymmetry and the coefficients of the primaries.

From these properties many others follow, for example that the correlators must take a veryconstrained form, and for the theory to be unitary all operators have to have scaling dimensions(eigenvalues of the dilatation operator D) greater than the unitarity bound.

For a theory to be a Conformal Field Theory (CFT):

• The theory must contain in the spectrum of operators a spin 2 tensor Tµν with dimension∆ = d, the spacetime dimension, or equivalently, the stress tensor must be conserved. Thiscertainly was not present in the Generalized Free Theory we studied in the last section, exceptwhen ∆ = d

2− 1 and we have an ordinary free theory.

We will discuss below why Tµν is so uniquely important, and how its special properties relate to theuniversality of gravity. For now it should at least be apparent from our discussion above that Tµν isrelated to implementing the conformal symmetries in a way that is manifestly local in spacetime(‘local’ because the symmetries are expressed in terms of Tµν(x), which is located at the point x).

6.2 Radial Quantization and Local Operators – The General Story

Let us start with some general comments on the Hilbert space construction in QFT18. This procedureis linked to the choice of foliation of spacetime with fixed time surfaces. Time evolution connectsstates on one surface to states on the other surfaces.

Each leaf of the foliation becomes endowed with its own Hilbert space. We create in states ψin byinserting operators or directly specifying a wavefunction in the past of a given surface. Analogously

18One can find a very similar discussion in Slava’s notes.

56

we deal with out states by inserting operators or directly specifying the wavefunction in the future.The overlap of an in and out states living on the same surface 〈ψout|ψin〉 is equal to the correlationfunction of operators which create these in and out states (or to the S-matrix element if the in andout states consist of well-separated particles).

In standard textbook discussions of QFT, states are defined on flat spacelike surfaces in Minkowskispacetime. The time evolution operator is ‘P0’, the momentum generator in the time direction. Sincethe momenta all commute, states can be classified according to their total momentum and energy.To summarize

• States are defined on flat spacelike surfaces in Minkowski spacetime

• Since the momenta all commute

[Pµ, Pν ] = 0 (6.8)

states can be classified according to their total momentum and energy.

• The time evolution operator is ‘P0’, the momentum generator in the time direction. ViaLorentz transformations we can consider other compatible (boosted) notions of P0.

In CFTs we additionally have the symmetries D = dilatations and Kµ = special conformaltransformations. Could still use standard picture, but more natural to think about states in a newway – Radial Quantization. CFT algebra:

[Mµν , Pρ] = i(ηµρPν − ηνρPµ), [Mµν , Kρ] = i(ηµρKν − ηνρKµ), [Mµν , D] = 0

[Pµ, Kν ] = −2(ηµνD + iMµν), [D,Pµ] = Pµ, [D,Kµ] = −Kµ (6.9)

Some implications

• Simultaneously diagonalize D and Mµν = angular momentum. Label states by ∆ or τ = ∆− `and angular momentum `, so have

D|τ, `J〉 = (τ + `)|τ, `J〉 (6.10)

• Pµ is raising, Kµ lowering operator wrt D.

• Define primary as state killed by Kµ. Get discrete spectrum.

• Radial quantization with Hamiltonian is D, unitarty evolution via eiDτ with τ = log r.

The vacuum |0〉 is annihilated by all the conformal generators, since we assume conformal symmetryis unbroken. Associated to each primary state is a primary operator O(0) with twist τ and angularmomentum `, so that acting on the vacuum

O(0)|0〉 = |τ, `〉 (6.11)

57

This is the local operator/state correspondence (isomorphism or identification).Note that radial quantization and the operator/state correspondence can be treated in a natural

way in the path integral formalism. In that case we define states by wave functionals on Cauchysurfaces, and the path integral allows us to evolve from one Cauchy surface to another. Radialquantization is then immediate. The operator/state correspondence follows if we evolve back intime towards a point. Then the wave functional at the origin is equivalent to some insertion of afunctional of the fields directly into the path integral, and this insertion is itself the desired operator.

Let us now see how the conformal transformation properties of O(0) completely determine thetransformation properties of O(x), and therefore define the operator everywhere. This is interestingbecause it means that once we know about the primary state O(0)|0〉 we can derive everything wewould like to know about the corresponding local operator at any point in spacetime. The conformalgenerators act on O(0) according to

[Kµ,O(0)] = 0 (6.12)

[D,O(0)] = −i∆O(0) (6.13)

[Mµν ,O(0)] = −iΣ`µνO(0) (6.14)

Along with the assumption that

[Pµ,O(x)] = −i∂µO(x) (6.15)

this entirely determines the action of the conformal algebra on the local operator O(x), assumingthe commutation relations of the conformal group. At a general point in flat Euclidean space, wehave the commutators

[D,O(x)] = −i(∆ + xµ∂µ)O(x) (6.16)

[Pµ,O(x)] = −i∂µO(x) (6.17)

[Kµ,O(x)] = −i(2xµ∆ + 2xαΣαµ + 2xµxα∂α − x2∂µ)O(x) (6.18)

[Mµν ,O(x)] = −i(Σµν + xµ∂ν − xν∂µ)O(x) (6.19)

where ∆ is the scaling dimension of O and Σµν is a finite dimensional spin matrix encoding theangular momentum representation of O.

For fun let us derive the first infinitesimal approximation of the first equation above. We knowfrom equation (6.15) that we can write

O(ε) ≈ O(0) + εµ∂µO(0) + · · ·= O(0) + iεµ[Pµ,O(0)] + · · · (6.20)

Now if we act the dilation operator on O(ε) we find

[D,O(ε)] ≈ [D,O(0)] + iεµ[D, [Pµ,O(0)]]

= −i∆O(0) + iεµ ([[D,Pµ],O(0)]− [Pµ, [O, D]])

= −i∆O(0) + iεµ ([−iPµ,O(0)]− i∆[Pµ,O(0)])

= −i∆O(0) + iεµ (−∂µO(0)−∆∂µO(0))

≈ −i(∆ + εµ∂µ)O(ε) (6.21)

58

so we have derived the first of the equations (6.16) when x = ε is near to 0.The AdS dual of radial quantization is pictured in figure 7 – it’s just ordinary quantization in

AdS global coordinates. Specifically, ‘ordinary quantization’ in global coordinates means that weuse the global time t to define our clocks, and so the dilatation operator D is the Hamiltonian (thegenerator of time translations). The Cauchy surfaces in AdS on which states are defined are just theconstant t surfaces, parameterized by the coordinates ρ and Ω.

6.3 General CFT Correlators and the Projective Null Cone

We already saw some non-trivial examples of the way that conformal symmetry acts on andconstrains CFT correlators. In general, all two-point correlators are completely determined up to anormalization, while 3-pt correlators are determined up to a finite number of coefficients (in thecase of three scalar operators, there is only one overall coefficient). Higher point correlators can onlydepend on the conformal cross ratios.

We can study the conformal transformation properties in an elegant fashion by using the projectivenull cone coordinates PA, which satisfy PAP

A = 0 and are identified under PA ∼ λPA. Since we havebeen thinking about CFTs from the AdS point of view, it is natural to understand CFT correlatorsas expressed in terms of the PA with reference to AdS.

Recall that we defined a CFT operator O in terms of an AdS field φ(XA) as

O(t,Ω) = limε→0

φ(t, ρ(ε; t,Ω),Ω)

ε∆(6.22)

where for e.g. scalars m2 = ∆(∆− d) relates the AdS mass and CFT dimension, and ρ(ε; t,Ω) =π2− εf(t,Ω) controls the way we approach the boundary of AdS. Furthermore, using our coordinate

identifications we can write

X0 = Rcosh t

cos ρ→ cosh t

εf(t,Ω)=

1

εP0 (6.23)

Xd+1 = Rsinh t

cos ρ→ sinh t

εf(t,Ω)=

1

εPd+1

Xi = R tan ρΩi →Ωi

εf(t,Ω)=

1

εPi

The choice of function f(t,Ω) here represents a choice of a section of the null cone. Note that withour favorite f = e−t we obtain

(P0, Pd+1, Pi) =

(e2t + 1

2,e2t − 1

2, etΩi

)(6.24)

as usual for Euclidean flat space.Now in general, instead of taking ρ = π

2− εf(t,Ω) we could have equivalently taken ρ(ε; t,Ω)→

π2− 1

λεf(t,Ω) for a constant λ > 0. This has precisely the effect of sending PA → λPA in our

definition. But we also know from the definition of O that this would transform

O(λPA) = λ−∆O(PA) (6.25)

59

for a conformal primary operator obtained from an AdS theory. Note that we can use a different λfor each operator in a correlator – we do not have to rescale them all in the same way.

Furthermore, one can reverse the logic, and show that this transformation rule must obtain forany CFT primary expressed in terms of the null projective cone, regardless of whether it comes fromAdS. It accords with all the results we have derived before by other means.

This puts powerful constraints on CFT correlation functions when we combine it with the factthat only Pi · Pj are conformally covariant objects. In particular, it enforces that

〈O1(P1)O2(P2) = 0 if ∆1 6= ∆2 (6.26)

and that this correlator takes the conventional form otherwise. Furthermore, in the case of 3-ptcorrelators of scalars it implies

〈O1(P1)O2(P2)O3(P3)〉 =C123

(P1 · P2)α12,3(P2 · P3)α23,1(P1 · P3)α13,2(6.27)

where we must have

αij,k =∆i + ∆j −∆k

2(6.28)

to achieve the correct scalings under Pi → λiPi for each i = 1, 2, 3. Finally, in the case of n-ptcorrelators this rescaling property enforces the fact that aside from an overall factor, we can onlyhave dependence on the conformal cross ratios uijkl. In particular we have

〈O(x1)O(x2)O(x3)O(x4)〉 =1

(P1 · P2)∆(P3 · P4)∆A(u, v) (6.29)

with

u =(P1 · P2)(P3 · P4)

(P1 · P3)(P2 · P4)and v =

(P1 · P4)(P2 · P3)

(P1 · P3)(P2 · P4)(6.30)

Note that if these operators are identical (as displayed), then the correlator will be symmetric undertheir interchange. This means that exchanging 2↔ 3 sends

u→ 1

uand v → v

u(6.31)

so we have that

A(u, v) =1

u∆A(

1

u,v

u

)(6.32)

where the prefactor comes accounting for the explicit prefactor in the correlator. Similarly, if wetake 2↔ 4 then we find

u→ v and v → u (6.33)

so that we have

A(u, v) = A (v, u) (6.34)

Either of these two simple identities defines the bootstrap equation once we express the correlator interms of conformal partial waves.

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6.4 Deriving the OPE from the CFT, and from an AdS QFT

A key property of all local quantum field theories is the existence of an Operator Product Expansion.In CFTs this expansion converges in a finite region, and so it can be used to make various exactstatements. Let us first understand19 what the OPE is and what it has to do with locality. Then wewill show that any field theory in global AdS gives rise to CFT correlators that satisfy an OPE. Theidea of the derivation is pictured in figure 9.

The OPE says that for any local operators φ1 and φ2 we have

φ1(x)φ2(0) =∑

O

C(x, ∂)O(0) (6.35)

This relation is derived via path integral methods in Weinberg V2 [13], although it can be easilyderived without any explicit action for the CFT. Let us review it, and then we will show that anylocal QFT in AdS will give rise to CFT correlators that satisfy an OPE. As we will see, the derivationof the OPE in AdS is very similar to the derivation in the CFT.

To derive the OPE in the CFT, we consider two operators O1(x1) and O2(x2). Now draw a circlearound x1 and x2, centered at some point x – the circle should have a radius r larger than |x− x1|and |x− x2| so that it contains both points, but does not contain an insertion of any other operators.Choose to radially quantize the CFT about x. If we imagine the radial evolution from x outwards,we start with the vacuum and then we eventually hit the operators O1 at x1 and O2(x2). Thus onthe circle of radius r, we have some state

|ψ12(r)〉 = O1(x1)O2(x2)|0〉 (6.36)

because the CFT was in its vacuum at x. This state will be some linear combination of all of thestates in the Hilbert space. By the operator state correspondence, there exists some local operatorOψ12(x) such that

Oψ12(y)|0〉 = |ψ12(r)〉 = O1(x1)O2(x2)|0〉 (6.37)

The operator Oψ12 will certainly not have a definite dimension or spin – it’s an extremely abstractobject. However, we can express Oψ12 as a sum over all of the primary operators of the theory andtheir descendants, and categorized according to their dimension and spin. This gives the OPE

O1(x1)O2(x2) =∑

∆,`

λ∆,`C∆,`(x1 − y, x2 − y, ∂y)O∆,`(y) (6.38)

Note that we are free to use translation symmetry to take x1 = y = 0 for convenience, in whichcase the function on the RHS just becomes f∆,`(x1, ∂). In fact, this function is entirely determinedby conformal symmetry, so the only non-trivial information in the OPE is the value of the OPEcoefficients c∆,`, which are labeled entirely by the dimension and spin of the primary operators inthe theory. However, there are still an infinite number of these OPE coefficients. If this is your first

19In Weinberg V.2 the OPE is derived from the path integral, and in Slava’s notes its derived from radial quantization.

61

sight of the OPE, then it probably seems very abstract, and in fact there’s much more to say aboutand do with it, but before any of that let’s discuss how it’s derived from QFT in AdS.

In fact, the AdS derivation mostly copies from the CFT version, as pictured in figure 9. The keypoint is that the Hilbert space of the theory lives on complete spacelike Cauchy surfaces in AdS. Ast→ −∞ in global AdS coordinates, the theory is in the vacuum. If we insert some operators O1(x1)and O2(x2) at some finite times, then we can always study a time t in the future of x1 and x2. Onthis time slice we will find a wavefunction ψ12(t), where the radius of the corresponding CFT circleis just r = et in some units. But this state ψ12(t) could just as well have been created by setting itup at some point arbitrarily far into the past. In the limit that we setup ψ12 in the infinite past, wecan just create it by acting with a local operator at the origin in the CFT. This local operator canbe written as a sum over primary operators of definite dimension (AdS energy) and spin, and thisexpansion reproducies the OPE for the CFT.

Now let us look at the OPE in a more refined way, in order to relate the OPE coefficients ofdescendants to the coefficients of their associated primary operator. For example, let us considersome particular term associated with a primary operator O

O1(x)O2(0)|0〉 =c

xp[O∆,`(0) + · · · ] |0〉+ other primaries (6.39)

One might immediately guess that the power p is fixed the dimensions ∆1, ∆2, and ∆. We can verifythis by acting on both sides with the dilatation operator (and ignoring other contributions), giving

DO1(x)O2(0)|0〉 = (∆1 + x · ∂x)O1(x)O2(0)|0〉+ ∆2O1(x)O2(0)|0〉= (∆1 + ∆2 − p)

c

xp[O∆,`(0) + · · · ] |0〉 (6.40)

However, we can also compute this directly as an action on O, giving

Dc

xp[O∆,`(0) + · · · ] |0〉 =

c

xp[∆O∆,`(0) + · · · ] |0〉 (6.41)

so we find that p = ∆1 + ∆2 −∆ for consistency. More interesting relations can be obtained byconsidering not just the primary O∆,` but also the descendants that make up the ellipsis. The nextterm will be

O1(x)O2(0) =c

xp[O∆,`(0) + κxµ∂µO∆,`(0) + · · · ] |0〉 (6.42)

and we can determine the coefficient c using conformal symmetry. Let us act on this equation withKµ. Note that this SCT annhilates O2(0) because its primary, so we find

KµO1(x)O2(0)|0〉 = (2xµ∆1 + 2xαΣαµ + 2xµxα∂α − x2∂µ)O1(x)O2(0)|0〉 (6.43)

= (2xµ∆1 + 2xαΣαµ + 2xµxα∂α − x2∂µ)

c

xp[O∆,`(0) + κxµ∂µO∆,`(0) + · · · ] |0〉

This must match with the action of Kµ directly on the RHS, but this is

Kµc

xp(O∆,`(0) + κxα∂αO∆,`(0) + · · · ) |0〉 =

c

xp(κxα[Kµ, [Pα,O∆,`]](0) + · · · ) |0〉 (6.44)

62

where we note that Kµ annihilates the primary operator O(0), and we have written the last partvery explicitly in terms of the conformal generators to emphasize that it can be evaluated just usingthe algebra.

To match the two sides, note that in equation (6.43) the first two terms in Kµ actually cancelagainst the action of the derivative operators on x−p, so that all terms proportional to O∆,`(0) dropout. Then for ` = 0 we are left with the first term as

2κ∆xµO∆,0 = (2∆1 − p)xµO∆,0 (6.45)

so we find that

κ =1

2+

∆1 −∆2

2∆(6.46)

for the first coefficient. In principle we can compute all subsequent coefficients this way, although itwould be rather laborious.

Finally, now that we have demonstrated that the appearance of descendant operators in theOPE is entirely determined by the appearance of primaries, let us consider how to compute theprimary OPE coefficients. Actually, this is easy, and also provides a way to compute C(x, ∂). Themethod is to take the OPE expansion and compute its correlator with a single operator O∆,`, giving

〈O∆,`(z)O1(x)O2(y)〉 =

⟨O∆,`(z)

∑

primary O

λ12∆,`C(x− y, ∂y)O(y)

⟩(6.47)

where in our current notation λ12∆,` is the OPE coefficient. Now note that CFT 2-pt functions

always vanish unless the operators have the same dimension, charge, and spin, so on the RHSwe have projected out the 2-pt function of a single primary operator O∆,`. On the LHS we justhave a CFT 3-pt function, and these are fixed up to a finite number of constants (in the case of ascalar-scalar-spin ` correlator there is a unique answer kinematically, so we only have one overallconstant). Considering the scalar case for simplicity, we find that

c12∆,`=0

1

(x− y)α12∆(z − y)α2∆1(x− z)α1∆2= λ12

∆,0C(x− y, ∂y)1

(z − y)2∆(6.48)

So we see that there is a very simple relationship between the coefficient c12∆,0 of the 3-pt function

and the OPE coefficient λ12∆,0. Furthermore, we can also determine the function C(x − y, ∂y) by

matching the kinematics on both sides of this equation.

6.5 The Conformal Partial Wave Expansion and the Bootstrap

We derived the OPE in the case of generalized free theory, and then we explained that in fact theOPE exists with a finite radius of convergence in any CFT. Crucially, we found that all coefficientsfor descendant operators in the OPE are determined kinematically in terms of the coefficients ofprimary operators. We also saw how one can derive the OPE directly by thinking about the AdSfield theory description.

63

Let us now see what happens when we apply the OPE to a 4-pt CFT correlator. This isstraightforward; we see that

〈φ(x1)φ(x2)φ(x3)φ(x4)〉 =∑

τ,`

λτ,`Cτ,`(x12, ∂1)〈Oτ,`(x2)φ(x3)φ(x4)〉

=∑

τ,`

λτ,`Cτ,`(x12, ∂2)∑

τ ′,`′

λτ ′,`′Cτ ′,`′(x34, ∂4)〈Oτ,`(x2)Oτ ′,`′(x4)〉

=∑

τ,`

λ2τ,` [Cτ,`(x12, ∂2)Cτ ′,`′(x34, ∂4) 〈Oτ,`(x2)Oτ,`(x4)〉]

=1

(x1 − x3)2∆(x2 − x4)2∆

∑

τ,`

λ2τ,`gτ,`(u, v) (6.49)

In the first line we applied the OPE in 12, while in the second we applied it in 34. In the third linewe simplified by noting that

〈Oτ,`(x2)Oτ ′,`′(x4)〉 ∝ δττ ′δ``′ (6.50)

so this term is only present when τ = τ ′ and ` = `′. In the last line we noted that

1

(x1 − x3)2∆(x2 − x4)2∆gτ,`(u, v) ≡ (Cτ,`(x12, ∂2)Cτ ′,`′(x34, ∂4)〈Oτ,`(x2)Oτ,`(x4)〉) (6.51)

is entirely determined by conformal symmetry – in other words, this conformal partial wave is apurely kinematic quantity. The conformal block coeffs are squares of OPE coeffs:

λ2τ,` > 0 (6.52)

This is how Unitarity shows up.Another equivalent way to think about it is via the insertion of complete set of states in the

middle. Organize according to irreps of conformal symmetry. This is the same thing as using theOPE, as is obvious from the radial quantization derivation. Contribution of a single irrep is aconformal partial wave or conformal block :

gτ,`(u, v)

(x213x

224)∆φ

= 〈φ(x1)φ(x2)

( ∑

τ,` irrep

|α〉〈α|)φ(x3)φ(x4)〉 (6.53)

This is an even easier way of explaining these conformal partial waves – it is just what we get whenwe insert 1 into the correlator as a sum over a particular set of states.

The conformal partial wave expansion is in all respects analgous to the partial wave expansion ofscattering amplitudes. The analogy connects angular momentum to angular momentum, while thetwist τ or dimension ∆ = τ + ` plays the role of the center of mass energy.

The conformal blocks are known (complicated) functions in many, but not all cases. For example,for CFTs in d = 2 dimensions the conformal partial waves are

gτ,`(z, z) = kτ+2`(z)kτ (z) + kτ+2`(z)kτ (z) (6.54)

64

structure constants of the schematic formX

k

f12kf34k(. . .) =X

k

f14kf23k(. . .) . (3.3)

The (. . .) factors are functions of coordinates xi, called conformal partial waves. They areproduced by acting on the two-point function of the exchanged primary field k with thedi↵erential operators C appearing in the OPE of two external primaries. Thus, they are alsofixed by conformal invariance in terms of the dimensions and spins of the involved fields.

f12k f34k

f14k

f23k

k

1

2 3

4

X

k

= k

1

2 3

4

X

k

Figure 1: The conformal bootstrap condition = associativity of the operator algebra.

The dream of the conformal bootstrap is that the condition (3.3), when imposed on four-point functions of suciently many (all?) primary fields, should allow one to determine theCFT data and thus solve the CFT. Of course, there are presumably many di↵erent CFTs,and so one can expect some (discrete?) set of solutions. One of the criteria which will helpus to select the solution representing the 3D Ising model is the global symmetry group,which must be Z2.

Our method of dealing with the conformal bootstrap will require explicit knowledge ofthe conformal partial waves. In the next section we will gather the needed results.

4 Conformal Blocks

In this paper we will be imposing the bootstrap condition only on four-point functions ofscalars. Conformal partial waves for such correlators were introduced in [7] and furtherstudied in [9, 10]; they were also discussed in [12]. Recently, new deep results about themwere obtained in [13–15]. Significant progress in understanding non-scalar conformal partialwaves was made recently in [43] (building on [44]), which also contains a concise introductionto the concept. Below we’ll normalize the scalar conformal partial waves as in [15]; seeAppendix A for further details on our conventions.

Consider a correlation function of four scalar primaries i of dimension i, which is fixedby conformal invariance to have the form [3]

h1(x1)2(x2)3(x3)4(x4)i =

x2

24

x214

1212

x214

x213

1234 g(u, v)

(x212)

12(1+2)(x2

34)12(3+4)

, (4.1)

6

Figure 10: This figure illustrates the bootstrap equation relating the conformal partial waveexpansion in two different channels. We see the explicit appearance of varoius OPE coefficients inthe different channels.

where u = zz and v = (1− z)(1− z) and

k2β(x) = xβ2F1(β, β, 2β;x) (6.55)

where this is the hypergeometric function. The reason for the notation of z and z is that if we takethe correlator

〈φ(0)φ(z)φ(1)φ(∞)〉 (6.56)

then we see that

u =x2

12x234

x213x

224

=(zz)∞2

12∞2→ zz (6.57)

and we obtain a similarly simple result with v. So z and z are literally complex coordinates in a twodimensional plane.

Recall that in the previous section we noted various identities between CFT correlators when weexchanged 2↔ 3 and 2 ↔ 4. The bootstrap equation uses this crossing symmetry to equate theconformal partial wave expansions in different channels:

u−∆φ + u−∆φ

∑

τ,`

Pτ,` gτ,`(u, v) = v−∆φ + v−∆φ

∑

τ,`

Pτ,` gτ,`(v, u) (6.58)

This is pictured in figure 10. Let us now see how the conformal partial wave decomposition works inour favorite example, that of a Generalized Free Theory. In the next section we will study whathappens when we add AdS interactions.

The GFT the 4-pt correlator can be written as

〈O(x1)O(x2)O(x3)O(x4)〉 =1

(x213x

224)∆

(1

(zz)∆+ 1 +

1

(1− z)∆(1− z)∆

)(6.59)

65

We want to equate this with the conformal block decomposition, so in other words we are looking for

1

(zz)∆

∑

τ,`

Pτ,` gτ,`(z, z) = 1 +1

(1− z)∆(1− z)∆(6.60)

where we have already canceled off the contribution of the identity operator on both sides. Now letus expand the RHS at small z and z; we find

1 +1

(1− z)∆(1− z)∆≈ 2 + ∆(z + z) +

∆(∆ + 1)

2(z2 + z2) + ∆2(zz) + · · · (6.61)

Notice that all of the powers that appear are integral. However, note that the terms on the LHSwith the conformal partial waves behave as

1

(zz)∆gτ,`(z, z) = (zz)

τ2−∆(z` + z` + · · ·

)(6.62)

So matching both sides around z, z ∼ 0 immediately implies that we must always have τ = 2∆ + 2nfor a generalized free theory. We have just re-discovered the fact that the only operators that canappear in the OPE of O(x)O(0) are

O(x)O(0) ∼ 1 + aO2(0) + bO∂2O + · · · (6.63)

so that they involve O∂2n∂µ1 · · · ∂µ`O and have dimension 2∆ + m. By matching both sides ofequation (6.62) we can compute the conformal partial wave coefficients (and therefore the OPEcoefficients) for a generalized free theory. When we add AdS interactions in the next section wewill see that the various operators get anomalous (shifted) dimensions proportional to the couplingconstants. Note that shifting τ produces (zz)γ ≈ 1 + γ log(zz), so anomalous dimension appear aslogarithms in CFT correlators.

7 Adding AdS Interactions

Now let us see how an interacting QFT in AdS spacetime give rise to an interacting C(F)T. Then wewill see how abstract CFT data like OPE coefficients and anomalous dimensions can be computedin AdS/CFT, and then we will translate these ideas into the much more mundane-sounding physicsof wavefunction overlaps and the binding energies between particles.

7.1 Anomalous Dimensions from Basic Quantum Mechanics

In the following sections we will discuss AdS Feynman diagrams and perform some computationsin λφ4 theory. For now we will discuss a simpler, old-fashioned perturbation theory approach [10]that basically amounts to an application of undergraduate quantum mechanics. This approach isactually easier and more efficient than Feynman diagrams in many cases.

66

Consider a situation where we have a free Hamiltonian D(0) that we have diagonalized, and weadd a small interaction Hamiltonian DI . Everyone knows that to first order in perturbation theory,the energy of a D(0) eigenstate |ψ〉 simply changes by

γψ =⟨ψ|DI |ψ

⟩(7.1)

We can apply this formula directly in the case of λφ4 theory in AdS in order to compute the energyshifts of 2-particle states in AdS. Recall that 2-particle states in AdS are created by the CFToperators [OO]n` where O = limρ→π/2 φ as usual. So computing the energy shifts of these states isequivalent to computing the ‘anomalous dimensions’ of these operators.

Let us consider λφ4 theory in AdS3 for simplicity. The (Lorentzian) action is

S[φ] =

∫

AdS

d3x√−g

(1

2(∇φ)2 − m2

2φ2 − λ

4!φ4

)(7.2)

We derive the Hamiltonian as usual, by defining canonical momenta and computing D = Pq − L.This gives the Hamiltian (Dilatation operator) for the free theory plus an interaction part, which is

DI(t) =λ

4!

∫

AdS

d2x√−gφ4(t, x) (7.3)

This interaction vanishes except for the scalar or ` = 0 partial wave, so [OO]n` do not receiveanomalous dimensions for ` > 0. This is directly analogous to the fact that λφ4 interaction onlycontributes to the ` = 0 partial wave when we study scattering in flat spacetime. Specializing to thecase of ` = 0 we want to compute the matrix element

γ(n) = 〈n, 0|DI |n, 0〉

=λ

4!

∫

AdS

d2x√−g 〈n, 0|φ4(x)|n, 0〉 (7.4)

where

|n, `〉 = [OO]n`|0〉 (7.5)

is the primary state corresponding to two particles in AdS without any total angular momentum.Now we can evaluate this matrix element directly. In order for it to be non-vanishing, two of theφ(x) operators must destroy the initial state particles while two destroy the final state particles, sowe find

γ(n) =λ

4

∫ 2π

0

dθ

∫ π2

0

dρsin ρ

cos3 ρ〈n, 0|φ2(ρ, θ)|0〉〈0|φ2(ρ, θ)|n, 0〉 (7.6)

However, note that these states are scalar primaries, and φ2(x) is a scalar operator, so by symmetrywe must have

〈0|φ2(ρ, θ)|n, 0〉 =1√2π

(eit cos ρ)2∆+2n (7.7)

67

since these are primary wavefunctions that are annhilated by Kµ. Note that this matrix elementwould vanish for ` > 0 – this is how one can see that λφ4 does not produce anomalous dimensionsat first order for ` > 0 operators. The correct normalization can be computed [10], although thecalculation is a bit non-trivial. I have simply written the correct result for AdS3. Performing theintegrals gives

γ(n) =λ

8π(2∆ + 2n− 1)(7.8)

so we have computed the anomalous dimension of all [OO]n` in λφ4 theory in AdS. As we haveemphasized before, anomalous dimensions are just AdS energy shifts due to interactions.

7.2 Perturbative Interactions and λφ4 Theory

Interacting QFTs in AdS can be studied in exactly the same way that we study such theories inflat spacetime. In particular, we can expand perturbatively in the coupling constants and computeusing AdS Feynman diagrams. To derive the Feynman rules we can use either an operator or a pathintegral formalism, as usual; the result are rules for propagators, which are just the 2-pt functions ofAdS fields, and vertices, which follow as direct generalizations of those of flat spacetime. A significanttechnical difference is that in AdS spacetime we lack a set of commuting translation operators, andso we cannot use the Fourier transform to momentum space to simplify computations. Conceptuallythis may almost be an advantage, but computationally it means that AdS Feynman diagrams aremuch more cumbersome to calculate.

Let us consider a standard example, that of λφ4 theory in AdS. We have an AdS path integral

Z =

∫Dφe−S[φ] (7.9)

where the Euclidean action is

S[φ] =

∫

AdS

dd+1x√−g

(1

2(∇φ)2 +

m2

2φ2 +

λ

4!φ4

)(7.10)

The bulk-to-bulk propagators have a simple expression in terms of the geodesic distance σ(X, Y )between X and Y :

G∆(X, Y ) = C∆z∆/2

2F1(∆, h,∆ + 1− h, z), (7.11)

where z = e−2σ.20 The factor C∆ = Γ(∆)/(2πhΓ(∆ + 1 − h)) with h = d/2. This propagator canbe obtained as usual as the solution to the Klein Gordon equation in Euclidean AdS with a delta

20This normalization of the bulk-to-bulk propagator differs from that in e.g. [10] by a factor of C∆. Other usefulrepresentations of the bulk-to-bulk propagator are

G∆(X,Y ) = C∆y∆/2

2∆ 2F1(∆

2,

∆

2+

1

2,∆ + 1− h, y) (7.12)

=C∆u∆ 2F1(∆,∆− h+

1

2, 2∆− 2h+ 1,− 4

u), (7.13)

where u = (X − Y )2 and y−12 = cosh σ

R = 1 + u2R2 .

68

function source, namely

(∇2X +m2

)G∆(X, Y ) = (2π)d+1δd+1

AdS(X − Y ) (7.14)

It is also given as a sum over modes if we compute

〈φ(X)φ(Y )〉 = G∆(X, Y ) (7.15)

using the expansion of φ(X) in terms of creation and annihilation operators with correspondingwavefunctions.

The Feynman rules for the theory are identical to the position space Feynman rules for λφ4

theory in flat spacetime, except that we replace flat space propagators with G∆(X, Y ), and weintegrate over AdS. For example, the 4-pt correlator of φ to first order in λ is

〈φ(X1)φ(X2)φ(X3)φ(X4)〉 = λ

∫

AdS

dd+1Y√−gG∆(X1, Y )G∆(X2, Y )G∆(X3, Y )G∆(X4, Y ) (7.16)

If we had a gφ3 theory instead we would have

〈φ(X1)φ(X2)φ(X3)φ(X4)〉 = g2

∫

AdS

dd+1Y1

√−g(Y1)G∆(X1, Y1)G∆(X2, Y1) (7.17)

×∫dd+1Y2

√−g(Y2)G∆(Y1, Y2)G∆(X3, Y2)G∆(X4, Y2)

Loops can be calculated in the same way, with an AdS position-space integral for every vertex, anda propagator for each line.

Now that we know how to compute correlators of fields in AdS, we can apply our usual dictionaryto compute the correlation functions of CFT operators. We need only apply our formula

O(t,Ω) = limε→0

φ(t, π

2− εf(t,Ω),Ω

)

ε∆(7.18)

for each of the AdS fields in an AdS correlator. Applying this formula to any of the examples above,we immediately notice the appearance of the function

GB∂(t,Ω;Y ) = limε→0

1

ε∆G∆

(t,π

2− εf(t,Ω),Ω;Y

)(7.19)

which we refer to as the bulk-boundary propagator. For example, to compute the CFT 4-pt correlatorat tree level in λφ4 theory in AdS, we use the formula

〈O(~r1)O(~r2)O(~r3)O(~r4)〉 = λ

∫

AdS

dd+1Y√−gGB∂(~r1, Y )GB∂(~r2, Y )GB∂(~r3, Y )GB∂(~r4, Y ) (7.20)

This is an example of how to use AdS/CFT to compute non-trivial CFT correlation functions.

69

Fortunately, the bulk-boundary propagator GB∂ is much simpler than the bulk-bulk propagatorG∆. In terms of the projective null cone coordinates P , we simply have

G(P,X) = limε→0

C∆e−σ(X,Y (ε))

2F1(∆, h,∆ + 1− h, e−2σ(X,Y (ε)))

ε∆, (7.21)

where in this limit Y → 1εP becomes the boundary point. In this limit the geodesic distance goes to

infinity, and we simply obtain

GB∂(P,X) =C∆

(2P ·X)∆=

1

Γ(∆− h+ 1)

∫ ∞

0

dt

tt∆e−2tP ·X (7.22)

where as usual

C∆ =Γ(∆)

2πhΓ(∆− h+ 1)(7.23)

The last formula in terms of a t integral is useful for computations.This formula for GB∂ was essentially forced on us by conformal symmetry – the bulk boundary

propagator could only depend on the conformal invariant P ·X, and the scaling relation O(λP ) =λ−∆O(P ) forces the power-law relation that we derived. In this normalization the 2-pt function is

〈O(P1)O(P2)〉 =C∆

(2P1 · P2)∆(7.24)

for an operator of dimension ∆. This could be derived by taking the point X → P2 on the boundaryof AdS.

Let us now try to compute the 4-pt correlator to first order in λφ4 theory. This is

〈O(P1)O(P2)O(P3)O(P4)〉 =λ

[2πhΓ(∆− h+ 1)]4

∫

AdS

dd+1Y4∏

i=1

∫ ∞

0

dtitit∆i e

−2tiPi·Y (7.25)

Now we can exchange the orders of integration to write

〈O(P1)O(P2)O(P3)O(P4)〉 =λ

[2πhΓ(∆− h+ 1)]4

4∏

i=1

∫ ∞

0

dtitit∆i

∫

AdS

dd+1Y e−2(∑i tiPi)·Y (7.26)

We can do the integral over AdS most easily using the Euclidean Poincare patch coordinates. Wecan use conformal symmetry to rotate so that the only non-vanishing component of Q is Q0, and we

have Y0 = 12

(z2+~r2+R2

z

), so that in this frame 2Q · Y = |Q|(1 + z2 + r2)/z, leading to

∫

AdS

dd+1Y e2Q·Y =

∫ ∞

0

dz

zz−d

∫dd~re−(1+z2+r2)|Q0|/z

= πh∫ ∞

0

dz

z(z|Q|)−he−(1+z2)|Q0|/z

= πh∫ ∞

0

dz

zz−he−z+Q

2/z (7.27)

70

where we note that Q20 = Q2 and we have now restored manifest conformal symmetry. This means

that

〈O(P1)O(P2)O(P3)O(P4)〉 =λ

[2πhΓ(∆− h+ 1)]4

4∏

i=1

∫ ∞

0

dtitit∆i

∫ ∞

0

dz

zz−he−z+

1z

∑4i,j=1 titjPi·Pj(7.28)

Now we can rescale ti → ti√z and factor out the z dependence completely. Then we find that

〈O(P1)O(P2)O(P3)O(P4)〉 ∝∫ ∞

0

4∏

i=1

dtitit∆i e

−∑4i,j=1 titjPi·Pj (7.29)

This last integral can be written using the Symanzik ‘star formula’. This formula arises by noting

e−x =

∫ i∞

−i∞dδ Γ(δ)x−δ (7.30)

Inserting this identity for each of the e−titjPi·Pj term and performing the ti integrals leads to

∫ ∞

0

n∏

i=1

dtitit∆i e

−∑ni,j=1 titjPi·Pj =

1

2

∫ i∞

−i∞

n∏

i<j

dδij2πi

Γ(δij)(2Pi · Pj)−δij (7.31)

where the δij are constrained by

n∑

j 6=i

δij = ∆i (7.32)

although in this case all ∆i = ∆, the dimension of O. Note that if we imagine that δij = pi · pj,then these constraints are just what we would find from momentum conservation plus the on-shellcondition p2

i = ∆i. One should think of the δij as analogous to the Mandelstam invariants sij for ascattering amplitude.

This last form for the correlator has transformed it into ‘Mellin space’, where the δij variablesparameterize the Mellin space. The Mellin amplitude is defined through

〈O(P1)O(P2)O(P3)O(P4)〉 =

∫ 4∏

i<j

dδij2πi

Γ(δij)(2Pi · Pj)−δijM(δij) (7.33)

where M(δij) is the Mellin Amplitude. Here we have found that a λφ4 interaction in AdS spaceleads to a Mellin amplitude

M(δij) = λ (7.34)

so it is simply a Mellin space-independent constant. This should remind you of the simplicity ofmomentum space scattering amplitudes.

71

We can also re-write the Mellin form by eliminating the extra δij variables, giving the non-trivialpart of the correlator as

A(u, v) =

∫ i∞

−i∞

dδ12dδ14

(2πi)2M(δ12, δ14)Γ2(δ12)Γ2(δ14)Γ2(∆− δ12 − δ14)u∆−δ12v−δ14 (7.35)

We see that the Mellin amplitude is just the transform in the conformally invariant cross-ratios. Forgeneral reference, the Mellin transform of a function f(x) is

M(s) =

∫ ∞

0

xs−1f(x)dx (7.36)

f(x) =

∫ i∞

−i∞

ds

2πix−sM(s) (7.37)

with the inverse also listed. Note that this is basically just a Fourier transform in the logarithm ofthe usual variables. So for the 4-pt correlator the Mellin amplitude is just a Mellin transform in thecross-ratios u and v.

What about the position-space correlator? In general, it is a special function referred to in theliterature as a ‘D-function’, which are usually written as D∆1∆2∆3∆4 as they depend on the externaldimensions. The dependence on the spacetime dimension d only comes about through an overallnormalization factor. The choice

D1111(z, z) =2Li2

(1z

)− 2Li2

(1z

)+ log(zz) log

(z(z−1)z(z−1)

)

z − z (7.38)

can be expressed in terms of logarithms and dilogarithms. The other D-functions with integer ∆i

can be computed in terms of this one by taking derivatives, but in general for non-integral ∆i thereis no such simple expression. Here zz = u and (1− z)(1− z) = v as usual.

7.3 Conformal Partial Waves and λφ4

Let us consider λφ4 theory in AdS, and see how the interaction effects the CFT. For concreteness,let us imagine that we are working in AdS4, so d = 3, and we have fixed the squared mass of the φfield to be m2 = ∆(∆− 3) = −2/R2, so that ∆ = 1. Then the correlator of the operator O obtainedfrom taking φ to the boundary is

〈O(x1)O(x2)O(x3)O(x4)〉 =1

x213x

224

(1 +

1

zz+

1

(1− z)(1− z)

)(7.39)

−λ Nx2

13x224

2Li2(

1z

)− 2Li2

(1z

)+ log(zz) log

(z(z−1)z(z−1)

)

z − z

where N is a normalization constant that just depends on our choice ∆ = 1 and d = 3. We chosethese parameters so that we could utilize the simplest non-triival D-function.

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In section 6.5 we discussed the conformal partial wave expansion of correlators, and the specificpartial waves that appear in the case of a generalized free theory. Now we would like to see howthe conformal partial wave expansion changes due to the λφ4 interaction. We are interested in thisbecause it tells us about the change in the OPE coefficients that are proportional to 3-pt functionssuch as

〈O(x1)O(x3)[OO]n,`(y)〉 (7.40)

where [OO]n,` is the primary operator that would have dimension 2∆ + 2n + ` and spin ` in thegeneralized free theory. It can be viewed schematically as

[OO]n,` ∼ O(↔∂)2n ↔

∂µ1 · · ·↔∂µ` O (7.41)

although there are a lot of combintorial coefficients and traces to remove to get the precise primaryoperator.

Furthermore, the correlator also tells us about the anomalous dimensions γ(n, `) of theseoperators – this is the shift in their dimension due to the interaction. Recall that these operatorsimply correspond to 2-particle states in AdS, and that the dilatation operator is the AdS Hamiltonian,so an anomalous dimension in this context is simply the interaction energy (attractive or repulsive)between the two particles. As we will see, this can be computed directly using old-fashioned quantummechanical perturbation theory.

Let us expand the correlator in z, z near 0, since this is the OPE limit. Ignoring the normalizationfactor and just keeping track of the parametric dependence on λ, we find

A(z, z) =1

zz+ 2 + z + z − λ

2(4 + z + z − (2 + z + z) log(zz)) + · · · (7.42)

where the ellipsis denotes higher order terms in z, z. Recall that conformal partial waves behave as

1

(zz)∆φgτ,`(z, z) = (zz)

τ2−∆φ

(z` + z` + · · ·

)(7.43)

The terms that are independent of λ are just what we have in a generalized free theory. But let usexamine the order λ terms more closely. The correlator contains

A(z, z) ⊃ (2− 2λ+ λ log(zz)) (7.44)

Recall that the OPE in a generalized free theory begins (at small z, z) with

O(z, z)O(0) =

[1

(zz)∆+

1

NO0,0

O2(0) +z

NO0,1

O∂zO(0) +z

NO1,0

O∂zO(0) + · · ·]

(7.45)

The terms we have found in the correlator A are of order z0z0, so we can identify them ascorresponding to the operator O2(0). In other words, this is the operator [OO]n,` with n = ` = 0.We see that in the generalized free theory the conformal block coefficient was simply 2, but we have

73

found that this has shifted from 2→ 2 + 2λ. This also means that the OPE coefficient has shifted –since the conformal block coefficient is the square of the OPE coefficient, we have

c0,0 =√

2−√

2

2λ (7.46)

so that (c0,0)2 gives 2 + 2λ, the conformal block coefficient (although recall that the precise numericalcoefficient isn’t meaningful since we didn’t keep track of the normalization N ).

Now let us consider the λ log(zz) term in A. Note that

λ log(zz) ≈ (zz)λ − 1 (7.47)

in perturbation theory in λ. In other words, we should interpret this logarithm as a shift in thescaling dimension of O2(0). From this logarithm we can compute the anomalous dimension γ(n, `)for n = ` = 0, namely

γ(0, 0) = (z∂z + z∂z) [λ log(zz)] = 2λ (7.48)

Note that the anomalous dimension is positive for positive λ, so the dimension of O2 has increased.This is due to the fact that the corresponding 2-particle state in AdS is experiencing a repulsiveforce.

Let us now revisit the utility of the Mellin amplitude representation of the correlator. We notedthat for the 4-pt correlator we can write the contour integral

A(u, v) =

∫ i∞

−i∞

dδ12dδ14

(2πi)2M(δ12, δ14)Γ2(δ12)Γ2(δ14)Γ2(∆− δ12 − δ14)u∆−δ12v−δ14 (7.49)

We have just seen that perturbative AdS interactions shift the OPE coefficients and operator/statedimensions in the CFT, and that these shifts can be extracted by looking at the coefficients of power-law monomials in A(u, v). Furthermore, we have seen that anomalous dimensions are associatedwith logarithms.

Note that a single pole in an integral of the form

∫ i∞

−i∞

dδ

2πi

1

δ + ∆u−δ = u∆ (7.50)

while a double pole gives

∫ i∞

−i∞

dδ

2πi

1

(δ + ∆)2u−δ =

d

d∆u∆ = u∆ log u (7.51)

So the OPE coefficients and anomalous dimensions in a CFT are associated with the residues ofpoles in the Mellin integrand. Much of the power of Mellin space is that it allows us to use complexanalysis for the study of AdS/CFT correlators.

74

7.4 Physical Discussion of AdS/CFT with Interactions

In perturbation theory the free propagator plays a crucial role. Let us examine various physicallimits of the bulk-to-bulk propagator G∆(X, Y ). In the short distance limit, it must agree with theflat spacetime propagator in d + 1 dimensions. This is the limit where z = e−2σ(X,Y ) ≈ e−2r/RAdS

where r RAdS, 1/m is the distance between X and Y . Note that

2F1(∆, h,∆ + 1− h, e−2r/RAdS) ∝ 1

Γ(∆)

(RAdS

r

)d−1

(7.52)

where I have dropped some factors of 2 and π, and approximated the hypergeometric function near1. Thus we find that in the very short distance limit

Gshort(r) ∝(RAdS

r

)d−1

(7.53)

which is what we would expect for a field in d+ 1 dimensional flat spacetime at very short distances.At very long distances z 1, and we find that

Glong(r) ∝ e−∆ r

RAdS (7.54)

which falls off very quickly at long-distances in AdS, due to the mass and AdS curvature. There isalso a more delicate intermediate regime, where ∆ RAdS

r 1 in which the usual Yukawa potential

is reproduced. This can be obtained from a saddle-point approximation to the hypergeometricfunction at large ∆.

Let us consider whether the interactions disturb the limit where we take AdS QFT operatorstowards the boundary. There is an analogous question in flat spacetime. When we construct theflat spacetime S-Matrix, a key physical point is that we can study initial and final states composedof asymptotically well-separated particles. This is actually rather subtle, and in fact there arewell-known IR divergences in four and fewer dimensions that must be treated with great care.

In AdS/CFT the question is whether when we take bulk operators φ(t, ρ,Ω) to the boundary inan interacting theory, the interactions effectively shut off. The answer is yes – in fact, interactionsshut off in AdS much more robustly than in flat spacetime, due to the AdS curvature. Recall that insection 2.2 we showed that even massless fields in AdS decrease exponentially fast at large distancesfrom their sources. This means that well-separated sources in AdS decouple very quickly at largedistances, i.e. as we approach the boundary of AdS.

In particular, we would like to see that as we take our AdS fields to the boundary, their interactionsare dominated by points deep within the bulk (i.e. not arbitrarily close to the boundary). Notethat when we approach the boundary we take φ(t, ρ(ε; t,Ω),Ω) with ε→ 0 but with fixed t,Ω. Thismeans that when we have a correlator

〈φ1(t1, ρ(ε; t1,Ω1),Ω)φ2(t2, ρ(ε; t2,Ω2),Ω2) · · ·φn(tn, ρ(ε; tn,Ωn),Ωn)〉 (7.55)

as we take ε→ 0, the geodesic distance σ(Xi, Xj) between Xi and Xj will always go to infinity. Inother words, any two distinct points on the boundary of AdS are infinitely far from each other. This

75

by itself does not prevent us from having interactions near the boundary, since we are rescaling theφi by ε−∆i to obtain CFT operators.

However, consider some interaction vertex (from a Feynman diagram) localized at Y in AdS. If Yfollows Xi as Xi → Pi on ∂AdS, then the vertex at Y will be further and further from all of the othervertices, and so bulk-to-bulk propagators connecting it to the rest of the AdS Feynman diagramwill be exponentially suppressed. This is the reason that non-trivial interactions do not follow ourbulk fields out to the boundary. Note that if all interaction vertices move together out towards theboundary then there is no suppression, but this uniform displacement does not lead to non-trivialinteractions at infinity. This is analogous to the fact that in flat spacetime, the integration over thecenter of mass coordinate just leads to an overall momentum conserving delta function enforcingtranslation invariance.

7.5 A Weinbergian Proof of Conformal Symmetry

We saw by explicit computations that free fields in AdS give rise to Generalized Free Theories, whichare conformal theories. Now we are studying AdS field theories with interactions, so how do we stillknow that the boundary theory is a conformal theory?

This is not so obvious. In a free theory we have the conformal symmetry generators D(0), P(0)µ ,

K(0)µ , and M

(0)µν . But when we add interactions we are changing the dilatation operators = bulk

Hamiltonian by

D = D(0) +DI (7.56)

where DI represents the effects of interactions. The free generators may have satisfied the conformalalgebra, but now the presence of DI will (naively) ruin it. In order to restore conformal symmetrywe need to deform the other generators.

If we are dealing with a local relativistic QFT in AdS spacetime, then the preservation of theconformal algebra including interactions is actually pretty easy to see. The point is that the AdSisometries21 are explicitly a symmetry of the action, and so we can use Noether’s theorem to computethe symmetry generators as classical or quantum operators. Then these operators also act naturallyon the fields when we take them to the boundary of AdS, φ → O, and the discussion of section7.4 shows that in fact the interactions decouple so that the CFT operators O transform just as wefound in the free AdS theory.

Let us make this argument in a more general and formal way. We expect the interaction tosatisfy

[DI ,Mµν ] = 0 (7.57)

so that we do not need to deform the AdS rotations, and we can continue to use angular momentumquantum numbers. This should not be surprising, since angular momentum is a discrete quantumnumber, so it would be difficult to imagine deforming it continuously. However, the generators P

(0)µ

21Note that in the case of AdS3 in the presence of gravity, we have more symmetries than just the AdS3 isometries,but this more subtle point can only be seen by studying the asymptotic symmetry group of the spacetime.

76

and K(0)µ do not commute with D(0) and so we should expect that they will need to be deformed by

P Iµ and KI

µ in the presence of the interaction DI . Specifically, if we consider Kµ we must have

[D(0) +DI , K(0)

µ +KIµ

]= −K(0)

µ −KIµ (7.58)

so we need to define KIµ and ensure that this commutation relation holds. We already know that

the zeroth order commutation relations work, so we need

[D(0), KI

µ

]+[DI , K(0)

µ

]+[DI , KI

µ

]= −KI

µ (7.59)

Let us assume that

DI(t) =

∫

AdS

ddx√−gV(t, x) (7.60)

where we have included the t dependence to emphasize that the integral is a spatial integral at somefixed time t. In the interaction picture, the time evolution of V is given by

∂tV(t, x) = −i[D(0),V(t, x)] (7.61)

So now let us look at the commutator of K(0)µ with DI . We worked these generators out explicitly

for AdS3, so specializing to that case note that

[K(0)± , DI ] = −i

∫d2x

sin ρ

cos3 ρe−it±iθ

(sin ρ∂t + i cos ρ∂ρ ∓

1

sin ρ∂θ

)V(t, ρ, θ) (7.62)

We can replace the first term with the commutator, while integrating the last two terms by partsgives

[K(0)± , DI ] = −

∫d2x√−g sin ρ e−it±iθ

([D(0),V(t, x)] + V(t, x)

)(7.63)

This means that if we define

KI± =

∫d2x√−g sin ρ e−it±iθV(t, x) (7.64)

then we will satisfy the desired commutation relations if the term [DI , KIµ] vanishes. This will be

possible if

[V(t, x),V(t, y)] = 0 (7.65)

This is the condition we wanted to derive – we can preserve the conformal symmetry if the interactiondensity commutes with itself outside the lightcone. Weinberg derived precisely this condition in orderto ensure the Poincare invariance of the flat spacetime S-Matrix; here we see that it is necessaryto preserve the conformal invariance of a theory living in AdS spacetime. The same condition issufficient to preserve the interacting commutators of the Pµ generators.

77

7.6 Effective Conformal Theory and the Breakdown of AdS EFT

Thus far we have been talking about quantum field theory in AdS as though it is immutable and UVcomplete. However, in most cases22 it is more sensible to view QFT as an effective field theory thatbreaks down at an energy scale Λ, or (more physically) at a short-distance scale 1/Λ. So it becomesnatural to ask how the cutoff scale Λ in AdS translates to the CFT. Roughly speaking, one gets theanswer immediately by noting that an AdS energy scale Λ is an eigenvalue of the AdS HamiltonianD, so it must translate into a dimension ΛRAdS in the CFT. An AdS field theory with a UV cutoffΛ corresponds to a CFT where we only keep states and operators with dimension ∆ < ∆Λ with∆Λ = ΛRAdS.

As a more detailed comment – to discuss EFT in an invariant way, we would like to talk aboutintegrating out states with large invariant mass. This means that if we have some AdS/CFT state|ψ〉 and some AdS operator that creates it, then we have a wavefunction

∇2AdS〈0|ΦAdS(X)|ψ〉 ∝M2〈0|ΦAdS(X)|ψ〉 (7.66)

But the ∇2AdS operator is just a combination of conformal generators, in fact it is the conformal

Casimir operator

∇2AdS = D2 +

1

2(K · P + P ·K) +MµνM

µν (7.67)

and so states of definite invariant mass have definite conformal Casimir, where the eigenvalues ofthis operator are M2 = ∆(∆− d) + `(`+ d− 2) for general spins. So integrating out the high massstates means integrating out operators/states with large conformal Casimir. Since single particlestypically have small spin, large Casimir naturally corresponds to large dimension ∆.

We know from experience that EFTs neglecting higher dimension operators are a good systematicapproximation for lower-energy physics. This translates into the statement that corrections tocorrelators involving operators of dimension ∆ from neglected higher dimension effects should besuppressed by powers of ∆/∆Λ. In fact, some recent results show that the approximation can beeven better than this. For example, consider the conformal partial wave expansion of a correlator

〈φ(x1)φ(x2)φ(x3)φ(x4)〉 =1

(x13)2∆(x24)2∆

∆Λ∑

∆,`

λ2∆,`g∆,`(u, v) (7.68)

where instead of summing over all ∆, `, we have restricted the sum to ∆ < ∆Λ. One can prove[14] that in the Euclidean region this results in an exponentially good approximation to the CFTcorrelator. This is better than one might have expected from effective field theory in AdS, wherecorrections are generally power-laws. The reason it is better is that we are assuming that the OPEcoefficients and operator dimensions are exactly correct for ∆ < ∆Λ, and that all corrections comefrom dropping the conformal partial waves with larger dimension. In an EFT description we do notsimply leave out the states with large energy – we also miss power-law corrections to the low-energy

22In the case of AdS/CFT this is always true, because once we involve gravity the QFT description can only be anapproximation.

78

states. This is (most-likely) why dropping high dimension partial waves gives a better approximationthan one might guess from EFT.

One can see AdS EFT directly by studying theories with massive particles, moving the cutoff,and integrating these heavy states out. Then the CFT correlators, OPE coefficients, and anomalousdimensions can be matched between the high-energy and low-energy description. This analysis wascarried out explicitly in [10].

8 On the Existence of Local Currents Jµ and Tµν

When we first learn QFT (or perhaps just classical mechanics), one of the earliest lessons is thebeautiful and important Noether’s theorem, which connects symmetries to conservation laws. Whenwe have a Lagrangian description, Noether’s theorem allows us to connect three different ideas:

1. the existence of a symmetry of the theory

2. the existence of a local current Jµ(x) (so called because it depends on a point in spacetime,and because we usually implicitly assume that it is insensitive to very distant physics) that isconserved, so that ∂µJµ(x) = 0

3. the existence of a charge (no spacetime dependence) Q that measures the conserved quantityassociated with the symmetry; if Jµ and Q both exist then we can usually write Q =

∫d3xJ0(x)

Global symmetry currents Jµ and the stress-energy tensor Tµν form an extremely important class ofoperators in CFTs. As we mentioned above, conventionally CFTs are defined to have a conservedTµν , although one can study ‘CTs’ or ‘almost-CFTs’ that have all the other features of a CFT. Solet us discuss these operators in more detail in order to understand their physical properties and tosee why they play such an important role. In particular, we will discuss

1. The difference between a simple conserved charge Q versus a local symmetry current. It ispossible to have Jµ without Q or a charge Q without a corresponding Jµ.

2. How local currents such as Jµ(x) and Tµν(x) generate symmetry transformations that producenew states, thereby telling us something about the Hilbert space of the theory.

3. The fact that when we gauge local symmetries, we identify all these different states, thereby‘mod-ing out’ by the symmetry. Gauge symmetries are really redundancies of the description.

Although we will discuss physics quantum mechanically in this section, all of our statements alsopertain to the classical phase space of classical theories, and in fact might be more natural andelementary in that context.

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8.1 An SU(2) Example

To understand these issues very concretely, let’s study a very simple free theory of a pair of complexbosons, (Φ1,Φ2) with an SU(2) symmetry. The action is just

S =

∫ddx

2∑

i=1

(∂µΦ†i∂

µΦi −m2Φ†iΦi

)(8.1)

where i = 1, 2. The local conserved current associated to this SU(2) global symmetry is the operator

Jµa (x) = iσija

(Φ†i (x)∂µΦj(x)− ∂µΦ†i (x)Φj(x)

)(8.2)

where σija are the SU(2) generators (Pauli matrices). Let’s check that it is conserved

∂µJµa (x) = iσija

(∂µΦ†i (x)∂µΦj(x) + Φ†i (x)∂µ∂

µΦj(x)− ∂µ∂µΦ†i (x)Φj(x)− ∂µΦ†i (x)∂µΦj(x))

= iσija m2(Φ†i (x)Φj(x)− Φ†i (x)Φj(x)) = 0 (8.3)

Note that the momentum conjugate to Φi is

Πi(x) ≡ δLδΦi

= Φ†i (8.4)

so that the time component of the current can also be written as

J0a(x) = iσija

(Φ†i (x)Π†j(x)− Πi(x)Φj(x)

)(8.5)

This means that the commutators of J0a (x) can be computed easily using the canonical commutation

relations, giving

[J0a(t, x),Φi(t, y)] = iδd−1(~x− ~y)σija Φj(t, x) (8.6)

so as we would expect, the current locally rotates the Φi field in SU(2) space. We can measure atotal charge using the integrated quantity

Qa =

∫dd−1xJ0

a(x) (8.7)

and note that it is time-independent using the conservation of Jµa and the vanishing of fields atinfinity.

Although Qa can be used to measure the total charge (in this case, an SU(2) representation)associated with a given state, it always adds up the contribution of the entire universe. This severelylimits the utility of Qa by itself.

However, the local current Jµa is much more powerful, because we can use it to turn one stateinto another. The point is that we can smear J0 over a small region surrounding some isolated

80

sub-system, so that it only transforms that sub-system, but leaves the rest of the universe alone.When you pick up an object and spin it around, you are (in some sense) taking advantage of theexistence of local translation and rotation currents. We will come back to this point directly belowwhen we discuss Tµν , but for now let us stick with our SU(2) current.

For example, say we have a state |ΨK〉composed of K well-separated Φ1 particles, located aroundx1, x2, · · ·xK . Then we can immediately construct an SU(2) matrix operator Ra via

Rij(t) = exp

[iπ

2

∫

Bi⊃xidd−1yJ0

1 (t, y)σ1

](8.8)

where the region Bi is some compact ball containing only the ith particle, and σ1 is the 1th or xthPauli matrix (chosen as a random eaxmple). Note that [Rij(t), Rkl(t)] = 0 since they are compactlysupported away from each other. Acting with Rij on the state |ψK〉 will take the ith particle from(1, 0) to (0, 1), but leave the others alone. In other words, out of the state |ΨK〉 with all the bosonsin the (1, 0) configuration, we can generate all SU(2) configurations for the isolated bosons! Thisobviously would not be possible with only the global charge Qa. Note that if we had only studiedan abelian current then all we could do is multiply the state with a phase, which is rather boring –this is why we chose an SU(2) current as our example.

An important feature of Rij is that it takes eigenstates of the Hamiltonian to other eigenstates.In other words, it guarantees that if a particle in the (1, 0) state is an eigenstate, then (0, 1) isan eigenstate as well. The conservation of Jµa is crucial for this, because it means that the timederivative of the Ri is

∂tRij = Rijiπ

2

∫

Bi⊃xidd−1y∂tJ

01 (y)σ1

= Rijiπ

2

∫

Bi⊃xidd−1y∂jJ

j1(y)σ1

= Rijiπ

2

∫

∂Bi

dd−2s njJj1(s)σ1 (8.9)

and this last quantity vanishes when it acts on a state where the particles are well-separated, becausethe current will vanish on the surface of the ball Bi. This means that operators like Rij are theexponential of local charges that are conserved in states where the particles are well-separated.

8.2 A Very Special Current – Tµν

The conserved energy-momentum tensor Tµν plays a special role because it generates the spacetimesymmetry transformations, including the conformal transformations in CFTs.

To understand more concretely what we can do with Tµν(x), let’s revisit the ultra-familiarsymmetry of rotations in the case of a free QFT. If we take a free φ field in 2 + 1 dimensions, wehave the action

S =

∫dtdrrdθ

1

2

((∂φ)2 −m2φ2

)(8.10)

81

In these specially chosen coordinates, rotations are just φ → φ + ε∂θφ, and we find that the 0component of the current is T 0

θ = rφ∂θφ. Note that since the canonical conjugate to φ is π(x) = φ(x),we can also write this as

T 0θ (t, x) = r∂θφ(t, x)π(t, x) (8.11)

so that it’s clear what the commutation relations of T 0θ (x) are with φ(y), namely

[T 0θ (t, x), φ(t, y)] = iδ2(x− y) [r∂θφ(t, x)] (8.12)

so clearly T 0θ generates rotations, as desired. The global charge is

Q =

∫drdθ rφ∂θφ (8.13)

However, this charge simply rotates the entire universe. So it isn’t very useful physically – we cannotobserve that we have rotated the entire universe.

Fortunately, we can make a local version using T µθ . Using a radius dependent function η(r), wecan now consider the rotation operator

R[η] =

∫drdθ η(r)(rφ∂θφ) (8.14)

As an example, we might have a gas of particles well-inside some sphere of radius R, along withsome far away gas. So if we choose η(r) = Θ(R− r) then we have a rotation that turns off at R –we are just rotating the region inside the sphere while holding the outside universe constant.

We would now like to check that by performing this transformation we have not changed theenergy of our state. Since ∂µJ

µ = 0, we see that

∂tR[η] = i [H,Q[η]] =

∫drdθ η(r)∂iJ

i(r, θ)

=

∫drdθ [∂rη(r)] Jr(r, θ)

=

∫dθ Jr(R, θ) (8.15)

where the last line follows because the r-derivative of η(r) is simply a delta function. But thisresult makes a lot of sense – we see that R[η] is time independent, and therefore commutes with theHamiltonian, as long as we are working in a state where Jr doesn’t see any particles at the surfaceof the sphere. So we can rotate everything inside a ball separately from the rest of the universe andobtain an equivalent but new state as long as there aren’t any particles on the surface of the ball.

So this is just a spacetime version of the SU(2) operator we defined in equation (8.8). Instead ofperforming an internal SU(2) rotation on everything within a ball, it performs a spacetime rotationon the contents of the ball, creating a physically distinct state in the Hilbert space.

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8.3 Separating Q and Jµ – Spontaneous Breaking and Gauging

We mentioned above that there are situations where a current Jµ exists but Q does not. This canhappen when the integral defining Q is not well-defined, because it doesn’t converge in infinite volume.A very physically important example of this phenomenon is when the symmetry is spontaneouslybroken. In that case the current Jµ still exists and is conserved, so ∂µJ

µ = 0, however Jµ does notannihilate the vacuum:

Jµ(x)|0〉 = |πJµ(x)〉 6= 0 (8.16)

This makes it obvious that if we try to integrate over all of (an infinite) spacetime, the result willbe ill-defined, assuming we have not also broken translational symmetry. This state |πJµ(x)〉 is aone-goldstone boson state. A more physical state is

pµFπ√

2p0

|πp〉 =

∫dd−1x eip·xJµ(x)|0〉 (8.17)

and up to a normalizaiton, this is just a state consisting of a single goldstone boson with momentum~p. The non-existence of Q corresponds with the fact that a goldstone boson state with ~p = 0 is notnormalizable. Note that current conservation implies that

−ipµpµFπ

|πp〉 =

∫dd−1x eip·x∂µJµ(x)|0〉 = 0 (8.18)

so that we must have

p2 = 0 (8.19)

for a one-goldstone boson state. In other words, we learn that goldstone bosons must be massless.This basically shows that Jµ(x) acts on the vacuum to create a one-particle state, so goldstonebosons really are single particles. One can read more about all of this in Weinberg V2 [13].

There are also situations where Q exists but Jµ does not. The example that is most relevantfor these notes is the ‘CT’ dual23 of non-gravitational d+ 1 dimensional field theories in AdS. Inthese d-dimensional boundary theories we have global symmetry charges associated with the fullconformal group, enlarging the Poincare generators, but these do not arise from the integral of alocal current (these local currents would be made out of Tµν if it existed, but it does not).

An alternate way to have a charge Q without Jµ is to ‘gauge’ the symmetry corresponding to Jµ.This means that we identify all of the states related by the action of Jµ, turning the symmetry intoa redundancy and eliminating many degrees of freedom. To be precise, in a gauge theory the variousstates related by the action of operators such as the Ri of equation (8.8) would all be viewed as thesame state. As an example where degrees of freedom really are eliminated from a low-energy theory,consider a complex field

φ(x) = (v + h(x))eiθ(x) (8.20)

23A theory with conformal symmetry but not a ‘Field Theory’.

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If v 6= 0 then θ(x) is a true goldstone boson degree of freedom. However, if we gauge the symmetryunder which θ(x)→ θ(x) + ε, then we elimnate this degree of freedom, as it is ‘eaten’ by the abeliangauge field Aµ(x).

In the case of gravity, when we gauge spacetime symmetries we remove local spacetime observablesfrom the exact quantum mechanical theory; the only well-defined observables are ‘holographic’ –they are associated with infinity. Let us be a bit more precise about this. The energy-momentumtensor Tµν is really the current for the d different charges Pµ, which correspond to translations (notrotations, or dilatations and special conformal transformations – these are associated with currentslike xνTµν). When we place a theory with energy-momentum tensor Tµν in a dynamical spacetime,we are therefore gauging, or redundantly identifying, the spacetime translations. Local translationsoccur when one object in spacetime moves with respect to another object. In general relativity,distances between objects are gauged, although obviously just as with spin one gauge theories, thegravitational field also changes under these gauge transformations (diffeomorphisms).

These ideas see a concrete realization when one studies membranes (or actually just particles) inspacetime. Particles, strings, and membranes can all be thought of as objects that spontaneouslybreak translation symmetry! If we have a d− 1 dimensional membrane in d-dimensional spacetime,then we can parameterize it using a goldstone boson field

πd(t, x1, · · · , xd−1) (8.21)

that tells us that the spacetime position of the membrane is (t, x1, · · · , xd−1, πd). There is a universallow-energy action for πµ called the DBI action

T

∫dd−1x

√1− ∂µπd∂µπd (8.22)

which just measures the world-volume of the membrane. The parameter T is the tension of themembrane. But the interesting point for us is that when we turn on gravity, thereby ‘gauging’ thetranslation symmetry, the goldstone boson πd gets ‘eaten’ by the gravitational field, just like astandard goldstone boson in a spin one gauge theory. How do we see this? Actually, it’s trivial – wecan always just choose a spacetime coordinate system built on top of the membrane so that themembrane is at πd = 0 by definition. With this gauge choice, obviously the membrane fluctationshave been absorbed into fluctuations of the spacetime metric!

These statements are also directly related to the well-known Weinberg-Witten theorem [15],which says that theories with massless spin-1 particles cannot have an associated conserved current,and that theories with massless spin 2 particles cannot have a conserved stress-energy tensortransforming appropriately under spacetime symmetries. The point with this theorem is thatgauging the symmetry in order to include a spin one or spin two massless particle necssarily altersthe Poincare transformation properties of the associated current.

The statement that gauge theories have fewer degrees of freedom may seem counter-intuitive,because the gauge theories we usually study have more degrees of freedom – either additional gaugebosons or gravitons (viewing gravity as a gauging of the spacetime symmetries). Essentially, this isthe case because we need to introduce a fluctuating gauge field Aµ or gravitational field gµν in orderto identify states related by symmetry without sacrificing locality.

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Finally, we should mention that in the case of theories such as QED and non-abelian gaugetheories, the existence of Q without Jµ is fairly trivial (sometimes physicists simply neglect itsexistence entirely). The reason is that this global charge is conserved, so once the universe existsin an eigenstate of Q the eigenvalue can never change. Thus we can view universes with differentvalues of Q as different super-selection sectors, or as different theories entirely – they can never ‘talkto each other’. Also, in theories such as Yang-Mills or QCD where charge is confined, a non-zerocharged state in an infinite spacetime would have infinite energy, so if we are only interested in finiteenergy states, then we must always have Q = 0, rendering Q completely trivial.

9 Universal Long-Range Forces in AdS and CFT Currents

The central point of this section is that spin one gauge fields Aa in AdS are dual to spin one conservedcurrents Jµ in the CFT, while the spin two graviton in AdS is dual to the spin two energy momentumtensor Tµν of the CFT. So we will begin by explaining how this correspondence works. Then ourmajor theme will be that the universality of the interactions of gauge bosons and gravitons in AdStranslates into the universality of the correlators of Jµ and Tµν in the CFT, due to Ward identities.

Some of the ideas from the previous section can be stated more formally in terms of Wardidentities for conserved currents in CFTs. We will derive these CFT identities and then we willexplain how they relate to the universality of long-range forces in AdS.

9.1 CFT Currents and Higher Spin Fields in AdS

Before we discuss symmetries and dynamics, let us note a few kinematic features of conservedcurrents in CFTs and massless particles in AdS. The basic point will be to note how the countingof degrees of freedom match between AdS and the CFT. This is essentially a quick version of theanalysis we went through in the first several sections of these notes for a scalar particle, where weshowed that one-particle scalar states in AdS correspond to the states created by O(0) acting onthe AdS/CFT vacuum. Here the point is that Jµ(0)|0〉 corresponds to a vector particle in AdSin its ground state, while Tµν(0)|0〉 corresponds to a spin two particle in AdS in the ground state.Conservation of the currents corresponds to massless particles in AdS, which have fewer degrees offreedom due to a gauge redundancy. If the AdS theory is free (corresponding to coupling g = 0 orGN = 0) then we just have Gaussian correlators for Jµ or Tµν . Let us give a quick explanation ofsome of these statements.

Conserved currents satisfy

∂νJνµ2···µ` = 0 or [P ν , Jνµ2···µ` ] = 0 (9.1)

We will assume that these are symmetric traceless tensors. At the level of the CFT states we have

P νJνµ2···µ` |0〉 = 0 (9.2)

In other words, this descendant state and all others that normally would have followed from furtherapplications of P µ vanish. This means that a large number of states that would have been presentfor a current that wasn’t conserved have been eliminated from the Hilbert space of the theory.

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For a spin 1 current Jµ, we have d − 1 independent primary DoFs, where we are comparingto the case of a scalar operator with 1 DoF. For a traceless spin 2 current Tµν , we naively haved(d + 1)/2 − 1 degrees of freedom, but another d are eliminated by the conservation condition,leaving us with only d(d− 1)/2− 1 DoFs. Similar counts can be applied to higher spin currents.

A simple calculation determines that ∆ = d − 2 + ` for conserved currents. This is the CFTscaling dimension of a massless spin ` field in AdS. Our prior derivation of the unitarity boundproves that if ∆ = d− 2 + ` then the current must be conserved. For the converse, note that since Jis primary we have

0 = [Kα, [Pβ, Jµ1···µ` ]] + [Jµ1···µ` , [Kα, Pβ]]

= [Kα, [Pβ, Jµ1···µ` ]] + 2∆JJµ1···µ`ηαβ + 2∑

i

ηαµiJµ1···β···µ` − ηβµiJµ1···α···µ` (9.3)

Now if we take the trace via ηβµ1 then the first commutator term vanishes by conservation, giving

0 = ∆JJαµ2···µ` − (d+ `− 2)Jαµ2···µ` (9.4)

where the −2 comes from one missing term when i = 1 and a single non-vanishing contribution fromthe first term in the summand, also when i = 1. So we have shown that ∆J = d− 2 + `.

The point of these observations is that they agree with the DoF counting for massless fields withspin – specifically, a massless gauge boson AA in AdSd+1 containts the same degrees of freedom asa conserved current Jµ in a CFTd, and a massless graviton hAB in AdSd+1 has the same degreesof freedom as the stress-energy tensor Tµν in a CFTd. In the case of gravitons and gauge bosons,the extra degrees of freedom between the massive and massless case are eliminated by the gaugeredundancy of either a gauge theory with Lie Group symmetry or the diffeomorphism redundancy ofGeneral Relativity. So to summarize, conserved spin ` CFT currents are dual to massless AdS fieldsof spin ` whose description necessarily involves extra redundancy in order to eliminate unphysicaldegrees of freedom.

Now let us see more explicitly how the mapping works for a simple example. Chapter 7 ofRaman’s notes [8] also give a nice discussion. First consider a free massive spin one boson withaction

S = −∫

AdS

dd+1X√−g

(1

4FABF

AB +1

2m2ABA

B

)(9.5)

where FAB = ∇AAB −∇BAA as usual, and indices are raised and lower with the AdS metric. Thistheory does not have a gauge symmetry. Alternatively, one can view this as a gauge theory that hasbeen higgsed and written in unitarity gauge.

However, as in any theory where the kinetic term comes from F 2, the component A0 does nothave a kinetic term (there is no term of the form ∂tA0 anywhere in the action). This means that A0

is non-dynamical, so we can use its equations of motion to eliminate it, after which point we willhave d+ 1− 1 = d separate degrees of freedom. Concretely, the AB field can create d distinct statescorresponding to a single particle at rest; these are the d states created by a general current Jµ(0) inthe CFT when it acts on the CFT vacuum.

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The equation of motion for AB is

∇MFMN −m2AN = 0 (9.6)

as usual. In the Poincare patch coordinates for e.g. AdS5 from the equations of motion we obtain

z3∂z

(Azz3

)= ∂µA

µ (9.7)

so that if we wish we can express the Az component in terms of the others, instead of eliminatingthe A0 component as is conventional. Explicitly, we have

Az(z, x) = z3

∫ z

0

dw∂µA

µ(x,w)

w3(9.8)

The utility of this method is that we can write

limz→0

Aµ(z, x)

z∆−1= Jµ(x) (9.9)

and so these components become the (not conserved) CFT current Jµ of dimension ∆. Note thatto get a finite result we scale Aµ by its twist τ = ∆ − `, not its dimension, as we approach theboundary. One can obtain similar results for massive tensor fields. But we also see what happenedto Az. We have

limz→0

Az(z, xµ)

z∆= lim

z→0

∂µAµ(z, xµ)

z∆−1= ∂νJν(xµ) (9.10)

Az is a redundant degree of freedom which becomes the (also redundant) descendant operator ∂νJνin the CFT.

Now let us consider the case of a massless gauge boson in AdS. In this case the action has thegauge redundancy under

AM(X)→ AM(X) +∇MΛ(X) (9.11)

for any scalar function Λ(X) on AdS. However, the field strength FMN is gauge invariant in theabelian case, and it transforms as an ordinary adjoint field in the non-abelian case. Thus we can useits components Fµz to directly define

Jµ(x) = limz→0

Fµz(z, x)

zd−3(9.12)

Note that here we have that τ = ∆− ` = d− 2, and we removed an additional power of z from thedenominator in this definition because of the derivatives in Fµz.

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Another natural way to obtain this result is to fix the gauge Az(z, x) = 0. There is a residualgauge symmetry – we can shift Aµ by ∂µΛ(x) independent of z – and so we can also take ∂µA

µ = 0in the directions orthogonal to z. This immediately implies that if we take

Jµ(x) = limz→0

Fµz(z, x)

zd−3

=∂zAµ(z, x)

zd−3

=Aµ(z, x)

zd−2(9.13)

and furthermore that we will have ∂µJµ(x) = 0 from our residual gauge condition, so Jµ(x) will be a

conserved current, as expected.A similar analysis applies to massive and massless linearized gravitational fields hMN in AdS.

See Raman’s notes [8] for the details.

9.2 CFT Ward Identities and Universal OPE Coefficients

Ward identities are a formal consequence of symmetries as applied to correlation functions in QFTs.They play an especially important role in CFTs because they give us a lot of information about theinteractions of currents Jµ and especially the energy momentum tensor Tµν with other operators inthe theory. In particular, Ward identities for the CFT currents

〈Jµ(y)O1(x1)O2(x2)〉 and 〈Tµν(y)O1(x1)O2(x2)〉 (9.14)

directly relate to the universality of gauge and gravitational forces in AdS.The Ward identity states that

∂

∂yµ〈Jµ(y)O1(x1)O2(x2) · · · On(xn)〉 = −i

n∑

m=1

δd(y − xm)〈O1(x1) · · · [GmOm(xm)] · · · On(xn)〉(9.15)

where Gm enacts (represents in the group theory sense) the symmetry transformation correspondingto Jµ on a given operator Om. If Om does not transform under this symmetry, then it does notcontribute to the right hand side. Let us give the path integral derivation of this Ward identity forthe conserved current Jµ(x).

To derive the identity, consider the path integral computation of a correlator

〈O1(x1) · · · On(xn)〉 =

∫[Dφ]O1(x1) · · · On(xn)e−S[φ] (9.16)

where the operators Oi are some general operators in the theory with fields represented by φ. Nowif we perform an infinitesimal symmetry transformation under which

δεOi = −iε(x)GiOi (9.17)

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then the action and the operators appearing within the path integral will both transform. Byassumption, the action would be invariant if ε(x) were a constant, and the measure of integration isinvariant, so we must have

〈O1(x1) · · · On(xn)〉 =

∫[Dφ] (1 + δε) (O1(x1) · · · On(xn)) e−S[φ]−

∫ddx∂µJµ(x)ε(x) (9.18)

Expanding to first order in ε tells us that

〈δε (O1(x1) · · · On(xn))〉 =

∫ddx ε(x)∂µ〈Jµ(x)O1(x1) · · · On(xn)〉 (9.19)

In order for this identity to hold for any infinitesimal function ε(x), we must have the Ward identityof equation (9.15).

If it exists, the conserved charge associated with Jµ is

Q =

∫dd−1xJ0(t, x) (9.20)

This charge will be time-independent because ∂µJµ = 0. We can also use the Ward identity to give

a formal proof that

[Q,Om] = −iGmOm (9.21)

for Lorentzian correlators (where it is more obvious what ‘time’ means). For this purpose letOm(tm, xm) have tm distinct from the other ti in any n-pt correlator. Now let us take the Wardidentity and integrate over the spatial y and between times ty ± ε for an infinitesimal ε. Let uschoose m = 1 for convenience. This gives

〈[Q(t1 + ε)O1(x1)] · · · On(xn)〉 − 〈[Q(t1 − ε)O1(x1)] · · · On(xn)〉 = −i〈G1O1(x1) · · · On(xn)〉 (9.22)

Since these are Lorentzian correlators, analytically continued from Euclidean spacetime, they will betime ordered. Since the other operators appearing in the correlator were completely arbitrary, thisidentity can be interpreted as equation (9.21).

Our proof of the Ward identity relied on a path integral for the CFT. One can also derive theWard identity more directly from the CFT axioms [16, 17] by expanding the OPE of Jµ(x)Oi(0)and constraining the possible terms that appear using conformal invariance.

Now let us consider the Ward identity for the correlator between an abelian spin one current andany number of operators. If we integrate the identity over all y, we find that the LHS vanishes, andso we must have that

∑

i

qi = 0 (9.23)

for the correlator to be non-vanishing, where Gi = qi for an abelian current. This also follows fromthe fact that the vacuum is uncharged, so Q|0〉 = 0. If we apply the Ward identity to a non-vanishing3-pt correlator with the current, we see

∂µ〈Jµ(y)O(x1)O†(x2)〉 = −iq[δd(y − x1)〈O(x1)O†(x2)〉 − δd(y − x2)〈O(x1)O†(x2)〉

](9.24)

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Recall that 3-pt functions such as 〈JOO†〉 are completely determined by conformal invariance upto some overall constant. Once the operator O is normalized, the Ward identity completely fixesthis constant to be the charge q of O. In other words, the Ward identity tells us that conservedcurrents must have a 3-pt coupling proportional to charge. Of course this 3-pt coupling also sets theOPE coefficient for the appearance of Jµ in the OPE of O(x)O†(0). The Ward identity is the CFTversion of the statement that massless gauge bosons must couple to a conserved charge.

Now let us consider the analogous statements for the energy momentum tensor Tµν . In thiscase we have the ‘obvious’ Ward identites associated with the conservation condition ∂µT

µν = 0which give information about translations. However, there are also various non-obvious identitiesassociated with rotations, dilatations, and special conformal transformations. The first also followsfrom conservation of T µν , but the latter are associated with the tracelessness condition T µµ = 0. Thisis the conservation condition for the scale current

Sµ(x) = xνTµν (9.25)

We already know that D is the charge associated with the scale current, and so it acts as [D,O(x)] =−i(∆ + x · ∂x)O(x). Thus we see that

∂µ〈Sµ(y)O(x1)O†(x2)〉 = 〈T µµ (y)O(x1)O†(x2)〉+ 〈yν∂µT µν(y)O(x1)O†(x2)〉= −i∆

[δd(y − x1)〈O(x1)O†(x2)〉+ δd(y − x2)〈O(x1)O†(x2)〉

](9.26)

−i[δd(y − x1)x1 · ∂x1〈O(x1)O†(x2)〉+ δd(y − x2)x2 · ∂x2〈O(x1)O†(x2)〉

]

We see that the final term on the first line is equal to the terms on the last line – this identity is justthe Ward identity for translations. This means that

〈T µµ (y)O(x1)O†(x2)〉 = −i∆[δd(y − x1)〈O(x1)O†(x2)〉+ δd(y − x2)〈O(x1)O†(x2)〉

](9.27)

This identity immediately tells us that the 3-pt correlation function 〈Tµν(y)O(x1)O†(x2)〉 has auniversal coefficient proportional to ∆. Note that this 3-pt function exists for all primary operators inthe theory, since Tµν generates the conformal transformations on all operators/states. This contrastswith the abelian current Jµ we considered above, which only acts on the sector of charged states. TheWard identity we have derived is the CFT version of the statement that gravity couples universallyto energy-momentum, whose role is played here by the dimension-charge ∆.

As a final comment, let us consider the 3-pt correlator

〈Tµ1ν1(x1)Tµ2ν2(x2)Tµ3ν3(x3)〉 (9.28)

In general d this correlator can involve various tensor structures, and it needs to be normalized. Itis natural to normalize the 2-pt function of Tµν via

〈Tαβ(x)Tµν(0)〉 =CTx2d

Iαβ,µν(x) (9.29)

with

Iαβ,µν(x) ≡ 1

2(IαµIβν + IανIβµ)− 1

dηαβηµν with Iab ≡ ηab − 2

xaxbx2

(9.30)

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A natural normalization of Tµν occurs because various integrals of Tµν give the conformal generators;e.g. Pµ which must act on the conformal primaries in a specific way. This fixes CT , a central chargeof the CFT. Roughly speaking, one can think of CT as counting the number of degrees of freedom ofthe theory. This follows intuitively because the direct product of two CFTs with central charges C1

and C2 yields a new CFT with central charge C1 + C2, since the energy-momentum tensors in thetwo CFTs are decoupled.

Once the energy-momentum tensor is normalized the coefficient of each tensor structure in its3-pt correlator gives dynamical information about the theory. Since all 〈TµνOO†〉 correlators mustbe proportional to ∆O, by looking at the universal energy-momentum 3-pt correlator we can fixthe proportional constant once and for all for a given theory. For more details of this see [17]. Thisproportionality constant is the inverse square root of a central charge of the CFT. This centralcharge is proportional to a power of MplRAdS in the AdS theory.

In general d there are several central charges, and they are more conventially discussed ascoefficients of the expansion of the ‘trace anomaly’, which is 〈T µµ 〉 evaluated when the CFT is placedin a general curved spacetime. For example, in d = 4 we have

〈T µµ 〉 =c

16π2WabcdW

abcd − a

16π2

(RabcdR

abcd − 4RabRab +R2

)− a′

16π2∇2R (9.31)

where the terms on the right hand side are various curvature tensors for the 4-d spacetime in whichthe CFT is living. Note that WabcdW

abcd = RabcdRabcd − 2RabR

ab +R2/4 is the square of the Weyltensor, and the terms proportional to a are the 4-d Euler density. Also, the quantity a′ is schemedependent, but both a and c are physically meaningful, and a is known to decrease under RG flows.These coefficients can also be computed directly from the 3-pt correlator of Tµν in flat spacetime,see [17]. In AdS5 theories where gravity has only an Einstein-Hilbert action term, it turns out thata = c, and so both are proportional to (M5RAdS)3, where M5 is the 5-d Planck scale.

9.3 Ward Identites from AdS and Universality of Long-Range Forces

In the previous section we saw how CFT Ward identities imply that conserved spin one currentsmust couple to conserved charges, while the stress-energy tensor must have couplings proportionalto the dimension ∆. In both cases by a ‘coupling’ we meant the magnitude of the universal 3-ptcorrelators

〈Jµ(y)O1(x1)O2(x2)〉 and 〈Tµν(y)O1(x1)O2(x2)〉 (9.32)

which also set the OPE coefficients for the appearance of Jµ and Tµν in the OPE of O1(x)O2(0).Note that these operators O1 and O2 were any operators in the theory, not just the operators dualto single-particle states in AdS. However, in the case where we have some AdS field theory involvinga gauge boson Aµ or a graviton hµν coupling to some other field φ, we will have some linearizedcouplings in AdS of the schematic form

q

∫dd+1X

√−g(φ†∂µφ− ∂µφ†φ)Aµ or√GN

∫dd+1X

√−g∂µφ∂νφhµν (9.33)

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Om(xm)

FAz

Figure 11: This figure illustrates the derivation of the Ward identity for a CFT current from Gauss’sLaw in AdS. Taking the surface integral towards the boundary of AdS gives the integral form of theWard Identity given in equation (9.34).

If we compute the AdS correlator 〈φ(X1)φ(X2)Aµ(X3)〉 or 〈φ(X1)φ(X2)hµν(X3)〉 and then sendXi → Pi on the boundary of AdS as usual, then we will recover the 3-pt CFT correlators of a currentor the stress-energy tensor from equation (9.32). The normalization of these 3-pt correlators willthen either be the charge or the energy in Planck units, for Aµ and Tµν , respectively.

Let us give a quick derivation/interpretation of the CFT Ward identity for a gauge field AMin AdS in order to better understand the physics. The Ward identity can be re-expressed as thestatement that

∫

Bm

ddy∂

∂yµ〈Jµ(y)O1(x1)O2(x2) · · · On(xn)〉 = −i〈O1(x1) · · · [GmOm(xm)] · · · On(xn)〉 (9.34)

where Bm is a d-dimensional ball containing the point xm and none of the others; we can make theball Bm as small as we like. Now the left hand side can be re-expressed as a surface integral

∫

∂Bm

dd−1sµ〈Jµ(y)O1(x1)O2(x2) · · · On(xn)〉 = −i〈O1(x1) · · · [GmOm(xm)] · · · On(xn)〉 (9.35)

where the normal to the surface of Bm is contracted with Jµ. Now let us use the definition of thecurrent from the bulk gauge field, equation (9.13), to write this as

limz→0

z3−d∫

∂Bm

dd−1sµ〈Fµz(y, z)O1(x1) · · · On(xn)〉 = −i〈O1(x1) · · · [GmOm(xm)] · · · On(xn)〉 (9.36)

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Finally, let us interpret this equation by viewing all of the operators as z → 0 limits of bulk fieldsφi(x, z). In that case, the surface integral, pictured in figure 11, is just

∫

∂Bm

dd−1sAEA(s, z) = qm (9.37)

where the integral is being performed at fixed small z in AdS. This is the integral of the AdS ‘electricfield’ EA = FAz around the state created by the mth operator. The other particles/blobs/states areirrelevant because they are infinitely far away in the limit that z → 0. So the Ward identity in theCFT just corresponds to Gauss’s law in AdS!

Now we will review an old argument of Weinberg that proves that in flat spacetime, masslessspin one particles must couple to conserved charges while massless spin two particles must couple toenergy-momentum. The argument also shows that higher-spin massless particles cannot couple in away that would lead to long-range forces. This classic theorem goes a long way towards explainingwhy the spectrum of elementary particles we observe ends at the spin two graviton. The argumentitself is independent of AdS/CFT, but it is probably the best general argument one can give in orderto explain why massless spin one and spin two particles couple in a universal way at long distances.

We begin by considering some scattering amplitude Mn(pi) for n particles in flat spacetime.Given this n-particle process, what is the amplitude for emitting an additional very soft photon withmomentum q, so that q · pi pj · pk? This question is of great importance because the exchange ofparticles with small energy and momentum is what leads to long-range forces.

If the soft photon is emitted from any external line in the scattering amplitude then we find acontribution

Mn(pi)n∑

m=1

empµm

pm · q − iεεµ (9.38)

where εµ is the polarization vector for the soft photon, and em is the coupling constant for themth particle. One might also wonder if the photon can couple to vectors made from the spin ofthe mth particle, but this will also not give a pole as q → 0 since multipole moments cannot givelong-distance effects. There will be other contributions from emission of the photon from internalparts of the diagram, but these will not have a pole as q → 0, so they will be subdominant in thesoft limit. The soft emission amplitude is dominated by contributions from long time periods oforder 1/q associated with the nearly free propagation of the initial and final state particles.

The pole we have found literally corresponds to a sum over contributions of the form

eiem∫dtnµmAµ(x(t)) − 1 ≈ iem

∫dt nµmAµ(x(t)) + · · ·

= iem ε · n∫dteiq·x(t) (9.39)

which is just the amplitude for a charged particle to source an electromagnetic field, with x(t) = tn,because the action for such charged particle is just the integral of Aµx

µ(t) along the world-line of

93

the particle. This gives the contribution

n∑

m=1

em ε · nm∫ ∞

0

dteitnm·q (9.40)

where the integral over time proceeds from the scattering process out to infinity, and nm is a unitvector pointing in the direction of pm. Performing the integral gives the soft emission amplitude.

Consider the Lorentz transformation properties of this scattering amplitude. In particular, notethat the polarization tensor εµ for a massless particle does not transform as a vector, instead ittransforms as

εµ → Λνµεν + α(q,Λ)qµ (9.41)

where α is some function. So in other words, the polarization tensor shifts by qµ under generalLorentz transformations. This means that to recover a Lorentz invariant S-Matrix we must have

Mn(pi)n∑

m=1

empµm

pm · q − iεqµ =Mn(pi)

n∑

m=1

em = 0 (9.42)

The sum of the charges of the scattering particles must vanish. In other words, a massless spin oneparticle must couple to a conserved charge.

Consider the Lorentz transformation properties of this object. In particular, note that thepolarization tensor εµ for a massless particle does not transform as a vector, instead it transforms as

εµ → Λνµεν + α(q,Λ)qµ (9.43)

where α is some function. So in other words, the polarization tensor shifts by qµ under generalLorentz transformations. This means that to recover a Lorentz invariant S-Matrix we must have

Mn(pi)n∑

m=1

empµm

pm · q − iεqµ =Mn(pi)

n∑

m=1

em = 0 (9.44)

The sum of the charges of the scattering particles must vanish. In other words, a massless spin oneparticle must couple to a conserved charge.

In the case of a massless spin two particle, the soft emission amplitude must be

Mn(pi)n∑

m=1

Gmpµmp

νm

pm · q − iεεµν (9.45)

where εµν is the polarization tensor of the graviton, and Gm are its coupling constants. Thepolarization tensor transforms as

εµν → ΛαµΛβ

ν εαβ + αµqν + ανqµ (9.46)

94

and so in order to have a Lorentz invariant S-Matrix, we must have

n∑

m=1

Gmpµmqµ = 0 (9.47)

But the only way this can be zero for all qµ without putting a restriction on the allowed momenta pµfor scattering is if Gm = G is a universal constant, so that the equation is satisfied due to momentumconservation. Thus a massless spin two particle must couple universally to energy-momentum.Repeating this argument for higher spin massless particles shows that they cannot couple in a waythat will give rise to long-range forces, which necessarily involve propagators with poles in the softq → 0 limit.

As promised, we have shown that massless particles must couple to conserved charges or momenta,meaning that they couple to both fundamental and composite particles/objects/states with universal3-pt vertices. Interpreted in AdS, these universal 3-pt vertices are just Feynman diagrams that giverise to universal correlators for Jµ and Tµν with any operators O1 and O2.

9.4 Cluster Decomposition and Long-Range Forces from the Bootstrap

We will prove that every CFTd in d ≥ 3 with a scalar operator φ must contain infinite sequences ofoperators Oτ,` with twist approaching τ → 2∆φ + 2n for each integer n as `→∞. We show howthe rate of approach is controlled by the twist and OPE coefficient of the leading twist operator inthe φ× φ OPE.

This result is interesting because it can be interpreted as a statement about superhorizon locality,or ‘cluster decomposition’, in AdS for general CFTs. The universally computable sub-leadingcorrections can be further interpreted as the interaction energies due to the exchange of gravitonsand gauge bosons.

9.4.1 Existence of Operators with τ → ∆1 + ∆2 + 2n at Large Spin

Let us begin by considering what naively appears to be a paradox. Consider the 4-point correlationfunction in a CFT with only Gaussian or Generalized Free Theory type correlators. These GTFsare the dual of free field theories in AdS, as we have discussed. We will study the 4-pt correlatorof a dimension ∆φ scalar operator φ in such a theory. By definition, in GTF the 4-pt correlator isgiven as a sum over the 2-pt function contractions:

〈φ(x1)φ(x2)φ(x3)φ(x4)〉 =1

(x212x

234)∆φ

+1

(x213x

224)∆φ

+1

(x214x

223)∆φ

,

=1

(x213x

224)∆φ

(u−∆φ + 1 + v−∆φ

). (9.48)

Since this is the 4-pt correlator of a unitary CFT, it has a conformal block decomposition in everychannel with positive conformal block coefficients. The operators appearing in the conformal block

95

decomposition are just the identity operator 1 and the “double-trace” operators On,` of the schematicform

On,` ∼ φ(∂2)n∂µ1 . . . ∂µ`φ, (9.49)

with known conformal block coefficients P2∆φ+2n,` and twists τn,` = 2∆φ + 2n. Factoring out anoverall (x2

13x224)−∆φ , the conformal block decomposition in the 14→ 23 channel reads

u−∆φ + 1 + v−∆φ = v−∆φ + v−∆φ

∑

n,`

P2∆φ+2n,` g2∆φ+2n,`(v, u), (9.50)

where the v−∆φ on the RHS is the contribution from the identity operator. If we look at the behaviorof the conformal blocks g2∆φ+2n,`(v, u), we notice a simple problem with this equation: it is knownthat the conformal blocks g2∆φ+2n,`(v, u) in the sum on the RHS each have at most a log u divergenceat small u, but the LHS has a u−∆φ divergence. Thus the LHS cannot be reproduced by any finitenumber of terms in the sum. To be a bit more precise, the conformal blocks have a series expansionaround u = 0 with only non-negative integer powers of u and at most a single logarithm appearing,so in particular we can write24

v−∆φg2∆φ+2n,`(v, u) = f0(v) + uf1(v) + u2f2(v) + . . .

+ log(u)(f0(v) + uf1(v) + u2f2(v) + . . .

). (9.51)

But this means that if the sum on the right-hand side of equation (9.50) converges uniformly, itcannot reproduce the left-hand side, which includes the negative power term u−∆φ and does notinclude any logarithms.

The simple resolution of this ‘paradox’ is that the sum over conformal blocks does not convergeuniformly near u = 0. In fact, the sum does converge on an open set with positive real u, but whenRe[√u] < 0 the sum diverges. So we must define the sum over conformal blocks for general u as the

analytic continuation of the sum in the convergent region. Crucially, the analytic continuation of thesum contains the power-law u−∆φ that is not exhibited by any of the individual terms in the sum.

Let us see how this works in a bit more detail, so that in particular, we can see that the sumover twists τ = 2∆φ + 2n at fixed ` converges in a neighborhood of u = 0, but the sum over angularmomentum diverges for u < 0. For the purpose of understanding convergence, we need only studythe conformal blocks when τ or ` are very large. In the very large τ limit with |u|, |v| < 1 the blocksare always suppressed by u

τ2 or v

τ2 . The conformal block coefficients are bounded at large τ [14].

This means that for small |u| and |v|, the sum over τ will converge. In fact, once we know that thesum converges for some particular u0, v0 we see that for u < u0 and v < v0, the convergence at largeτ becomes exponentially faster.

Now consider the ` dependence. At large ` and fixed τ the crossed-channel blocks in the|u| |v| 1 limit behave as

gτ,`(v, u) ≈ `12 2τ+2`

√π

vτ2K0

(2`√u) `√u1≈ 2τ+2`−1v

τ2e−2`

√u

4√u

, (9.52)

24See for example equation (2.32) in [18]. The sum over powers of (1− v) can be performed explicitly to find ahypergeometric function, whose singularities at v ∼ 1 are known.

96

where K0 is a modified Bessel function. Notice that for Re[√u] > 0 there is an exponential

suppression at very large `, but for Re[√u] < 0 there is an exponential growth. Note also that at

small v the lowest twist terms (n = 0) will dominate.Now, the mean field theory conformal block coefficients in any dimension d are [19]

P2∆φ+2n,` =

[1 + (−1)`

](∆φ − d

2+ 1)2

n(∆φ)2n+`

`!n!(`+ d2)n(2∆φ + n− d+ 1)n(2∆φ + 2n+ `− 1)`(2∆φ + n+ `− d

2)n, (9.53)

where the Pochhammer symbol (a)b ≡ Γ(a+ b)/Γ(a). In particular, for n = 0 and at large even ` wecan approximate

P2∆φ,``1≈ q∆φ

√π

22∆φ+2``2∆φ− 3

2 , (9.54)

where q∆φis an `-independent prefactor.25 Thus the sum in Eq. (9.50) at large ` and |u| |v| 1

takes the form

v−∆φ

∑

n, large `

P2∆φ+2n,` g2∆φ+2n,`(v, u) ≈ q∆φ

∞∑

large even `

`2∆φ−1K0

(2`√u). (9.55)

This sum converges at large ` for positive real√u, and so we will define it by analytic continuation

elsewhere in the complex u plane. As can be easily seen by approximating the sum with an integral,the result reproduces the u−∆φ power-law term on the left-hand side of equation (9.50), as desired.Thus we have seen that general power-laws in u are reproduced by conditionally convergent large `sums in the conformal block decomposition, with a power-law dependence on ` producing a relatedpower of u as u→ 0.

In fact, we can generalize this argument from GFT to any CFT in d ≥ 3 dimensions. The pointis that the 4-pt correlator

〈φ(x1)φ(x2)φ(x3)φ(x4)〉 =1

(x213x

224)∆φ

(u−∆φ + u

τ∗2−∆φ (· · · )

). (9.56)

where the · · · are a sum over terms that are finite in the u→ 0 limit, and the twist τ∗ ≥ d2− 1 by

the CFT unitarity bound (and in fact τ∗ = d− 2 typically, due to Tµν). So in fact in the limit thatu→ 0 the leading term u−∆φ from the identity operator can be separated from all other terms, andit can only be reproduced by the towers of operators we found in GFT in the limit `→∞. Onecan make the argument very rigorous by using the positivity properties of the conformal blocks andconformal block coefficients that follow from unitarity.

9.4.2 Properties of Isolated Towers of Operators

So far, we have made no assumptions about the precise spectrum of operators that accumulate at thespecial values τ ∼ 2∆φ + 2n. However, it is interesting to consider the case where an accumulation

25Explicitly, q∆φ=(

8Γ(∆φ)2

).

97

point 2∆φ + 2n is approached by a single tower of operators O2∆φ+2n,` with ` = 0, 2, . . . , which areseparated by a twist gap from other operators in the spectrum at sufficiently large spin. This occursin every example we are aware of, including all theories with a perturbative expansion parametersuch as 1/N (see Appendix ??) or the ’t Hooft coupling in e.g. N = 4 SYM, and we will alsodiscuss some non-perturbative examples at the end of this subsection. It would be very interestingto identify any CFTs without operators with twists near 2∆φ + 2n for every sufficiently large valueof `.

With this additional assumption, we will be able to characterize subleading corrections to thebootstrap equation in the small u limit. On the left-hand side, these corrections come from theoperators Om with minimal nonzero twist. Thus, we have the approximate relation

1 +2∑

`m=0

Pmuτm2 fτm,`m(0, v) ≈

∑

τ,`

Pτ,` vτ2−∆φu∆φfτ,`(v, u), (9.57)

which is valid up to subleading corrections in u in the limit u→ 0. We have assumed that `m ≤ 2because higher spin operators either have twist greater than that of the energy-momentum tensor or,as argued in [?, ?], they are part of an infinite number of higher-spin currents that couple as if theywere formed from free fields. Dolan and Osborn [?] have given a formula in general d appropriate forthe conformal blocks corresponding to Om exchange on the left-hand side, where we have expandedat small u. This is

fτm,`m(0, v) = (1− v)`m2F1

(τm2

+ `m,τm2

+ `m, τm + 2`m, 1− v). (9.58)

It will be important that this hypergeometric function can be expanded in a power series at small vwith terms of the form vk(ak + bk log v). The logarithms will be related to the ‘anomalous dimensions’that emerge at large `.

Now we expect there to be a finite separation between the lowest twist τm and the other twistsin the theory. To prove this, we consider two cases separately, that of τm < d − 2 and that ofτm ≥ d− 2. In the former case, Om must be a scalar operator due to the unitarity bound. We willassume that there are a finite number of scalar operators with dimension below any given value,26

which immediately implies that the twist of Om must have a finite separation from the other twistsin the theory. To have a non-vanishing 4-pt correlator φ must be uncharged, so in the absence oflower twist scalars we have τm = d − 2, because the energy-momentum tensor will appear in theφ(x)φ(0) OPE. We can then apply the Nachtmann theorem, which says that minimal twists must benon-decreasing functions of `, to conclude that τm is separated from the other twists in the theory.Note that, crucially, unless φ is a free scalar field we have τm

2−∆φ < 0, so the sub-leading powers

of u grow as u→ 0. Taken together, these comments imply that there is a limit u→ 0 where theexchange of the identity operator plus a finite number of Om dominates the left-hand side of thebootstrap equation.

Assuming the existence of an operator with twist approaching 2∆φ + 2n for each `, we wouldlike to constrain the deviation of their conformal block coefficients δP2∆φ+2n,` from MFT and their

26This assumption would follow, for instance, from the assumption that the CFT has a well-defined partitionfunction at non-zero temperature, or that the four-point function of the energy-momentum tensor is finite.

98

anomalous dimensions g(n, `) ≡ ∆On,` − 2∆φ − 2n− `. This is possible because the Om contribute adominant sub-leading contribution at small u, with a known v-dependence that can be expanded ina power series with integer powers at small v. The fact that we have only integer powers vn andvn log v multiplying u

τm2 on the left-hand side of equation (9.57) means that the right-hand side can

reproduce these terms only with the conformal blocks we just discovered above, namely those with

twists approaching the accumulation points τ(n, `) = 2∆φ + 2n + γ(n, `) so that vτ(n,`)

2−∆φ is an

integer power in the `→∞ limit.In fact, expanding Eq. (9.58) for the Om conformal blocks at small v gives

fτm,`m(v) =Γ(τm + 2`m)(1− v)`m

Γ2(τm2

+ `m)

∞∑

n=0

((τm2

+ `m)n

n!

)2

vn[2(ψ(n+ 1)− ψ

(τm2

+ `m

))− log v

],

(9.59)where ψ(x) = Γ′(x)/Γ(x) is the Digamma function, and (a)b = Γ(a+ b)/Γ(a) is the Pochhammersymbol. The details of this formula are not especially important, except insofar as it makes explicitthe connection between the coefficients of vn and vn log v in the series expansion. We will nowsee that the vn terms must come from δP2∆φ+2n,` while the vn log v terms are a consequence ofγ(n, `). This means that the ‘anomalous dimension’ and the correction to the conformal blockcoefficients must be related at large `. These quantities were seen to be related [20] to all orders inperturbation theory [19] in the presence of a 1/N expansion, so our result extends this relation to anon-perturbative context.

The vn log v terms in equation (9.59) can only be reproduced by expanding vτ2−∆φ in γ(n, `) in

the large ` conformal blocks. For simplicity let us consider the situation where there is only oneoperator accumulating near 2∆φ + 2n for each `, and that the conformal block coefficients approachPMFT

2∆φ+2n,`. In this case we can write the RHS of the crossing relation as

∑

`

PMFT2∆φ+2n,`

[γ(n, `)

2log v

]`

12 4`√πz∆φK0(2`

√z)vn(1− v)∆φF (d)(2∆φ + 2n, v), (9.60)

where we have used a Bessel function approximation to k2`(1− z) discussed in Appendix ??.27 Inorder for the sum to produce an overall factor of u

τm2 ∼ z

τm2 , we must have power law behavior in

γ(n, `) at very large `,

γ(n, `) =γn`τm

, (9.61)

with a coefficient γn related to the OPE coefficient Pm of the leading twist operator Om in equation(9.57). In large N theories, the coefficients γn are suppressed by powers of 1/N . However, we stress

that all we need in order to expand vγ(n,`)

2 in log v in the large ` sum is the property that it is powerlaw suppressed as `→∞, which is true even if the coefficients γn are O(1) or larger.

27Strictly speaking, the Bessel function approximation breaks down for ` 1/√z, so we are here implicitly using

the fact that the sum including the MFT coefficients is dominated by the region of fixed `2z, where the approximationis valid. If the reader is concerned about this, one can instead write these sums using the hypergeometric functionk2`(1− z). However, we find the Bessel function formulae to be useful for explicitly doing computations using theintegral approximations to these sums.

99

The integer powers of v in equation (9.59) must then be reproduced by

∑

`

PMFT2∆φ+2n,`

[δP2∆φ+2n,` +

1

2γ(n, `)

d

dn

]vn`

12 4`√πz∆φK0(2`

√z)(1− v)∆φF (d)(2∆φ + 2n, v), (9.62)

where the γ(n, `) ddn

piece comes from expanding the τ dependence of the conformal block in small

γ(n, `). Again requiring that we correctly produce the overall uτm2 ∼ z

τm2 behavior, we must have

δP2∆φ+2n,` =cn`τm

(9.63)

to leading order in 1/` in the large ` limit, with a coefficient cn related to γn.As an example, it is particularly simple to do this matching explicitly for the leading twist tower

with n = 0. In this case, matching the log v and v0 terms gives the relations

γ0 = −Pm2Γ(∆φ)2Γ(τm + 2`m)

Γ(∆φ − τm2

)2Γ( τm2

+ `m)2, c0 =

[ψ(τm

2+ `m

)+ γ + log 2

]γ0, (9.64)

where γ is the Euler-Mascheroni constant. It is important to note the relative sign between γ0 andPm (which is strictly positive), since this is required in order to satisfy the Nachtmann theoremasymptotically at large `. It is then straightforward to continue this matching to higher orders in v.

10 AdS Black Holes and Thermality

Let us begin with a thought experiment. Consider a CFT living on the Lorentzian cylinder R×Sd−1,and let us slowly heat it up. Since the dilatation operator serves as the Hamiltonian, as we increasethe temperature the CFT will be in a state characterized by larger and larger operator/statedimensions.

Since the AdS Hilbert space is identical to that of the CFT, we can interpret our hot CFT as athermal state in AdS. But what will this state consist of? At low temperatures we will just have athermal gas made up of the light particles in AdS. Due to the AdS geometry, these particles willmostly move around near the center of AdS, with only occasional excursions further away. Thismeans that as we increase the temperature, we will be cramming more and more energy into a regionof roughly fixed size. In the presence of dynamical gravity, this cannot go on forever – eventually, atsome critical temperature Tc, the hot gas will collapse to form a black hole in AdS. Our thoughtexperiment shows that black holes in AdS must correspond to a hot CFT!

Now let us consider the thermodynamics more carefully. We will use the microcanonical ensemble,fixing the total energy of the system, as we will see that the microcanonical and canonical ensembleslook somewhat different in equilibrium, due to the existence of a phase with negative specific heat.

Let us nevertheless use the notation T to denote the average energy of a graviton in the gas ofgravitons. In that case the occupation number of the n, ` mode will be

1

1− e−β(∆+2n+`)≈ RAdST

∆ + 2n+ `(10.1)

100

when this ratio is large. For gravitons ∆ = d, and there will be of order T d such modes with∆ + 2n+ ` < T , so we expect that

Etot ∼ T d+1RdAdS (10.2)

for a gas of gravitons with T 1/RAdS. This is what we would expect for a box of size RAdS

in d + 1 dimensions. As we increase T , Etot will eventually grow so large that its correspondingSchwarzschild radius will be of order RAdS, at which point the gas will collapse to form a large blackhole. In d+ 1 dimensional flat spacetime the Schwarzschild radius would be

RS = (Gd+1Etot)1d−2 (10.3)

where Gd+1 = (Md+1)1−d is the Newton constant and Md+1 is the d+1 dimensional Planck mass. If weare studying a homogeneous gas, collapse will first occur on the largest scales, because gravitationaleffects are cumulative with energy. This suggests that a black hole only forms once RS ∼ RAdS,when the energy Etot is within its own Schwarzschild radius, so that

Tcollapse ∼(Gd+1R

2AdS

)− 1d+1 (10.4)

For example, in the case of AdS4 this would be a temperature√M4/RAdS, at an intermediate energy

between the AdS scale and the Planck scale.However, this analysis in the microcanonical ensemble has been rather misleading. First of

all, in most examples considered in the literature there are other regimes aside from the gas ofgravitons and the large AdS black hole – there is also a Hagedorn stringy regime, and a regime wheresmall black holes (much smaller in size than RAdS) can be produced. Furthermore, the parametricrelationships change due to the presence of extra dimensions of RAdS size. Finally, this analysiswas misleading because in actuality, the large AdS black holes have such a large entropy that inthe canonical ensemble they start to dominate already at T ∼ 1/RAdS. In other words, the hugenumber of different AdS black hole states is able to make up for their e−βE Boltzmann factor.

10.1 The Hagedorn Temperature and Negative Specific Heat

Let us briefly discuss the Hagedorn phenomenon, since it is interesting and simple. The canonicalensemble partition for any physical system is

Z[β] =∑

ψ

e−βEψ (10.5)

where as usual β = 1/T , and the sum is over all quantum mechanical states ψ in the theory. Thesum can be re-organized as

Z[β] =∑

E

eS(E)−βE (10.6)

101

where eS(E) is the number of states with energy E – this defines the entropy as a function of E. Notethat if S(E) grows faster than β∗E for some β∗ in the limit that E →∞, then this sum divergesand partition function is ill defined!

To understand this phenomenon better, note that the expectation value of the (total) energy is

〈E〉 = − ∂

∂βlogZ[β] (10.7)

Let us consider what happens if we have the borderline case S(E)→ β∗E in the limit that E →∞.Then we find

Z[β] ≈∞∑

E

e(β∗−β)E =1

β∗ − β(10.8)

so we find that the expectation value of the energy is

〈E〉 =1

β∗ − β(10.9)

Thus we see that as β → β∗, the expectation value of the energy diverges. Reversing the logic, ittakes an infinite amount of energy to achieve the temperature T∗ = 1/β∗. This T∗ is the Hagedorntemperature, and it is the maximum possible temperature for the system. Hagedorn behavior isrelevant because it is the thermodynamic behavior of a gas of strings. Thus if a stringy regime existsin AdS, then there will be a range of temperatures showing a Hagedorn behavior.

There is an even more extreme behavior, where S(E) grows more rapidly than β∗E for any fixedβ∗. For example, in d+ 1 dimensional flat spacetime, black holes have an entropy

S(RS) ∼ Rd−1S

Gd+1

(10.10)

which translates into

S(E) ∼ (Gd+1)1

2−dEd−1d−2 (10.11)

This means that S(E) grows with a power of E greater than one. This implies the famous statementthat flat space black holes have a specific heat

∂

∂T〈E〉 < 0 (10.12)

Increasing the temperature actually decreases the black hole mass, because the black hole entropydominates the free energy. This is relevant to AdS because this is the behavior of ‘small’ AdS blackholes, with RS RAdS. So these states can also contribute to the partition function in AdS/CFT.

102

10.2 Thermodynamic Details from AdS Geometry

To make all of this more precise, let us study the Euclidean CFT on S1 × Sd−1, following [21]. Nowwe will be using the canonical ensemble, so we will be studying states that minimize the free energy,which includes the effects of entropy. Compactifying the Euclidean time direction with period βcorresponds to studying the partition function of the theory on Sd−1 at a temperature T = 1/β, asusual. Note that the Sd−1 also has some radius β′, and since we are studying a CFT, the physicsonly depends on the ratio of β/β′. For convenience we can fix β′ = 1, although one can also find ascaling limit [21] where β′ →∞, so that we get a Euclidean CFT on S1 ×Rd−1.

We can write Euclidean AdS in the coordinates

ds2AdS =

(r2

R2AdS

+ 1

)dt2 +

dr2

r2

R2AdS

+ 1+ r2dΩ2 (10.13)

This coordinate system follows immediately from our usual global coordinates by taking r =RAdS tan ρ. We have introduced this particular coordinate system so that we can naturally write theAdS-Schwarzschild solution

ds2Sch =

(r2

R2AdS

+ 1− ωdM

rd−2

)dt2 +

dr2

r2

R2AdS

+ 1− ωdMrd−2

+ r2dΩ2 (10.14)

where M is the physical mass of a black hole, and the constant

ωd =16πGd+1

(d− 1)Vol(Sd−1)(10.15)

is written in terms of the d+ 1 dimensionsal Newton constant Gd+1. There is a horizon at the largestroot r+ of

r2

R2AdS

+ 1− ωdM

rd−2= 0 (10.16)

The physical space ends at r+, since after this point the metric is no longer Euclidean. The metricfor the full S1 × Sd−1 will only be smooth if we must avoid conical singularities in t. In the vicinityof r+, we can write r = r+ + δr and the metric looks like

ds2 ≈ δr

(dr2

+ + (d− 2)R2AdS

R2AdSr+

)dt2 +

1

δr

(R2AdSr+

dr2+ + (d− 2)R2

AdS

)d(δr)2 + r2

+dΩ2 (10.17)

We can define a new coordinate x = 2√δr, in which case we see that we have the metric for a cone

in x and t. To avoid a conical singularity at x = 0, we must take the periodicity in t→ t+ β with

β =4πR2

AdSr+

dr2+ + (d− 2)R2

AdS

(10.18)

103

This result means that the temperature 1/β is fixed in terms of the black hole mass M . Furthermore,β has a maximum as a function of r+, so we see that for β > βmin, which means

T < Tmin =1

RAdS

√d(d− 2)

2π(10.19)

there is no thermodynamically stable black hole. In other words, when T < Tmin the thermody-namically stable state of the spacetime will be a gas of particles (and perhaps strings and smallblack holes) in otherwise empty AdS, while for T > Tmin we can also have the AdS-Schwarzschildspacetime describing a large black hole.

The phase transition between these two states is called the Hawking-Page phase transition. Wecan compute the transition temperature by studying the difference in Free Energies between AdSand the AdS-Schwarzschild solution. It is computed in [21], and the result is

F =Vol(Sd−1)(R2

AdSr2+ − rd+1

+ )

4Gd+1(dr2+ + (d− 2)R2

AdS)(10.20)

We see that in the large r+ limit the free energy of AdS-Schwarzschild dominates. The two freeenergies are equal when r+ = RAdS, so that we have a phase transition temperature of

Tc =1

RAdS

d− 1

2π(10.21)

In other words, heating up the AdS/CFT system above this temperature (in the canonical ensemble)will result in an equilibrium state consisting of a large AdS black hole.

11 Final Topics

11.1 Various Dictionaries

In these notes we have defined CFT operators directly in terms of limits of bulk fields, via

O(t,Ω) = limε→0

φ(t, ρ(ε; t,Ω),Ω)

ε∆(11.1)

where ρ(ε; t,Ω) = π2− εf(t,Ω) and the function f(t,Ω) determines the geometry of the boundary on

which the CFT lives. This allowed us to compute correlators for CFT operators O in terms of limitsof AdS correlators using AdS Feynman diagrams.

In fact, there are two AdS/CFT ‘dictionaries’, and our choice has been the less common onein the literature. Our choice has the great advantage that it makes it obvious how to identify theHilbert spaces of the AdS and CFT theories, and it also seems more immediate and intuitive to me.The moreconventional dictionary has the advantage that it is easier to use when considering finitedeformations of the CFT.

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The conventional dictionary involves computing the AdS path integral, where the bulk fieldsφ(t, ρ,Ω) → φ0(t,Ω) for some classical field configuration φ0(t,Ω) on the boundary of AdS. Thisgives a partition functional

ZAdS[φ0] =

∫

φ0

Dφe−SAdS [φ] (11.2)

Alternatively, we could imagine computing a CFT partition function

ZCFT [φ0] =

∫

CFT

Dχe−SCFT [χ]+∫ddxφ0(x)O(x) (11.3)

where O(x) is a CFT operator and φ0(x) is a classical source for this operator. The claim of theusual AdS/CFT dictionary is that

ZAdS[φ0] = ZCFT [φ0] (11.4)

In particular, this means that we can compute CFT correlators by differentiating ZAdS with respectto φ0 and then setting it to zero, giving

〈O(x1) · · · O(xn)〉 =δ

δφ0(x1)· · · δ

δφ0(xn)ZAdS[φ0]

∣∣∣∣φ0=0

(11.5)

This form of the dictionary is particularly useful because it tells us how to compute the partitionfunction of the CFT for finite values of the sources φ0, which we can then interpret as couplingconstants, since in effect, φ0O has been added to the CFT action.

Let us see why these two dictionaries are equivalent, following [22]. To make the argument, letus use the Poincare patch for convenience. We will introduce two reference surfaces, one at somez = ε and another at z = `, with ε ` 1. The bulk path integral can be broken up as

ZAdS[φ0] =

∫Dφz<` e−Sz<`[φ]

∫Dφz=`

∫Dφz>` e−Sz>`[φ]

=

∫DφΨUV [φ0, φ; ε, `]ΨIR[φ, `] (11.6)

where φ(x) is the field φ(`, x) at the surface z = ` (so it only depends on x, not z), and we havewritten the path integrals over the intervals [ε, `] and [`,∞] in terms of a ‘UV’ and ‘IR’ wavefunction.Only the UV wavefunction knows about φ0. Our usual dictionary is defined by

〈O(x1) · · · O(xn)〉 = lim`→0

`−n∆

∫DφΨIRφ(x1) · · · φ(xn)ΨUV

∣∣∣∣φ0=0,ε=0

(11.7)

The other dictionary based on partition functions will be equivalent if

lim`→0

∫DφΨIR

[δ

δφ0(x1)· · · δ

δφ0(xn)ΨUV

]

φ0=0

∝ lim`→0

`−n∆

∫DφΨIRφ(x1) · · · φ(xn)ΨUV

∣∣∣∣φ0=0

(11.8)

105

The ‘UV’ region plays a simple role – it is only present so that we can differentiate ΨUV to obtainthe field φ at the intermediate surface at z = `.

We have previously discussed the fact that AdS interactions become irrelevant as we approachthe boundary, so we can assume that ΨUV is computed purely from the free theory. In this case wecan just calculate it directly. It is

ΨUV [φ0, φ] =

∫ φ(`)=φ

φ(ε)=φ0εd−∆

Dφ exp

[−1

2

∫ `

ε

ddxdz√−g

((∇φ)2 −m2φ2

)](11.9)

This is just a Gaussian integral, so we can evaluate it by solving the equations of motion for φ withthe correct boundary conditions and then inputting the solution back into the action. The equationsof motion

∇2φ−m2φ = 0 (11.10)

will have two solutions, which we can refer to as Gεk(z, x) and G`

k(z, x). One can compute these Gk

in terms of bessel functions in the case of pure AdS. We choose a linear combination of solutions sothat Gε

k(ε, x) = 1 and Gεk(`, x) = 0, and vice versa for G`

k(z, x). The label k is a momentum labelfor the xµ coordinates, although in principle it could be any label associated with a complete basisfor solutions. Now the solution for φ(z, k) (we have gone to momentum space for convenience) withthe desire boundary conditions is simply

φ(z, k) = φ0(k)εd−∆Gεk(z) + φ(k)G`

k(z) (11.11)

When we evaluate the action with this solution it vanishes up to a total derivative term, and so theresult only involves boundary terms of the form φ∇zφ evaluated at ε and `. Terms quadratic in φ0

or φ will not contribute when we differentiate ΨUV and then set φ0 → 0. Thus the relevant termswill be

ΨUV [φ0, φ] ∝ exp

[−1

2

∫ddk φ0(−k)φ(k)

(ε−∆∇zG

`k(ε)− `−d∇zG

εk(`)

)](11.12)

where we note that the z derivatives of the G solutions can be non-vanishing on both ends of theinterval. Evaluating the derivatives for the actual solutions gives

ΨUV = exp

[−(2∆− d)`−∆

∫ddxφ0(x)φ(x) + · · ·

](11.13)

where the ellipsis denotes other terms that either vanish more quickly as `→ 0 or that are quadraticin φ0 or φ. So we see that

(1

ΨUV

δΨUV

δφ0

)

φ0=0

= `−∆(2∆− d)φ(x) (11.14)

as desired, and the equivalence between the two dictionaries has been proven.

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11.2 Breaking of Conformal Symmetry

In AdS/CFT, the conformal symmetry of the CFT is realized geometrically as the group of AdSisometries. If we want to use holography to study theories that are not conformally invariant, thenwe must study quantum field theories in spacetimes with less symmetry than AdS. In order to dominimal violence to the conformal symmetry, we can study Poincare invariant theories that areapproximately conformal in some regime, for example at very high or very low energies. However, weshould note that recently AdS/CFT type ideas have been used to attempt to study toy non-relativisticsystems in order to make contact with condensed matter physics.

Since we wish to preserve Poincare symmetry, it is most natural to view AdS in the coordinates

ds2 =1

z2

(dz2 + ηµνdxµdxν

)(11.15)

Now we would like to break conformal symmetry. The laziest option is to simply declare, by fiat,that there exist membranes, conventionally called either the ‘UV Brane’ and the ‘IR Brane’, that endspacetime at some finite positions zUV zIR. In this case the AdS isometries will only be brokenby the presence of these membrane, in the same way that a brick wall breaks translation invariance.

Note that the positions zUV and zIR transform under dilatations

D = z∂z + xµ∂µ (11.16)

as lengths, so they specify a short and long distance scale, respectively. These obviously correspondto energy scales µUV = 1/zIR and µIR = 1/zIR.

Let us first consider the case when zIR =∞, so that the IR brane is absent and we only have aUV brane. If we like we can add whatever d dimensional quantum field theory we like by includingan action localized to the UV brane. We can (roughly) interpret the UV Brane as setting a UVcutoff µUV for the CFT, and the fields localized to the UV brane represent some other theory thatcouples to the CFT. In particular, when we quantize gravity in this space, the boundary conditions(or UV brane action) for the graviton will be important. If we do not impose Dirichlet boundaryconditions then the graviton will have a normalizable zero mode, and so our theory will included-dimensional gravity.

Now let us consider an IR brane at zIR in the absence of a UV Brane. In this case there canbe no dynamical gravity, or any other degrees of freedom aside from the CFT. However, what wedo have is some kind of conformal symmetry breaking associated with the energy scale µIR. Sincethis IR brane comes out of nowhere, we do not have any particular insight into the nature of theconformal symmetry breaking in the CFT. However, what we can say is that the IR brane willfluctuate in position (do to gravitational effects if nothing else), and in the absence of other fieldsthese fluctuations will be a massless goldstone boson mode. This mode is often called the radion ordilaton; it is the goldstone boson of broken conformal symmetry.

We can also put d-dimensional quantum fields directly on the IR brane. These can be veryroughly interpreted as composite or emergent degrees of freedom that arise when conformal symmetryis broken. Again, this is rather ad hoc from the CFT point of view.

With both an IR and a UV brane, we have both a UV cutoff for the CFT and conformalsymmetry breaking in the IR. In this case the dynamics become a bit more interesting, because we

107

can stabilize the position zIR by giving fields in the ‘slice’ of AdS appropriate boundary conditions.The idea is that the configuration of the AdS fields will depend on zIR, and therefore the total bulkenergy will depend on zIR. Minimizing this energy determines the equilibrium value of zIR. Theoriginal implementation of these ideas was called the Goldberger-Wise mechanism. It was interestingbecause this setup, when applied to weak-scale physics, is called a Randall-Sundrum model, and ithas the potential to explain the hierarchy problem.

One can consider more involved and also much more physically plausible models of conformalsymmetry breaking where the AdS metric itself is modified to a metric of the form

ds2 =1

A(z)2

(dz2 + ηµνdxµdxν

)(11.17)

where A(z) is some function that asymptotes to A(z) ∼ z for z ∼ 0. Such a model representsa theory that is approximately conformal in the UV, where z → 0, but that breaks conformalsymmetry in a more complex and interesting way. Such models have been studied in great detail inthe literature, both phenomenologically and as precise string theory backgrounds.

11.3 String Theory on AdS5 × S5 and the N = 4 Theory

The most famous and well-studied AdS/CFT duality is between type IIB Supergravity or StringTheory on AdS5 × S5 and the N = 4 supersymmetric Yang-Mills theory.

The type IIB theory is a chiral theory with (2, 0) supersymmetry in 10 dimensions. The masslessparticle spectrum includes the graviton with 35 components, two gravitinos with 56 componentseach, two 2-form tensors with 28 components each, two Weyl spinors with 8 components, two scalars,and a self-dual 4-form with 35 components. The only parameters of the IIB string theory are thestring coupling gs and α′, which has units of length2 and sets the string length scale.

The N = 4 SYM theory is the maximally supersymmetric 4-d gauge theory. The particle content(at weak coupling) includes an SU(N) gauge boson, four adjoint fermions, and six adjoint scalars.The parameters of the theory are N and the ’t Hooft coupling λ = g2N where g is the gauge coupling.It turns out that this theory is exactly conformal, and so the coupling λ can take any value. Actually,the coupling g combines with a possible θ-angle to form a complex coupling τ . Both theories are selfdual under SL(2, Z) transformations of τ or gs; these transformations combine S and T duality.

The IIB Superstring Theory in AdS5 × S5 is identical to N = 4 SYM theory via the AdS/CFTduality. This means there must be a symmetry and SL(2, Z) preserving identification of theparameters, this is

τ ≡ 4πi

g2YM

+θ

2π=

i

gs+

χ

2π(11.18)

The coupling α′ is dimensionful, but of course we can eliminate its dimensions using RAdS, so thatwe have

α′

R2AdS

∼ 1√gsN

=1√λ

(11.19)

108

This means that the energy scale setting the mass of stringy states will be λ1/4. So in order to havea good (effective) field theory description in AdS5 × S5 where the stringy states decouple, we musttake

λ 1 (11.20)

which means that the gauge degrees of freedom in the N = 4 SYM theory are very strongly coupled.We also have that

GN ∼1

N2(11.21)

so in order to have weak low-energy gravity we must take

N 1 (11.22)

However, we could have easily anticipated this from our previous discussions – the CFT must havea large central charge in order to have weakly coupled AdS gravity, and the central charge of theN = 4 theory is proportional to N2.

Devoted readers may be confused about the presence of the S5 factor – our spacetime is not pureAdS, but instead is an AdS compactification. In place of pure AdS, we have the metric

ds2 =1

z2

(dz2 + ηµνdxµdxν

)+R2

AdSdΩ25 (11.23)

Due to the AdS5 part of the geometry, the states are still organized according to the 4 dimensionalconformal group. When we take a bulk field φ(X) to the boundary, we find that the radius of theS5 shrinks to zero relative to the AdS directions. So we are left with a 4 dimension CFT, not a9 dimensional theory. However, the S5 is still relevant as we can have arbitrary KK modes fromdecomposing fields in terms of harmonics on the 5-sphere. The geometry of the S5 is reflected in theN = 4 theory as the SU(4) = SO(6) R-symmetry group, which rotates the fermions and scalars.All of the states in the theory can be classified according to the full N = 4 superconformal group,and the R-symmetry sits inside this as a subgroup. More generally, the two theories both have anSU(2, 2|N ) global symmetry; in the SYM theory it is superconformal symmetry while on the gravityside it is geometrized in AdS5 × S5.

Only gauge invariant SYM operators will be primary. The single-trace operators, correspondingto one-particle or one-string states in AdS at large N , can be constructed as objects such as

OI1···Ikµ1···µ` = Tr[FµνD

µψaΦ8DνDµ1 · · ·Dµ`ψaΦI1 · · ·ΦIk

](11.24)

where we include any number of scalars, gauge bosons, fermions, and derivatives contracted withinan SU(N) trace. We have also given the operator some Lorentz indices, so it is a higher spin tensor,and also some SU(4) = SO(6) R-symmetry indices from the SO(6) indices of the scalars. TheLorentz indices correspond to spin in AdS5, as we have come to expect; the R-symmetry indices tellus about the motion of the state on the S5 part of the geometry.

109

There are an exponential number of single-trace operators of this form. If all of them havetheir naive scaling dimensions, then we will have an AdS theory with a number of particle typesexponential in the particle mass! However, at large ’t Hooft coupling λ almost all of these operatorsget a large ‘anomalous dimension’ of order λ1/4, becoming massive string states. The only operatorsthat do not receive large anomalous dimensions are protected by the superconformal symmetryand current conservation. This has been proven, with the exact spectrum computed in detail viaintegrability methods. At this point, integrability (combined with some bootstrap-based insightsabout AdS locality) can basically be viewed as a proof of AdS/CFT for the case of the N = 4 SYMtheory. However, it is not known how to extend integrability results, in detail or as a rough picture,to other large N gauge theories.

11.4 Open Problems

Probably the largest and most open-ended unsolved problem is to find all complete theories ofquantum gravity in AdS. AdS/CFT has transformed this into the question of identifying andunderstanding all CFTs. To emphasize how open this problem is – currently there are no knownnon-supersymmetric CFTs that can be described by weakly coupled effective field theory in AdS.But there are an infinite number of non-supersymmetric effective field theories in AdS. Do none,few, most, or almost all of them have a UV completion?

There are various sub-problems to the above. For example, can one classify the examples withsome given amount of supersymmetry? Can one show that some/most/all examples involve stringtheory in the UV? Can we show that there aren’t any global symmetries in quantum gravity, byshowing that if CFT correlators are exactly constrained by a symmetry then there must exist acorresponding current? Can we find families of CFTs related by the change of a bulk parameter,such as the hierarchy MplRAdS, so that in particular we can take a flat space limit RAdS →∞ in acontinuous way?

Further problems involve black holes. In AdS/CFT, unitarity is obvious, so the famous black holeinformation paradox has become the question of how to reconstruct effective field theory observablesacross a black hole horizon using CFT operators. Beyond the information paradox, there is thequestion of why CFTs contain black hole type states (small and large) in their large dimensionspectrum. Can one compute and understand the OPE coefficients of these states from the CFT side?Can we extract the expected BH entropies, and their formation and evaporation rates? Becauseblack holes are such generic objects in gravitational theories, one might expect to find a similarlygeneric story in (the correct class of) CFTs.

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