The Kerr/CFT correspondenceChapter 1: Introduction hole with rotation, i.e. the Kerr black hole, and a conformal eld theory. The main motivation to study such a correspondence is that

The Kerr/CFT correspondence

Teresa Bautista

Thesis for the

Master's in Theoretical Physics

Supervised by

Georey Compère, Erik Verlinde

University of Amsterdam

ITFA

December 18, 2012

Abstract

Kerr black holes are realistic models of astronomical black holes. A good understanding

of their microscopic structure is of utmost interest. In this thesis we review the basics of

the Kerr/CFT correspondence [1], which is a duality between the near-horizon quantum

states of the extremal Kerr black hole and a Conformal Field Theory. The geometry

in the near-horizon area of extremal Kerr is the so-called NHEK, with isometry group

SL(2,R)×U(1). In this background geometry, boundary conditions are found for which

the algebra of surface charges enhances the U(1) to one copy of the Virasoro algebra

with central charge c = 12J/~, J being the angular momentum of the black hole. The

gravitational rotating degrees of freedom can then be described by a chiral 2-dimensional

CFT. The chemical potential associated to these degrees of freedom is a dimensionless

temperature Tφ = 1/2π. The Cardy formula is then used to reproduce the Bekenstein-

Hawking entropy of the extremal Kerr black hole. Finally, we present some results for

the near-horizon geometry of the extremal 5-dimensional Cveti£-Youm black hole, which

ideally would reduce to NHEK after an appropriate compactication.

i

Table of contents

Abstract i

1 Introduction 1

2 2d Conformal Field Theory 5

2.1 The conformal group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Conformal Field Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.3 The Cardy formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3 Anti-de Sitter spacetime 29

3.1 Causal structure and conformal boundary . . . . . . . . . . . . . . . . . . 31

3.2 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.3 Euclidean AdS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4 3-dimensional Einstein gravity with Λ < 0. 39

4.1 Asymptotic symmetries and surface charges . . . . . . . . . . . . . . . . . 40

4.2 Thermal AdS3 and the BTZ black hole . . . . . . . . . . . . . . . . . . . . 47

4.3 The Brown-Henneaux construction . . . . . . . . . . . . . . . . . . . . . . 51

5 The AdS/CFT correspondence 57

5.1 Symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.2 AdS/CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

5.2.1 AdS3/CFT2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.3 The microscopic BTZ entropy . . . . . . . . . . . . . . . . . . . . . . . . . 70

6 The Kerr black hole 73

6.1 M2 > a2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

6.2 M2 < a2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6.3 M2 = a2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

7 Near-horizon geometries 83

iii

TABLE OF CONTENTS

7.1 Reissner-Nordström . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

7.2 NHEK . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

7.3 Entropy of the extremal Kerr black hole . . . . . . . . . . . . . . . . . . . 92

8 The Kerr/CFT correspondence 95

8.1 Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

8.2 Virasoro algebra and central charge . . . . . . . . . . . . . . . . . . . . . . 100

8.3 The Dual CFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

8.4 Entropy matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

9 Cveti£-Youm 107

9.1 Non-BPS branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

9.2 BPS branch . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

10 Conclusions 115

A Hamiltonian formulation of General Relativity 117

iv

CHAPTER 1

Introduction

Black holes have long been object of intense research since they constitute the

paradigmatic lab for probing a theory of quantum gravity. Somehow paradoxically,

they are both very simple and very complex systems. Its simplicity is due to the 'No

Hair Theorem', which says that a black hole is completely specied by its mass M ,

angular momentum J and electrical charge Q. Once a star collapses to form a black hole,

the resulting gravitational eld contains no information on the details concerning the

original star besides the aforementioned charges. Therefore, one could compare it with

a structureless elementary particle. Understanding the simple macroscopic aspects of a

black hole is the aim of classical General Relativity.

The complexity of a black hole is due to its high number of microstates or degrees of

freedom. This is reected into a very high entropy, much higher than the entropy of the

star before the collapse. In this respect, a black hole is to be regarded as a statistical

ensemble, which in general is also specied by its conserved quantum numbers such

as energy, spin and charge. Understanding this complex microstructure is the aim of

quantum gravity. One of the main goals of this thesis is to acquire some knowledge on

this microscopic and statistical description for the extremal Kerr black hole.

Let's start by reviewing some of the basic aspects of black holes. A black hole is a

spacetime solution of the Hilbert-Einstein action that contains a region which is not

in the backward lightcone of future timelike innity. The boundary of this region is a

stationary null surface called event horizon, therefore the event horizon causally separates

the inside from the outside of the black hole. In the 70's, Bardeen, Carter and Hawking

established three important laws on the black hole mechanics:

• Zeroth Law: for stationary black holes, the surface gravity κ is constant on the

event horizon. This is obvious for spherically symmetric horizons but is generally

1

Chapter 1: Introduction

true for non-spherical spinning black holes.

• First Law: a balance equation is satised, namely dM = κ8πGdA+µdQ+ΩdJ , where

M is the mass and A the area of the black hole, µ is the electrical chemical potential

and Ω the angular one, corresponding to the angular velocity on the horizon.

• Second Law: the net area in any process never decreases, ∆A ≥ 0. Allowed processes

are the infall of matter in the black hole or the coalescence of two black holes. A

disallowed one in 4 dimensions is the fragmentation of black holes into multiple

ones.

These laws resemble strikingly the three laws of thermodynamics for a body in thermal

equilibrium. Hawking and Bekenstein discovered that this is more than just a resemblance

and that actually there is a profound connection between the geometry of a black hole and

its thermodynamics. In 1973, Bekenstein proposed that a black hole must have entropy,

motivated by the gedankenexperiment of throwing a bucket of water into a black hole.

Noting the formal analogy between the area of the black hole and entropy drawn by the

second law, he concluded that the entropy has to be proportional to the area of the black

hole.

If a black hole has got energy and entropy, then it must have a temperature and like any

hot body, it must radiate. Luckily, in 1975 Hawking showed that it is possible for a black

hole to radiate when taking quantum eects into account, namely due to the constant

creation and annihilation of particle-antiparticle pairs in the vacuum near the horizon1.

Hawking showed that the emitted spectrum is thermal with temperature

TH =~κ2π.

With this relation between temperature and surface gravity, the black hole mechanical

laws become the laws of thermodynamics. Using this temperature expression and the

rst law of thermodynamics dM = THdS, one deduces (for Schwarzschild black holes,

i.e. when J = Q = 0) the area law for the entropy of a black hole

S =A

4G~

where the area A is expressed in Planck units. These expressions for the Hawking

temperature and the entropy in terms of geometrical parameters of the black hole are

generally true.

1As the black hole radiates, it loses mass and its horizon area decreases, thereby seemingly providing

an explicit violation of the area theorem. However, the entropy in the Hawking radiation increases,

providing a way out. Bekenstein argued that the relevant quantity is a generalized entropy that accounts

for the entropy of the black hole as well as that of the radiation.

2

More fundamentally, the entropy contains information about the microscopic structure

of a system, which is reected through Boltzmann's law S = k log(d), where d is the

degeneracy or total number of degrees of freedom given an energy of the system. The

Bekenstein-Hawking entropy is also a measure of the number of microstates for a black

hole and therefore is a valuable piece of information which we are going to derive for

dierent cases throughout this thesis.

As the entropy scales like the area rather than the volume, it violates our naive intuition

of thermodynamical extensivity of the entropy. This area scaling leads to the idea of

holography: quantum gravitational theories must have a lot fewer degrees of freedom

that non-gravitational ones. Fundamentally then, gravity has to be dierent than other

theories. This idea culminated in the Holographic Principle [2], established by 't Hooft

and Susskind in 1993, according to which gravitational theories can be characterized by

a quantum eld theory with one spacelike dimension less. The Holographic Principle

materializes into the so-called AdS/CFT dualities [3], from which the best understood

examples are the AdS5/N = 4 super Yang Mills theory and the AdS3/CFT2

correspondences. These dualities establish that the fundamental gravitational quantum

states in Anti-de Sitter backgrounds can be identied with states in a dual eld theory

with conformal invariance living in the boundary of the background geometry.

Although we haven't made clear yet what we mean by gravitational quantum degrees of

freedom, it is obvious that the interesting quantum dynamics happen in the area near

the horizon of the black hole. In other words, to count the entropy only the near-horizon

geometry of the black hole is relevant. By means of a near-horizon limit, this geometry

can be singled-out from the asymptotic region of the black hole. It constitutes a system

with its own dynamics and thermodynamics, these reecting the quantum microstates

that live on the horizon.

Extremal black holes play an important role in this context. These are dened as

stationary black holes with vanishing Hawking temperature, which translates into certain

conditions satised by their charges. The relevance of extremal black holes is due to the

AdS factors that their near-horizon geometries exhibit. Because of this, they constitute

the ideal scenario where to make use of the aforementioned AdS/CFT dualities. An

important statement of these dualities is the equivalence between the partition functions

of the two dual theories. On the gravitational side, it is not known what the quantum

microstates of the black hole are, therefore it is hard to write the partition function in

the canonical approach. The hope is that knowledge from the dual CFT can help identify

these microstates.

The goal of this thesis is to understand the Kerr/CFT correspondence [1]. This is a

correspondence between quantum gravity on the near-horizon region of an extremal black

3

Chapter 1: Introduction

hole with rotation, i.e. the Kerr black hole, and a conformal eld theory. The main

motivation to study such a correspondence is that astrophysical black holes are generically

rotating and electrically neutral. The microstructure of the Kerr black hole is therefore

of obvious interest. Moreover, near-extremal black holes were claimed to be observed, for

example GRS1915− 105 [4] or MCG− 6− 30− 15 [5], which are respectively 98% and

99% close to extremality. Experiments probing the physics close to the event horizons

of these black holes can denitely shed some light on the eects of quantum gravity in

the near-horizon region. Hence, the Kerr/CFT correspondence has the potential to make

contact between theoretical black holes and the real world ones.

This thesis consists of two main parts. The rst part, corresponding to the rst four

chapters, contains the necessary background. In chapter 2 we introduce the basics of

any conformal eld theory: the conformal group, the Virasoro algebra and the central

charge. We nally derive the Cardy formula. In chapter 3 we thoroughly go over the

dierent metrics, isometries and peculiarities of Anti-de Sitter spacetime. In chapter 4

we start with some general formalism on asymptotic symmetries and surface charges and

continue to focus on asymptotically AdS spacetimes in 3 dimensions, its thermalization

and the BTZ black hole. Finally we present the the asymptotic symmetry group for

asymptotically AdS3 spacetimes, the algebra of charges and the Brown-Henneaux central

charge. In the last chapter of the rst part we give a little introduction to the AdS/CFT

correspondence. As an example, we use it to derive the microscopic entropy of the BTZ

black hole.

In the second part of the thesis we focus on the Kerr/CFT correspondence. In chapter

6 we go into some detail explaining the features of the Kerr black hole and its dierent

regimes. In chapter 7 we compute the near-horizon geometry of the extremal Kerr black

hole, the so-called NHEK geometry. First we do so for the Reissner-Nordström black

hole, which serves as an easy introduction. We close the chapter with the computation of

the Bekenstein-Einstein entropy for extreme Kerr. In chapter 8 we present the celebrated

correspondence. We rst nd a generalized temperature for NHEK which describes the

rotational degrees of freedom. We then nd the asymptotic symmetry group and the

central charge. Finally we compute the microscopic entropy and emphasize the matching

with the macroscopic computation. In the last chapter of the thesis we present some

calculations done for the ve-dimensional Cveti£-Youm black hole. In particular, we try

to nd the conditions, in terms of its four charges, in which this black hole becomes

extremal. It turns out to exhibit two extremality branches, a BPS and a non-BPS branch.

Finally we nd the near-horizon geometry for the two branches.

4

CHAPTER 2

2d Conformal Field Theory

Symmetry principles are of outstanding importance in physics thanks to the understand-

ing and simplications they provide. In quantum eld theory, the Poincaré invariance

plays a major role. It is natural then to look for possible generalizations of this invariance

hoping that they will provide new insights. An interesting generalization is the addition

of scale invariance, which links physics at dierent scales. Field theories exhibiting both

invariances are called Conformal Field Theories, and they have an important role in

many dierent contexts in physics, most notably in statistical physics, where they oer

a description of critical phenomena, and string theory and holography, where they are

an essential element in the AdS/CFT correspondence.

In this section we will review some of the main aspects of 2-dimensional CFT's. We

will start with the conformal group, then we will move on to discuss the content of a 2-

dimensional CFT, the generators of the symmetries, the appearance of the central charge

and the Virasoro algebra. Finally, we will discuss some of the thermal aspects, computing

the asymptotic growth of states and the entropy of the theory, for which we will derive

the Cardy formula.

Some of the main references about CFT's are [6] and [7]; other sources recommended for

studying are [8] and [9].

2.1 The conformal group

The conformal group, in any dimension d, is the group of coordinate transformations

x→ x′ that leave the metric invariant up to a rescaling factor

gµν(x)→ g′µν(x′) = Ω(x)gµν(x). (2.1)

5

Chapter 2: 2d Conformal Field Theory

These transformations therefore preserve the angle between two vectors. We will now

restrict to the case of at spacetime, gµν = ηµν , of signature (p, q). To examine the

conformal generators one performs an innitesimal transformation xµ → x′µ = xµ +

εµ and imposes that the relation (2.1) is satised. In this way, one nds the following

dierential equation for the vectors εµ

Lεηµν(x) =2

d(∂ · ε)ηµν(x), (2.2)

which depends on the number of dimensions d of the theory. This is called the conformal

Killing equation and the vectors that satisfy it are conformal Killing vectors. It can

obviously be generalized to any spacetime, with the partial derivative replaced by the

covariant one, and holds for any nite conformal generator. Examining (2.2), one nds

that for d > 2, εµ can be at most quadratic in x and so there are only four inequivalent

coordinate transformations: translations, rotations, scale transformations and special

conformal transformations. The rst two are generated by the Poincaré group, therefore

a subgroup of the conformal group. There are a total of 12(d + 2)(d + 1) generators and

the conformal group is isomorphic to SO(p+ 1, q + 1).

Conformal algebra in 2 dimensions

The case of d = 2 exhibits special features and is the one we are interested in. In two

dimensions, (2.2) for the two innitesimal transformation parameters εµ become the

Cauchy-Riemann equations, therefore the conformal transformations are holomorphic

and anti-holomorphic transformations in the complex plane.

z → f(z), z → f(z). (2.3)

where in the (z, z) coordinates the metric becomes ds2 = dzdz1.

Assuming that the innitesimal transformations ε(z), ε(z) admit Laurent expansions

around z, z = 0, the generators corresponding to these transformations are ln =

−zn+1∂z, ln = −zn+1∂z and satisfy the so-called Witt algebra

[lm, ln] = (m− n)lm+n, [lm, ln] = (m− n)lm+n, [lm, ln] = 0. (2.4)

Therefore the conformal algebra in the case d = 2 is innite dimensional. However,

all we have inferred up until here is local. Since we haven't imposed that conformal

1This holds for both Euclidean and Lorentzian signature of the original metric. Ultimately, we mostly

encounter Minkowskian metrics ds2 = −dτ2 +dσ2, for which the holographic coordinates can be thought

of as the light-cone coordinates from the original space-time, z, z = σ±τ . However, normally it is simpler

to work with Euclidean backgrounds, for which the time coordinate is analytically continuated t = −iτ ,yielding z, z = σ ± it. Most literature develops the theory in Euclidean signature.

6

2.1 The conformal group

transformations are invertible and map the whole Riemann sphere to itself (innity has

to be added to the complex plane if we want to impose all transformations to have

an inverse), strictly speaking one can not give these transformations a group structure.

That's why we only use the word algebra for the local generators.

Global conformal transformations

One can distinguish then the global transformations, also called projective transforma-

tions. These do form a group, the so-called special conformal group, and are therefore

well dened on the Riemann sphere and non-singular at both z → 0,∞. The subset of

generators that satisfy these properties is l0,±1, l0,±1. These generators form the algebra

sl(2,R)× sl(2,R), subalgebra of the local Witt algebra,

[l1, l0] = l1, [l0, l−1] = l−1, [l1, l−1] = 2l0

[l1, l0] = l1, [l0, l−1] = l−1, [l1, l−1] = 2l0

This subset of generators closes under the Lie bracket precisely because they correspond

to the transformations on the complex plane that are encountered in higher dimensions,

as follows from their denition : l1, l1 generate special conformal transformations; l−1, l−1

generate translations; l0 + l0 generates dilations and i(l0 − l0) generates rotations (i.e. if

we express z = reiθ, the two latter correspond to −r∂r and −∂θ respectively). Therefore,the special conformal group is isomorphic to SO(2, 2). This can also be shown by doing

combinations of the previous generators and going back to the original coordinates of

the Minkowskian metric, (τ, σ) = 12(z ∓ z). One then nds the conformal Killing vectors

that perform translations, rotations, dilations and special conformal transformations on

the original space time

−l−1 − l−1 = ∂σ = iPσ − l−1 + l−1 = ∂τ = iPτ

i(l0 − l0) = −i(τ∂σ + σ∂τ ) = Lτσ

l0 + l0 = −τ∂τ − σ∂σ = −iDl1 + l1 = −2τσ∂τ − (τ2 + σ2)∂σ = −iKσ l1 − l1 = −(τ2 + σ2)∂τ − 2τσ∂σ = iKτ

These generators obey the following brackets

[D,Pµ] = iPµ [D,Kµ] = −iKµ

[Kµ, Pν ] = 2i(ηµνD − Lµν) [Kρ, Lµν ] = i(ηρµKν − ηρνKµ)

[Pρ, Lµν ] = i(ηρµPν − ηρνPµ)

7


where it is implicit that ηµν = diag(−1, 1). These generators can be expressed in a

compact form Jab such that Jab = −Jba, where a, b = −1, 0, τ, σ, as

Jµν = Lµν J−1,µ =1

2(Pµ −Kµ)

J−1,0 = D J0,µ =1

2(Pµ +Kµ) (2.5)

which explicitly obey the commutation relations of the so(2, 2) algebra

[Jab, Jcd] = i(ηadJbc + ηbcJad − ηacJbd − ηbdJac) (2.6)

where ηab = diag(−1, 1, diag(ηµν)). (Notice that, in the case of Euclidean signature, the

boost generator Lµν would become the generator of rotations with an extra minus sign

for the second term and ηab = diag(−1, 1, 1, 1)).

Since SO(2, 2) ∼= SL(2,C)/Z2 then the special conformal group can also be parametrized

in the following way

f(z) =az + b

cz + d| a, b, c, d ∈ C and ad− bc = 1 (2.7)

with the Z2 xing the sign freedom to replace all the parameters by minus them-

selves.

2.2 Conformal Field Theory

A Conformal Field Theory is a eld theory that is invariant under conformal transfor-

mations. This means that the physics of the theory looks the same at all length scales,

only angles play a role. If the theory has no preferred length scale, there can be nothing

in the theory like a mass or a Compton wavelength. Therefore, CFT's only contain

massless excitations. The lack of a length scale also precludes the existence of a non-

trivial S-matrix, since it does not allow the standard denition of asymptotic states.

The main content of a CFT, therefore, is not the mass spectrum and S-matrices, but

correlation functions, the elds and the behavior of certain operators under conformal

transformations.

The interpretation of a conformal transformation in a CFT depends on whether the

metric is regarded as a xed background or as a dynamical eld. When the metric is

dynamical, the transformation is a dieomorphism and represents a gauge symmetry

of the theory2. If the background is xed, the conformal transformation represents a

2This is for example the case of string theory in the Polyakov formalism, where the conformal

transformations are residual gauge transformations that can be undone by a Weyl transformation.

8


global physical symmetry and conserved currents are associated to it through Noether's

theorem. In the following, we will use the latter approach, concentrating on 2-dimensional

eld theories with a xed at background metric.

An important consequence of conformal invariance is that the trace of its stress-energy

tensor vanishes in the quantum theory in at space in any dimension. In general, many

theories have this feature at the classical level. However, at the quantum level, the need

of a cuto to regulate the theory spoils scale invariance and the vanishing of the trace

is hard to preserve. In CFT's, this follows from the fact that the variation of the action

under a scale transformation is precisely proportional to the trace.

A 2-dimensional CFT theory contains an innite set of elds, including all its derivatives.

Among these, there are the so-called quasi-primary elds which, under global conformal

transformations (z, z)→ (f(z), f(z)), transform as tensors of weight (h, h)

Φ(z, z)→ Φ′(z, z) =

(∂f

∂z

)h(∂f∂z

)hΦ(f(z), f(z)). (2.8)

This expression is the generalization of the transformation law for the metric and it

means that Φ(z, z)dzhdzh is invariant under conformal transformations. (h, h) are real-

valued and are called the conformal weights or conformal dimensions of the eld. In a

unitary CFT, all operators have h, h ≥ 0, as we will see later on. These weights tell us

how operators transform under rotations and scalings. We will see this explicitly when

we study the operators that implement conformal transformations on the elds.

Only in 2-dimensional CFT's there exist the so-called primary elds, which transform as

in (2.8) for all conformal transformations. Primary elds are in particular quasi-primary

elds. Fields not transforming in this way are called secondaries and they can be expressed

as linear combinations of the quasi-primaries and their derivatives. Derivatives of elds,

for example, in general have more complicated transformation properties. In general, the

elds in a conformal eld theory can be grouped into families [φn] each of which contains

a primary eld and an innite set of secondary elds (including its derivative), called its

descendants. We will go back to these when we build the Hilbert space.

The theory is covariant under conformal transformations, in the sense that correlation

functions satisfy

〈N∏i=1

Φi(zi, zi)〉 =N∏i=1

(∂f

∂z

)hiz=zi

(∂f

∂z

)hiz=zi

〈N∏j=1

Φj(f(zj), f(zj))〉 (2.9)

This covariance property leads to very specic restrictions on the form the point functions

can take. Imposing it on innitesimal transformations of the elds yields dierential

9


equations which specify the analytical dependence on the points zj , zj . In particular,

the two and three-point functions are completely xed up to a constant. Higher-point

functions are not fully determined but constraints can be derived, the so-called Ward

identities, which encode the conformal covariance of the theory (in curved Riemann

surfaces, there exists a conformal anomaly that spontaneously breaks the invariance under

the full conformal group. In that case, the Ward identities encode only the covariance

under the unbroken subgroup).

The theory must contain a vacuum state which is invariant under the global conformal

group (the global group SL(2,C), for example, is the unbroken subgroup by conformal

anomaly on the sphere).

Radial quantization

Now let's have a look at the quantization procedure and Hilbert space of a conformally

invariant theory. We begin with at Minkowski spacetime coordinates (τ, σ), with the

spacelike coordinate being compactied on a circle, σ ∼ σ + L. This denes a cylinder

in this coordinates. We perform a Wick rotation (−iτ, σ) which gives the spacetime

a Euclidean signature. The light-cone coordinates become ζ, ζ = ∓iτ + σ, so the 2-

dimensional Minkowski space notions of left and right-moving turn, in Euclidean space,

into purely holomorphic and anti-holomorphic dependence on the coordinates ζ, ζ. Then,

one can perform the conformal transformation z = e2πiζ/L, z = e−2πiζ/L, which maps

the cylinder to the complex plane where equal time surfaces become circles centered in

the origin. Innite past and future are mapped to the points z = 0,∞ on the plane.

Therefore, time evolution becomes radial evolution; the Hamiltonian, regarded as the

time translation operator, is mapped to the dilatation operator. The Hilbert space is then

built up on surfaces of constant radius. This procedure for dening a quantum theory on

the plane is known as radial quantization. In this scheme for example, conserved charges

will be computed by contour integrals around the origin of the complex plane, since they

are dened as integrals over a xed time hypersurface.

The conformal generators on the complex plane ln = −zn+1∂z, ln = −zn+1∂z translate

under the inverse conformal map into the generators of conformal transformations

on the cylinder, which in the coordinates (ζ, ζ) are also holomorphic transformations

(ζ, ζ) → (f(ζ), f(ζ)) for the same reasoning discussed before. The generators become

ln = i L2πe2πinζ/L∂ζ , ln = −i L2πe

−2πinζ/L∂ζ .

Stress-energy tensor and OPE

Back to the complex plane, given the set of isometries one uses the Noether theorem

to derive conserved currents jµ and charges Q. The charges generate the innitesimal

conformal transformations on the elds according to δεΦ = ε[Q,Φ]. The stress-energy

tensor Tµν would correspond to the Noether current associated to translations; therefore

10


it is conserved. It is also symmetric (rather, it can be symmetrized) and traceless. This

last property, which is only true in at space, follows from scale, rotation and translation

invariance. These properties lead to a vanishing vacuum expectation value of the trace of

the stress-energy tensor which imply that the trace is identically zero. Since the variation

of the metric is δgµν = Λ(x)gµν , the variation of the action is proportional to the trace of

the stress-energy tensor. Therefore, this trace being zero implies conformal invariance and

vice-versa. However, it is worth noticing that in classical conformal eld theories of higher

dimension than 2, conformal invariance doesn't imply a vanishing of the trace. In general,

one can not draw this implication in the opposite sense, since the proportionality function

Λ(x) between the variation of the action and the trace is not an arbitrary function but

follows from the conformal Killing equation.

Because of the aforementioned properties, the stress-energy tensor acquires holomorphic

dependence. Tracelessness implies Tzz = 0 and the divergenceless property implies Tzz =

T (z), Tzz = T (z).

The general Noether current associated to conformal transformations results from the

product of the stress-energy tensor with an innitesimal conformal Killing vector jµ =

Tµνεν . Using the aforementioned properties of the stress-energy tensor one can prove that

this current is indeed conserved (in particular, one can think of the tracelessness condition

as a requirement for the conservation of the dilatation current jµ = Tµν xν). One can also

think of it in the following way: because T (z) is conserved, then T (z)ε(z) is also conserved,

for every holomorphic function ε(z). Therefore, as we will see, the theory acquires an

innite set of conserved charges Qn ≡ Ln (associated to every vector εn = −zn+1) which

gives rise to the analog of the local conformal algebra in 2 dimensions.

The charges associated to these currents are dened as the 0-th component of the current

integrated over a xed time-slice. This would be the computation on the cylinder. As

we mentioned before, when mapping to the complex plane this corresponds to contour

integrals on concentric circles. Therefore the charges are computed with

Qε =1

2πi

∮dzε(z)T (z), Qε =

1

2πi

∮dzε(z)T (z). (2.10)

As mentioned, these charges generate the innitesimal conformal transformations z →z + εn(z) and its antiholomorphic counterpart on the elds through their equal-time

commutator.

δε,εΦ(w, w) = [Qε +Qε,Φ(w, w)]

=1

2πi[

∮dzε(z)T (z),Φ(w, w)] +

1

2πi[

∮dzε(z)T (z),Φ(w, w)]

11


However, products of operators in Euclidean space radial quantization are only well-

dened if the operators are time-ordered. The analog of time ordering in radial

quantization is radial ordering, implemented by the operator R dened as

R(A(z)B(w)) =

A(z)B(w) for |z| > |w|B(w)A(z) for |z| < |w|

(2.11)

(with relative minus sign in the case of fermionic operators). Then, the equal-time

commutator of the spatial integral of a local operator j0 with a local operator Φ(τ, σ)

becomes the contour integral of the radially ordered product. In the case of the variation

of the elds, this one becomes

δεΦ(w, w) =1

2πi(

∮|z|>|w|

−∮|z|<|w|

)(dzε(z)R(T (z)Φ(w, w)))

=1

2πi

∮wdzε(z)R(T (z)Φ(w, w)) (2.12)

where we have only showed the holomorphic contribution. In the last line the in-

tegral is around w due to the deformation of the contours. This variation has to

equate the innitesimal version of the transformation for a primary eld (2.8), which

is h∂ε(w)Φ(w, w) + ε(w)∂Φ(w, w). From the resulting equation, the short distance

singularities of the product of T and T with Φ can be derived

R(T (z)Φ(w, w)) ∼ h

(z − w)2Φ(w, w) +

1

z − w∂wΦ(w, w)

R(T (z)Φ(w, w)) ∼ h

(z − w)2Φ(w, w) +

1

z − w∂wΦ(w, w) (2.13)

where ∼ means equality modulo expressions regular as w → z. This operator product

expansion (from now on we will drop the symbol R) denes the notion of a primary

eld Φ(w, w) of conformal weight (h, h) since it encodes its conformal transformation

properties. It can also be thought of dening the quantum stress-energy tensor and

encoding information about the correlation functions.

The Virasoro algebra and the central charge

One can now have a look at the particular case of the conserved charges associated to

the generators of the local conformal transformations εn = −zn+1, as we have briey

mentioned before. These charges,

Ln =1

2πi

∮dzzn+1T (z), Ln =

1

2πi

∮dzzn+1T (z) (2.14)

are called the Virasoro generators and coincide with the Fourier modes of the stress-

energy tensor on the cylinder. Fourier expansions can in general be written for conformal

12


operators and are called mode expansions. For the stress-energy tensor

T (z) =∑n∈Z

z−n−2Ln, T (z) =∑n∈Z

z−n−2Ln. (2.15)

It is worth mentioning, since we will encounter it later on, that the mode expansion of

the stress-energy tensor on the cylinder diers from that on the plane. In general, the

mode expansion depends on the Riemann surface in which the conformal eld theory

is described. In general, a conformal eld φ(z, z) of conformal dimension h, may be

expanded on the complex plane as follows

Φ(z) =∑n∈Z

φnz−n−h. (2.16)

When going to the cylinder, using the transformation rule (2.8), the mode expansion

becomes

Φcyl(ζ) =∑n∈Z

φne−niζ =

∑n∈Z

φnz−n. (2.17)

Since the Virasoro generators implement the 2-dimensional conformal transformations

on operators, then L0 ± L0 are the generators of dilations and rotations respectively.

The innitesimal action of these two generators on a primary eld would yield the

variations

δΦ = [L0 ± L0,Φ(w, w)] = (h± h)Φ(w, w) + (w∂ ± w∂)Φ(w, w)

which obviously coincides with the innitesimal version of (2.8) δε,εΦ(w, w) = (h∂ε(w) +

ε(w)∂)Φ(w, w) with ε(w) ∝ w. From here we can give a more physical meaning to

the conformal weights. If an operator is an eigenstate of dilations and rotations, then

their eigenvalue under rotations, i.e. their spin, is s = h − h. Their eigenvalue under

dilations is ∆ = h+ h. This is the so-called scaling dimension, which is the dimension

that is usually associated to elds and operators by dimensional analysis. For example,

derivatives increase the dimension of an operator by one. However, the dimension that

elds have in the classical theory is not necessarily the same they have in the quantum

theory.

When the Hilbert space is built up using radial quantization, the Hamiltonian corresponds

to the dilation operator and hence is expressed in terms of the Virasoro generators as

H = L0 + L0. Analogously, spatial translations would be generated by the rotation

generator, therefore the momentum operator results in P = L0− L03. These expressions

can exhibit some prefactor depending on the theory. For example, on a cylinder of circle

L they include a prefactor of 2πL .

3Many sources introduce a factor i in the denition of the momentum operator.

13


We want now to determine the OPE of two stress-energy tensors. This can be computed

by performing two conformal transformations, but it can also be deduced from general

arguments. Tµν is an operator of weight (2, 0), and similarly Tµν has weight (0, 2). This

follows from its scaling dimension being ∆ = 2 because the energy is obtained by

integrating over space and from its spin being s = 2 because it is a symmetric 2-tensor.

This means that the TT OPE takes the form

T (z)T (w) = ...+2

(z − w)2T (w) +

1

z − w∂T (w) + ...

This expansion contains in principle higher-pole terms because T is not a primary but

a quasi-primary eld since it cannot follow from derivatives of other elds. Each term

in the expansion must have scaling dimension ∆ = 4, so extra terms must be of the

formOn

(z − w)n, with ∆[On] = 4− n.

But in a unitary CFT there are no operators with h, h < 0 as we will prove later, so

the most singular term is of order n = 4 and must have a constant as the numerator.

Therefore we write

T (z)T (w) ∼ c/2

(z − w)4+

2

(z − w)2T (w) +

1

z − w∂T (w) (2.18)

with an analogous expression for the antiholomorphic counterpart and T (z)T (w) =

regular. A term of order n = 3 cannot be introduced since it would brake symmetry

under the exchange of z with w. The fourth-order term can be though of as a measure

of how much T (z) diers from being a primary eld. In fact, a secondary eld can be

dened as having higher than the double pole singularity (2.13) in its operator product

expansion. The constants c, c in the fourth-order singularity term are called the left and

right central charges respectively and their value depends on the particular theory under

consideration. c is in principle an independent constant, but the two charges turn out to

be the same in some theories, for example for the free boson and the free fermion.

Using this OPE, one can compute the algebra of commutators satised by the operator

modes Ln, Ln

[Ln, Lm] = (

∮dz

2πi

∮dw

2πi−∮

dw

2πi

∮dz

2πi)zn+1T (z)wm+1T (w) =

=

∮0

dw

2πi

∮w

dz

2πizn+1wm+1(

c/2

(z − w)4+

2T (w)

(z − w)2+∂T (w)

z − w) (2.19)

where the commutator of the integrals is evaluated by rst xing w and deforming the

dierence between the two z integrations into a single contour around w. The result,

14


also for the antiholomorphic part, is an innite dimensional algebra, called the Virasoro

algebra

[Ln, Lm] = (n−m)Ln+m +c

12n(n2 − 1)δn+m,0

[Ln, Lm] = (n−m)Ln+m +c

12n(n2 − 1)δn+m,0

[Ln, Lm] = 0

Every conformally invariant quantum eld theory determines a representation of this

algebra with some value for the central charges. The Virasoro algebra is therefore isomor-

phic to the Witt algebra (2.4), satised by the generators of conformal transformations

on the holomorphic coordinates, but with a central extension that comes from taking

into account quantum eects. The subset of L0, L±1 and their antiholomorphic cousins

satisfy the sl(2,Z) algebra. They constitute the global conformal group, under which the

vacuum of the theory must be invariant (meaning that it must be annihilated by these

operators). For CFT's dened on non-at Riemann surfaces, c, c signal the presence of a

conformal anomaly, a non-zero trace of the stress-energy tensor, breaking the conformal

invariance such that only the global conformal group remains an exact symmetry group,

this is the symmetry the Ward identities reect.

With the OPE of the stress-energy tensor with itself one can also compute its variation

under an innitesimal conformal transformation

δεT (z) =1

2πi

∮zdwε(w)T (w)T (z) =

c

12∂3z ε(z) + 2∂zε(z)T (z) + ε(z)∂zT (z) (2.20)

This transformation can be integrated to a nite one as

T (z) = (∂z′

∂z)2T ′(z′) +

c

12z′, z, where f(z), z =

f ′′′

f ′− 3

2(f ′′

f ′)2 (2.21)

where T ′ ≡ Tz′z′ and the operation , is called the Schwartzian derivative. This one

vanishes for transformations of the global conformal group; that's why the stress-energy

tensor is a quasi-primary, it is SL(2,C) primary but not Virasoro primary.

Physical meaning of the central charge

The central charges are key to characterize a CFT. Roughly speaking they measure

number of degrees of freedom in the CFT. For example, in the free scalar theory, c =

c = 1, while if the theory contains D non-interacting free scalar elds, c = c = D. They

do not need to be an integer, though, but they are positive in all unitary theories since

〈T (z)T (w)〉 = c/2(z − w)4.

15


Let's look for a more accurate physical meaning. In (2.21), the extra term in the

transformation that impedes T to be a primary eld does not depend on T itself, giving

the same contribution to the expectation value of the energy in all states. Therefore, it

aects only the zero mode energy and we can say that the central charge is related to

the Casimir energy of the system.

To exemplify this, we are going to consider a CFT dened on a cylinder of circle L, which

as shown is parametrized with ζ, ζ = σ ∓ iτ and mapped to the complex plane through

the conformal transformation z = e2πiζ/L, z = e−2πiζ/L. It is worth mentioning already

at this point that the cylinder is a very natural geometry where to dene a CFT. On

the one hand, we have already seen that the intuition for doing manipulations on the

complex plane frequently comes, through radial quantization, from referring things back

to the cylinder. On the other hand, as we will see, thermal CFT can only be dened on a

cylinder. Another common construction is on the torus, which will become very important

in the next section. CFT's can in general be dened in any Riemann surface. Actually, in

the context of string theory it becomes natural to do so, since perturbative expansions

run over Riemann surfaces with increasing genus. Like this, tree level corresponds to a

CFT on the Riemann sphere (or complex plane), 1-loop diagrams correspond to a CFT

on the torus, and so on.

Back to the cylinder, we can use (2.21) to compute what the stress-energy tensor becomes

under the conformal map from the plane to the cylinder. It becomes

T cyl(ζ) = (2π

L)2[−z2T plane(z) +

c

24]. (2.22)

The Hamiltonian on the cylinder is dened as the Noether charge that follows

from integration over a spacelike surface of the 0-th component of the stress-energy

tensor

H =

∫dσTττ = −

∫dσ(Tζζ + Tζζ). (2.23)

Supposing that the ground state energy vanishes when the theory is dened on the plane,

i.e. 〈T plane〉 = 0, then the ground state energy on the cylinder is

E0 = −π2(c+ c)

6L(2.24)

This is the Casimir energy on a cylinder. We can explicitly see now that E goes to 0

when L → ∞. Thus, imposing a periodicity condition on a coordinate of at space to

obtain the cylinder, changes the vacuum energy. We will see this same idea popping up

in other contexts in chapter 4.

16


The Weyl Anomaly

In the classical theory, the stress-energy tensor is traceless. In the quantum theory though,

things are more subtle. The trace vanishes if the space is at, but it doesn't in curved

backgrounds. This holds not only for its vacuum expectation value but also for any state

in the theory, therefore 〈Tr(T )〉must depend only on the background metric, and through

a local and 2-dimensional quantity. The only candidate is the Ricci scalar R. The factor

of proportionality turns out to be the central charge, as

Tr(T ) = − c

24πR = − c

24πR. (2.25)

This occurs because the curvature introduces a macroscopic scale in the system which,

despite preserving scale invariance, is reected in this anomaly for the trace of the stress

tensor4.

Representations of the Virasoro algebra

We now want to build the Hilbert space of a CFT. The states will have to organize into

representations of the Virasoro algebra of charges. As mentioned before, the Hamiltonian

of the theory in terms of the charges is H = L0 + L0. Let's start by considering a state

|ψ〉 that is an eigenstate of L0 and L0 as

L0|ψ〉 = h|ψ〉, L0|ψ〉 = h|ψ〉.

so we'll refer to h and h as the energy eigenvalues.By acting with the Ln operators we

can get further states with shifted eigenvalues

L0Ln|ψ〉 = (LnL0 − nLn)|ψ〉 = (h− n)Ln|ψ〉.

This tells us that Ln are raising and lowering operators depending on the sign of n. When

n > 0, Ln lowers the energy and when n < 0 it raises the energy. If the spectrum is to be

bounded below, there must be some states which are annihilated by all Ln, Ln for n > 0-

Such states are called primary states. In the language of representation theory they are

also called highest weight states, they are the states of lower energy. Representations

of the Virasoro algebra can now be built by acting on the primary states with raising

operators. This results in an innite tower of states, the so-called descendants. From an

4This is called the Weyl or trace anomaly because in 2 dimensions, the metric can always be put in

the form gµν = e2wδµν , by which the Ricci scalar results into R = −2e−2w∂2w. Therefore, the trace

takes dierent values on backgrounds related by a Weyl transformation w and depends only on the Weyl

factor.

17


initial primary state |ψ〉, the tower starts

|ψ〉L−1|ψ〉

L2−1|ψ〉, L−2|ψ〉

L3−1|ψ〉, L−1L−2|ψ〉, L−3|ψ〉

...

The whole set of states is called a Verma module. They are the irreducible representations

of the Virasoro algebra. Therefore, to derive the spectrum of the whole theory we just

need to know the spectrum of primary states. The vacuum state |0〉 is annihilated by all

the lowering operators but also by L0, meaning it has h = 0. This state preserves the

maximum number of symmetries.

The energy eigenvalues h, h that here label states turn out to be related to the conformal

weights that label operators. This is due to the State-Operator map, which is a map

between states and local operators. The existence of such map is a priory surprising

because of the dierence between states and operators in terms of their nature, states

are dened over an entire spatial slice while local operators live on a single point. The

key point of this construction is that the distant past in the cylinder gets mapped to a

single point z = 0 in the complex plane, so a state on the cylinder in the far past can be

related to specifying local information at the origin of the plane. Let's see this in a bit

more detail.

In a eld theory, states are wavefunctionals that depend on the eld operators Ψ[φ(σ)].

The two-point function or propagator between two functionals is given by the path

integral at initial and nal xed congurations of the elds

G(φi, φf ) =

∫ φf=φ(τf )

φi=φ(τi)Dφe−S[φ].

Then a general state can be written by integrating over all possible initial congurations

at time τi and weighting each such conguration with the initial wavefunctional

Ψf [φf (σ), τf ] =

∫Dφi

∫ φf=φ(τf )

φi=φ(τi)Dφe−S[φ]Ψi[φi(σ), τi].

If we map this formulation on the conformal plane, the states are dened on circles of

constant radius and evolution is governed by the dilation operator. If the initial state is

at ri, the path integral of the propagator integrates over all eld congurations with xed

initial and nal boundary conditions φi = φ(ri), φf = φ(rf ), and the nal state results

18


as an integral over all initial congurations at ri. If we now take the initial state back to

the far past, the initial condition seats at r = 0 and we must integrate over the whole

disk up to rf . The initial wavefunctional in the integrand represents the weighting of the

path integral at the point z = 0 so it behaves exactly like a local operator inserted at the

origin. Therefore, inserting a general local operator O(z = 0) in the integrand denes a

state in the theory, a dierent state for each operator

Ψ[φf ; r] =

∫ φf=φ(r)

De−S[φ]O(z = 0). (2.26)

In particular, by inserting the identity operator into the path integral, we can create the

vacuum of the theory. Now it's easy to relate primary states with primary operators,

which are related one-to-one through the State-Operator map. Consider the state |O〉built from inserting a primary operator O into the path integral at z = 0. If we look at

the action of the Virasoro generators on this state

Ln|O〉 =

∮dz

2πizn+1T (z)O(z = 0) =

∮dz

2πizn+1(

hOz2

+∂Oz

+ ...),

where the path integral is implicit, we can see the eect of various generators on the

state |O〉:

• Ln|O〉 = 0 for all n > 0. This is true only for primary operators and means that

the state |O〉 is a primary. From here the bijective correspondence.

• L0|O〉 = h|O〉, which establishes that the conformal weight of the operator coincides

with the energy eigenvalue of the corresponding primary state.

• L−1|O〉 = |∂O〉. This is to be expected since L−1 is the translation generator.

From this correspondence follows that the most important content of a CFT is the

spectrum of weights of primary operators, since this coincides with the spectrum of

energy and angular momentum of the states of the theory dened on the cylinder.

Unitarity

So far we haven't imposed the condition of unitarity that normally quantum eld theories

exhibit and which entitles probability conservation. This condition follows in Minkowski

signature spacetime if the Hamiltonian is hermitian. In the case of our CFT dened on

the cylinder, the energy density follows from

H = Tττ = −(T (ζ) + T (ζ)) = (2π

L)2∑n

(Lne−2πinζ/L + Lne

−2πinζ/L).

19


where the Ln's are assumed to be the ones on the cylinder. For the Hamiltonian to be

hermitian it is then required that

Ln = L†−n.

This condition, together with requiring that there are no negative-norm states in the

theory, leads to the following conditions for the weights and central charge

• h ≥ 0. This follows from looking at the norm |L−1|ψ〉|2 = 〈ψ|L+1L−1|ψ〉 =

〈ψ|[L+1, L−1]|ψ〉 = 2h〈ψ|ψ〉 ≥ 0. The only state with h = 0 is the vacuum.

• c > 0. This follows from the norm |L−n|0〉|2 = 〈0|[LnL−n]|0〉 = c12n(n2 − 1) ≥ 0.

These are the two conditions for h, h and c that we have used in previous deriva-

tions.

2.3 The Cardy formula

In this section we proceed to compute the degeneracy of states and the entropy of a

CFT, which is given by the Cardy formula. In the previous section, we have seen how

the central charge provides an extra contribution to the vacuum energy. We will now

show that the central charge is also related to the density of high energy states. As we

will see, a thermal CFT can only be consistently dened on a cylinder. If the CFT is

originally dened on a cylinder, thermalization then yields a torus background topology.

The computation of the partition function on the torus oers certain advantages due

to some of its topological properties and therefore is going to be our arena. Let's rst

introduce the torus.

The torus and modular invariance

The torus is a closed Riemann surface with genus g = 1 and Euler number χ = 0, it is

therefore at. In the coordinates of the complex plane z, z, the at metric is ds2 = dzdz.

The torus is a quotient of the complex plane, meaning that z ∼ z+1 ∼ z+τ are identied.

Another way of representing the torus is by choosing a set of coordinates in which these

identications are trivial. If one does z = σ1 + τσ2, z = σ1 + τσ2 the identications

become σ1 ∼ σ1 + 1, σ2 ∼ σ2 + 1. The line element becomes ds2 = |dσ1 + τdσ2|2 which

is normally multiplied by 1/=(τ) so that the volume is normalized to 1. τ is called the

complex structure or modulus of the torus and it cannot be changed by innitesimal

dieomorphisms or Weyl rescalings.

However, it does transform under the so-called Modular transformations, which turn out

to be symmetries of the torus. Due to the periodicity conditions, it is obvious that the

transformations T : τ → τ + 1, X : τ → ττ+1 leave the torus invariant. Products

20


of these two transformations generate the whole group of modular transformations. A

common set of generators is given by T and T−1XT−1 ≡ S : τ → −1/τ . The most

general transformation can be parametrized by

τ ′ =aτ + b

cτ + d→ A =

(a b

c d

)detA = 1, a, b, c, d ∈ Z

Such transformations form the group SL(2,Z)/Z2 = PSL(2,Z), the group of the previous

matrices modulo a change of sign of its entries which doesn't aect the transformation.

In the development of the Cardy formula, we will see how modular invariance is a key

point.

The Cardy formula and the asymptotic growth of states

Let's proceed now with the computation of the degeneracy of states of a CFT. In the

microcanonical ensemble, the entropy is essentially the logarithm of the density of states

ρ(E). One way to obtain it is by manipulating the partition function.

In general, the partition function can be expressed as a path integral with periodic

boundary conditions

Z = Tr[e−βH ] =

∫dq〈q, β|q, 0〉E =

∫[dq]P e

−SE

where the subscript E indicates the Euclidean action. In other words, this integral runs

over all periodic paths on (0, β). In the path integral language then, β acquires the role

of Euclidean time. Therefore, a nite temperature of the CFT leads to periodic time

evolution, and the background topology becomes that of a cylinder of circle 1/T . If the

CFT is initially dened on a cylinder, then giving it a nite temperature yields the

topology of a torus The points on the base of the cylinder are therefore identied with

those of the top circle. and the resulting torus has =(τ) = β/L. For the identications to

be analogous to those that dene the torus though, we also have to allow for a <(τ) ≡σ/L 6= 0, i.e. the time evolution along the cylinder is not strictly vertical but twisted,

states are translated an amount σ in the spatial direction. This twisting is therefore

performed by the momentum operator P . The partition function we have to compute

then becomes

Z = Tr[e−βHeiσP ] (2.27)

where Lτ = σ + iβ (the L factor is introduced for later convenience but entitles no lack

of generality). We recall now that for a CFT on a cylinder5 of circle L, H and P can be

5The Virasoro generators of a CFT on the cylinder and on a torus are the same since the identications

that generate the torus from the cylinder are not a conformal transformation.

21


expressed in terms of the Virasoro generators. This follows from H being the generator of

time translations, therefore using the ground state energy (2.24) (corrected with a factor

1/2π)

Hcyl = −∂τ +E0

2π= −2π

L(z∂z + z∂z)−

π(c+ c)

12L=

2π

L[L0 + L0 −

c+ c

24].

This can also be seen by starting from H = 2πL (Lcyl0 + Lcyl0 ) and P = 2π

L (Lcyl0 − Lcyl0 ), and

expressing the Virasoro generators on the cylinder in terms of the Virasoro generators

on the plane using the transformation of the stress-energy tensor (2.22)

T cyl(ζ) = (2π

L)2[−z2T (z) +

c

24] = −(

2π

L)2[∑n∈Z

Lnz−n − c

24].

Since the expansion on the cylinder doesn't carry the conformal weight in the exponent,

as in (2.17), T cyl(ζ) = −(2πL )2

∑n∈Z L

cyln e−2πinζ/L and it follows that 6

Lcyln = Ln −c

24δn,0.

The partition function then becomes

Z(τ, τ) = Tr[e−βHeiσP ] = Tr[e2πiτLcyl0 e−2πiτ Lcyl0 ] = Tr[qL0−c/24qL0−c/24] (2.28)

where q ≡ e2πiτ , q ≡ e−2πiτ .

Parallel, we consider an auxiliary partition function in which we do not account for the

central term appearing in the Virasoro generators in the cylinder and we expand the

trace taking into account that h, h are the lowest eigenvalues of the Virasoro generators

L0, L0

Z ′(τ, τ) = Tr[qL0 qL0 ] =

∞∑nR,nL=0

ρ(nR, nL)qnR+hqnL+h. (2.29)

ρ(nR, nL) is the degeneracy of descendant states of level nR for L0 and level nL for L0.

This degeneracy can be obtained from Z ′(τ, τ) by contour integration

ρ(nR, nL) =1

(2πi)2

∮dq

qnR+h+1

dq

qnL+h+1Z ′(τ, τ)

where τ and τ can be treated as independent variables. If the CFT is chiral, the two

sectors will be excited independently and therefore the degeneracy factorizes ρ(nR, nL) =

6The minus sign in this denition is added for convenience. To avoid confusion with some of the

suggested references, some of them dene the conformal map from the plane to the cylinder as z = e2πζ/L,

where ζ includes the i factor. In that case, there is a relative minus sign in the transformation of the

stress-energy tensor.

22


ρ(nR)ρ(nL) which translates into the factorization of the partition function Z ′(τ, τ) =

Z ′(τ)Z ′(τ). Therefore, we can write

ρ(nR) =1

2πi

∮dq

qnR+h+1Z ′(τ), ρ(nL) =

1

2πi

∮dq

qnL+h+1Z ′(τ). (2.30)

We will now restrict to the chiral sector, the antichiral one follows analogously. We will

also use nR ≡ n.

To compute these degeneracies, it would be convenient to make use of the modular

properties that follow from a derivation in the torus. A CFT on a torus is modular

invariant. This follows from Nham's proof [10] that conformal invariance is a sucient

condition for modular invariance. The key result from Cardy was to use the modular

invariance of the partition function (2.28) [11].

This point is a bit subtle since the partition function (2.28) is actually not modular

invariant. Modular invariance has to be satised by physically-meaningful quantities,

like scattering amplitudes. For example, in bosonic string theory, the vacuum energy, i.e.

the partition function of the vacuum, at rst loop is a sum over all inequivalent torae

of the intermediate partition function computed for every torus. These intermediate

partition functions, which are similar to the partition function we encounter in this

derivation, aren't modular invariant either. The integral of these partition functions

over the fundamental domain of the modular parameter brings an additional factor in

the measure that yields a modular invariant result, as is expected since an energy is

a physical quantity. From this we can conclude that although not invariant, partition

functions on the torus can be assumed to be modular covariant, which allows for a

change in the partition function of a τ -dependent prefactor. For the rest of the derivation,

covariance would actually be enough, but to keep it simple we will use the full invariance.

However, leaving any physics aside, it should be mentioned that modular invariance is

normally postulated for conformal eld theories and therefore it is assumed for non-

physical quantities. This is the approach taken by many authors, also in the review by

Carlip [11]. Both the derivation and applicability of the Cardy formula seem to still have

some question marks. For example, Cardy proved the formula only for theories with

c < 1 [12]. However, the formula is generalized and used for theories with all c's without

rigorous proof. Ultimately, one could argue that as long as it yields the expected results

(agreement with other entropy results), the best approach is to assume its correctness

and applicability and hope for an eventual understanding of why this is so.

The auxiliary partition functions Z ′, Z ′ are not modular invariant. However, one can

notice that Z ′(τ) = qc/24Z(τ) and the same for the antichiral sector. We can now use

modular invariance under the transformation S to write

Z ′(τ) = qc/24Z(τ) = qc/24Z(−1/τ) = e2πiτc/24e2πic/24τZ ′(−1/τ),

23


and introduce it to (2.30)

ρ(n) =

∮dτe−2πiτ(n+h)e2πicτ/24e2πic/24τZ ′(−1/τ), (2.31)

where a change is performed to the integrating variable. To evaluate this integral, the

best we can do is a saddle point approximation when n→∞.

Let's recall that the saddle point method is used for approximating an integral of the

form∮g(z)enf(z)dz by deforming the contour in the complex plane to pass near the

saddle point of the integrand in roughly the direction of steepest descent. The saddle

point satises f ′(z∗) = 0. For the approximation to be valid, n has to be very large,

which ensures that the saddle point contribution grows as n→∞ (i.e. making the slopes

around it steeper). In this limit, the phase varies very rapidly. g(z) then is a slow varying

prefactor that accounts for the region far away from the saddle point that is not inuenced

by n. The advantage of this conguration is that g(z) can be approximated by g(z∗) and

f(z) by f(z∗) − |f ′′(z∗)||z − z∗|2/2 which leads to a Gaussian integral. However, in the

following we will cut the approximation of f(z) to 0-th order.

Looking at our integral (2.31), we could a priori speculate

f(n, τ) = −(n+ h)τ +c

24τ +

c

24τ

g(τ) = Z ′(−1/τ) =∑

ρ(n)e−2πi(n+h)/τ .

where we have left the factor 2πi out and we don't read a factorized dependence on n

in f(τ). The rst thing we have to check is if indeed g(τ) has a slow variation near the

saddle point. In the limit of n large, this one seats at τ∗ ≈ i√

c24n ≡ iε. Substituting this

into g(τ)

g(iε) = Z ′(i/ε) =∑

ρ(n)e−2π(n+h)/ε.

If h vanished, then this function would approach a constant ρ(0) in the limit ε going

to 0, leading to a slow variation. However, if h 6= 0 then g(τ) varies rapidly near the

saddle point and the approximation is not valid. This hints to the denition of a new

function

Z ′(τ) = q−hZ ′(τ) =∑

ρ(n)e2πinτ

such that the identications for the integrand functions become

f(n, τ) = −(n+ h)τ +c

24τ +

c

24τ− h

τ

g(τ) = Z ′(−1/τ) =∑

ρ(n)e−2πin/τ .

24


All we have to do now is evaluate f(τ) on the saddle point. Taking the limit of large n,

f(τ∗) ≈ −i√

(c−24h)n6 . The degeneracy of states results in

ρ(n) = ρ(0)e2π√

(c−24h)n6 .

ρ(0) will not pose a problem since, after taking the logarithm, the constant term is

negligible compared to the term proportional to√n. The Cardy formula is

ρ(n)n→∞−−−→ e2π

√(c−24h)n

6 = e2π√ceffn

6 (2.32)

and the same follows for the antichiral degeneracy. The notation of ceff is due to the

fact that theories with h 6= 0 show this shift in the central charge.

It has to be noticed that, although this formula is general, it has to be satised that

c > 24h for the derivation to hold. The entropy then follows from taking the logarithm,

which taking into account both chiral sectors is

S = 2π(

√ceffnR

6+

√ceffnL

6) (2.33)

The Cardy formula and the entropy

In the previous derivation, the Cardy formula yields the degeneracy of states as a function

of the eigenvalues of L0 and L0. It is therefore appropriate for computations in the

microcanonical ensemble. The canonical version of the formula, a bit less involved, also

makes use of modular invariance but only deals with thermodynamical quantities and

yields the entropy at the limits of low and high temperature.

First of all, it is worth realizing that, although we just assigned a temperature for the

CFT in the previous section, this is not a trivial point. In fact, a priory it would seem that

this is not possible since a temperature would introduce an energy scale and therefore

a length scale in the theory. However, for a CFT dened on a cylinder, the circle L

introduces a scale and still does not interfere with the conformal invariance. Therefore,

thermal CFT can not be dened on the complex plane. Then, since the only energy scale

of the theory is L−1, a natural denition for the physical temperature is

T =1

LT (2.34)

where T becomes a dimensionless temperature independent of the macroscopic param-

eters of the theory. Naturally, if L → ∞ recovering the complex plane, the physical

temperature goes to 0. We will see that the entropy of a CFT for large values of T only

depends on this dimensionless quantity.

25


Now let's nd the entropy. We start by considering the CFT dened on the Euclidean

torus. By extension from the Euclidean cylinder, the coordinates are (t, σ) and their

periodicities are t ∈ [0, β) and σ ∈ [0, L) (here σ does not refer the real part of the

modulus of the torus). As shown, the periodicity of the Euclidean time coordinate t

follows from the temperature of the theory β = 1/T 7. To absorb the explicit length scale

L, we rescale the coordinates as

t =1

Lt, σ =

1

Lσ. (2.35)

Their periodicities become t ∈ [0, β) with β = β/L = T−1 and σ ∈ [0, 1). For convenience

we redene the partition function Z(β) = Tr[e−βH ] = Tr[e−LβH ] ≡ Z(β).

Now, let's analyze the partition function at the two limits of the temperature. At low

temperatures, i.e. at β → ∞, the trace is dominated by the energy of the ground state.

As computed in the previous section in (2.24), the vacuum energy in the cylinder is E =

−π2c/6L (restricting to the chiral sector). The partition function at low temperatures

becomes

Z(β) ≈ eπ2cβ/6. (2.36)

Now, due to modular invariance we can interchange the spacelike and timelike coordinates

without changing the partition function, i.e. the torus is the same no matter how you

assign the periodicities to the two coordinates. Therefore, the timelike coordinate acquires

periodicity σ ∈ [0, 1) and the spacelike coordinate t ∈ [0, β). To compare with our original

partition function, we want the spacelike coordinate to have the range [0, 1). Therefore

we rescale the coordinates again

σ′ =1

βσ, t′ =

1

βt.

The timelike coordinate acquires periodicity σ′ ∈ [0, 1/β). The partition function now

depends on the new periodicity and, due to modular invariance, it holds that

Z(1/β) = Z(β). (2.37)

Here, it can be explicitly seen that modular invariance allows to relate the partition

function at low and high temperature. Now, we can easily use this last expression to

explore the limit to high temperature β → 0

Z(1/β) ≈ eπ2c6β . (2.38)

7We mentioned in the previous paragraph that the temperature could be consistently dened due to

the length scale introduced by the circle of the cylinder, therefore through the periodicity of the spacelike

coordinate. In the torus, the temperature follows from the periodicity of the Euclidean time coordinate.

This doesn't have any relevance thanks to modularity of the torus

26


To compute the entropy at the limits, we recall that the partition function can be written

in terms of the free energy as Z(β) = e−βF . Therefore, at high temperatures, we can

write

F ≈ − π2c

6Lβ2= −π

2c

6LT 2.

To compute the entropy,

S = −∂F∂T|L = −L∂F

∂T= −∂F

∂T

where we have dened F ≡ LF . Therefore, the entropy satises

S =π2

3cT .

This is the Cardy formula in the canonical ensemble. As anticipated, it does not depend

on the macroscopic scale L of the theory. In the context of the Kerr/CFT correspondence,

we will also see that the entropy only depends on the dimensionless temperature dened

for the Kerr black hole vacuum.

We have to pay attention, though, to a last couple of important points. When using the

Casimir energy for the lowest expectation value of the Hamiltonian, we are assuming

that the lowest eigenvalue of L0 is 0. However, this may not be so in a general CFT,

so we should account for it in the same way that, in the previous section, in (2.29) we

expanded the auxiliary partition function in qn+h. This h 6= 0 in the expansion was the

one responsible for the shift of the central charge to ceff in the nal result (that was

actually coming from ensuring a slow variation of the integrand function g(τ) in the

saddle point approximation).

Also, we have to generalize the previous formula to a general one that includes both chiral

sectors. One should take into account that, since the two chiral sectors are independent,

a priori they should also exhibit independent temperatures as the chemical potential

describing left and right degrees of freedom. In the previous derivation of the entropy,

we have regarded the temperature in general as being the inverse of the time periodicity,

without assigning to it any chiral character. Since we have argued previously that the

temperature is the inverse of the modular parameter τ of the torus in which the partition

function is being computed, one could think of dening the two temperatures as =(τ) =

1/TL and =(τ) = 1/TR. However this would not yield independent temperatures. A

better way to regard them is as the periodicities of the two light-cone coordinates rather

than time, which responds more intuitively to the left-right movement. Therefore, if β is

the periodicity of Euclidean time and σ is the periodicity of the spacelike coordinate, the

periodicities of the light-cone coordinates are β ± σ. The temperatures can be dened

27


then as 1/TL = β + σ and 1/TR = β − σ, giving the relation

1

T=

1

2(

1

TL+

1

TR).

Therefore, taking into account the contribution of each sector, the most general form of

the Cardy formula is

S =π2

3(ceff TL + ceff TR). (2.39)

28

CHAPTER 3

Anti-de Sitter spacetime

Anti-de Sitter is a maximally symmetric spacetime, that is, a spacetime with the highest

degree of symmetry possible in a certain number of dimensions n. Maximally symmetric

manifolds have the same curvature everywhere and in every direction, which translates

into translational and rotational isometries. This implies the existence of n(n + 1)/2

isometries and therefore Killing vectors. These manifolds, thus, are characterized by the

sign of the curvature R, the signature of the metric and the number n of dimensions.

For Lorentzian signature, the corresponding maximally symmetric spacetime with

positive curvature is de Sitter spacetime, and for negative curvature is Anti-de Sitter

spacetime.

In general relativity, Anti-de Sitter spacetime is the solution to Einstein's equations in

the absence of any ordinary matter or radiation, it is a vacuum solution. The inherent

curvature of this spacetime is then accounted for by means of a negative cosmological

constant Λ such that Tµν = −Λgµν , which entails a negative vacuum energy density.

The n-dimensional Anti-de Sitter spacetime is dened in the (n + 1)-dimensional at

embedding

ds2 = −dX20 +

n−1∑i=1

dX2i − dX2

n (3.1)

by the hyperboloid equation

−X20 +

n−1∑i=1

X2i −X2

n = −l2 (3.2)

where l2 = −1/Λ is the radius of curvature. To solve this constraint one can introduce

dierent parametrizations on the hyperboloid. One set of these are the so-called Poincaré

29

Chapter 3: Anti-de Sitter spacetime

coordinates (t, u, xi), dened by

X0 =u

2[1 +

1

u2(l2 − t2 +

n−2∑i=1

(xi)2)]

Xn =lt

u

Xi =lxi

u, i = 1, ..., n− 2

Xn−1 =u

2[1− 1

u2(l2 + t2 −

n−2∑i=1

(xi)2)]

and t, xi ∈ R, u > 0. In these coordinates, the metric becomes

ds2 =l2

u2(−dt2 + du2 + d~x2). (3.3)

A related metric that we will encounter later on is the one obtained doing r = 1/u

ds2 = l2(dr2

r2+ r2(−dt2 + d~x2)). (3.4)

This metric is conformal to Minkowski spacetime, therefore its conformal diagram has the

triangular structure corresponding to the right half of the Minkowski rectangular diagram

(since u only takes on positive values). However, with these coordinates only half of the

hyperboloid is covered, the so-called Poincaré patch. One can instead introduce global

coordinates (τ, ρ,Ωi), dened by

X0 = l cosh ρ cos τ,

Xn = l cosh ρ sin τ

Xi = l sinh ρΩi, i = 1, ..., n− 1;∑i

Ω2i = 1 (3.5)

which lead to the metric

ds2 = l2(− cosh2 ρdτ2 + dρ2 + sinh2 ρdΩ2n−2) (3.6)

With ρ ≥ 0 and 0 ≤ τ < 2π, these coordinates cover the whole hyperboloid once. In

the limit ρ → 0 the metric behaves as ds2 ' l2(−dτ2 + dρ2 + ρ2dΩ2), which means

the hyperboloid has the topology of S1 ×Rn−1. An often encountered metric is the one

obtained by performing the change sinh ρ = y such that y ≥ 0. The metric is

ds2 = l2(−(1 + y2)dτ2 +dy2

(1 + y2)+ y2dΩ2

n−2). (3.7)

30

3.1 Causal structure and conformal boundary


To analyze the causal structure of AdS it is convenient to perform a conformal

compactication, i.e., nd a change of the global coordinate ρ that brings the boundary

to a nite value. Doing tan θ = sinh ρ, with 0 ≤ θ < π/2, the metric becomes

ds2 =l2

cos2 θ(−dτ2 + dθ2 + sin2 θdΩ2

n−2). (3.8)

The rst thing one can notice is that this spacetime contains closed timelike curves,

since the timelike coordinate τ is angular. However, this is not an intrinsic property of

the spacetime but rather a consequence of how we derived the metric from a particular

embedding. So, one can unwrap the circle S1 allowing τ to run over the entire real

line obtaining in this way the universal covering space of AdSn. Once obtained a causal

spacetime, we can use the conformally rescaled metric ds2 = −dτ2 + dθ2 + sin2 θdΩ2 to

study the causal structure and asymptotics of AdS. The conformal diagram is a lled

cylinder, with the timelike coordinate τ in the vertical axis and the θ coordinate in the

perpendicular one. The revolution coordinate around the τ axis corresponds to the rst

angular coordinate of the Sn−2 sphere and every point is a Sn−3. Therefore, a spacelike

hypersurface of constant τ is conformally mapped to a (n− 1)-hemisphere; the equator,

at θ = π/2, is the boundary of the space at a time-slice.

Let's rst give a closer look to the diagrams of the 2 and 3-dimensional cases. The

metric for AdS2 in global coordinates is conformal to ds2 = −dτ2 + dθ2. In this

case, the coordinate θ ranges from −π/2 to π/2 since S0 consists of two points. The

conformal diagram is represented in gure 3.1. Points on the vertical central line are at

the spatial origin, while the left and right-hand-side boundaries represent timelike surfaces

corresponding to spacelike and null innity and have topology of R1. Since there are no

conformal transformations that make timelike innity nite (without reducing the whole

diagram to a point), it is represented by the disjoint points i+ and i−.

In the conformal diagram null geodesics draw triangular regions. A single one of these

regions corresponds to the Poincaré patch, covered by the aforementioned coordinates

(t, u) or (t, r). This follows from imposing u > 0 on the change of coordinates from

Poincaré to global coordinates1. The Poincaré patch acquires this triangular shape since

the AdS metric (3.3) is conformal to (half) Minkowski and the global coordinates

transformation is almost a conformal compactication. However, the triangular patch

1The change of coordinates is t = sin τcos τ−sin θ

= (√

1 + y2 sin τ)/r, r = 1u

= cos τ−sin θcos θ

=√1 + y2 cos τ + y, where we have used the change tan θ = −y with an additional minus sign that

doesn't change the metric. We will use this change of coordinates later on in chapter 7).

31


Figure 3.1 On the left, conformal diagram for AdS2. Every diamond-shaped region is a

Poincaré patch. Dotted lines represent timelike geodesics. On the right, conformal diagram

for AdS3, which adds a circle to that of AdS2. We can also see the Poincaré patch limited

by the two null hyperplanes.

in the diagram of global coordinates does not exhibit the innities and spatial origin at

the same points the Minkowski diagram does.

Timelike geodesics never reach the boundary at innity, since leaving from a point on

the origin and moving radially outward, they eventually converge at the origin a ∆τ = π

interval later.

In 3 dimensions AdS3, conformal to ds2 = −dτ2 + dθ2 + sin2 θdφ2, adds a circle to the

AdS2 metric. The diagram results into a lled cylinder and is shown in gure 3.1. The

cylindrical boundary still corresponds to spatial and null innities and is also timelike;

it has the topology of R× S1. In this case, the Poincaré patch in the conformal diagram

corresponds again to the area comprised between two null hyperplanes intersecting the

cylinder at π/4 with respect to the θ axis.

The timelike character of the boundary, with topology R×Sn−2, is an important feature

of AdS and plays a key role in the AdS/CFT correspondence. This is because it prevents

AdS from having a well-posed Cauchy problem. Let's see what this means. In general,

to have a good understanding of the causality structure of a spacetime it is necessary

to analyze the initial-value problem or Cauchy problem [13]. That is, to determine

the evolution of matter elds on a xed background spacetime, it is rst necessary

to determine in which region of the spacetime manifold one has to specify the initial

conditions. An important concept is the future domain of dependence D+(S) of a subset

32


S of the spacetime manifold.2 This is dened as the set of all points p in the manifold

such that every past-directed inextendible causal curve through pmust intersect S. Causal

curve refers to one that is timelike or null everywhere. The boundary of D+(S) is called

the future Cauchy horizon H+(S) and is a null surface. Analogously one can dene the

past domain of dependenceD−(S) and its past Cauchy horizonH−(S). The point of these

concepts is clear: since information cannot propagate faster than light, then information

(initial data for matter elds) specied on S is sucient to determine the situation (value

of the elds) at any other point in the whole domain of dependence D(S). Finally, we

dene a Cauchy surface to be the (closed achronal) surface whose domain of dependence

is the entire manifold. Then, a spacetime has a well dened initial-value formulation if

it has a Cauchy surface: from information dened on it one can predict what happens

in the entire spacetime. If a spacetime has a Cauchy surface it is said to be globally

hyperbolic.

Having dened the key concepts, let's analyze what happens in AdS. Because innity

is timelike, there exists no Cauchy surface in the space and so the space is not globally

hyperbolic. While there exist spacelike surfaces that cover the whole space, surfaces of

constant τ for example, one can nd null geodesics which never intersect these surfaces,

namely the null geodesics starting at spatial innity. Therefore, such a surface has a

domain of dependence bounded by the two null geodesics starting at spatial innity at

the same τ . Given initial data on such a surface, one cannot predict any further than the

Poincaré patch dened by the two corresponding aforementioned null geodesics. Points

beyond this region are aected by information owing in from innity. Therefore, to

make the Cauchy problem well posed on AdS one has to specify boundary conditions at

innity, i.e. at θ = π/2.

Conformally at boundary

Another interesting feature that will play an important role in the correspondence is

the relation between the boundary of AdS and Minkowski spacetime. It turns out that

the conformal boundary of AdSn is identical to the conformal compactication of the

(n − 1)-dimensional Minkowski space. The latter starts with the Minkowski metric in

spherical coordinates

ds2 = −dt2 + dr2 + r2dΩ2n−3.

The conformal compactication is performed on the time and radial coordinate through

the change of coordinates t±r = tan (τ ± φ)/2, with which the metric becomes conformal

to the so-called Einstein static universe

ds2 = −dτ2 + dφ2 + sin2 φdΩ2n−3 = −dτ2 + dΩ2

n−2.

2The subset S has to be closed and achronal, meaning that no two points in S can be connected by

a timelike curve.

33


Because of the ranges of the (t, r) coordinates, the (τ, φ) coordinates are restricted to

the interior of the famous triangular region 0 ≤ φ < π and |τ | + φ < π. As before, one

can analytically continue outside of the triangle to the maximally extended space with

τ ∈ R. This is exactly the conformal boundary of AdSn, the metric (3.8) is conformal to

the (n− 1)-dimensional Einstein static universe at θ = π/2.

3.2 Isometries

Since AdSn spacetime is dened in an (n+1)-dimensional at embedding with signature

(−,−,+, ...,+), it is invariant under the group SO(2, n−1). Looking at (3.6), the metric

is invariant under translations in τ and the rotations of Sn−2, which are represented by

the subgroups SO(2) and SO(n − 1) of the isometry group. The norm of the Killing

vector ∂τ is everywhere negative and constant in the conformally rescaled AdS. Since

the metric is diagonal, τ is called the global time coordinate of AdS.

On the other hand, if we look at the metric in the form (3.3), the manifest symmetries

are the dilatation transformation (t, u, ~x) → (ct, cu, c~x) (with c > 0) of the subgroup

SO(1, 1) and the Poincaré transformations on (t, ~x) with the subgroup ISO(1, n − 2).

This is consistent with the aforementioned relation between the conformal boundary of

AdSn and conformal Minkowski spacetime. In other words, AdSn can be thought of as an

(n− 1) dimensional Minkowski with a warp factor that depends on an additional radial

coordinate, u. Therefore, the isometry group of (n− 1) dimensional Minkowski, acts on

the boundary coordinates (t, ~x).

AdS2, with 3-dimensional isometry group SO(2, 1) ∼= SL(2,R) has got 3 obvious Killing

vectors, ξ1 = ∂t, ξ2 = ∂τ = ( 12r2 + t2+1

2 )∂t − tr∂r and the dilation Killing vector ξ3 =

t∂t − r∂r.

The isometry group of AdS3 is SO(2, 2) ∼= SL(2,R) × SL(2,R). As a maximally

symmetric space it has 6 independent Killing vectors, which correspond to the 6 Lorentz

symmetries, two rotations and four boosts, of the 4-dimensional embedding (since the

translations don't leave the hyperboloid equation invariant). Therefore, in terms of the

embedding coordinates, the Killing vectors are

ξ1 = X2∂1 −X1∂2 ξ2 = X0∂3 −X3∂0

ξ3 = X1∂0 +X0∂1 ξ4 = X2∂0 +X0∂2

ξ5 = X1∂3 +X3∂1 ξ6 = X2∂3 +X3∂2.

We can perform a change of coordinates and express them in the global coordinates

34

3.2 Isometries

(τ, φ, r)

ξ1 = ∂τ ξ2 = ∂φ

ξ3 = − r√1 + r2

cosφ sin τ∂τ −√

1 + r2

rsinφ cos τ∂φ +

√1 + r2 cosφ cos τ∂r

ξ4 = − r√1 + r2

sinφ sin τ∂τ +

√1 + r2

rcosφ cos τ∂φ +

√1 + r2 sinφ cos τ∂r

ξ5 =r√

1 + r2cosφ cos τ∂τ −

√1 + r2

rsinφ sin τ∂φ +

√1 + r2 cosφ sin τ∂r

ξ6 =r√

1 + r2cos τ sinφ∂τ +

√1 + r2

rcosφ sin τ∂φ +

√1 + r2 sinφ sin τ∂r (3.9)

These Killing vectors can be combined to give the representation in which the algebra

becomes exactly sl(2,R) × sl(2,R). For completion, we give these combinations, that

result into the following Killing vectors ξi3

ξ1 =i

2(ξ1 + ξ2) =

i

2(∂τ + ∂φ)

ξ2 =i

2(ξ1 − ξ2) =

i

2(∂τ − ∂φ)

ξ3 =1

2[ξ3 + iξ4 + i(ξ5 + iξ6)] =

ei(τ+φ)

2(

ir√1 + r2

∂τ +i√

1 + r2

r∂φ +

√1 + r2∂r)

ξ4 =1

2[−ξ3 + iξ4 + i(ξ5 − iξ6)] =

e−i(τ+φ)

2(

ir√1 + r2

∂τ +i√

1 + r2

r∂φ −

√1 + r2∂r)

ξ5 =1

2[ξ3 − iξ4 + i(ξ5 − iξ6)] =

ei(τ−φ)

2(

ir√1 + r2

∂τ −i√

1 + r2

r∂φ +

√1 + r2∂r)

ξ6 =1

2[−ξ3 − iξ4 + i(ξ5 + iξ6)] =

e−i(τ−φ)

2(

ir√1 + r2

∂τ −i√

1 + r2

r∂φ −

√1 + r2∂r)

satisfying the algebra

[ξ1, ξ3] = −ξ3, [ξ1, ξ4] = ξ4, [ξ3, ξ4] = 2ξ1

[ξ2, ξ5] = −ξ5, [ξ2, ξ6] = ξ6, [ξ5, ξ6] = 2ξ2.

3This set of Killing vectors ξi can also be obtained by solving the Killing equation Lξgµν = 0 using

the ansatz ξµ(τ, φ, r) = fµ(r)g(τ ± φ).

35


3.3 Euclidean AdS

It turns out that some computations in eld theory, like computations of time-

ordered correlation functions4, are more practical in the Euclidean version of AdS

spacetime.

Since AdS has a global time coordinate τ , on which the metric doesn't depend, the

continuation to Euclidean signature is straightforwardly implemented by performing a

Wick rotation τ → iτE . Using the change of coordinates (3.5), n-dimensional Euclidean

AdS can be dened as the hyperboloid equation X20 − ~X2 − X2

E = l2 embedded in

R1,n. We can also obtain the same space, i.e. the whole hyperboloid, by Wick rotating

the Poincaré t coordinate, even though the Poincaré coordinates only cover half of the

hyperboloid in the Lorentzian case.

From (3.3), the metric results in

ds2E =

l2

u2(dt2E + du2 + d~x2) (3.10)

where u > 0. Euclidean AdSn is topologically a n-dimensional disk. In this representation,

the boundary is conformal to Rn−1, at u = 0, together with an added single point at

u = ∞ (this is because the prefactor 1/u2 of the metric (3.10) shrinks ∞ to a point),

which is equivalent to the sphere Sn−1. In terms of the whole space, this added point at

u = ∞ compacties AdSn to an n-dimensional disk. This (n − 1)-dimensional spherical

boundary is the Euclidean version of the conformal compactication of Minkowski space

being the boundary of Lorentzian AdSn.

An equivalent way to describe Euclidean AdSn space is through an open ball∑n

i=1 y2i < l2

in the Euclidean space Rn. Euclidean AdSn can be identied with this ball with the

metric

ds2 =4∑n

i=1 dy2i

(1− |y|2/l2)2. (3.11)

The closed ball∑n

i=1 y2i ≤ l2 corresponds to the compactied Euclidean AdSn, with

again the sphere Sn−1, dened as∑n

i=1 y2i = l2, as the conformal boundary. The metric

(3.11) on the open ball does not extend over the boundary because of its singularity.

To get a metric which is also dened on the boundary one can multiply the original

4In at space, to continue a Green function 〈A1(x1, t1)...An(xn, tn)〉 from Lorentzian to Euclidean

signature, we let A(x, t) → A(x, 0)e−HτE . In a theory with energy bounded from below, the Euclidean

space Green function is guaranteed to converge only if the operators are time ordered, i.e. τE,j > τE,j+1,

since it will carry the exponentials e−H(τE,j−τE,j+1). The same holds in AdS space, Euclidean Green

functions are related to time-ordered Lorentzian ones by means of a Wick rotation. Green functions of

free elds on AdS have been computed using this method.

36

3.3 Euclidean AdS

metric with a function f2 which is positive on the open ball and has a zero on the

boundary, the most natural choice being (1− |y|2/l2)2. As f is not unique, the extended

metric ds2 = f2ds2 is only well-dened up to conformal transformations. Therefore, the

boundary Sn−1 only has a conformal structure, which is preserved by the action of the

isometry group, SO(1, n) in this case.

37

CHAPTER 4

3-dimensional Einstein gravity

with Λ < 0.

In this chapter we will focus on asymptotically Anti-de Sitter spacetimes in 3 dimensions.

In general, low-dimensional models are a useful source of inspiration because they allow

to reach understanding in an easier setting than the higher dimensional cases. However,

we will see that the study of 3 dimensional gravity is not only academic, 3-dimensional

AdS spacetimes play a relevant role in the description of the microstructure of higher

dimensional systems, such as the D1-D5-P system [8] and the 4-dimensional Kerr black

hole.

The motivation to have a proper denition of an asymptotically AdS spacetime comes

from the need of writing down a gravitational partition function. Geometries that

asymptote to AdS, in the sense that they have the AdS-metric behavior at spacelike

innity and the same boundary topology, appear very often as near-horizon geometries

of black holes or factors of these. The states contributing to the gravitational partition

function of these black holes are excitations of their near-horizon AdS factors. Hence, the

eventual goal of identifying and counting microstates of a black hole requires a partition

function where the geometries that are summed over are constrained to be asymptotically

AdS. Intuitively, the vacuum AdS solution can be regarded as the ground state and

excitations may arise from inserting elds, matter or black holes on this background

geometry. The 3-dimensional black hole, the BTZ black hole [14], plays an important

role here as the saddle point of the action, which carries the maximum contribution to

the entropy of the Hilbert space. It is clear then that we need to rene the denition of

asymptotically AdS such that it includes the right excitations.

The spotlight is therefore translated to the asymptotics, which motivates the arrival on

39

Chapter 4: 3-dimensional Einstein gravity with Λ < 0.

scene of the formalism of asymptotic symmetries and global charges. These are bijectively

related to the allowed geometry excitations and so nding them and their algebra is of

capital importance. The computation of this algebra for the case of 3-dimensional AdS

spacetimes was done in 1986 by Brown and Henneaux in [15], understanding their work

is the main aim of this chapter. They showed that the algebra of charges is a full Virasoro

algebra and computed the central charge. Although their computation was at the classical

level, the appearance of this typically-conformal algebra constitutes the rst hint to a

gravity/CFT duality, which will be the topic of next chapter. There we will see how this

duality allows to compute the entropy of the black hole through the use of the Cardy

formula.

Another aspect that is important in order to give this gravitational phase space a

statistical treatment is the need to thermalize it. We have already seen how temperature

can be assigned to a CFT by making the Euclidean time on the cylinder periodic. We

will use a similar idea for the geometries of interest. Modular invariance will also play

a role here as thermal AdS and the BTZ black hole are related through a modular

transformation.

4.1 Asymptotic symmetries and surface charges

Most common black holes are asymptotically at, as going towards spatial innity the

gravitational eld is less and less intense and the curvature tends to zero. This idea can

be generalized to an asymptotic curvature that is constant though not necessarily zero.

Therefore, spacetimes can be asymptotically at, de Sitter or Anti-de Sitter.

It is now natural to wonder about the remaining symmetry structure of a spacetime that

only resembles AdS asymptotically. This issue is going to be the focus of this section. The

relevance of the asymptotic symmetries and the role they play in the correspondence will

keep getting clear in further sections. For now, we will introduce the main ingredients:

boundary conditions, surface charges and the asymptotic symmetry algebra. Later, we

will focus on asymptotically AdS spacetimes.

Asymptotic Killing vectors

Symmetries are reected by Killing vectors. Let's recall that a Killing vector of a

metric satises the Killing equation Lξgµν = ∇µξν + ∇νξµ = 0. Also, as introduced

in (2.2), a conformal Killing vector satises the conformal Killing equation Lξgµν(x) =2d(∇ · ξ)gµν(x), where d is the dimension of the spacetime. They generate conformal

transformations on the spacetime. Now, the current context leads to dening yet a

third class of vectors, the asymptotic Killing vectors. In a way, they have to reect

40


the condition of a dieomorphism that only represents a symmetry asymptotically, so

the Killing equation has to be relaxed. One way to formulate this, is by expressing the

spacetime metric in the following way

gµν(xa) = gµν(xa) + hµν(xa) = gµν(xa) +O((xr)−pµν ) (4.1)

where xr represents the coordinate whose limit to∞ corresponds to the boundary, gµν(xa)

represents a background metric (in our case the AdS metric) and hµν(xa) uctuations on

top of it that obviously have to decay asymptotically. Then, an asymptotic Killing vector

is such that it preserves the metric at innity in the sense that this one transforms under

the Lie derivative into a metric of the same form (4.1), the asymptotic Killing equation

being Lξgµν = O((xr)−pµν )

A set of boundary conditions corresponds to a specic set of integer values for pµν .

Once the boundary conditions are xed, one can compute the asymptotic Killing vectors.

Dierent boundary conditions may be relevant in dierent physical contexts.

Surface charges

It has to be noticed that the construction of the asymptotic Killing vectors doesn't

involve any knowledge of the dynamics of the spacetime, except for requiring that the

background metric is a solution of the Einstein equations. As long as the boundary

conditions are reasonable, a solution for the asymptotic Killing vectors can be found

without any reference to general relativity. It would seem then that the boundary

conditions can be chosen rather arbitrarily. However, this is not exactly so due to the

generators associated to the asymptotic symmetries. These generators are the so-called

surface charges, which have been extensively studied in General Relativity because of the

importance of symmetries and conserved charges. Total energy and angular momentum

are canonical examples of surface charges, the generators associated to the Killing vectors

∂t and ∂φ. The derivation of surface charges can be done in the Hamiltonian or in the

Lagrangian formulation. In this thesis we will use the former, for which an introduction

to the Hamiltonian formulation of GR and the ADM formalism has been presented in

appendix A. In this framework, the theory of surface charges was developed in [16], [17],

[18] and [15]1.

In the Hamiltonian formalism, the surface charges appear as the canonical generators of

the asymptotic symmetries of the theory; with each symmetry ξ is associated a phase

space function Qξ which generates the corresponding transformation of the canonical

variables. This generator diers from a linear combination of the constraints Hµ of the

1In the Lagrangian framework, several approaches have been developed, among which there is the

covariant formalism of Barnich, Brandt and Compère [19], [20], [21], which in particular was the one

used in the original paper of the Kerr/CFT correspondence.

41


canonical formalism by a surface term which is such that Qξ has well dened functional

derivatives. Let's see this in more detail.

In appendix A we have shown that the GR Hamiltonian generically has the form

Hgrav =

∫Σt

ddy√q(NH+NaHa) +

∮∂Σt

dd−1θ√γ(NHbdy +NaHbdya ). (4.2)

In the standard canonical formalism, the Hamiltonian generates time translations. The

Hamiltonian (4.2) generalizes this idea generating evolution along the ow vector tµ =

Nnµ + Naeµa . To see this more clearly, let's rewrite the integrand in the full-spacetime

basis doing

H = nµHµ and Ha = eµaHµHbdy = nµQµ and Hbdya = eµaQµ.

The rst two quantities are the Hamilton and momentum constraint functions and vanish

on shell. The two latter do not necessarily vanish on shell. Writing the integrand in this

basis gives the more intuitive

Hgrav =

∫Σt

ddy√qHµtµ +

∮∂Σt

dd−1θ√γQµtµ.

It is now natural to generalize this Hamiltonian to other generators by replacing the ow

vector tµ by a generic deformation ξ, thus dening the canonical generator of Lξ (Lie

transport) as

Qξ =

∫Σt

ddy√qHµξµ +

∮∂Σt

dd−1θ√γQµξµ (4.3)

in such a way that Qt = Hgrav. This generator depends on the elds and their canonically

conjugate momenta. One is typically interested in charges of solutions to the eld

equations, for which Hµ = 0. The piece that remains is the boundary term of Qξ

Qξ =

∮∂Σt

dd−1θ√γQµξµ (on-shell) (4.4)

Because this on-shell charge only contains the boundary piece it is called the surface

charge. The purpose of this boundary term in the gravitational Hamiltonian is to yield

a Hamiltonian with well-dened functional derivatives such that it is a well dened

generator of surface deformations through the Poisson bracket. The terms coming from

the variation of the boundary piece exactly cancel the unwanted surface terms from the

42


variation of the bulk piece when one assumes that the variation of the canonical eld

vanishes on the boundary; as explained in appendix A, when δN = δNa = δqab = 0,

where qab is the induced metric on the boundary. However, in the context of the above

generalization, where we have substituted the ow vector for a general transformation

ξ, we wish to allow for a non-vanishing variation of the eld on the boundary. In

particular, we want it to acquire the variation dictated by the boundary conditions,

i.e. allow for δqab = hab = O((xr)−pµν ). Obviously, this requirement makes sense when

the transformation ξ is an asymptotic Killing vector.

Allowing for this variation of the induced eld on the boundary implies that we cannot

use the canonical expressions (A.4) for Hbdy, Hbdya to write Qµ. Instead, we need to vary

the bulk piece, keep all the surface terms that emerge and equate them to the variation

of the boundary term. This will dene the surface charge on-shell. We proceed to do this

without specifying all the terms. The generic form for the bulk variation will include the

two terms corresponding to the variation of the canonical variable qab and its conjugate

momentum pab

δ

∫Σt

ddy√qHµξµ =

∫Σt

ddy√q(...)abδqab + (...)abδp

ab+

∮∂Σt

dd−1θ√γ....

Now, Q is dened such that its variation satises

δ

∮∂Σt

dd−1θ√γQµξµ = −

∮∂Σt

dd−1θ√γ....

We conclude that

δQξ = −∮∂Σt

dd−1θ√γ... (on-shell). (4.5)

In order to nd Qξ we need to functionally integrate this relation, which is not an easy

task and whose details we will skip. A relevant point to notice is that this integration

determines the central charge up to an integration constant. At the level of Hamiltonian

classical mechanics, the appearance of this constant is the cause of a central term in the

algebra of surface charges. We will see that this also happens for our case.

As follows from the above derivation, the surface charges heavily depend on the boundary

conditions chosen. These have to be such that the surface charges are well-dened, nite

and conserved, since we are dealing with a spatial boundary. Well-dened refers to the

fact that the surface charges have to be dierentiable. A surface charge Q[g], functional

of the canonical variable gµν(xa), is dierentiable if its variation can be brought to the

form

δQ =

∫δQ

δgµν(xa)δgµν(xa)dnx (4.6)

43


for any allowed variation δg of the metric eld and for functional derivatives δQ/δgµν(x)

that are regular functions of x. Dierentiable therefore also means that the charges are

integrable.

If the boundary conditions are too restrictive, they will rule out all interesting excitations.

If they are too permissible, they become inconsistent because transformations preserving

them lead to ill-dened or innite charges. In general, there is a narrow window of

consistent and interesting boundary conditions. There is no universal procedure to dene

the boundary conditions. The one that we will present in 4.3 for AdS3 consists of

promoting all exact symmetries to asymptotic symmetries and then acting with these

on solutions that want to be accounted for to generate tentative boundary conditions.

Then these are restricted to admit nite, well-dened and conserved charges. In 8.2, the

approach for nding the boundary conditions for the extreme Kerr black hole near-horizon

geometry will be rather based on the analysis of non-trivial rotating dynamics and the

hope of a Virasoro extension of the isometry algebra. In general, imposing reasonable

and consistent boundary conditions requires an analysis of the asymptotic dynamics of

the theory.

Asymptotic symmetry group

Associated to a set of boundary conditions comes the so-called asymptotic symmetry

group (ASG). This is dened as the set of allowed symmetry transformations modulo the

set of trivial ones. Allowed dieomorphisms refers to the ones that preserve the boundary

conditions, while trivial ones, also called pure gauge, are those that act trivially at innity,

i.e. whose surface charge vanishes. The ASG can contain the isometry group as a subgroup

and even extend to an innite number of generators. Although it is called a 'group' the

asymptotic symmetries form a Lie algebra, let's see this for AdS.

Asymptotically AdS spacetimes and the algebra of charges

We want to focus on asymptotically AdS spacetimes, i.e. we want to identify the phase

space of solutions that are allowed deformations of an AdS background metric. Naively,

we look for metrics that for large r become (in global coordinates (3.7))

ds2 → l2(−r2dt2 +dr2

r2+ r2dΩ2

n−2) (4.7)

However, we need a more concrete denition of AdS asymptotics in terms of the ASG and

surface charges. In other words, we need to make sense of what we mean by reasonable

boundary conditions, the precise way in which the solutions should approach (4.7) for

large r. This was done by Henneaux and Teitelboim in [17], where they presented the

natural conditions for asymptotically AdS4 spaces. The hope is that this strategy can be

applied to other dimensions and spaces.

44


According to their denition in 4 dimensions, the boundary conditions imposed at spatial

innity should meet the following three requirements:

• they should be invariant under the action of the exact isometry group SO(2, 3).

• they should make the surface charges associated with the generators of SO(2, 3)

nite and have well-dened functional derivatives. Also, the charges should obey

the so(2, 3) algebra.

• they should be such that the phase space contains the background solution, i.e.

AdS and all other solutions that are reasonable and of physical interest. In the

case of AdS3, this means for example the BTZ black hole.

In the paper, the authors propose a set of boundary conditions which meet the above

requirements. For these boundary conditions, they also present an explicit expression for

the so(2, 3) charges in terms of the canonical variables and they show that they are nite

and close under the Dirac bracket.

A year after this paper was published, Brown and Henneaux showed in [18] that the

Poisson bracket2 of the generators of two asymptotic symmetry transformations is also

a dierentiable generator in any dimensions and background. The theorem, called the

representation theorem, encodes two pieces of information. First, that the Poisson bracket

is dierentiable, which is due to the fact that it contains no unwanted surface terms

in its variation and therefore has well dened functional derivatives. Second, that the

transformation generated by the Poisson bracket preserves the boundary conditions.

This theorem clearly points at the existence of an algebra of charges in general, that

is isomorphic to the ASG Lie algebra but that may or may not be the same as the exact

isometry algebra. In the case of 3-dimensional asymptotically AdS spaces, this algebra

was explicitly presented in the later paper by the same authors [15]. In this paper, they

showed that the ASG is actually enhanced with respect to the exact isometry group to

an innite dimensional algebra and that the surface charges actually satisfy a projective

representation of the asymptotic symmetry algebra, i.e. it acquires a central extension.

As already mentioned, intuitively we would expect this from the integration constant up

to which the charges are dened. This extension is due to a residual term that is left after

2The Dirac bracket is a generalization of the Poisson bracket to treat systems with constraints in the

Hamiltonian formalism, such as the Hamilton and momentum constraints presented in appendix A. The

two types of brackets are related to each other through the addition of a term containing the metric of

constraints. Throughout this thesis, the two brackets are going to be used without distinction.

45


identifying the generator of the Lie bracket of asymptotic symmetries3. Accordingly, the

Dirac bracket of the charges of two given asymptotic symmetries diers from the charge

associated to the Lie bracket of the two symmetries by a constant term, in the sense that

it does not depend on the canonical variables

Qξ[q], Qη[q] = Q[ξ,η][q] + Cξη, (4.8)

where we have explicitly written the functional dependence of the charges on the canonical

eld and Cξη is the central term. To choose the arbitrary integration constant in the

denition of the charges, these are dened such that

Qξ[q] = 0 ∀ξ ∈ ASG.

Now, the central term can be obtained by evaluating (4.8) on the background geometry,

since it does not depend on the metric on which (4.8) is evaluated. In this case, Q[ξ,η][q] =

0. The bracket doesn't vanish since it actually represents the change in the rst charge

under the transformation implemented by the second charge Qξ[q], Qη[q] = δηQξ[q].

Since we are evaluating these charges on the background metric, then Qξ[q] vanishes

before the deformation by η and the central charge reduces to the charge evaluated on

the metric deformed by η4

Cξη = Qξ[Lη q] = −Qη[Lξ q]. (4.9)

The central charge rules out when the asymptotic symmetries are exact symmetries of

the background metric. In this situation, the charges evaluated for the background are

invariant, i.e. the bracket vanishes, since the background itself is unchanged under the

symmetry transformation. Then it follows that Cζη = 0. It also has to be noticed that the

central charge does not explicitly depend on the details of the boundary conditions, in the

sense that other boundary conditions leading to the same asymptotic dieomorphisms

will give the same central charge.

In section 4.3 we will focus on this paper [15] since it is the best known example of the

above formalism. We will reproduce the asymptotic symmetry group of AdS3 and the

central charge for its surface charge algebra.

3This is eventually related to the integrand of the gravitational action not being coordinate invariant.

The action of Einstein gravity integrates the Ricci scalar, which contains second derivatives of the

metric. If this action is integrated by parts, the integrand will depend on the Christoel symbols, which

are not coordinates invariant. This will lead to a surface term on the action when performing a change

of coordinates.4Since the charge not only depends on the metric, other variation terms should in principle be

taken into account. The variation of the asymptotic Killing vector δηξ = [ξ, η] leads to Q[ξ,η][q], which

also vanishes. Other contributions are subleading in 1/xr, which vanish as the asymptotic limit to the

boundary is taken.

46

4.2 Thermal AdS3 and the BTZ black hole


In this section we are going to focus on 3 dimensions. Studying 3 dimensional gravity

is interesting because it has proven to be a useful model from which to learn about

higher dimensional theories. What is special about 3 dimensions is that pure Einstein

gravity seems trivial. In n spacetime dimensions the canonical variable, the induced

metric, has n(n − 1)/2 independent components. The solutions for this metric follow

from the variational principle on the Hilbert-Einstein action, and therefore, as seen in

appendix A, they have to satisfy the Hamilton and momentum constraints, which are n

equations. The total number of degrees of freedom results n(n−3)/2, which vanishes in 3

dimensions. Since the gravitational eld contains no local dynamical degrees of freedom,

the spacetime away from sources is locally equivalent to the empty space solution of

Einstein's equations, AdS3. Matter, which is assumed to be localized, has no inuence

on the local geometry of the source free regions, and therefore can only eect the global

geometry of the spacetime.

In 1992, Bañados, Teitelboim and Zanelli found a black hole solution to 3-dimensional

pure Einstein gravity with Λ < 0, the BTZ black hole [14], [22]. This black hole solution

exists because, although 3 dimensional gravity is locally trivial, globally it is not. This

means that it can only dier from the vacuum AdS3 solution topologically.

The BTZ black hole is given by the metric

ds2 = −(r2 − r2

+)(r2 − r2−)

l2r2dt2 +

l2r2

(r2 − r2+)(r2 − r2

−)dr2 + r2(dφ+

r+r−lr2

dt)2 (4.10)

with −∞ < t < ∞, 0 < r < ∞ and 0 ≤ φ < 2π. The two values of the radius are given

by

r2± = 4lG(Ml ±

√(Ml)2 − J2). (4.11)

The two constants M and J are the surface charges associated with asymptotic time

displacement and rotational invariances, respectively. G is the Newton constant in 3

dimensions. The lapse function N(r) vanishes for the two values of the radial coordinate

r±; the black hole event horizon corresponds to r+. In order for the horizon to exist one

must haveM > 0 and |J | ≤Ml. In the case |J | = Ml, both horizons coincide constituting

an extremal BTZ black hole. Using the area law, the entropy can be computed and then

the Hawking temperature follows, resulting in

S =2πr+

4G, T =

∂S

∂M|−1J =

r2+ − r2

−2πl2r+

. (4.12)

It can be seen that the Hawking temperature indeed vanishes at extremality.

47


The BTZ black hole shows many similar properties to its analog in 3+1 dimensions, the

Kerr black hole that we will thoroughly present in chapter 6. And not only geometrically,

also the proposed Kerr/CFT correspondence resembles a lot the duality established

for the BTZ black hole. It is therefore a very interesting object to study. Intuitively,

the dominant contributions in this metric at r → ∞ are the same as global AdS3.

More precisely, as we will see in 4.3, the boundary conditions on AdS3 were originally

chosen such that they include the BTZ metrics as solutions, therefore making the BTZ

solution asymptotically AdS by construction. Next, we show how the space of solutions

is thermalized and how the BTZ metric can be derived from global AdS3.

Thermal AdS

In the context of quantum gravity, where the gravitational eld gµν has to be quantized,

the background metric on which asymptotic deformations act can be thought of as the

vacuum of the phase space of solutions. In this phase space, the other metric solutions,

reached through perturbations hµν on the background, are quantum excitations. In our

case of interest, AdS3 would be the vacuum and the BTZ metrics, quantum excitations5

of the Hilbert space, dierent values of the charges (M,J) representing dierent energy

excitations. Since the vacuum is supposed to be the most symmetric state, excitations can

be reached by doing coordinate identications that reduce the symmetry. Thermalization

of the phase space also follows from time-coordinate identications, as we have shown in

the context of the CFT partition function, where to assign a temperature to the CFT

we had to dene it on a torus. In this section we follow the derivation presented in the

appendices of [23]

Since we want to thermalize AdS3, we are going to work with its Euclidean metric. As

explained in chapter 3, Euclidean AdS3 is dened as a 3-dimensional hyperboloid in a

at embedding as

X20 +X2

1 +X22 −X2

3 = −l2.

The previous condition can be equivalently stated by saying that a point in AdS3 is

represented by an SL(2,R) matrix X, X = X01+Xaγa, a = 1, 2, 3, where

γ1 =

(0 1

1 0

), γ2 =

(1 0

0 −1

), γ3 = i

(0 −1

1 0

),

such that detX = l2. The metric can then be expressed as

ds2 = Tr[dX2].

5We will see later on, in 5.3, that identifying the actual vacuum upon which the BTZ excitations are

built is a bit more involved.

48


One then introduces the global parametrization (3.5) for Xi

X0 = l cosh(1/r) sinh t, X3 = l cosh(1/r) cosh t

X1 = l sinh(1/r) cosφ, X2 = l sinh(1/r) sinφ, (4.13)

where t is the Euclidean time, to nd the AdS3 metric in global coordinates

A way to introduce arbitrary identications of the coordinates in the metric is by doing

the following

X ≡ u−1Xu,

the matrix u being parametrized again as u = u01+ uaγa.

We now have a look at a special way of parameterizing the hyperboloid, namely

X =

(eu/2 0

0 e−u/2

)( √1 + r2 r

r√

1 + r2

)(eu/2 0

0 e−u/2

). (4.14)

Because of the introduction of the exponentials, we must identify u ≡ u + i2π in order

to cover the hyperboloid only once. Through the denition

u = t+ iφ (4.15)

the Euclidean metric of AdS3 is recovered and the wanted periodicity of φ ≡ φ + 2π is

satised. Now, because of the form in which (4.14) is parametrized, we see that an extra

identication in the time coordinate can be performed by dening another quotient of X

as

X ≡

(eβ/2 0

0 e−β/2

)X

(eβ/2 0

0 e−β/2

)(4.16)

which leads to a second u cycle u ≡ u + β, the time cycle which determines the

periodicity of t. Through the same argumentation used in the partition function of a

thermal CFT in 2.3, the periodicity in the time coordinate is to be identied with the

temperature of the spacetime as t ≡ t+1/T . This was argued in the context of the toroidal

geometry, which is dened from at space by two consecutive non-trivial identications of

both the timelike and spacelike coordinates. In the partition function language, the tilting

of the identication of the torus using the modular parameter τ (not to be confused with

the Lorentzian time coordinate) was implemented by an angular potential introduced

in the evolution operator. This converted the partition function to the form (2.27) Z =

Tr[e−βHeiσP ]. This angular potential can be introduced in the periodicity of the time

coordinate by generalizing it to the complex temperature

t+ iφ ≡ t+ iφ+ β − iσ := t+ iφ+ β, where β =1

T− iσ.

49


or it can be interpreted as an extra identication of the coordinate φ, which could be

implemented by a matrix (e−iπτ 0

0 eiπτ

)(4.17)

with 2πτ = σ + iβ. In conclusion, by modding out with the identications (t, φ) ≡(t+β, φ−σ), on top of φ ≡ φ+2π, we have given AdS3 a nite temperature and angular

momentum6.

BTZ as a quotient of AdS3

We now turn to the BTZ black hole. We will show that it also follows from AdS3 by

dening quotients on the original metric. The BTZ Euclidean metric follows from doing

the Wick rotation t→ it and JE = iJ in (4.10). In this metric again, φ ≡ φ+ 2π. Now,

we dene a quotient of (4.14) as

X ≡

(e−iπτ 0

0 eiπτ

)X

(eiπτ 0

0 e−iπτ

)

This induces the identication u ≡ u + 2πiτ , which again denes another cycle in the

u-plane, independent of u ≡ u + i2π. Then, the Euclidean BTZ metric follows from

identifying the previous cycle with the space cycle, namely the identication for the

angular coordinate φ. This can be done by dening

u = iτ(φ− it).

Then the cycle u ≡ u + i2π becomes the time cycle, leading to a periodicity for the

time coordinate t ≡ t+ i2π/τ . If we dene

τ = −|r−|+ ir+, (4.18)

where

r2± = 4G[M ±

√M2 + (JE/l)2],

and a new dimensionful radial coordinate r7 in terms of the r coordinate in (4.14) as

follows

r2 =r2/l2 − τ2

2

|τ |2

6There is an i factor in the denition of the angular momentum operator that brings this sign dierence

in the angular momentum of the spacetime.7Notice that in the BTZ metric (4.10), the radial coordinate is dimensionful and therefore r± carry

an extra l2 factor with respect to what we dene here. Since we identify the temperature with coordinate

periodicities, it is better to use dimensionless coordinates so that we can use modular invariance.

50

4.3 The Brown-Henneaux construction

(note that r− = i|r−| is purely imaginary), then the metric of AdS3 parametrized as

(4.14) becomes that of Euclidean BTZ after dropping the tilde of r. The periodicity of

the time coordinate has to be identied with the complex temperature β = 1T − iσ.

Therefore, the physical temperature follows from the real part of i2π/τ

β =i2π

τ=

2πr+

r2+ − r2

−+ i

2πr−r2

+ − r2−

⇒ T =r2

+ − r2−

2πr+,

which agrees with what we found at the beginning of the section.

Therefore, both thermal AdS3 and the BTZ black hole follow from the identication

u ≡ u + 2πiτ , together with the natural one u ≡ u + 2πi, but they dier in the way

these two are assigned to the time and space component. We have already seen that

interchanging the time and space coordinates is natural in the case of modular invariance.

In the thermal picture, we can interpret the τ parameter of the identications with the

modular parameter of the AdS3 torus. Therefore, it is obvious that thermal AdS3 and

the BTZ are related through a modular transformation S : τ → −1/τ since the time

periodicities in the two cases are t ≡ t − 2πiτ and t ≡ t + 2πi/τ . We say that thermal

AdS3 corresponds to the torus with the Euclidean time coordinate in the uncontractible

cycle of the torus, a time-slice of the AdS3 cylinder can shrink to a point since it is

a vacuum solution. The BTZ on the other hand has the spacelike coordinate in the

uncontractible cycle and the timelike in the contractible one. The BTZ black hole's

Penrose diagram resembles the AdS3 diagram asymptotically, acquiring the two timelike

boundaries at innity. The event horizon of the black hole is located between the two

boundaries, preventing the constant-time slices to be shrinkable to a point. The torus

then has to be built up by rst identifying the singularities and then identifying the

timelike boundaries. The Euclidean timelike coordinate can shrink to a point because it

is periodic8.


In this section we will focus on the asymptotic symmetry group of AdS3 and its surface

charge representation. The 1986 paper by Brown and Henneaux [15] is the reference where

this was discussed in detail. In this paper they presented a choice of boundary conditions

and computed the asymptotic symmetry group for an AdS3 background. As already

mentioned in section 4.1, they showed that the Poisson/Dirac bracket algebra of the

chargesQξ is isomorphic to the Lie algebra of the innitesimal asymptotic symmetries but

8Actually, that's what happens in the horizon of Euclidean black holes, where the time circle shrinks

to a point due to the vanishing the lapse function. The temperature is therefore computed with the

periodicity of the time coordinate at r →∞.

51


including a central extension, therefore representing a copy of the full Virasoro algebra.

Finally, they computed the central charge.

Asymptotic symmetry group of AdS3

The rst step in nding the ASG is to dene a set of boundary conditions at spatial

innity. In the original paper, the authors considered two dierent choices and showed

that they lead to two dierent ASG's. Their rst natural guess was to restrict the space

of solutions to the family of metrics of the BTZ black hole. This restriction serves as

the boundary condition on the metric perturbations. They found that the asymptotic

symmetry group associated to these boundary conditions is R× SO(2) and the charges

are precisely the energy and angular momentum of locally at 2+1 gravity9.

Then they aim to weakening the restriction to the BTZ metric so that the ASG is

enlarged to coincide with the AdS3 isometry group SO(2, 2). Now, this new requirement

determines the new boundary conditions in the following way: if the AdS3 isometry group

has to be part of ASG, then the metric obtained by deforming the BTZ metric with an

AdS3 isometry must be asymptotically AdS. By computing the transformed metric for

all the Anti-de Sitter group, the following boundary conditions are generated10 htt = O(1) htφ = O(1) htr = O(1/r3)

hφφ = O(1) hφr = O(1/r3)

hrr = O(1/r4)

(4.19)

Geometries respecting these boundary conditions are said to be (locally) asymptotically

AdS3. Having determined these boundary conditions, an analysis of the Lie transforma-

tion equations shows that the asymptotic symmetries are described by vector elds that

can be parametrized as

ξt = lT (t, φ) +l3

r2T (t, φ) +O(1/r4)

ξr = rR(t, φ) +O(1/r)

ξφ = Φ(t, φ) +l2

r2Φ(t, φ) +O(1/r4) (4.20)

satisfying

lT,t(t, φ) = Φ,φ(t, φ) = −R(t, φ), lΦ,t(t, φ) = T,φ(t, φ) (4.21)

T (t, φ) = − l2R,t(t, φ), Φ(t, φ) =

1

2R,φ(t, φ). (4.22)

9Remember that the formalism of surface charges is the one required in general relativity to dene

conserved quantities, it cannot be done in the standard Noether method since there is no global concept

of time and space.10In section 5.1 we will see a dierent origin for these boundary conditions, they are motivated by

relaxing by one order the fall-o of the metric variations under the isometries.

52


For the above vectors, the O(1/r4) terms in the t, φ components and O(1/r) term in

the r component represent pure gauge transformations, so the ASG will consist of the

vectors (4.20) quotient those diering by these fall-o terms.

Now, equations (4.21) are the Cauchy-Riemann equations with an indenite metric, i.e.

the conformal Killing equations in two dimensions. Once a solution for T (t, φ),Φ(t, φ)

is selected, the remaining functions R(t, φ), T (t, φ), Φ(t, φ) are determined. Therefore,

the asymptotic symmetry group is isomorphic to the pseudo-conformal group in 2

dimensions. This can be explicitly seen by Fourier decomposing the conformal Killing

equations (4.21) and (4.22). The asymptotic Killing vectors acquire an index, reecting

the innite dimension of the algebra

ξtn = l(1− n2l2

2r2)e−in(t/l+φ)/2, ξrn =

irn

2e−in(t/l+φ)/2, ξφn = (1 +

n2l2

2r2)e−in(t/l+φ)/2.

ξtn = l(1− n2l2

2r2)e−in(t/l−φ)/2, ξrn = − irn

2e−in(t/l−φ)/2, ξφn = −(1 +

n2l2

2r2)e−in(t/l−φ)/2

(4.23)

The above asymptotic Killing vectors satisfy the local conformal algebra

[ξn, ξm] = i(n−m)ξn+m [ξn, ξm] = i(n−m)ξn+m [ξn, ξm] = 0 (4.24)

We see then that the SL(2,R)R×SL(2,R)L algebra satised by the AdS3 isometry group,

which here would be again satised by the subgroup with n = −1, 0, 1, is enhanced to

the Witt algebra.

Surface-charge algebra and central charge

Having dened the boundary conditions and derived the ASG, it is time to compute

the algebra of surface charges. Up to this point, the asymptotic Killing vectors dened

asymptotic symmetries under the Lie transport. In the canonical formalism, due to the

ADM foliation of spacetime (see appendix A) these Killing vector elds become the

allowed asymptotic deformations of the constant-time hypersurfaces. The hypersurface

deformation vector elds ξµ, µ = (⊥, a) correspond to the full-spacetime Killing vectors

but expressed in the tangent/normal basis ξµ∂µ = ξ⊥nµ + ξaeµa . The components then

satisfy ξ⊥ ≡ ξ = Nξt, ξr = ξr +N rξt, ξφ = ξφ +Nφξt, the lapse and shift functions

N,Na appearing through to the normalization of the normal vector nµ and denition of

the ow vector tµ.

The spacelike hypersurfaces Σt are described by the induced metric qab, the canonical

variable, and its conjugate momentum pab, on which the Hamiltonian generator depends.

Therefore, the boundary conditions on the full-spacetime metric must be converted into

boundary conditions on the canonical variables qab and pab. For the induced metric, we

53


can simply pull-back onto Σt the boundary conditions (4.19) δqab = hab = eµaeνbhµν , then

for the conjugate momentum they can be computed using the usual canonical denition,

and they are(hφφ = O(1) hφr = O(1/r3)

hrr = O(1/r4)

) (δpφφ = O(1/r5) δpφr = O(1/r2)

δprr = O(1/r)

). (4.25)

Now, these boundary conditions on the canonical variables have to be introduced in the

expression of the surface charge. First of all, we have to nd an explicit expression for

this. As explained in section 4.1, the surface charge is dened through the surface terms

arising from the variation of the bulk piece of the generator of an asymptotic symmetry

(4.3). Therefore, we have to nd this surface terms. First, let's write the bulk term of (4.3)

in the normal/tangent basis as Hµξµ = Hξ +Haξa. Now, the Hamilton and momentum

constraints H,Ha were found in appendix 4.1 to be

H = −R− 2Λ

2κ+ 2κ(pabpab −

1

d− 1p2), Ha = −2∇bpab.

The variation of the bulk piece of the surface charge (4.3) can be computed to give

δ

∫Σt

ddy√qHµξµ =

∫Σt

ddy√q(...)abδqab + (...)abδp

ab −∮∂Σt

dd−1θ√γrc2ξaδpac+

+1

2κGabcd(ξ∇dδqab −∇dξδqab) + (2ξapbc − ξcpab)δqab, (4.26)

where we introduced Gabcd ≡ qc(aqb)d− qabqcd. The surface charge density is then dened

such that its variation, as in (4.5), equates minus the surface term of (4.26). In the paper

[15], using the boundary conditions (4.25), the integrated charge is presented to be

Qξ[q] =

∮∂Σt(∞)

dd−1θ√γra

1

2κGabcd(∇bhcd − hcd∇b)ξ + 2pabξb+O(h2) (4.27)

where the barred quantities depend on the background metric of global AdS3 and ∂Σt(∞)

means the r →∞ limit of the boundary of the hypersurface.

Once the surface charges are obtained, the next important step is to show that they form

an algebra. As mentioned in section 4.1, this is done, in the original paper, with the help

of the theorem in [18] that states that the Poisson bracket of two well-dened generators

is also a well-dened generator. The authors then compute the algebra explicitly showing

that the Poisson bracket generator is indeed the generator of the Lie bracket of the two

original deformations. We will not reproduce this computation here, but we will proceed

to compute the central charge of the algebra.

54


Let's recall that the vectors of the ASG could be parametrized as (4.23). We will denote

the charges associated to the vector elds ξn, ξn by Qn ≡ Ln, Qn ≡ Ln respectively.

Knowing that the asymptotic Killing vectors satisfy the Witt algebra and given that the

surface charge is linear in the asymptotic deformation, the explicit algebra of the surface

charges is

Ln, Lm = (n−m)Ln+m+Cn,m, Ln, Lm = (n−m)Ln+m+Cn,m, Ln, Lm = Cn,m.

The central term can then be computed using (4.9), the charge (4.27) has to be evaluated

for the Lie-deformed AdS3 metric. In the paper, the authors showed that the central term

is of the form Cn,m = c12n(n2 − 1)δn+m

11. The resulting central charge is

c = c =3l

2G(4.28)

with the Cn,m = 0. Therefore, the algebra of charges constitutes the full Virasoro

algebra.

The appearance of this Virasoro algebra clearly points at a description of Einstein gravity

with Λ < 0 in 3 dimensions in terms of a 2-dimensional CFT, which constitutes the

rst step towards a more general AdS3/CFT2 duality. In a CFT, the central charge

represents the Casimir energy of the vacuum with respect to the plane, this vacuum

being annihilated by the generators of the global conformal group. This picture can also be

drawn for the phase space of AdS3, where the vacuum metric is annihilated by the charges

L0,±1, L0,±1 which constitute therefore the subalgebra without the central extension. This

central charge, in the way it was originally found in the Brown-Henneaux construction, is

classical since it appears because of the ambiguity in dening the canonical generators. We

will see in section 5.2.1, in the context of the actual AdS3/CFT2 correspondence, how the

Brown-Henneaux central charge is reproduced from the Weyl anomaly of the dual stress

tensor dened on the boundary. However, the central term in a CFT Virasoro algebra is a

quantum eect. The Brown-Henneaux central charge acquires the ~ factor and therefore

becomes quantum when the Dirac brackets are replaced by quantum commutators.

It is important to notice the symmetry enhancement unique of 3 dimensions, the exact

isometry group being enhanced to an innite algebra of asymptotic symmetries and of

charges. In 4 dimensions, the Henneaux-Teitelboim denition of asymptotically AdS4

spaces in [17] required for the boundary conditions to be such that the algebra of charges

reproduces the exact AdS symmetry group SO(2, 3). This requirement being satised

11 This dependence of the central term on the indices can also be derived from the Bianchi identity

for the charges. This translates into a similar identity for the central term, namely (m − p)Cn,m+p +

(p − n)Cm,p+n + (n −m)Cp,n+m = 0. This enforces the central term to be of the form Cn,m ∝ (C1n +

C2n3)δn+m.

55


for both 3 and 4 dimensions, one would expect it to hold in any dimensions. It turns out

that not only this is indeed so, but there is also a systematic implementation. In [24] it

was shown how a bijective correspondence between exact and asymptotic Killing vectors

can be found in any dimensions without requiring a set of boundary conditions consistent

with well-dened charges. The equations that determine the asymptotic Killing vectors

follow from a naive but intuitive constraint on the fall-o of the metric variations and

the result is compatible with the denition, a posteriori, of nite charges; we will address

this correspondence in 5.1. Automatically then, the algebra of charges reproduces the

conformal group in (d− 1) dimensions SO(d− 1, 2), and the dual description of gravity

in AdSd in terms of a (d−1)-dimensional CFT generalizes to arbitrary dimensions.

56

CHAPTER 5

The AdS/CFT correspondence

Brown and Henneaux made a rst step towards the AdS/CFT correspondence. An

important aspect of their arising dual description that we didn't mention so far is where

the dual CFT seats. Since the gravitational degrees of freedom, represented by the global

charges, are located at the asymptotic region, the CFT can be thought of as living in the

very boundary of the AdS spacetime. This is possible because the conformal boundary

of AdSn is the (n − 1)-dimensional Minkowski spacetime, as shown in section 3.1. The

arising dual description is therefore holographic, it is established between degrees of

freedom living in the bulk and a theory living in the boundary.

This idea of bulk physics being holographic is more than just a particular feature of the

Brown-Henneaux construction; it was actually elevated to a principle by 't Hooft and

Susskind in 1993, the so-called Holographic Principle [2]. According to this principle,

any gravitational theory in d + 1 dimensions should have a description in terms of a

quantum eld theory in d dimensions without gravity. The motivation for this principle

comes from the fact that quantum gravitational theories must have a lot fewer degrees of

freedom than non-gravitational theories due to the area law. Fundamentally then, gravity

has to be dierent than other theories in which the number of degrees of freedom scales

like the volume and therefore respond to our idea of thermodynamical extensivity of the

entropy.

The AdS/CFT correspondence provides an explicit and precise example of the Holo-

graphic Principle. This correspondence was proposed by Maldacena in 1997 in [25] and

states that, for certain conformal eld theories with gauge group SU(N), their large

N limit in d dimensions can be described in terms of supergravity on the product of

a d + 1-dimensional AdS space with a compact manifold. The next year, Witten went

a step further and made this correspondence more precise by establishing the exact

57

Chapter 5: The AdS/CFT correspondence

connections between observables of both theories [26]. Therefore, the analogy established

by Brown and Henneaux between 3-dimensional pure gravity and a 2-dimensional CFT

is actually just one aspect of a much broader correspondence between quantum gravity

and eld theories. The Brown-Henneaux construction would belong to the AdS3/CFT2

correspondence, which is one of the few well-established examples. Another one is the

correspondence between Type IIB string theory on AdS5 and N = 4 Super Yang-Mills

theory in the 4-dimensional boundary. Good reviews on the AdS/CFT correspondence

are [27] and [3], and on AdS3/CFT2 are [28] and [23].

These dualities become very useful in the context of black holes as a tool to count

their microstates. As we have already mentioned, most black holes have a near-horizon

geometry that is locally an AdS geometry, like the BTZ, the Reissner-Nordström, the

Kerr black holes as well as some string theory black holes. The microstates of the black

hole, understood as a thermal ensemble, are degrees of freedom believed to live on the

horizon, which are to be identied with the asymptotically AdS excited states of the

near-horizon factor. Therefore, the duality allows to reproduce the black hole degrees of

freedom counted by the Bekenstein-Hawking entropy with the dual CFT of the near-

horizon geometry. This is the main aspect we will focus on in the case of the Kerr/CFT

correspondence, i.e. reproduce the Bekenstein-Hawking entropy for the extremal Kerr

black hole with a speculated dual CFT.

In this chapter we will try to give some ideas on what the AdS/CFT correspondence

is, with special focus on the AdS3/CFT2 which is the main reference for the Kerr/CFT

correspondence. At the end, we will present the example of how this duality was used to

reproduce the Bekenstein-Hawking entropy for the BTZ black hole.

5.1 Symmetries

Following up with the approach and discussion we presented in the last chapter, we

are rst going to address the matching of asymptotic symmetries in the AdS bulk and

conformal symmetries in the boundary in any dimensions. We have already seen how

for AdS3, the group of exact Killing vectors enhances to an innite Witt algebra of

asymptotic ones that therefore becomes isomorphic to the conformal symmetry algebra

of the boundary. This enhancement is unique of 3 dimensions, as it is unique for 2-

dimensional CFT's, but it also includes a one to one correspondence between the 6

exact Killing vectors in the bulk and the subset of asymptotic ones with n = 0,±1. This

correspondence turns out to hold also in any dimension as was shown by Barnich, Brandt

and Claes in [24]. They found this correspondence by constraining the fall-o conditions

58

5.1 Symmetries

of the metric variations in a systematic way in any dimension. We present here their

result.

Let's recall that the AdSn metric in global coordinates looks like

ds2 = l2(−(1 + r2)dτ2 +dr2

(1 + r2)+ r2dΩ2

n−2) (5.1)

where we can write the spherical factor as dΩ2n−2 =

n−1∑a=2

fady2a with f2 = 1, fa =

sin2 y2 sin2 y3... sin2 ya−1, a = 3, ..., n − 1. First of all we have to look at the fall-o

conditions of the exact Killing vectors. The Killing vectors of the Lorentz transformations

of the at embedding are ξp = cpqXq, where Xq are the coordinates in the embedding

and the matrix cpq is antisymmetric and with only two non-vanishing entries. The Killing

vectors of AdSn then follow from the pull back of the former to the hyperboloid

ξα = cpq∂Xp

∂xαXq, (5.2)

where xα = τ, r, ya; a = 2, ..., n− 1. Working out explicitly these expressions in terms of

the dierent cpq, the asymptotic fall-o conditions at spatial innity r →∞ are

ξr → O(r), ξa → O(1), a = 0, 2, ..., n− 1. (5.3)

In the following, we will use Landau's O-notation. We assign a function ξ(r) an

asymptotic degree |ξ|, which characterizes the behavior of the function near the boundary.The notation ξ → O(rm) means |ξ| ≤ m and ξ → o(rm) means |ξ| < m. We also

assign an asymptotic degree to dierential operators, so that the partial derivative has

asymptotic degree opposite to the corresponding coordinate. Like this, ξ → O(rm) implies

∂µξ → O(rm−1). Also, we assume that a = 0, 2, ..., n − 1 unless the τ index is shown

explicitly, in which case a = 2, ..., n− 1.

Now, knowing the fall-o conditions of the exact Killing vectors, we can determine the

fall-o of the Lie variation of the metric Lξgαβ . Explicitly

Lξgrr → O(r−2), Lξgar → O(r), Lξgab → O(r2). (5.4)

A somehow natural denition for asymptotic Killing vectors is then those vectors that

behave like the exact Killing vectors at innity, i.e. like (5.3), and satisfy the restricted

metric fall-o conditions

Lξgrr → o(r−2), Lξgar → o(r), Lξgab → o(r2). (5.5)

59


Obviously, asymptotic Killing vectors that fall-o as

ξr → o(r), ξa → o(1) (5.6)

automatically satisfy the constraints (5.5) and are therefore considered trivial. We want

then to compute the equivalence classes of asymptotic Killing vectors modulo trivial ones.

Non-trivial ones can then be parametrized according to their fall-o condition

ξr → rR(τ, y) + o(r), ξτ → T (τ, y), ξa → Φa(τ, y). (5.7)

The constraints (5.5) then lead to the following equations for the previous functions

R = −∂τT∂aT = fa∂τΦa

∂τT = ∂aΦa +

∑b<a

Φb cot yb

fb∂aΦb + fa∂bΦ

a = 0, a 6= b,

where in the last three equations, the summation convention for repeated indices does

not apply.

Now we turn to the boundary, which is conformally at. The boundary induced in the

metric by, which we denote g′ab, is the Einstein Static Universe metric which follows from

(5.1) by taking the large r limit ds′2 = −dτ2 +n−1∑a=2

fady2a. The conformal Killing vectors

of this metric satisfy the conformal Killing equation (2.2), which now reads

Lξg′ab =2

n− 1D′cξ

cg′ab. (5.8)

These equations for the τ and ya components of the conformal vectors turn out to be

equivalent to the last three of the equations (5.1), which is easy to see once the equations

are developed explicitly. Therefore, for every conformal Killing vector of the boundary

there is an (equivalent class of) asymptotic Killing vector of the bulk. Since the conformal

algebra of the boundary is isomorphic to the exact isometry algebra of the bulk, there is a

one-to-one correspondence between exact, non-trivial asymptotic and conformal Killing

vectors in any dimension n ≥ 3.

In the particular case of n = 3, equations (5.1) become

R(τ, φ) = −∂τT (τ, φ) ∂τT (τ, φ) = ∂φΦ(τ, φ) ∂φT (τ, φ) = ∂τΦ(τ, φ). (5.9)

60

5.2 AdS/CFT

Turning to the local 2 dimensional conformal algebra, this was built up with the

generators ln = −zn+1∂z (we just consider the holomorphic sector). If we rewrite these

in the original coordinates of the Minkowski metric (τ, φ)

ln = −(τ + φ)n+1

2(∂τ + ∂φ) (5.10)

it is straightforward that these conformal Killing vectors satisfy the last two equations

of (5.9). Therefore, each one of these vectors is in one-to-one correspondence with an

asymptotic Killing vector of AdS3, or rather its equivalence class, a complete set of these

being dened by

Tn(τ, φ) = −(τ + φ)n+1

2, Φn(τ, φ) = −(τ + φ)n+1

2, Rn(τ, φ) =

(n+ 1)(τ + φ)n

2.

Therefore, it also follows from this formalism that the correspondence asymptotic/conformal

Killing vectors translates to the enhanced algebras in 3 dimensions. It has to be noted

that equations (5.9) are exactly the equations (4.21) that parametrized the asymptotic

Killing vectors in the Brown-Henneaux construction. This already shows the agreement

between the two procedures. In the paper the authors also showed how the asymptotic

Killing vectors so found lead to nite global charges.

Summing up, one can nd a one-to-one correspondence between exact Killing vectors,

asymptotic Killing vectors and conformal Killing vectors on the boundary. The fact that

SO(2, d − 1) acts on AdSd as a group of ordinary or asymptotic symmetries and on its

boundary as a group of conformal symmetries means that there are two ways to get a

physical theory with SO(2, d− 1) symmetry: in a eld theory with (or without) gravity

on AdSd, and in a conformal eld theory on its boundary.

5.2 AdS/CFT

Now we move on to build some intuition and basic ideas about the AdS/CFT

correspondence. We will refer this to the setting with which Maldacena originally

presented the correspondence [25], the duality between N = 4 U(N) super Yang Mills

theory and strings on AdS5 × S5. The source of reference we follow here is his review

[27].

First we have to understand the relevance of the large N limit by presenting the setting

of the eld theory. Consider a eld theory with a large number N of elds that transform

into each other under a symmetry such as SO(N) or U(N). In the case of U(N), the

61


elds are hermitian matrices M (either gauge elds or matter elds transforming in the

adjoint representation). The Lagrangian is of the generic form

L =1

g2Tr[(∂M)2 +M2 +M3 + ...]

and the action is U(N) invariant. First note that the eective coupling constant is g2N .

These comes from analyzing the Feynman rules for this theory. Each propagator (P)

contributes with a factor g2 and each vertex (V) with a factor 1/g2. Each closed line

(CL) contains a sum over the gauge index and therefore contributes with a factor N .

Therefore, each diagram contains the prefactor

(g2)P−VNCL (5.11)

Because of the two U(N) indices of the elds, the lines of the diagrams have to be

doubled, and therefore the diagrams can be drawn on a two-dimensional surface. Some

diagrams can be drawn on a plane or a sphere, the so-called planar diagrams, and others

have to be drawn on surfaces of higher genus due to the intersections of the lines or strips.

It is possible then to do the above counting of powers of the coupling constant in terms

of the elements of the surface (F: faces, E: edges, V: vertices), namely

NF−E+V (g2N)p = N2−2h(g2N)p,

where h is the genus of the surface and p means a certain power. The sum of all planar

diagrams gives

N2(c0 + c1(g2N) + c2(g2N)2 + ...) = N2f(g2N),

where ci are numerical coecients containing the momentum integrals and details of

the diagram. The full partition function then sums over all genus, as the string theory

perturbation expansion, and becomes

logZ =

∞∑h=0

N2−2hfh(g2N). (5.12)

The motivation to consider the large N limit becomes clear now, since then only the

planar diagrams contribute. This is called the 't Hooft limit, more specically

N →∞, λ ≡ g2N = xed,

where λ is called the 't Hooft coupling constant. These theories are expected to be

described in terms of strings, which was suggested by 't Hooft in [29]. We will not get

into the details, but intuitively, the sum over genus in (5.12) reminds us of the string

theory perturbation expansion, in which diagrams also have a width (because strings

62

5.2 AdS/CFT

are extended objects) and loops correspond to the holes of the Riemann surface that

represents each diagram. In the 't Hooft limit, the string theory is dened to be whatever

results from summing the planar diagrams. The larger λ is, the larger the number of

diagrams that contribute, which become dense on the sphere.

Starting with a 4-dimensional gauge theory, we might expect naively to nd bosonic

strings moving in 4 dimensions. But we know bosonic strings are only consistent in 26

at dimensions. The reason for this inconsistency comes fromWeyl symmetry gab → eφgab(typical from the classical Polyakov action with which worldsheet string theory is built

up) not being a symmetry quantum mechanically. In the quantum theory, the change in

the action consists of an integral of terms which depend on the φ from the conformal

factor as

δS =26−D

48π

∫1

2(∇φ)2 +R(2) + µ2eφ

To keep Weyl invariance at the quantum level either D = 26, which is not the theory

we want, or we integrate over φ so that it doesn't appear anymore. This integration is

like adding an extra dimension, and this is what we are going to consider. This is also

consistent with the Holographic Principle, according to which we expect a dual to the

gauge theory containing gravity in one dimension more. So we will be looking at strings

in 5 dimensions.

Now we should specify the space where the strings move. It should have 4-dimensional

Poincaré symmetry, so the metric must have the form

ds2 = w(z)2(dx21+3 + dz2), (5.13)

where we have used dieomorphism invariance to set the coecient of dz2 equal to that

of dx2. Suppose we are dealing with a conformal eld theory, N = 4 Yang Mills is an

example. Then x→ λx should be a symmetry. But in string theory, the string length or

the string tension yield a scale for the theory. So the only way a rescaling of x can be

implemented is at the level of the metric, as an isometry of (5.13). This implies that z has

to change accordingly z → λz and that w = R/z, the metric resulting into 5-dimensional

AdS space. So in a way we have reversed here the approach presented in the last chapter.

Instead of beginning with an AdS geometry, then moving to the asymptotics and nally

nding a dual conformal theory in the boundary; here we have started with a conformal

theory and shown that a dual string theory description requires an extra dimension,

therefore it will live in the bulk, and that conformal invariance requires this bulk to have

an AdS geometry.

N = 4 U(N) super Yang Mills and strings on AdS5

Consider a theory with 4 supersymmetries in 4 dimensions. This theory has a unique eld

content but there is the freedom of choosing the gauge group and coupling constants.

63


This theory is also conformal invariant. The 't Hooft coupling is λ = g2YMN and with

this, a modular parameter can be associated to the theory τYM = i2π/g2YM

1. The theory

then is said to have an S-duality under which τYM transforms into −1/τYM .

On the string theory side, we look for a solution to the type IIB supergravity equations

of motion. A solution is AdS5 × S5, with radius of both the sphere and AdS

lAdS = (4πgsN)1/4ls ∼ N1/4lpl, (5.14)

where gs is the string coupling and 2πl2s = 2πα′ is the inverse of the string tension. The

parameter N here is the ux of the F5 form through the S5 and is also related to the

number of D3 branes, let's see how.

There is an argument that relates these two theories which relies on looking at the near-

horizon geometry of D3 branes. This one turns out to be AdS5 × S5. Excitations living

in this horizon geometry have very small energies as seen from the asymptotic region.

At low energies, therefore, only these excitations will survive, focusing the interest in the

near-horizon region. Also at low energies, the eld theory in N D3 branes is N = 4 U(N)

Yang Mills theory. So we see a connection between these two theories, as two alternative

but equivalent descriptions.

This connection materializes through the relation of the two coupling constants be-

ingi

g2YM

= τYM = τ =i

gs(5.15)

This notation emphasizes that both theories have an SL(2,Z) duality symmetry and

that the relation between the two is also some sort of duality: in the large N limit, λ

kept xed but large, the Yang Mills theory is strongly coupled (eventhough gYM is small)

while the string theory is weakly coupled, and vice versa. This is because from (5.14)

and (5.15) follows that

λ = g2YMN ∼ gsN ∼ (

lAdSls

)4.

When λ >> 1, then the radius of the bulk geometry lAdS is much smaller than ls, the

intrinsic size of the graviton. Therefore, supergravity is a good approximation, which

again is the gs → 0 limit of string theory. This comparison of the coupling constants of

the two sides also implies that physical quantities will depend only on the size of AdS in

string units, and not on α′.

1This modular parameter exhibits also a real part which depends on another parameter of the theory.

64

5.2 AdS/CFT

Partition functions

One of the aims of the correspondence is to be able to compute and compare the partition

functions of both sides. For the gravity side, the partition function is written in an AdS

background. Because this is not a globally hyperbolic geometry, boundary conditions have

to be specied for the elds and therefore the partition function depends on these. If the

string theory reduces to supergravity, then we can approximate the partition function

with the value of the classical action2

ZAdS [φ(~x, z) |z=0= φ0(~x)] ∼ e−N2Sclas[φ]+O(α′) × (Quantum corrections),

where O(α′) are the stringy corrections that correct the gravity action into the classical

string action. One of the main statements of AdS/CFT is that for each eld in the bulk,

there is a corresponding operator in the dual eld theory. For example, the graviton is

dual to the stress-energy tensor on the boundary. Even further, it establishes the following

relation between the bulk partition function and the generating function

ZAdS [φ(~x, z) |z=0= φ0(~x)] = 〈e∫d4xφ0(~x)O(~x)〉eld th.

So the boundary conditions of the elds in the bulk act as sources to generate

correlation functions of the corresponding operators. Innitesimal changes in the

boundary conditions then correspond to the insertion of an operator on the CFT side.

This particular step was shown by Witten in 1998 [26], establishing exact relations

between the observables of both the conformal eld theory and supergravity . He also

showed that the conformal dimensions of operators in the CFT are given by the masses

of the corresponding particles in supergravity.

5.2.1 AdS3/CFT2

The AdS3/CFT2 correspondence is another well-established example of the AdS/CFT

correspondence. This is the relevant correspondence in the context of this thesis because

the near-horizon geometry of the extremal Kerr black hole consists of quotients of warped

AdS3. Unfortunately, here we will just make some comments on the dual stress tensors

and partition functions; for a good introduction see [28].

2The dependence on N2 in the exponent and not on g2N is due to the same reason why physical

quantities depend only on lAdS and not on α′. A way to think about this is to choose units where

lAdS = 1, then α′ = 1/√g2YM and so all gravity computations depend only on N .

65


Boundary stress tensor

We have already seen that both pure AdS3 and the BTZ black hole are solutions of the

Einstein-Hilbert action

IEH =1

16πG

∫d3x√g(R− 2

l2). (5.16)

Other excitations on the background metric also have to be solutions of this action. From

the point of view of the AdS/CFT correspondence it is very important to have a well-

dened action because it is the weight of excitations in the gravitational partition function

that has to match with the dual CFT one. Therefore we need to ensure a well-dened

variational principle.

To address this issue it is convenient to work in Gaussian normal coordinates, which yield

a metric with an isolated term for the radial coordinate η

ds2 = dη2 + gijdxidxj , (5.17)

where gij is the induced metric on hypersurfaces of constant η and is an arbitrary function

of the other two coordinates. Evaluated in (5.17), the action (5.16) acquires a boundary

term after an integration by parts, namely

Ibdry = − 1

8πG

∫∂M

d2x√gTrK,

where K is the extrinsic curvature dened as Kij = 12∂ηgij which amounts for the change

in gij as one moves in the η direction (see appendix A). So dened, the variation of the

boundary term would spoil a variational principle in which gij is xed in ∂M but not

its normal derivative. This motivates the introduction for the Gibbons-Hawking term

IGH = −Ibdry which cancels the boundary term exactly.

We proceed now with the variational principle and we consider the variation of the action

with respect to gij . The variation will consist of the term proportional to the equations

of motion, that therefore vanishes on-shell, and another boundary term

δ(IEH + IGH) = − 1

16πG

∫∂M

d2x√g(Kij − TrKgij)δgij . (5.18)

A boundary stress tensor can be identied then from its denition δI = 12

∫∂M d2x

√gT ijδgij

as

T ij = − 1

8πG(Kij − TrKgij).

This is the stress tensor living in the boundary since it is derived from the variation

with respect to the induced metric on the boundary and it is coordinate independent.

66

5.2 AdS/CFT

This stress tensor encodes the asymptotic gravitational excitations, namely the surface

charges. Given that the latter satisfy the Virasoro algebra, then naturally this boundary

stress tensor is dual to the stress tensor of the CFT living on the boundary.

Now, asymptotically AdS spacetimes grow at leading order as r2 or equivalently as e2η/l

at innity, subleading terms growing at most as e−2η/l (this follows from the boundary

conditions in this particular set of coordinates). An expansion can be written for the

metric

gij = e2η/lg(0)ij + g

(2)ij + ... (5.19)

g(0)ij is the conformal boundary metric, dened only up to Weyl transformations induced

by a redenition of η. It is then natural to aim for a variational principle in which

g(0)ij is held xed while the subleading terms in (5.19) are allowed to vary. However,

the variation of the action (5.18) depends explicitly on δg(2)ij and therefore spoils the

variational principle. This is solved by adding another counterterm

Ict = − 1

8πGl

∫∂M

d2x√g.

This term included, the on-shell variation of the action becomes of the form

δI =1

2

∫d2x

√g(0)T ijδg

(0)ij , (5.20)

with

Tij =1

8πGl(g

(2)ij − Tr(g

(2))g(0)ij ), (5.21)

where indices are raised and lowered with the induced metric. This is the AdS3 stress-

energy tensor. It is important to note that its trace is non-vanishing

Tr(T ) = − 1

8πGlTr(g(2)).

The subleading terms in the metric expansion can be computed by solving Einstein's

equations, and the g(2) turns out to satisfy Tr(g(2)) = l2R(0)/2. We see therefore that

the trace of the stress tensor is proportional to the Ricci scalar. This matches the

Weyl anomaly of a CFT stress tensor which in (2.25) was shown to be of the form

Tr(T ) = − c24πR. From here, the central charge can be read and it exactly yields the

Brown Henneaux central charge c = 3l/2G.

So we see that the boundary stress tensor has the right CFT properties. Form its

expression (5.21), holomorphic and antiholomorphic components can be derived by doing

the change of coordinates g(0)ij dx

idxj = dwdw with w ≡ w + 2π. These two components

67


then can be used to write down the conformal charges and in particular the Virasoro

generators as in (2.14)

Ln −c

24δn,0 =

∮dwe−inwTww, Ln −

c

24δn,0 =

∮dwe−inwTww (5.22)

These would of course be one-to-one with the global charges found by Brown and

Henneaux. In the CFT theory, the states must transform in representations of the

ASG.

Dual partition functions

The main point of the AdS3/CFT2 correspondence is the equivalence between the

partition functions

ZAdS3 = ZCFT2 . (5.23)

In the high energy regime, the gravity partition function is dominated by the contribution

of a single large BTZ black hole. In this limit, the asymptotic growth of states in the CFT

given by the Cardy formula, agrees with the asymptotic growth from the gravity side. If

instead of taking the high energy limit of (5.23) we want the general case, the left hand

side will have to incorporate the contributions of all the solutions in the asymptotically

AdS3 phase space. These geometries appear in the path integral weighted by its Euclidean

action. For example, a bulk geometry that must be accounted for is one whose topology

is a three-dimensional solid torus, including all possible excitations on top of it, i.e.

particle, string and brane states that can wind around the solid torus. After all the

contributions have been taken into account, one can hope to match the exact CFT

partition function.

Let's analyze now the two limits separately. We start by considering the boundary being

a torus of parameter τ and with the line element written in complex coordinates as

ds2 = dwdw, with w ≡ w+2π ≡ w+2πτ . ZCFT can either be evaluated as a path integral

on the torus or in the canonical formulation, as we have shown to give (2.28)

Z(τ, τ) = Tr[e2πiτL0−c/24e−2πiτ L0−c/24]. (5.24)

We have already seen the the imaginary part of τ plays the role of inverse temperature

and the real part a chemical potential for angular momentum. Now let's consider ZAdS .

Writing a canonical formula for it as (5.24) is not possible because we lack a description

of the Hilbert space content, i.e. we can't identify the gravitational eigenstates. At low

energies, the leading state consists of a gas of particles moving on AdS and at suciently

high energies we encounter black holes. The latter, though, are not individual states of

the Hilbert space because they have an entropy on their own, so we can not add their

68

5.2 AdS/CFT

contribution to the trace of Z straightforwardly. In the path integral formulation, though,

we can easily introduce these black hole solutions by weighting them with their action.

Since they imply the main contribution at high energies, they are the saddle points of

the action. Therefore, each partition function is best written in each formalism and their

matching is supposed to shed some light on which are the relevant gravitational degrees

of freedom.

We focus now on the gravitational side. The simplest saddle point is thermal AdS3. To

evaluate its action we want to integrate (5.20). Since we know the result has to depend

on τ we have to introduce the periodicity in the metric. This can be done by introducing

the coordinate

z =i− ττ − τ

w − i− ττ − τ

w

where w was dened such that induced metric on the boundary g(0)ij dx

idxj = dwdw with

w ≡ w + 2π ≡ w + 2πτ3, so the new coordinates have z ≡ z + 2π ≡ x + 2πi. Like this,

τ appears in the metric and the variation of the action can be computed to be (after

converting back to w coordinates

δI = 4π2i(−Twwδτ + Twwδτ).

Since thermal AdS3 is the ground state, it is invariant under the isometry group

SL(2,R) × SL(2,R), and therefore is annihilated by the global Virasoro generators,

L0,±1, L0,±1 for this geometry. Using this in (5.22) we nd

Tww = − c

48π, Tww = − c

48π,

which yields the action

Ithermal =iπ

12(cτ − cτ). (5.25)

Therefore, the low temperature behavior of ZAdS as =(τ)→∞ is

lnZAdS(τ, τ) = − iπ12

(cτ − cτ). (5.26)

At high temperatures, the main contribution to the partition function comes from black

hole solutions, i.e. the BTZ. We have shown in 4.2 how thermal AdS with modular

parameter τ is the same as the BTZ black hole with parameter τ ′ = −1/τ . Therefore we

can conclude that

IBTZ =iπ

12(cτ ′ − cτ ′) = − iπ

12(c

τ− c

τ). (5.27)

3 In terms of global coordinates it corresponds to the change w = φ+ it/l, which is as we did in (4.15)

to dene thermal AdS.

69


Therefore, the high temperature behavior =(τ)→ 0+ of the partition function is

lnZAdS(τ, τ) =iπ

12(c

τ− c

τ). (5.28)

All this has been derived in the gravity side, where c is the Brown-Henneaux central

charge. We can use this now to compute the entropy for the black hole at high

temperature. From (5.24) we can write in the saddle point approximation

lnZ = S + 2πiτ(L0 −c

24)− 2πiτ(L0 −

c

24). (5.29)

From the two previous results it follows that

L0 −c

24=

1

2πi

∂ lnZ

∂τ= − c

24τ2, L0 −

c

24=

1

2πi

∂ lnZ

∂τ= − c

24τ2, (5.30)

from which the entropy reads

S = 2π

√c

6(L0 −

c

24) + 2π

√c

6(L0 −

c

24). (5.31)

This is the Cardy formula that gives the hight temperature behavior of the entropy of a

CFT. The derivation of it in the CFT side used the modular covariance of the partition

function, which in the gravity side is also used to compute the action of the BTZ from

that of thermal AdS3. Therefore, the high-temperature entropy of both agrees provided

that the central charges also agree.

5.3 The microscopic BTZ entropy

We present now the example in which the AdS3/CFT2 correspondence is used to

reproduce the Bekenstein-Hawking entropy of the BTZ black hole. The near-horizon

geometry of the BTZ black hole exhibits a locally AdS3 geometry (more precisely a circle

bration over an AdS2) [30], then the black hole entropy (4.12) can be microscopically

computed with the dual CFT. This was done in the paper [31] by Strominger, where

agreement between the two entropies was found.

Dening the vacuum

Let's recall that both the BTZ and the AdS3 metrics are solutions of the same Einstein-

Hilbert action. In section 4.2, we have shown how both thermal AdS3 and the BTZ

black hole follow from modding out the AdS3 solution. We have argued that since the

vacuum, intuitively considered the global AdS3 solution, is supposed to be the state with

70

5.3 The microscopic BTZ entropy

maximum symmetry, excitations of this are to be reached by doing identications that

reduce this symmetry by changing the topology of the space. However, one can wonder

if the BTZ solution with zero mass and angular momentum could also be considered the

vacuum of the theory. As we will see, both of them can be considered ground states, a

feature that, although requiring supersymmetry, has a nice interpretation in the CFT

side.

If we set J = 0, the two horizon radius of the BTZ black hole become r+ = 8Ml2G and

r− = 0, and the BTZ metric (4.10) becomes parametrized as

ds2 = −(r2

l2− 8MG)dt2 +

dr2

( r2

l2− 8MG)

+ r2dφ2. (5.32)

From this metric follow both the M = 0 BTZ black hole and the [global AdS3 metric

when doing M = −1/8G. The rst is like global AdS3 in Poincaré coordinates but with

an identication in the angular coordinate. So the black hole spectrum is achieved from

excitations of the M = 0 metric and the M = −1/8G solution is isolated, since trying to

extend to black hole metric by varying M , naked singularities are encountered between

M = 0 and M = −1/8G.

Now, to set up the CFT picture, we rst have to x the Virasoro charges for the BTZ

excitation. It is convenient to choose the additive constants in L0 and L0 so that they

vanish for the M = J = 0 black hole. One then has

M =1

l(L0 + L0), J = L0 − L0. (5.33)

In a supersymmetric conformal eld theory the Ramond ground state has M = 0. We

can identify this ground state with the zero-mass black hole. The Neveu-Schwarz ground

state, which is not supersymmetric, has a mass shift

L0 = L0 = − c

24. (5.34)

Using the result for the Brown and Henneaux central charge for the charge algebra in

AdS3 (4.28), we can nd the resulting mass M = −1/8G, which coincides with that of

AdS3.

Entropy

Now, to account for the entropy of the black hole we want to count the number of

excitations of the AdS3 vacuum with mass M and angular momentum J . We want to

do this in the semiclassical regime, where M is large and the cosmological constant in

Planck units is small, i.e. l G. Then, the central charge satises c 1 and according

to (5.33) it must hold that

nR + nL c,

71


where nR, nL are the eigenvalues of the zero Virasoro modes. In the dual CFT, this

corresponds to the high-energy regime, precisely where the Cardy formula for the

degeneracy of states (2.32) holds. Therefore, the entropy follows from

SBTZ = 2π

√cnR

6+ 2π

√cnL6

where we don't introduce a shift for the central charge because the states are excitations

of the Ramond ground state. Using the value of the central charge (4.28) and equations

(5.33) for the Virasoro eigenvalues, we obtain

S = π

√l(lM + J)

2G+ π

√l(lM − J)

2G=

2πr+

4G.

This is exactly the BTZ Bekenstein-Hawking entropy found from computations of the

metric in (4.12). In the CFT side, the Cardy formula counts the degeneracy of states for

the high energy regime. As the energy increases, the higher modes get more and more

populated. The interpretation of this on the gravity side is that a BTZ black hole is

formed.

72

CHAPTER 6

The Kerr black hole

In the rst part of this thesis we have developed the necessary background to address

the central topic of the project presenting the basic ingredients of the well established

dualities between gravity and eld theories. In this second part, we will focus on the

very topic, the correspondence between quantum gravity in the near-horizon region of

extremal Kerr black holes and conformal eld theories. The logical rst step in this feat

is to study the Kerr black hole.

Kerr black holes are black holes with rotation. Thus, the Kerr solutions to Einstein's

equations have to be asymptotically at, stationary and axisymmetric around the axis

of rotation. Stationarity implies the existence of a Killing vector eld K that is timelike

near ∞. If t is the time coordinate, then K = ∂/∂t and the metric doesn't depend on t.

Axisymmetry implies the existence of a Killing vector eld R which is spacelike near ∞and for which all orbits are closed. If φ is the azimuthal angle coordinate, then R = ∂/∂φ

and the metric components do not depend on it either.

From the uniqueness theorems [32] for black holes, in particular from the Carter-Robinson

theorem, follows that if a vacuum spacetime is asymptotically-at, stationary and non-

singular on and outside an event horizon, then it is also axisymmetric and must belong

to the two-parameter Kerr family, being the parameters the mass M and the angular

momentum J of the black hole. The Kerr metric in Boyer-Lindquist coordinates (t, r, θ, φ)

is

ds2 = −(∆− a2sin2 θ)

Σdt2 − 2asin2 θ

(r2 + a2 −∆)

Σdtdφ

+sin2 θ(r2 + a2)2 −∆a2sin2 θ

Σdφ2 +

Σ

∆dr2 + Σdθ2, (6.1)

73

Chapter 6: The Kerr black hole

where

Σ = r2 + a2 cos2 θ (6.2)

∆ = r2 − 2Mr + a2 (6.3)

These coordinates turn out to be oblate spheroidal coordinates:

• surfaces of constant r are confocal ellipsoids whose major axis are 2√r2 + a2 and

whose minor axis is 2r, where a is the focal distance of the ellipsoids and is xed

for all of them. The range of this coordinate is 0 ≤ r ≤ ∞. r = 0 corresponds then

to the disk x2 + y2 ≤ a2, z = 0

• surfaces of constant |η| = | cos θ| form hyperboloids of one sheet (formed by

revolution of a hyperbola around the conjugate axis). The upper half of the

hyperboloid corresponds to angle θ and the lower half to angle π − θ. Thus,

−1 ≤ η ≤ 1.1

• surfaces of constant φ are planes through the z axis, so that φ is the angle measured

on the (x, y) plane with respect to the x axis.

As aforementioned, the metric depends on the two parameters M,J through a = J/M ,

so a is the angular momentum per unit mass of the black hole as measured from

innity.

As a → 0, the metric reduces to the Schwarzschild solution, spherically symmetric,

static, vacuum solution. If both M → 0 and J → 0, the metric reduces to Minkowski

spacetime (with the spatial part in ordinary polar coordinates if we put a = 0 and in

oblate spheroidal coordinates if we keep a nite).

Let's now look for the singularities of the Kerr metric. Through inspection of equation

(6.1), one nds a singularity at Σ = 0, which is only the case when r = 0 and θ = π/2.

This locus corresponds to the ring x2 + y2 = a2, the boundary of the r = 0 disk. This

ring is a real curvature singularity, since the scalar RabcdRabcd diverges on it, and it is

the only curvature singularity the solution has.

However, (6.1) shows two other coordinate singularities. One of these is the usual rotation

axis coordinate singularity at θ = 0, π. The other coordinate singularity occurs when

∆ = 0. This one can be written in the following form ∆ = (r − r+)(r − r−), where

r± = M ±√M2 − a2. From this expression follows that the features of this singularity

1A hyperboloid is specied by two-parameters, it depends on the angle θ and the distance d between

the origin and a vertex. In the oblate spheroidal coordinate system, however, these two parameters are

constraint in a way that an angle θ determines only one hyperboloid. This way, |η| = 1 → d = 0 and

corresponds to the conjugate axis (hyperboloid of eccentricity e = 1); η = 0→ d = a and corresponds to

the locus x2 + y2 ≥ a2, the (x, y) plane except for the inner region of the disk corresponding to r = 0

74

6.1 M2 > a2

depend on the relation between the parameters M and a, leading to the three dierent

regimes M2 > a2, M2 < a2 and M2 = a2. The last regime leads to the extremal black

hole, in which we are particularly interested. For now, we will start discussing the rst

regime and derive some of the features of the Kerr black hole.

6.1 M2 > a2

This is the most general situation. In this case, r = r± are real zeros of ∆ and

they constitute indeed two coordinate singularities. The surfaces with r = r± are null

hypersurfaces that can be crossed only in the ingoing direction by future directed timelike

lines. Therefore r+ constitutes the event horizon2 and r− is a Cauchy one3. In the area

between horizons, r becomes the timelike coordinate and so future directed lines have

to move in the direction of decreasing r. If a → 0, then r− becomes 0 and coincides

with the singularity, which is no longer a ring but the Schwarzschild pointlike singularity.

r+ becomes the Schwarzschild event horizon at r = 2M . To extend the metric across

these surfaces we can perform the so-called Finkelstein transformations. One can notice

that for null paths approaching the surfaces r = r± (at xed θ, φ), the progress in t as

compared to the progress along the r direction is divergent. From equation (6.1)

dt

dr=

Σ

∆→∞ when r → r± (6.4)

This suggests to search for a new coordinate which compensates for this divergence

encompassing with the variation along r. This is what the Tortoise coordinate r∗ does.

Solving (6.4) we nd t(r) and thus we can dene r∗ such that

dr∗ = ±dt+ const = ±r2 + a2

∆dr + const (6.5)

where we have put θ = 0. Now, the Finkelstein extension consists of dening an ingoing

null coordinate v and an angle coordinate ~φ such that

dv = dt+ dr∗ = dt+r2 + a2

∆dr

d~φ = dφ+a

∆dr

2The event horizon is the locus where hypersurfaces of constant r become null. Therefore we look

for the radius where the normal vector to these hypersurfaces ∂µr has zero norm. This happens at

grr(rH) = 0.3We have commented on Cauchy surfaces in 3.1.

75


The angle transformation is such that it cancels the terms ∼ 1/∆ that bring the

coordinate singularities in the horizons. In these coordinates, the Kerr metric turns into

the form

ds2 = 2drdv−2a sin2 θdrd~φ+ Σdθ2 +1

Σ((r2 + a2)2 −∆a2 sin2 θ) sin2 θd~φ2

− 4aMr

Σsin2 θd~φdv − (1− 2Mr

Σ)dv2 (6.6)

With this form, the metric is well behaved over the ranges −∞ < v, r <∞ and 0 ≤ θ <π, 0 ≤ ~φ < 2π. Both the rotation axis and ring singularities are still present.

This is of course not a maximal extension, since we could also have performed the

extensions by dening an outgoing null and angular coordinates with which the Kerr

metric turns into the same form (6.6) but replacing v by −v and ~φ by −~φ. The metric

will then be well dened over a patch analogous to the one of the ingoing extension but

in the opposite direction.

It is possible to build a maximal extension of the Kerr solution by combining Finkelstein

extensions. To cover all the resulting extension, one has to transform to Kruskal-type

coordinates in an analogous way as it is done for the Schwarzschild solution. These

coordinates will cover the crossover points where the horizons intersect themselves which

correspond to 2-dimensional spacelike hypersurfaces in the full 4-dimensional picture.

The Kruskal transformation introduces both ingoing and outgoing null coordinates and

convenient angles; these result into two null coordinate systems, each of which makes

the resulting metric be well dened in the neighborhood of r = r+, r = r− and

the intersections with themselves respectively. We are not going to write down these

steps explicitly (see [33]), we will only emphasize the existence of such coordinate

transformations and the subsequent well dened metric over the whole maximally

extended manifold.

The resulting conformal diagram is shown in gure 6.1. Since the Kerr spacetime

is asymptotically at, the innities will form the triangular pattern characteristic of

the Minkowski Penrose diagram. The horizon r+ lays in the same position as the

Schwarzschild event horizon r = 2M , closing the diamond-shaped region I where t is

the timelike coordinate. Region II will also show a diamond-shaped area, extending from

r+ to r− with r being the timelike coordinate. Finally, across r− seats region III, in

which t is again the timelike coordinate and the timelike ring singularity lays. Beyond

this one, the analytical extension to negative values of r leads to another asymptotically

at spacetime which makes region III also look diamond-shaped.

Let's move on to discuss the symmetries of the Kerr solution. As pointed out, the Kerr

metric is stationary and axisymmetric. This leads to the existence of the Killing vectors

76

6.1 M2 > a2

Figure 6.1 Penrose-Carter diagram of Kerr of the axis θ = 0. Region I corresponds to the

region between the conformal innities and r+; region II to the area between horizons;

and region III corresponds to the area of r ≤ r−, containing the ring singularity and

the analytically extended asymptotically at region. The dotted lines represent r = ctt

hypersurfaces, and the singularity with a zig-zag line. The coordinates (t, r) are only well

dened in region I. (Figure extracted from [33].)

K = ∂/∂t and R = ∂/∂φ. Any linear combination of these is also a Killing vector, and

this exhausts the whole set of Killing vectors of the Kerr spacetime. R is spacelike near

∞ and becomes 0 on the axis of symmetry. Its orbits are closed curves around this axis.

On the other hand, K is the only Killing vector that is timelike at arbitrarily large values

of r. However, it is not timelike everywhere outside the horizon r+. We can look for the

hypersurface in which K becomes null. Since

K = ∂/∂t→ KµKµ = − 1

Σ(∆− a2 sin2 θ),

it corresponds to re = M +√

(M2 − a2 cos2 θ) and satises re ≥ r+, the equality holding

on the locus θ = 0, π. This hypersurface is called the ergosurface or the static limit

surface, it is the boundary of the region in which particles can travel along an orbit of

77


the Killing vectorK and so remain static with respect to innity4. This surface is timelike

except at the locus θ = 0, π where it is null. Thus, except for these two points, particles

can cross this surface in both ingoing and outgoing directions. In the limit a → 0, the

ergosurface goes to lay on top of the outer event horizon and K is timelike everywhere

outside it.

The area between the stationary limit surface and the event horizon is called ergosphere.

K is spacelike in this area and it still is so at r+. Although particles can still scape from

this region, they can't stay static with respect to innity. The rotation of the black hole

causes the rotation of the spacetime around it, an eect called frame dragging. Particles

have to rotate with the black hole, and only photons on the stationary limit surface can

remain at rest with respect to it.

As explained, K is not null on the event horizon r+. This is due to the fact that the

Kerr spacetime is stationary but not static. However, there exists a Killing vector that

becomes null on top of it and so for which the event horizon is a Killing horizon. This

one corresponds to Q = K + ΩHR, where ΩH is the angular velocity of the event

horizon.

The Hawking temperature, surface gravity and angular velocity of the horizon5 are

TH =~κ2π

=~(r+ −M)

4πMr+, (6.7)

ΩH =a

2Mr+. (6.8)

These are related by the rst law of thermodynamics to the Bekenstein-Hawking entropy

as

S =A

4G~=

2πMr+

~G. (6.9)

4In [13], an observer moving along K is called stationary and the ergosurface the stationary limit

surface. This is not the wording we adopt here. Any observer who moves along a world line of constant

(r, θ) with uniform angular velocity sees an unchanging spacetime geometry in his neighborhood. Hence,

such an observer can be thought of as stationary relative to the local geometry. If and only if his angular

velocity is zero (moves along a world line of constant (r, θ, φ)) will he also be static relative to the black

hole's asymptotic Lorentz frame (relative to distant stars). Therefore we call the ergosurface the static

limit surface5 The way the angular velocity is computer is the following. Let us consider a photon emitted in

the φ direction at some radius r at some plane of xed θ. The line element results in ds2 = 0 =

gttdt2+2gtφdtdφ+gφφdφ

2. The angular velocity of the photon is then Ωφ = dφdt

= − gtφgφφ±√

(gtφgφφ

)2 − gttgφφ

.

These are the two velocities the photon can have in the vicinity of the black hole. Then, we can dene

the angular velocity of the event horizon, ΩH , to be the minimum angular velocity of a particle at the

horizon.

78

6.2 M2 < a2

6.2 M2 < a2

In this case, the radius of the horizons would become complex, therefore there would be

no such horizons. This would lead to a naked singularity as can be seen in gure 6.2.

Figure 6.2 In this case the singularity is not hidden by any horizons and becomes naked.

(Figure extracted from [33].)

Penrose suggested in 1969 the Cosmic Censorship Hypothesis [13] according to which, in

any situation arising from the gravitational collapse of an astrophysical object starting

from a well behaved initial situation (an asymptotically at spacetime that is non-singular

on some initial spacelike hypersurface), the singularities which result are hidden form

the outside by an event horizon. In other words, naked singularities which can both be

approached from and seen from outside cannot arise naturally from a well behaved initial

situation6. This Hypothesis, as the name says is not proven, but if naked singularities

were to exist, then the future would cease to be predictable from data given on an initial

spacelike hypersurface.

6This hypothesis does not exclude the visibility of preexisting singularities, such as the big bang or

the white holes, which cannot be reached in the future by a timelike trajectory, in this sense they are

not naked.

79


6.3 M2 = a2

This is the most interesting case in the context of this thesis. Under this condition, the

two horizons coalesce on top of each other r+ = r− = M leading to a degenerate Killing

horizon. According to (6.7), the Hawking temperature vanishes in this case leading to

an extremal Kerr black hole. Both properties can be used to dene an extremal black

hole. According to the cosmic censorship hypothesis a critical case such as the extremal

black hole should represent a physically unattainable limit, but one which is approachable

and therefore of considerable interest. In fact, no physical process is known that would

make an extremal black hole out of a non-extremal one, although spontaneous creation

of extremal black holes is allowed. If one attempts to send nely-tuned particles or waves

into a near-extremal black hole in order to approach extremality further, the window of

allowed parameters that allows to do so becomes smaller and smaller when approaching

extremality. On the other hand, a non-extremal black hole can easily be reached form an

extremal one by just sending a massive particle inside.

Figure 6.3 Conformal diagram of the θ = 0 axis of extremal Kerr. One can see the

degenerate horizon and the only remaining regions I and III. (Figure extracted from [33].)

80

6.3 M2 = a2

The extremal black hole metric follows from imposing the extremality condition on (6.1).

In this case, one can also introduce Finkelstein type extensions to go over the horizon

singularity, although there is no way to carry out a Kruskal type extension because the

transformation would contain singularities. However, it turns out that Kruskal extensions

are quite unnecessary since the Finkelstein extensions can be tted together to form

the maximally extended manifold. That's mainly because the Kruskal coordinates were

introduced to cover the crossover points where the horizons intersect themselves, but in

the extremal case these crossover points don't appear anymore. The Penrose diagram of

the extremal Kerr black hole can be seen in gure 6.3.

At extremality, the angular velocity becomes ΩH = 12M and the entropy S = 2πJ

~ , where

Newton's constant disappears because actually J = Ga2. A special feature of the extremal

geometry is that the proper distance along a t-constant slice from any point outside the

horizon to the horizon is logarithmically divergent. Like this, the geometry down to the

horizon is often called a throat.

81

CHAPTER 7

Near-horizon geometries

As we have already mentioned, the regions close to the horizons of extremal black holes

exhibit exact or asymptotic AdS geometries. These geometries can be thought of as

complete vacuum spacetimes on its own right, totally decoupled from the asymptotics

of the black hole solution. In this chapter we will derive the near-horizon geometry of

the 4-dimensional extremal Kerr black hole. This geometry, despite not containing an

exact AdS, acquires warped AdS3 factors at slices of constant θ and exhibits an eective

timelike boundary. Therefore, it is a candidate for acquiring a dual CFT description. We

will start, though, deriving the near-horizon geometry of the Reissner-Nordström black

hole. This one shows many similar features to Kerr but allows for a simpler treatment,

so it serves as an easy and useful reference. We will also address another subtle feature

of extremal black hole geometries, namely the discontinuity of the extremal limit.

7.1 Reissner-Nordström

The Reissner-Nordström spacetime represents the solution for a charged black hole. As

for the Kerr black hole, this solution exhibits two horizons, r = r± and the metric can

be written as

ds2 = −(r − r−)(r − r+)

r2dt2 +

r2

(r − r−)(r − r+)dr2 + r2dΩ2

2. (7.1)

The horizons depend on the charges of the black hole as seen from innity, the mass

M and the electric and magnetic charges Q and P , then r± = M ±√M2 −Q2 − P 2.

As in the case of Kerr then, M2 ≥ Q2 + P 2 is required for avoiding naked singularities

and the equality leads to an extremal black hole, whose horizons lay on top of each

83

Chapter 7: Near-horizon geometries

other. In the limit that Q and P go to zero, the inner horizon r− would approach the

r = 0 singularity leading to a Schwarzschild black hole. As in the Kerr spacetime, one

can perform Finkelstein extensions to cover the coordinate singularities of the horizons

in (7.1) and nally go to Kruskal-type coordinates to cover the maximal extension of

the solution, which results into a Penrose diagram of the same kind as the Kerr black

hole. However, the dierent nature of the singularity r = 0, which is pointlike in this

case, doesn't allow to analytically continue to negative values of the coordinate r and

therefore the diagram doesn't extend to another at space past this singularity. The

conformal diagram for the extremal black hole results from the non-extremal one making

the two horizons coincide, exactly as we showed for the Kerr black hole.

We proceed now to the computation of the extremality and near-horizon metrics. The

extremality limit is analogous to that discussed for the Kerr black hole. One simply

starts with the non-extremal Reissner-Nordström metric (7.1) and makes the two event

horizons coincide r+ = r− = M = Q ≡ ρ. To perform the near-horizon limit, we have

to perform a change of coordinates which includes a parameter λ that goes to zero when

zooming into the horizon. A change of coordinates we can do is

r =r − ρλρ

t =λt

2ρ.

When approaching the horizon one performs the limit λ → 0 and therefore the

coordinates will be kept xed. One can see that points a nite distance away from the

horizon will end up at ∞; points innitesimally close to the horizon will correspond

to nite values of the new coordinates. The resulting near-horizon metric in these

coordinates is

ds2 = ρ2(−r2dt2 +dr2

r2) + ρ2dΩ2

2. (7.2)

In the rst two terms we can recognize the AdS2 metric in Poincaré coordinates and the

last term corresponds to a two-sphere, both of radius ρ. Therefore, the near-horizon region

of the extremal solution approximates locally AdS2 × S2, this approximation becoming

exact only on the horizon.

This Anti-de Sitter asymptotic behavior can easily be seen by comparing the conformal

diagrams of the extremal Reissner-Nordström black hole and AdS2. For this comparison,

it is useful to use a slight variation of the near-horizon limit introduced above. One can

include the limiting parameter λ in the new near-horizon coordinate such that this goes

to zero in the limit1. Doing λ = r−ρρ2 one obtains the same metric form of (7.2) with the

only dierence that now the metric is singular on the horizon (notice that in both cases

1In this case no change for the timelike coordinate is required since there is no limiting parameter

that has to be canceled after the change of coordinates.

84


λ can be both positive and negative, meaning a near-horizon limit from outside or inside

the horizon respectively). One can now look for the locus of points corresponding to the

horizon λ = 0 in the AdS2 diagram using the change of coordinates from Poincaré to

global coordinates, which is

t =sin τ

cos τ − sin θ, λ =

cos τ − sin θ

cos θ. (7.3)

The extremal horizon, λ = 0, lays in the points τ = π/2 − θ, 3π/2 + θ of the AdS2

conformal diagram. Therefore, the area innitesimally close to the null horizons dening

the Poincaré patches in the AdS2 conformal diagram looks like the area innitesimally

close to the horizon of the extremal Reissner-Nordström black hole. This can be seen in

gure 7.1.

Figure 7.1 Comparison of the conformal diagrams of the extremal Reissner-Nordström

black hole on the left and of AdS2 on the right. The near-horizon region of the extremal black

hole in Poincaré coordinates, doing the limit from outside the horizon, leads to a Poincaré

patch of AdS2. If the limit were done from region III inside the horizon (λ < 0), then one

would reach the complementary Poincaré patch right above. (Figure extracted from [33].)

The appearance of this AdS2 factor in the near-horizon solution leads to the conclusion

that there should exist some type of dual 1-dimensional CFT description. This

AdS2/CFT1 correspondence is not completely understood yet [34].

Discontinuity of the extremality limit

Subtleties when taking limits can be relevant in general. An interesting feature of the

limit to extremality is that it is not continuous [35]. While in the extremal black hole the

two horizons are on top of each other, taking the limit with an extremality parameter

85


leads to a new spacetime in the area in between them. This parameter is introduced in

a change of coordinates such that when it goes to zero, both the extremality and near-

horizon limits are taken simultaneously. We will here analyze this issue separately for the

regions I and II of the conformal diagram for the Reissner-Nordström black hole2.

• Region I

We will start by introducing new coordinates and extremality parametrization.

First, a variable ε will parametrize the deviation from extremality in the following

way

r− = ρ− ε, r+ = ρ+ ε (7.4)

so that ε =√M2 −Q2 and ρ = M stays xed and between the horizons. The limit

ε → 0 will then be the limit to extremality. Next we introduce new timelike and

spacelike coordinates (ψ, χ) in the following manner

r = ρ+ ε coshχ ψ =ε

ρ2t (7.5)

These coordinates cover all of region I; χ = 0 corresponds to r = r+, χ = ∞ to

future/past null innity and −∞ < ψ <∞. The non-extremal Reissner-Nordström

metric in these coordinates is

ds2 = ρ2[− sinh2 χ

(1 + ερ coshχ)2

dψ2 + (1 +ε

ρcoshχ)2dχ2 + (1 +

ε

ρcoshχ)2dΩ2

2] (7.6)

Having introduced both ε and χ, there are two options. On the one hand, we can

make ε → 0 and coshχ → ∞ such that r+ = r− = ρ but r stays a nite distance

away from the horizon. This would eectively be the same as the extremality limit

just taken before, which would lead to the extremal Reissner-Nordström metric

(7.2) (after a change of coordinates) and thus to the asymptoticAdS2×S2 spacetime

in the near-horizon region.

On the other hand, we may just take the limit ε→ 0 while keeping χ nite. This is

equivalent to taking both the extremality and near-horizon limits simultaneously.

In this case, the metric (7.6) becomes

ds2 = ρ2[− sinh2 χdψ2 + dχ2 + dΩ22]. (7.7)

This metric happens to correspond again to AdS2×S2 but no to the entire manifold.

The transformations

coshχ =cos τ

cos θtanhψ =

sin τ

sin θ(7.8)

2 The treatment for region III however, is parallel to that for region I.

86


bring the AdS2 factor of (7.7) to the usual global form (3.8) of the AdS2 metric and

with them we can nd out the covered patches. For region I of the (non)extremal

Reissner-Nordström black hole, the ranges for the global coordinates become

−π/2 ≤ τ ≤ π/2 and 0 ≤ θ ≤ π/2, and r+, limiting region I, corresponds to

the locus τ = ±θ. This corresponds to region I of a Poincaré patch of the AdS2

diagram, see gure 7.2.

Figure 7.2 This diagram displays the dierent regions of the Poincaré patch of the

AdS2. They are drawn according to the encountered ranges of the coordinates and to clarify

the analogies with the dierent regions of the Reissner-Nordström and Kerr black holes that

we will be addressing.

This result is somewhat surprising. If we start from the extremal black hole, the

near-horizon limit yields the whole of AdS2 (or the Poincaré patch if expressed in

Poincaré coordinates), while if we start from region I of the non-extremal black

hole, the limiting metric only covers the region I of the Poincaré patch of AdS2.

• Region II.

Since region II is the area between the two horizons in the non-extremal case,

performing the extremality limit by coalescence of the two horizons simply shrinks

this region until it disappears. However, as done previously, we can introduce

coordinates in region II such that the limit to extremality and the near-horizon

one are taken simultaneously. First, we introduce the extremality parameter ε

r− = ρ− ε, r+ = ρ+ ε (7.9)

and then new timelike and spacelike coordinates (χ, ψ)

r = ρ− ε cosχ, ψ =ε

ρ2t (7.10)

that will cover the whole of region II with 0 < χ < π and −∞ < ψ < ∞.

As done previously, by introducing these in the Reissner-Nordström non-extremal

87


metric (7.1), and taking the limit ε → 0, which is equivalent to taking both the

extremality and near-horizon limits simultaneously, the metric of region II becomes

ds2 = ρ2[−dχ2 + sin2 χdψ2 + dΩ22] (7.11)

Again, this corresponds to a patch of AdS2 × S2. The transformations

cosχ =cos τ

cos θtanhψ =

sin θ

sin τ, (7.12)

that bring it to the global form of AdS2, determine the patch covered by the limiting

metric in the conformal diagram. The ranges become 0 ≤ τ ≤ π and −π/2 ≤ θ ≤π/2. r− corresponds to the locus τ = ±θ and r+ to the locus τ = ±θ + π. This

is region II of the Poincaré patch in the AdS2 conformal diagram (notice that

comparing the two regions II one is rotated π/2 with respect to the other).

The emergence of this spacetime is unexpected since region II in the extremal black

hole seems to disappear. However, it does not. One can compute the spacetime

distance between horizons using the metric in coordinates (χ, ψ) on region II before

taking the limit. This distance will be the proper time τ elapsed on a trajectory of

constant nite ψ, thus

∆τ = ρ

π∫0

dχ(1− ε

ρcosχ) = πρ (7.13)

Since it does not depend on the extremality parameter ε, this distance stays nite

when taking the limit, and so the horizons stay a constant physical distance apart.

Both the black hole solution and the near-horizon geometry AdS2 × S2, also called

compactication solution, follow from the same Einstein-Maxwell equations3, diering

only by boundary conditions. The appearance of these two solutions when considering

the extremality limit of the black hole results in this limit not being continuous. On the

one hand, considering points in regions I and II a nite proper distance away from the

horizon and then taking the limit, one obtains an extremal black hole. On the other hand,

the region II between horizons, together with the near-vicinity of the horizons in regions

I and III form the whole of the compactication solution when the limit is taken. The

dierence comes from the fact that these two limits are not taken in the same exact way.

In the rst case, one only considers extremality, either by just putting r+ = r− in the

non-extremal metric or taking the limit without zooming into the near-horizon region

(ε→ 0 and χ→∞). This is regardless of what the geometry at the region innitesimally

3In general, besides the black hole solutions, it is possible to nd static solutions to the Einstein-

Maxwell theory, with or without cosmological constant, that are a product of maximally symmetric

spaces.

88

7.2 NHEK

close to the horizon of this solution would be. In the second case, the extremality limit

is taken simultaneously with the near-horizon limit, so we are only considering solutions

innitesimally close to the horizon. In other words, one can say that the two limits do

not commute.

The motivation for doing this analysis, besides the surprising appearance of a discon-

nected spacetime, is that it may help to understand better the entropy of extremal black

holes, as degrees of freedom could be stored in this additional space arising in the region

between horizons. We will discuss this argument further in the next section when doing

the same analysis for the more interesting Kerr black hole.

7.2 NHEK

Let's move on to discuss the Kerr near-horizon limit. The main reference is the paper by

Bardeen and Horowitz [36]. We start o the Kerr extremal metric and we want to zoom

in down the throat. As before, we will introduce a parameter λ which will go to zero in

the limit, and new dimensionless coordinates in the following manner

r =r −MλM

t =λt

2Mφ = φ− t

2M(7.14)

Since 1/2M is the angular velocity of the horizon in the extremal case, the new set of

coordinates corotates with the horizon. So, we proceed to introduce these coordinates in

the extremal metric and perform the limit. The result is the metric

ds2 = 2a2Ω2(θ)

[dr2

r2− r2dt2 + dθ2 + Λ2(θ)(dφ+ rdt)2

](7.15)

where

Ω2(θ) =1 + cos2 θ

2Λ(θ) =

2 sin θ

1 + cos2 θ. (7.16)

This is the so-called NHEK, the near-horizon extremal Kerr geometry [36]. It solves

the Einstein equations since it follows from a change of coordinates of a solution of the

equations. As in the Reissner-Nordström case, this limiting metric is not asymptotically

at. However, in this case we cannot recognize in it the exact AdS2 × S2 factor. The

rst two terms correspond to AdS2 but there is a mixing term between dφ and dt due

to rotation; on the equator, it becomes a twisted product of AdS2 and a circle of radius

2a.

Along the θ-axis, the NHEK metric becomes

ds2 = 2a2

[dr2

r2− r2dt2

], (7.17)

89


which is exact AdS2. A quick look at the conformal diagram of the Kerr extremal metric

6.3, which is done for θ = 0, suggests the existence of this AdS2 spacetime arising near the

horizon in the same suggestive way it arose for extremal Reissner-Nordström, recall gure

7.1. The degenerate event horizon would correspond again to the locus θ = ±τ + π/2 of

the AdS2 conformal diagram.

The coordinates in (7.15) are the analogous to the Poincaré coordinates in AdS2 and only

cover part of NHEK. A change to global coordinates can be performed, transforming r

and t to coordinates (τ, y) as is done for AdS2, using (7.3) (with r instead of λ) and

the change tan θ = −y (an extra minus sign is introduced in this change for y, but that

doesn't change the form of the AdS2 metric). The change for the coordinate φ then follows

from choosing the new axial angle coordinate φ such that gφy = 0. The transformation

is

r =√

1 + y2 cos τ+y t =

√1 + y2

rsin τ φ = φ+ln

[cos τ + y sin τ

1 +√

1 + y2 sin τ

], (7.18)

and the NHEK metric becomes

ds2 = 2a2Ω2(θ)

[−(1 + y2)dτ2 +

dy2

(1 + y2)+ dθ2 + Λ2(θ)(dφ+ ydτ)2

]. (7.19)

These coordinates, with −∞ < τ, y <∞, cover the whole of NHEK.

The NHEK geometry has enhanced symmetries with respect to the Kerr solution. It

has its same discrete t− φ reection symmetry, its Killing vectors K and R, but it also

has the dilation symmetry of AdS2, r → cr, t → t/c, which actually amounts to the

possibility of rescaling the near-horizon parameter λ with a positive constant. Finally,

(7.19) is independent of τ , so ∂τ is also a Killing vector. Since this one corresponds to

symmetry under the global time translation of AdS2, the NHEK geometry has got the

isometries of AdS2 and the axisymmetry, its isometry group is SL(2,R)× U(1) and the

Killing vectors4

ξ1 = ∂t, ξ0 = t∂t − r∂r

ξ−1 = (1

2r2+t2

2)∂t − tr∂r −

1

r∂φ, L0 = ∂φ, (7.20)

where I have suppressed the hats on the Poincaré coordinates. Surfaces of constant τ are

always spacelike, thus τ is also a global time function of NHEK and there are no closed

timelike curves in the spacetime. However, it must be noted that the Killing vector ∂τ

4The Killing vector ξ−1 doesn't exactly correspond to the global time translation but rather to

ξ−1 = ∂τ − 12∂t.

90

7.2 NHEK

is not timelike everywhere in the NHEK spacetime. We will come back to this later

since this becomes important in the construction of the correspondence. Regarding the

coordinate y, surfaces of constant y are always timelike. The boundaries at y = ±∞ are

also timelike since certain timelike geodesics, characterized by high angular momentum,

can reach innity in nite coordinate time τ (although with innite ane parameter).

Timelike geodesics with zero angular momentum behave like geodesics in AdS2 and

never reach innity. These boundaries correspond to the boundaries of the AdS2 factor,

so eectively they are 1-dimensional. Because of this, one would expect a 1-dimensional

dual, a quantum mechanical theory. Due to the similarities between NHEK and AdS2×S2

it could also be expected their dual theories to be similar. However, as already pointed

out, the AdS/CFT correspondence in two dimensions is poorly understood as compared

to the higher dimension versions. Luckily, it turns out that in slices of constant θ, NHEK

becomes quotients of AdS3.

If we analyze the NHEK geometry, there exists a θ = θ∗ such that Λ2(θ∗) = 1. In this

case, the NHEK metric becomes

ds2 = 2a2Ω2(θ∗)

[−r2dt2 +

dr2

r2+ (dφ+ rdt)2

](7.21)

This is a quotient of the AdS3 metric, since it would correspond to Poincaré coordinates

but with the φ periodically identied φ ∼ φ+2π. If θ∗ < θ < π−θ∗, then Λ2(θ) > 1, and

if 0 < θ < θ∗ or π−θ∗ < θ < π, then Λ2(θ) < 1. So near the equator there is a stretched

AdS3 quotient and near the poles there is a squashed AdS3 quotient, becoming AdS2

for θ = 0. These 3-dimensional θ slices are called warped AdS3, obtained by bration

of S1 over AdS2. At θ = θ∗ the spacetime acquires the AdS3 SL(2,R)R × SL(2,R)Lisometry. At the other values of theta, the SL(2, R)L is broken to the U(1) of ∂φ.

The appearance of these AdS3 factors in the near-horizon region of the extremal Kerr

black hole is a hint that some correspondence with a 2-dimensional conformal eld theory

could be drawn. Moreover, we have seen in chapter 4 that a quotient of AdS3 is related

to the nite temperature partition function of the dual conformal eld theory. We will

see in the next chapter that the periodicity of φ allows for a denition of a temperature

Tφ for the extremal Kerr vacuum and therefore for the dual theory.

At this stage, one can still wonder which of the two, AdS2 or AdS3, is more relevant

for developing a holographic dual of the extreme Kerr appears. On the one hand, the

AdS2 factor pops up exactly and its isometry group is contained in the one of NHEK.

The isometry group of AdS3 is broken, although as we will see in the next chapter,

it is the broken left sector that is uplifted to a Virasoro algebra according to the

proposed correspondence. The timelike boundaries are eectively 1-dimensional since

they correspond to the ones of AdS2. To be able to establish a correspondence between

91


Killing vectors in the bulk and conformal Killing vectors in the boundary, this AdS

metric has to be conformally rescaled. However, this cannot be done in NHEK since it

is not the product of an AdS factor times a (maximally symmetric) manifold and does

not approach it asymptotically either. One can rescale the whole metric changing the

Poincaré coordinate r to u = 1/r. This leads to a conformal factor 1/u2, as in (3.3), and

the boundaries are situated at u = 0. However, because of this the conformal metric is

singular on them. One can still argue, though, that the Killing vectors of NHEK (7.20)

are conformal Killing vectors of the conformally rescaled metric.

7.3 Entropy of the extremal Kerr black hole

The Hawking-Bekenstein entropy of the extremal black hole can now be computed again

with the area law but using the near-horizon geometry, since thanks to its dilation

symmetry we can move the surface Σ to the near-horizon region without changing the

integral. Using NHEK, the entropy becomes

S =π

2G~

∫ π

0dθ2a2Ω2(θ)Λ(θ) =

2πJ

~. (7.22)

The ~ will disappear latter when we quantize the angular momentum, which is an integer

multiple of ~. This entropy obviously reproduces the extremality limit of (6.9). The main

goal of the Kerr/CFT correspondence is to reproduce this result with a microscopic

counting of degrees of freedom of the dual theory. As presented for the AdS3/CFT2

correspondence, we will appeal to the Cardy formula.

Discontinuity of the extremality limit and entropy discrepancies

For the possible implications it may have for the entropy and the identication of

the microscopic degrees of freedom, it is interesting to analyze the subtleties of the

extremality limit for the Kerr black hole. We will compute this limit from the region

between horizons.

We start o with the non-extremal metric (6.1) and we introduce an extremality

parameter ε and coordinates (χ, ψ) in the region II

r− = ρ− ε r+ = ρ+ ε

r = ρ− ε cosχ, ψ =ε

ρ2t, ϕ = φ+ tf(a)

As before, ρ = M is xed between the horizons and ε =√M2 − a2. For the Kerr case, we

also have to introduce a change for the angular coordinate. We expect the function f(a)

multiplying the time coordinate to be somehow related to the angular velocity, in the same

92

7.3 Entropy of the extremal Kerr black hole

way as for obtaining NHEK the new coordinates corotate with the black hole. Therefore,

we also assume it depends on the parameter a. We can nd this function by imposing

that the coecients of the terms of order O(ε−1) after the change of coordinates, vanish

so as to avoid divergences when taking the limit ε → 0. It follows that f(a) = −aρ2+a2 ,

which equals the familiar extremal angular velocity in the limit.

Introducing this function and taking the limit ε → 0, which amounts to perform both

the extremality and near-horizon limits simultaneously, the resulting metric is

ds2 = 2a2Ω2(θ)[−dχ2 + sin2 χdψ2 + dθ2 + Λ2(dϕ− cosχdψ)2] (7.23)

Where we have made the change ψ → ψ/2. So again, a spacetime appears in a region

which seemingly shrinks to zero at extremality. The rst two terms in the brackets

are already familiar, since they appeared in metric (7.11), so they correspond to AdS2.

Therefore, to recover NHEK in the global form (7.19), the change for the coordinates

(χ, ψ) follows from combining (7.12) with the change y = − tan θ. The transformation

for the axial angular coordinate then follows from imposing the invariance of the mixing

term dϕ− cosχdψ = dφ+ ydτ . The change of coordinates is5

cosχ =√

1 + y2 cos τ tanhψ =−y√

1 + y2 sin τϕ = φ+ ln

√y cos τ − sin τ

y cos τ + sin τ

Since the change of coordinates from (χ, ψ) to global coordinates is the same as for

AdS2, the same analysis done for the Reissner-Nordström black hole follows and the

AdS2 factor of the metric (7.23) would only cover region II. So the limit to extremality

is also discontinuous for the Kerr black hole.

In this case, though, the compactication solution is not a direct product of maximally

symmetric spaces but a twisted one. The disconnected spacetime in between horizons

being regarded as an AdS2 patch may therefore not lead to such neat interpretations

as in the Reissner-Nordström case, since in terms of the dual microscopic counting it is

more convenient to regard the NHEK solution as being quotients of AdS3, as we will see.

In fact, the ranges and conditions derived for the global coordinates that select region II

of AdS2 also restrict the mixing term. Nevertheless, we can still state that subtleties in

the limit remain and that there is a spacetime arising in between the degenerate horizons

that is not found in the extremal black hole solution and that covers part of NHEK.

5For completion, the change from (7.23) to the Poincaré form of NHEK (7.15) is

cosχ =1− (t2 − 1)r2

2rtanhψ =

1− (t2 + 1)r2

2tr2ϕ = φ+ ln

√(1 + tr)2 − r2

(1− tr)2 − r2

93


In [35], the appearance of the compactication solution at extremality was used to

propose an explanation to the discrepancy in the entropy resulting from dierent counting

methods. Besides dual microscopic counting and string theory counting, another way to

determine the entropy is using semi-classical methods6. However, these seem to lead

to a vanishing entropy at extremality, as opposed to the microscopic countings that

reproduce the Hawking-Bekenstein result exactly. It was suggested, for the Reissner-

Nordström black hole, that this discrepancy might be solved by assigning the vanishing

entropy computed with the semiclassical method to the extremal black hole solution and

the classical one to the compactication solution AdS2 × S2.

Their argument is based on the fact that microstate counting is done in global AdS,

which has dierent boundary conditions, and that even the near-horizon geometry of

the extremal one only approximates AdS2 × S2 locally. So from this point of view, the

microstate counting is not done for the original black hole.

In spite of accounting for the entropy discrepancy, this interpretation leads to other

problems. If extremal black holes have indeed vanishing entropy, then the area law is

automatically invalidated in this limit, since extremal black holes have a nite horizon.

Moreover, extremal black holes can become non-extremal by sending massive particles

inside. In this transition, the entropy would have to jump from zero to a nite value, and

there is no explicable origin for this sudden increase. It is also in contradiction with the

philosophy of the AdS/CFT correspondence, which precisely identies the Bekenstein-

Hawking entropy of the black hole with the one obtained microscopically from the dual

compactication solution.

After this dissertation, it seems that everything we can assure is that the compactication

solution becomes relevant when trying to identify the statistical degrees of freedom,

whether these being located on the near-horizon region of an extremal black hole or inside

the degenerate horizon. It would be disappointing though if this additional disconnected

spacetime had no role in this identication.

6The semi-classical calculation evaluates the gravitational path integral in the saddle-point

approximation of the Euclideanized action. This one receives boundary contributions, meaning

contributions from the horizon, which are the ones that yield the Bekenstein-Hawking entropy. In the

extremal black hole, however, the horizon seats at an innite proper distance away and this point is

removed from the Euclidean manifold, giving no contribution to the action and therefore yielding a

vanishing entropy.

94

CHAPTER 8

The Kerr/CFT correspondence

Finally, time has arrived to address the actual Kerr/CFT correspondence. It was

originally presented by Guica, Hartman, Song and Strominger in 2008 in [1]. Good

reviews are [37], [38]. The aim of this correspondence is to give some insight in the

quantum gravity in the vicinity of an extremal Kerr black hole. Mainly, what we are

interested in is to count Kerr quantum microstates so as to reproduce statistically the

Bekenstein-Hawking entropy. This is indeed achieved with the proposed correspondence

in [1]. The exact proposal is the existence of a duality between the NHEK geometry and

a chiral half of a 2-dimensional conformal eld theory. The authors showed that there

exist consistent boundary conditions for the NHEK geometry that allow for a Virasoro

algebra of charges. Contrary to what one would expect, this Virasoro doesn't arise from

an enhancement of the SL(2,R) factor of the exact symmetry group of NHEK but rather

from the U(1) of ∂φ. We see therefore the relevance of the rotational character of the

geometry. This rotation will also allow for a denition of a quantum vacuum and therefore

a temperature for this geometry. Having thermalized the spacetime and given the algebra

of charges, a CFT duality follows rather straightforwardly. Reading the central charge

from the algebra and using again the Cardy formula, the Bekenstein-Hawking entropy of

the extremal Kerr black hole is successfully reproduced.

Along the lines of this chapter, we will build each one of the elements required to address

this entropy computation. Obviously, there are many other important aspects besides the

search for an entropy agreement. For example, we will see how subtleties arise when trying

to dene the thermodynamical properties of the extremal Kerr. It turns out that actually,

the temperature associated to it can only be understood as an extremal limit of the

non-extremal geometry. This, among others, motivates the study of near-extremal near-

horizon geometries and the dualities that can be derived for these. Another interesting

95

Chapter 8: The Kerr/CFT correspondence

aspect is the exact content of the dual CFT. A general feature of the dualities built for

near-horizon geometries is that they only exhibit one Virasoro instead of the full algebra

associated to both left and right movers typical of exact AdS spacetimes. This poses

the question of what implications this has for the chirality of the dual CFT, whether it

should be interpreted as a chiral CFT or as a chiral half of a 2-dimensional CFT. We

will touch upon these subtleties although unfortunately without much detail.

8.1 Thermodynamics

In the previous chapter we showed that the near-horizon region of an extremal rotating

black hole is an isolated geometry and can be thought of as an independent spacetime.

Also, it is an isolated thermodynamical system. In this section, we derive a generalized

temperature Tφ for the near-horizon region. The aim is clear: we hope to reproduce the

Bekenstein-Hawking entropy of the extreme Kerr black hole in terms of the microstate

degeneracy of a CFT at nite temperature. The Hawking temperature cannot be taken to

be this temperature since it vanishes at extremality. A new eective temperature has to

be identied, which succeeds to characterize the near-horizon thermodynamics associated

to the spinning degrees of freedom. In the following development we will see how this

temperature pops up as a chemical potential that leads to a non-trivially populated Kerr

vacuum.

Let's start by writing the rst law for a non-extremal black hole

δS =1

TH(δM − ΩδJ) (8.1)

where the angular velocity is the chemical potential conjugated to the angular momen-

tum. At extremality then,M = Mext(J) = ΩextδJ . Since the entropy (7.22) only depends

on the angular momentum, one can dene the chemical potential

1

Tφ=dSextdJ

= 2π (8.2)

where we have dropped the ~ because quantum mechanically the angular momentum of

the black hole J is quantized as J = ~m.

This temperature can also be expressed as the extremality limit of (8.1) by rst

introducing extremal variations of the mass

1

Tφ= lim

TH→0

Ωext − Ω

TH= − ∂Ω/∂r+

∂TH/∂r+|r+=M (8.3)

96

8.1 Thermodynamics

The role of this chemical potential becomes clear in the context of the quantum eld

theory living in the surroundings of the extremal Kerr black hole. In general, quantum

elds around a black hole are in a thermal state, that is a density matrix with Boltzmann

weight factor 1. This quantum matter is expected to interact with the degrees of freedom

relevant for the statistical description of the black hole, i.e. the extreme Kerr microstates,

since they are supposed to be on the horizon. Because of this interaction, the temperature

of the quantum eld theory around the horizon can be identied with the temperature

describing the ensemble of Kerr microstates.

An important step when building a quantum eld theory is to nd the appropriate

vacuum. However, dening a vacuum for the Kerr black hole becomes problematic. This

is because of the lack of a global timelike Killing vector, a general issue when it comes to

dening quantum eld theories in curved spacetimes. A good introduction to the topic

can be found in [41], here we just briey build some intuition on the diculties of dening

the vacuum state.

In at spacetime, the most general solution to the Klein-Gordon equation follows from

doing a linear combination of a complete orthonormal set of modes, such as plane waves

uk = exp(ikx ± iwt). The choice for these modes is natural because plane waves are

eigenvectors of ∂t, a global timelike Killing vector of Minkowski spacetime. We talk about

positive or negative frequency modes depending on whether the eigenvalue is positive

or negative. Any scalar eld, then, can be written as an expansion of planewaves, the

coecients of the expansion being the creation and annihilation operators a†k, ak that

become quantized through the equal-time commutation relations. It is straightforward

then to dene the vacuum as the state that is annihilated by all annihilation operators.

In the same way, any other state has got an unambiguous number of particles determined

by the action of the number operator N = a†kak.

In curved spacetime, the procedure should be the same. We would like to nd a complete

orthonormal set of solutions to the Klein-Gordon equation, which in curved spacetime

acquires an extra term proportional to the the Ricci scalar becoming (−m2−ξR)φ = 0.

However, in a general curved spacetime there is no global timelike Killing vector, so there

is no natural way to select a complete set of modes with which to expand the scalar elds.

Therefore, in general dierent sets of modes uk, uk can be used for expanding the

solutions, leading to dierent sets of creation and annihilation operators. The dierent

sets of modes can be related through the so-called Bogoliubov transformations, which

translate (through equating the scalar eld expansions in both sets of modes) into

1A good reference for generalities of the density matrix formalism is [39] and for quantum eld theories

in black hole backgrounds is [40].

97


transformations of the creation and annihilation operators associated to each set as

ak =∑q

(α∗kqaq − β∗kqa†q).

These transformations mix the creation and annihilation of both sets of modes. This

has got important implications, namely that ambiguity appears in the denition of

the vacuum. Two dierent vacua's |0〉, |0〉 can be dened as being annihilated by

the two sets of annihilation operators ak, ak. However, because of the Bogoliubov

transformations, they will not be annihilated by the other set of annihilation operators.

So the vacuum states become many-particle states from the point of view of another

mode expansion.

Going back to our case, the Kerr geometry has no global timelike Killing vector.

Therefore, globally there is no quantum state with all the desired properties of a vacuum.

Frolov and Thorne, though, dened a vacuum [42] using the generator of the horizon

Q = ∂t + Ωext∂φ. This Killing vector, becomes timelike just outside the horizon at any

value of θ, until the so-called speed of light surface, where an observer must move at the

speed of light in order to corotate with the black hole (it is not the same as the stationary

limit surface limiting the ergosphere)2.

Then, quantum states are expanded in the eigenstates of the horizon generator, the

positive frequency modes being dened as e−iwt+imφ. For example, one may write

Φ =∑w,m,l

φwmle−iwt+imφfl(r, θ) (8.4)

These modes are the natural generalization from the ones of at spacetime to spacetimes

with rotation. As mentioned, quantum elds around a black hole are in a thermal state.

If we want to generalize the Boltzmann factor e−w/TH to a rotating black hole, we have

to add angular momentum as e−(w−mΩ)/TH . When taking the extremal limit making

TH → 0, it would seem the Boltzmann factor becomes trivial. However, states with

w = mΩext contribute to the density matrix. These states are precisely the ones that

correspond to nite energy excitations in the near-horizon region. Let's see this. When

approaching extremality, one can perform the change of coordinates (7.14) in order to

zoom close to the horizon. A scalar eld, by denition, transforms as Φ(xµ) = Φ(xµ)

under a coordinate change. This translates into the eigenstates as

e−iwt+imφ = e−iwt+imφ (8.5)

2This Frolov-Thorne is the analog of the Hartle-Hawking vacuum [ref], the vacuum dened in the

region outside the horizon of a Schwarzschild black hole. A Schwarzschild black hole does have a globally

timelike Killing vector, namely ∂t, and the vacuum outside the horizon is a density matrix ρ = e−w/TH

at the Hawking temperature TH .

98

8.1 Thermodynamics

and the near-horizon parameters become

m = m, w = Ωext(wλ+ m) (8.6)

Finite energies w in the near-horizon region at extremality correspond then to eigenstates

which at the asymptotic region satisfy w = mΩext. For the same reason, the charge

associated to the horizon generator Q vanishes at extremality. Q becomes after the near-

horizon change of coordinates (7.14) Q = λ/2M∂t. Since global charges are linear in

the excitation, QQ = λ/2MQ∂t . At λ → 0, nite energy excitations of the near-horizon

geometry amount to no net charges in the asymptotic region.

Finally, introducing the condition w = mΩext in the Boltzmann factor and taking the

extremality limit, one nds

exp(−w −mΩ

TH) = exp(−m(Ωext − Ω)

TH)→ exp(−m

Tφ) (8.7)

where in the second step we have used the denition (8.3). Therefore, despite a vanishing

Hawking temperature, the vacuum is not trivially populated at extremality. So the

temperature Tφ can be thought of as an eective temperature to explain the non-trivial

population of the Frolov-Thorne vacuum, a generalized temperature of the near-horizon

region.

It is worth noticing that the near-horizon limit in this development is actually not so

relevant. It was only used to associate the eigenstates contributing to the density matrix

of the vacuum as nite energy excitations from the near-horizon region point of view,

at extremality. Other than that, the thermodynamical properties were derived for the

Kerr black hole, using the timelike vector expressed in the Kerr coordinates. However,

the region where this quantum eld theory is dened, from the horizon up to the speed

of light surface, is close enough to the horizon.

An important remark must be made at this point. It turns out that the speed of light

surface approaches asymptotically the horizon at extremality. As mentioned, for non-

extremal black holes the horizon-generator is timelike just outside the horizon. This can

be seen by replacing r = r+ + ε in the norm of the vector. To rst order, the norm is

negative for all values of θ. At extremality though, the norm also vanishes at rst order

but at second order becomes |Q|2 = ε2(sin4 θ+ 8 sin2 θ− 4)/4Σ. This norm becomes null

at sin2 θ0 = 2√

5 − 4. For θ0 < θ < π − θ0 the norm is positive, therefore the horizon

generator becomes spacelike around the equator. This obviously poses diculties for

the Frolov-Thorne vacuum to be well dened. Out of extremality, instabilities disappear

due to the existence of a conserved energy related to the timelike Killing vector. At

extremality, there is no such conserved quantity and therefore one would in principle

99


assume the appearance of instabilities. We will not enter into details of this issue, the

reader is referred to [38] where it is explained why instabilities eventually do not appear,

allowing to dene the vacuum. In any case, the Frolov-Thorne temperature Tφ can be

always understood as the extremal limit of a non-extremal temperature, and this is how

it is formally regarded.

Finally, there is still another way we can see this temperature popping up. We have seen

that when thermalizing a spacetime, the temperature can be read of from the periodicity

of the conjugate coordinate to the Hamiltonian. In this case, the horizon generator has

acquired the role of the time translation operator and the Hamiltonian has incorporated

the rotation term accordingly. Therefore, as appearing in the Boltzmann factor of the

partition function, the temperature follows from the inverse of the periodicity of Q =

∂t + Ωext∂φ which is 2π because φ ≡ φ+ 2π.

8.2 Virasoro algebra and central charge

Boundary conditions and ASG

In [1], the proposed boundary conditions for the deviation hµν from the NHEK

background metric on the boundary y =∞ werehττ = O(y2) hτφ = O(1) hτθ = O(1/y) hτy = O(1/y2)

hφφ = O(1) hφθ = O(1/y) hφy = O(1/y)

hθθ = O(1/y) hθy = O(1/y2)

hyy = O(1/y3).

(8.8)

The deviations hττ and hφφ are of the same order as the leading terms in NHEK (7.19),

so this case diers from the AdS3 boundary conditions presented in section 4.3, where

all deviations are subleading. Since NHEK is not asymptotically AdS, these boundary

conditions have to be rather motivated from the dynamics of the theory. From a physically

intuitive point of view, the two main features that were required to satisfy a priori were

the existence of one copy of the Virasoro algebra in the ASG and the reection of the

corotating degrees of freedom described by a non-vanishing Tφ.

The most general dieomorphisms that preserve the boundary conditions (8.8) are of the

form

ξε = (−yε′(φ) +O(1))∂y + (C +O(1

y3))∂τ + (ε(φ) +O(

1

y2))∂φ +O(

1

y)∂θ,

where ε(φ) is an arbitrary smooth function and C is an arbitrary constant. The subleading

terms indicated above represent pure gauge transformations, trivial dieomorphisms.

100

8.2 Virasoro algebra and central charge

That is, deformations whose components only have these subleading terms have

vanishing generator. Now, we can separate the global time component from the above

dieomorphisms to obtain

ζε = ε(φ)∂φ − yε′(φ)∂y.

The appearance of this ∂φ and φ dependence respond to the will of describing corotational

degrees of freedom. Since φ ∼ φ + 2π, we can Fourier expand ε(φ) (as was done for the

asymptotic Killing vectors (4.23) in the AdS3 case) doing εn(φ) = −e−inφ. Denotingζn ≡ ζ(εn), the Lie brackets of these vector elds satisfy the Witt algebra

i[ζn, ζm] = (n−m)ζn+m. (8.9)

Now, it has to be noticed that, as opposed to the AdS3 case, this algebra is an

enhancement of the U(1) factor of the isometry group of NHEK and not of the SL(2,R) as

would be expected by analogy. In the case of AdS3 there is a one-to-one correspondence

between the exact Killing vectors and the asymptotic Killing vectors with n = 0,±1,

meaning that they satisfy the same algebra and that they have the same dependence in

t+ φ. In the NHEK case, the exact Killing vectors ξ0,±1 satisfy the sl(2,R) algebra and

so does the subgroup ζ0, ζ±1 of (8.9). However, ζ0 = ∂φ is the generator of the U(1),

and therefore the algebra (8.9) extends from the rotational factor. The two sets of Killing

vectors are totally unrelated, due to the separation of the τ component in the general

expression of the dieomorphism. Actually, in the original paper the authors mention

that the choice of boundary conditions was initially motivated by the assumption that a

non-trivial Virasoro algebra would exist with a zero mode proportional to ∂φ. Non-trivial

meaning that the Virasoro algebra would extend the SL(2,R), therefore an angular zero

mode not being natural a priori. The boundary conditions they found therefore did not

meet this requirement.

The remaining dieomorphism ∂τ = ξ−1 is one of the generators of the exact SL(2,R) of

the isometry group. Also ξ0 = t∂t − r∂r preserves the boundary conditions, but not the

third generator ξ1 = ∂t. Therefore, we see that the ASG doesn't completely include the

isometry group3. Both ξ−1 and ξ0 are pure gauge, though, the former is so by the choice

of the extremality condition as we explain in next section.

Algebra of charges and central charge

The ansatz (8.8) has to be validated by checking if the charges associated to the ASG

preserving these boundary conditions are nite, well-dened and conserved. In [1], it

3A question that is still not clear is whether an enhancement of the boundary conditions could

be found such that all SL(2,R generators could act on the geometry or even have non-zero charges

associated. So far no such solution has been found.

101


was shown that the Virasoro generators are nite and well-dened around the NHEK

background. This follows from inspection of the integrands of the variations of the

charges. It was shown in [43] that the Virasoro generators are conserved and well-dened

around any asymptotic solution given that one additionally regularizes the charges using

counter-term methods. Therefore, up to some technical details, it can be claimed that

the ansatz (8.8) is a consistent choice of boundary conditions.

Regarding the charge Q∂τ , this one not only has to be nite but we also want it to vanish.

This charge δQ∂τ ≡ ER measures the deviation from extremality M2 − J of the black

hole. Since we want to narrow down to extremal black holes, we want to restrict our ASG

to the subspace in which ER = 0. This restriction is compatible with having a subspace

of solutions associated to ζε since these asymptotic Killing vectors commute with ∂τ .

That's why the ∂τ excitation becomes trivial.

We will not go into more details about the exact expressions of the charges and the

proofs that they are well-dened, nite and conserved. We move on to discuss the more

interesting algebra of charges. As already extensively shown in chapter 4, the surface

charges, under the Dirac bracket, obey the same algebra as the asymptotic symmetries,

up to a possible central extension. If the charges associated to the ζn are Qn, the algebra

of charges is then

Qn, Qm = Q[ζn,ζm] + Cζn,ζm .

Let's recall that the central term Cn,m can be computed trough the charge associated

to ζn evaluated on the Lie variation of the background metric along ζm, as presented

in (4.9). Introducing the NHEK geometry (7.19) and the boundary conditions (8.8), the

central term is

Cn,m = −i(n3 + 2n)δn+mJ.

We are interested in reading the central charge from the central term identifying this withc

12n(n2−1)δn,m. According to the correspondence principle in semi-classical quantization,

Dirac brackets translate into commutators of quantum operators as ·, · → − i~ [·, ·].

According to this rule, the central term in the algebra acquires a factor of 1/~ when

operator eigenvalues are expressed in units of ~. To explicitly read o the central charge

c the following dimensionless quantum charges have to be dened

~Ln ≡ Qn +3J

2δn.

The quantum charge algebra is then

[Ln, Lm] = (n−m)Ln+m +J

~n(n2 − 1)δn+m.

102

8.3 The Dual CFT

The central charge reads

cL =12J

~. (8.10)

The central charge therefore depends on the angular parameter of the black hole4. In

the Brown-Henneaux central charge (4.28), the dependence on the parameters of the

black hole whose near-horizon geometry exhibits the AdS3 factor comes through the

AdS length l, which depends on these parameters explicitly. The space of solutions with

ER = 0 in NHEK with the boundary conditions (8.8) must form representations of one

copy of the Virasoro algebra with central charge 12J/~, in this sense a duality with a

chiral CFT can be drawn: there exists a correspondence between quantum gravity around

the near horizon Kerr geometry and a chiral 2d CFT. NHEK is then associated with a

thermal state of the chiral 2d CFT at a temperature Tφ = 1/2π. The interest of this

correspondence is that, as opposed to the examples presented before, this 2d CFT arises

from the limit of an observable astronomical system such as a 4-dimensional rotating

black hole.

8.3 The Dual CFT

An important feature that the CFT dual to NHEK inherits from its ASG is that one

chiral sector is frozen, taken to be the right chiral one. In the CFT language, this means

that states are not charged under the right-moving Virasoro generators. This translates

into a vanishing right central charge, and thermodynamically also into a vanishing right

temperature. This establishes an important dierence between the case of NHEK and

that of exact-AdS3 gravity, where both sectors are excited, the right and left central

charge are the same c = c = 3l2G and there are left and right temperatures describing

degrees of freedom moving in both directions.

This turns out to be a general feature of the gravity/CFT dualities established for the

near-horizon geometries of extremal black holes. This was shown in [30], where it was

argued that the near-horizon limit performed for the extremal black hole acts in the dual

CFT as a chiral limit that freezes one sector.

4If we were to rewrite the integral for the central charge as it would follow from (4.9), we would see

that this in fact reads

cL =3

~GN

∫ π

0

dθ2a2Ω2(θ)Λ(θ).

From this we learn that, in more general cases, the central charge can be read from the θ-dependent

charges of the near-horizon geometry. For black holes with electric and magnetic charges, these will also

appear in the near-horizon geometry and therefore in the central charge.

103


Let's see this for the BTZ black hole. We showed in 5.2.1 that BTZ black holes are dual

to thermal ensembles in the dual 2-dimensional CFT. To thermodynamically describe its

left and right-moving degrees of freedom, a left and right temperatures can be dened in

the fashion we showed in 2.3 for a general CFT. For the BTZ, the modular parameter

was dened in (4.18) as τ = −|r−|+ ir+. The BTZ temperatures result into

TL =r− + r+

4πl, TR =

r+ − r−4πl

. (8.11)

Its energy and angular momentum were shown to be

L0 −c

24= M − J, L0 −

c

24= M + J (8.12)

Recalling the expressions for the two horizons of the BTZ (4.11), this black hole is

extremal when M = J . Therefore we see how the right movers are in the ground

state, with TR = 0 and L0 = c/24, while the left moving temperature and L0 are

arbitrary.

Back to the general case, thermal ensembles in a single chiral sector of the CFT are

then dual to extremal black holes. NHEK is also associated to a thermal state of the

chiral CFT, however it is not completely analogous to the extremal BTZ but to its

near-horizon geometry. Doing the extremality limit on the BTZ we have put the right

temperature to zero and the right movers to the ground state, but we still have the whole

SL(2,R)×SL(2,R) isometry group of the extremal BTZ acting as a global geometry on

the boundary CFT. Having TR = 0 doesn't mean that the right sector is not populated,

in the same way that TH = 0 for extremal Kerr didn't lead to a trivial Boltzmann

factor for the vacuum. One can now take the near-horizon geometry of the extremal

BTZ black hole. This consists of an S1 bration over AdS2 (called a self dual orbifold

of AdS3) [30]. In the case of NHEK, the same geometry is found at θ-slices but in

a warped fashion due to the θ-dependence of the bration radius. Both near-horizon

geometries have isometry group SL(2,R)×U(1). In [30], consistent boundary conditions

were presented for the near-horizon geometry of the extremal BTZ such that the U(1)

enhances to a Virasoro algebra. Therefore a dual 2-dimensional CFT exists for the near-

horizon geometry of extremal BTZ, in a completely analogous way as for NHEK in the

Kerr/CFT correspondence. The near-horizon limit for the BTZ, breaks the SL(2,R)Lto the U(1). The remaining SL(2,R)R isometry factor now acts trivially on the space

of physical states in the boundary, as opposed to the case of the extremal black hole

geometry, where the whole isometry group acts on the boundary.

This also has implications for the vacuum of the theory. There is no natural SL(2,R)×SL(2,R) invariant geometry in the boundary conditions of NHEK which could be dual

to the vacuum state of the dual CFT. As well, for the BTZ case, the gravity dual to

104

8.4 Entropy matching

the vacuum state (global AdS3) does not belong to the phase space dened in the near-

horizon limit of the extremal black hole, which has the M = J = 0 BTZ as ground state

(see 5.3). It is not understood yet why there is no natural SL(2,R)× SL(2,R) invariant

geometry in gravity at all that is dual to the vacuum state.

Therefore, the dual CFT to NHEK has to be understood as a chiral half of a 2-dimensional

CFT (rather than a chiral CFT, which is a holomorphically factorized CFT), reached

after a limit that freezes one sector 5 on which SL(2,R) does not act as a global symmetry.

It has to be noticed that, so presented, the extremal BTZ geometry does not necessarily

follow from a near-horizon limit of an asymptotic geometry as NHEK does, so there is

no analog for the extremal Kerr black hole in the BTZ example.

Despite its warped AdS3 factors, its chiral half 2-dimensional CFT and its analogy

with the BTZ/CFT correspondence, NHEK still diers in certain aspects to exact AdS3

gravity. First of all, we already mentioned that the boundaries of NHEK are eectively

those of AdS2, so eectively 1-dimensional. Moreover, the cylinder picture typical of

AdS3 wherein excitations occur satisfying the appropriate boundary conditions does not

really follow for NHEK. In the AdS3 background, matter excitations can be consistently

implemented, with matter elds that appropriately decay at innity. Massless excitations

can seat close to the boundary and are described with the stress tensor of the dual CFT

living in the boundary. And also black holes can be found in the cylinder, like the BTZ.

This cannot be done for NHEK. Intuitively, if matter is sent very close to the horizon, it

will fall into the black hole and bring it out of extremality, which would automatically

destroy the geometry. As for massless excitations, since the boundaries are 1-dimensional

their behavior is more similar to massless excitations in AdS2.

8.4 Entropy matching

The goal now is to reproduce the entropy of extremal Kerr using the Cardy formula for

the dual CFT. First of all, we have to identify a temperature for the excited states. In

section 8.1 we derived the temperature Tφ = 1/2π of the scalar quantum elds living in

the QFT in the near-horizon region of the extremal Kerr black hole. This temperature,

the chemical potential associated to the angular momentum, is to be identied with the

rotational degrees of freedom of the black hole itself due to the thermalization of the

near-horizon region, the quantum elds around the black hole and its microstates are

interacting. Now, since we identify the excitations along ∂φ with the excited states of the

5In the original paper of the Kerr/CFT correspondence, two temperatures TL and TR were presented,

as following from the Boltzmann factor of the Frolov-Thorne vacuum. However a geometrical derivation

is not presented (i.e. in terms of a modular parameter), and their expression seems to follow only from

the wish of having one vanishing at extremality.

105


left sector of the dual CFT, whose right SL(2,R)R sector is frozen, then these excited

states are described by a thermal density matrix with temperatures

TL = Tφ, TR = 0.

We can now use the Cardy formula in its canonical form (2.39), which again reads S =π2

3 (ceff TL+ ceff TR), to compute the entropy of the CFT. The TL from our CFT is indeed

the dimensionless temperature that enters the formula as pointed out in 2.3. Using the

central charge (8.10) and TL, the entropy found is

SCFT =2πJ

~≡ SextKerr (8.13)

which exactly agrees with the Bekenstein-Hawking entropy found for the extremal Kerr

black hole (7.22).

However, this is surprising mainly because the value of the temperature TL does not fall

into the 'high temperature' regime of applicability of the Cardy formula, which would

require TL 1. In [37] it was argued that a sucient condition for the applicability

of Cardy is that the temperature is large relative to the energy gap of the spectrum of

excitations. This implies that a large number of degrees of freedom are excited. The gap

for the extremal Kerr black hole is known to be small thanks to semiclassical reasoning:

if the gap were to be suciently large, the Hawking calculation of the radiation of the

black hole wouldn't be valid, since the black hole cannot radiate with energies less than

the gap.

Alternatively, one can think that the agreement of these two entropies is a hint that the

Cardy formula has a larger range of applicability. The approach then should be to take

it for granted and try to understand why this is so. Also, we do not know the exact

properties of the CFT. Therefore we cannot know if it also satises the requirements of

the Cardy formula, such as modular invariance or unitarity (i.e. c > 0). Acquiring more

insight on the content of the dual CFT would help understand why the Cardy formula

succeeds.

106

CHAPTER 9

Cveti£-Youm

In this last chapter we are going to compute the near-horizon geometry for the extremal

Cveti£-Youm black hole. The Cveti£-Youm black hole is a charged and rotating 5-

dimensional solution [44]. Its electrically-neutral limit is the Myers-Perry black hole [45].

The solution is described by 4 charges: the mass M , two angular momenta Jφ, Jψ and

the electric charge Qe. Since this is the 5-dimensional brother of the Kerr black hole, one

can hope that its near-horizon geometry at extremality reduces to the NHEK geometry

after an appropriate compactication. Having a 5-dimensional model of NHEK would be

interesting since dualities for 5-dimensional black holes are better understood. This is

the main motivation for doing this computation.

The Cveti£-Youm metric is presented in terms of the parameters m, l1, l2, s, c where s =

sinh δ, c = cosh δ. The charges in terms of the parameters are

M = (π

4G5)3m(1 + 2s2)

Jφ = (π

2G5)m(l1c

3 − l2s3)

Jψ = (π

2G5)m(l2c

3 − l1s3)

Qe =1

4π

∫S3∞

(F − F ∧A√3

) = −2√

3πmcs (9.1)

The metric is expressed in the coordinates t, r, φ, ψ, θ the rst one being timelike, the

second radial and the three last ones being angular directions. The metric has got three

commuting Killing vectors K = ∂t, R = ∂φ, P = ∂ψ.

The steps to follow are rst to nd the special values of the charges at which the black

hole is extremal. Secondly nd the angular velocity at extremality. And nally nd the

107

Chapter 9: Cveti£-Youm

near-horizon metric. We will nd that there are two branches of extremality, meaning

two sets of extremality conditions such that the only way to go from the near horizon

geometry of one branch to the other is by rst going back to the original geometry and

then impose the second extremality condition. These two branches are the non-BPS and

the BPS branch, the names will be justied at due time.

9.1 Non-BPS branch

The extremality condition follows normally from the double poles of grr. The values of

r for which grr diverges are the radius of the horizons. These coincide at extremality

leading to a double pole. This is how we can nd what is the condition the parameters

have to obey to fulll extremality.

The zeros of the denominator of grr are found at

r2± =

1

2(2m− l21 − l22)±

√(2m− l21 − l22)2 − 4l21l

22

Both solutions are positive, so they are both solutions of r2. At extremality r+ = r−, so

the condition for the parameters is:

2m = (l1 ± l2)2 ∀δ

The solution with the minus sign can actually be disregarded because it yields a complex

value for the horizons r2ext = −l1l2 (we can take both l1 and l2 to be positive without

loss of generality).

Let's now translate the extremality condition to the charges. We have to nd expressions

that relate the four charges without the parameters. Since M and Qe do not depend on

the angular parameters, they relate to each other through

M2 =3

16G25

(3π2m2 +Q2e). (9.2)

If we sum the angular momenta charges in (9.1), at extremality we get an expression

which does not depend on the angular parameters

Jφ + Jψ =πm3/2

√2G5

(c3 − s3).

Now, working out the factor (c3−s3) from the previous equation writing s, c as a function

of m,M , the relation between the charges at extremality is

(aφ + aψ)2 =25

33πG5M −

1

2 · 3πQ2e

G5M− 1

42 · 33/2π

Q3e

(G5M)2

108

9.1 Non-BPS branch

where ai ≡ Ji/M is the angular momentum per unit mass1.

The case δ = 0 corresponds to reproducing the extremal 5-d Myers-Perry black hole

M =3πm

4G5

Jφ + Jψ =πm3/2

√2G5

→ (Jφ + Jψ)2 =25G5

33πM3.

Regarding the angular velocities of the black hole at extremality, two of them Ωφ and Ωψ

have to be computed (since the rotation group in 4 dimensions is SO(4) = SO(3)⊗SO(3),

so there are two casimirs and so two planes of rotation). The way to compute them is

through the vanishing of the norm of the horizon generator ξ = ∂t + Ωφ∂φ + Ωψ∂ψ on

the horizon. More precisely, it holds that gµνξν = 0, which yields two equations for the

velocities. They turn out to coincide and to be equal to

Ωφ = Ωψ =1

(l1 + l2)(c3 − s3).

We now proceed to compute the near-horizon geometry. The proposed change of

coordinates is

r = r+ + r0ερ =√l1l2 + r0ερ, τ =

ε

r0t

φ = φ+ Ωt, ψ = ψ + Ωt

where ε is the near-horizon parameter and the angular velocity is the one at extremality.

The new coordinates are dimensionless. The introduction of the constant r0 is justied

so that we can impose the nal geometry to be of the form

ds2 = Γ(θ)[−ρ2dτ2 +dρ2

ρ2+ α2(θ)dθ2 +

∑ij

γij(θ)(dφi + kiρdτ)(dφj + kjρdτ)]. (9.3)

where φ1 = φ and φ2 = ψ. This is the general form that the near-horizon geometry of

a stationary, extremal 5-dimensional black hole exhibits [46]. This is a generalization of

the form in 4 dimensions and follows from symmetry arguments, more precisely from

assuming two U(1) axial symmetries and the already familiar SL(2,R). The aim is

1The units convenient for general relativity belong to a length system called geometrized units, in

which G4 = 1 and c = 1. Mass, length and time all have units of mass. The dimensions of GD · ρm are

always the same in any dimension D. In 5 dimensions then [G5] = M . The parameters then m, l1, l2 have

dimensions [m] = M2, [li] = M and of course [s] = [c] = 1.

109


therefore to nd the metric functions Γ(θ), α(θ), γij(θ), ki. The entropy of the extremal

black hole depends on the rst three of these functions according to

S =1

4GN~

∫Σvol(Σ) =

π2

GN~

∫dθΓ(θ)3/2α(θ)

√γ11(θ)γ22(θ)− γ12(θ)2.

where θ ∈ [0, 2π).

After performing the change of coordinates and taking the near-horizon limit, the metric

functions result in

Γ(θ) =Σ0 + 2ms2

4, α = 2

γ11 =4(1− x2)

(l2 + (−l2 + l1)x2 + (l2 + l1)s2)3(l1 + (l2 + l1)s2)(l2 + (l1 − l2)x2 + (l2 + l1)s2)2+

+(1− x2)[−2l2l1(l2 + l1)c3s3 + l21c2(l2 + (l1 − l2)x2 − (l2 + l1)s4)−

−l22s2(l2 + (l1 − l2)x2 + (l2 + l1)s2(2 + s2))]

γ22 =4x2

(l2 + (−l2 + l1)x2 + (l2 + l1)s2)3l2l1(−l2 + l1)x4 + (l2 + (l2 + l1)s2)3−

x2[−l22(2l1 + l2(−2 + c2)) + l2(4l22 − 3l2)s2 + 2l2l1(l2 + l1)c3s3

+l22(l2 + l1)(2 + c2)s4 + l2(l2 + l1)s6]

γ12 =4x2(1− x2)

(l2 + (−l2 + l1)x2 + (l2 + l1)s2)3l2l1(l2 − l2x2 + l1x

2)

−(l2 + l1)s3[(l22 + l21)c3 + l2l1(2 + c2 + s2)s]

where Σ0 = l1l2+l21 cos2 θ+l22 sin2 θ and we have relabeled x = cos θ to shorten the results.

The expressions for k1, k2 and r0 are not presented because at the moment of nishing

this thesis, no satisfactory results were found. From the system of equations that follows

for these functions is obvious that a solution exists for them. The relevant conclusion is

that such functions exist and can be solved analytically, so there is indeed a solution of

the form (9.3). Since the entropy can be computed with the γij functions this result is

enough to proceed further with computations of the thermodynamical functions.

We can analyze this geometry in more detail and compare it with NHEK. First of all,

this geometry also has the AdS2 factor explicitly. This factor is again warped with the

two S1 of the angular coordinates. In NHEK however, the functions Ki were equal to

one. Again this geometry would be a quotient of this warped products because of the

angular coordinates having φi ∼ φi + 2π. At θ = 0, γ11 = γ12 = 0 but γ22 doesn't vanish.

Therefore, as opposed to NHEK even at this axis there is warping between the AdS2 and

one S1. The same happens at θ = π/2 where γ22 = γ12 = 0 but γ11 6= 0. This is also

dierent from the NHEK case, where Λ(θ) (7.16) only vanished in the rotational axis.

110

9.2 BPS branch

At other slices of constant θ, the radius of the warping will be bigger or smaller than

1 depending on the values of the parameters c, s, l1, l2, and also stretched and squashed

products will be found.

The entropy can be computed with the above functions. Although solution was found

for the general case, it is not compact at all, which suggests that simplications have to

be found. However, for the case δ = 0, which reduces to the Myers-Perry black hole, the

entropy is computed very easily to give

Sδ=0 =π2

2G5~√l1l2(l2 + l1)2. (9.4)

This result agrees with the extremal entropy of the Myers-Perry black hole found in the

literature2 [47].

9.2 BPS branch

The second extremality branch corresponds to doing δ → ∞ and m, l1, l2 → 0. These

conditions yield nite values for the charges. ForM to be nite and since δ →∞, then m

needs to go to 0 such that the product ms2 is nite. From the expressions of the angular

momentum charges follows that lis also has to be nite.

We can now justify the name of the branches. We saw that the relation between the mass

M and charge Qe of the black hole is given by the expression (9.2). If we put m→ 0, the

relation becomes M = (√

3/4G5)Qe. This is called the BPS bound and it is satised by

the so-called BPS black holes, which appear in the context of supersymmetric theories.

All supersymmetric black holes are extremal but this doesn't hold the other way around.

The fact that BPS black holes saturate the bound (meaning that other extremal black

holes will satisfy M > (√

3/4G5)Qe) is a sign of enhanced supersymmetry. We are not

going to give any insight about the nice properties of BPS black holes, the reader is

referred to [8].

For this branch, the charges at extremality in terms of the new parameters a ≡ ms2, b ≡

2The explicit expression in the reference given diers from the result found here from numerical

factors that we believe come from the dierent normalizations of the charges of the black hole.

111


s(l1 + l2) are

M =3π

2G5a

Qe = −2√

3πa

Jφ =πa

2G5b

Jψ = − πa

2G5b

If we apply this extremality condition on the metric and we perform the change of

coordinates r2 → r2 − 2a, the metric becomes

ds2 = −(1− 2a

r2)2dt2 +

1

(1− 2ar2 )2

dr2 + r2dθ2 − 4ab

r2(1− 2a

r2)(sin2 θdtdφ+ cos2 θdtdψ)+

sin2 θ(r2 − 4a2b2 sin2 θ

r4)dφ2 + cos2 θ(r2 − 4a2b2 cos2 θ

r4)dψ2 − 2

4a2b2 cos2 θ sin2 θ

r4dφdψ.

At extremality, the horizon is located at r =√

2a and both angular velocities vanish.

Therefore, there is no ergoregion and the horizon generator is globally timelike outside

the horizon. This facilitates the computation of the near-horizon geometry. The change

of coordinates is

r =√

2a+ r0ερ

t =r0

ετ

Again we introduce r0 so that we can impose the desired form (9.3) to the metric.

After the change of coordinates and having taken the limit ε → 0, the results are the

following

Γ = a/2 α2 = 4

γ11(θ) = 4 sin2 θ − 2b2 sin4 θ

a, γ22(θ) = 4 cos2 θ − 2b2 cos4 θ

a, γ12(θ) = −2b2 cos2 θ sin2 θ

a

In this case the results are much simpler than for the previous branch. We see that the

functions Γ and α do not depend on θ. Again, at θ = 0, γ11 = γ12 = 0 while γ22 6= 0

in general (it would vanish if 2a = b2). At θ = π/2 the opposite would happen just as

before. At other θ-slices, warped products of AdS2 with the two S1 are found, which will

be stretched or squashed. The special values of the angles at which the γij become 1 are

in this case very easy to compute, these are at

sin2 θ∗ =a±

√a2 − ab2/2b2

for γ11, cos2 θ∗ =a±

√a2 − ab2/2b2

for γ22,

sin2 θ∗, cos2 θ∗ =1±

√1 + a/2b2

2for γ12.

112

9.2 BPS branch

We can now proceed to compute the entropy of this extremal black hole using this near-

horizon geometry

S =π2a

G5~

√2a− b2.

The dependence on√

2a− b2 suggests an upper bound for the angular parameters

determined by the mass, which makes sense given the intuition that we have from the

extremality bound and which in the previous branch was 2m = (l1 + l2)2 (although recall

that this entropy is already for the extremal black hole).

113

CHAPTER 10

Conclusions

The Kerr/CFT correspondence establishes a duality between the microstates living in

the near-horizon geometry of an extremal Kerr black hole and a chiral half of a 1+1-

dimensional CFT. The NHEK geometry becomes warped AdS3 at θ-constant slices. Its

isometry group is SL(2,R)×U(1), from which the rst factor (corresponding to the AdS2

symmetries) acts trivially on the states of the boundary and the U(1) enhances to one

copy of the Virasoro algebra with central charge c = 12J/~. This reects the relevanceof the rotational degrees of freedom. This near-horizon geometry is thermalized with a

left-moving temperature Tφ = 1/2π. This accounts for a non-trivial Boltzmann factor in

the Frolov-Thorne vacuum dened for the near-horizon area of the Kerr black hole. This

vacuum is dened with the generator of the horizon Q = ∂t+Ω∂φ, which is timelike close

enough to the horizon. Tφ is identied with the temperature of the left-moving degrees

of freedom on the dual CFT. The right moving temperature vanishes and the right chiral

sector is frozen. Finally, the Cardy formula reproduces successfully the entropy for the

extremal Kerr black hole.

In view of this, the Kerr/CFT correspondence manages to t the basic ingredients of the

puzzle. However, we also have to recap its weaknesses. First of all, not all the isometry

group of NHEK belongs to the asymptotic symmetry group. One of the generators of the

SL(2,R) doesn't act on the boundary conditions, and therefore is not a global symmetry

on the boundary. This poses diculties in nding a vacuum for the dual theory, which

is already challenged by the fact that the ASG only contains one copy of the Virasoro

algebra. Secondly, the dual eld theory is not a genuine 2-dimensional CFT. Also, the

Frolov-Thorne vacuum cannot be dened at extremality because of the horizon generator

not being globally timelike. Therefore Tφ, despite having a natural interpretation, can

only be regarded as an unreachable extremal limit of a non-extremal temperature.

115

Chapter 10: Conclusions

Regarding the Cardy formula, this is used for a Tφ ∼ 1, which is outside its applicability

regime of high temperature. A better understanding of why it succeeds to reproduce

the macroscopic entropy would probably help understand better the microstructure and

thermodynamics of the black hole. And last but not least, NHEK does not allow for

matter excitations, which make the geometry unstable. This challenges the identication

of the content of the dual CFT, since in exact AdS3, the masses of matter elds have a

clear CFT interpretation as the conformal dimension of the dual operators.

Regarding the computations of the near-horizon geometry of the extremal Cveti£-Youm

black hole, it seems that this consists of warped products of an AdS2 factor with two S1 at

θ-slices for both branches at extremality. For the rst branch we obtain satisfactory results

for the metric functions Γ, α, γij and with these the entropy for the neutrally-charged case

can be computed. A compact result is found that agrees with the results in the literature.

However, the result for the other metric functions was not found yet. Also, nding a

compact result for the entropy in the general case can probably be done without much

diculties. For the second branch, we also only found partial results regarding the metric

functions although the entropy could be computed for the general case. Having entropy

results is interesting from the point of view of the possible holographic correspondences

for these extremal black holes since eventually they can allow to check the agreement

with the degeneracy of states of a dual CFT.

The very next step in this computation would be to nd the missing metric functions

in order to have the complete geometries. Once all the metric functions are found, it

would be interesting to nd a compactication of one of the angular coordinates to see if

NHEK can be reproduced. This should come from the rst branch of the extremal black

hole with δ = 0. Having the general result for the electrically charged black hole would

allow to generalize the compactication to the Kerr-Newman black hole, the electrically-

charged generalization of Kerr for which a correspondence has also been proposed (see

[38]). Another interesting analysis to do is the isometries of the geometries found here

and to check if there is an easy identication with the symmetries of NHEK.

116

APPENDIX A

Hamiltonian formulation of

General Relativity

In this appendix we will present the Hamiltonian formulation of gravity using the

Arnowitt-Deser-Misner (ADM) formalism. This formulation is necessary to dene the

surface charges that become important in chapter 4. As we will see, these charges

appear as the surface terms that the Hamiltonian requires in order to have a well-dened

variational principle. The main goal of this appendix is to reach the gravity Hamiltonian,

but for that we will need to introduce some other objects.

In this formalism, spacetime is foliated into an innite stack of constant time hypersur-

faces, the Hamiltonian implementing the evolution from one to another. The Hamiltonian

is also useful for nding generalized notions of conserved quantities, such as the ADM

mass and angular momentum.

The content presented in this appendix can be found in [48], although we follow the

shorter presentation done in [49].

Preliminaries

In the formulation we will introduce, we will be dealing with time-constant hypersurfaces

of a spacetime M . To deal with these hypersurfaces, it is necessary to introduce some

objects that distinguish between the degrees of freedom of the embedding space and

those intrinsic to the hypersurface. In the following we will use Greek indices to indicate

spacetime coordinates µ, ν, ... = 1, ..., d + 1, Roman indices from the beginning of the

alphabet to indicate coordinates on the hypersurfaces a, b, ... = 1, ..., d in the constant-

time foliation. Since we will have a second (embedded) foliation, we will use for this

Roman indices from the middle of the alphabet i, j, ... = 1, ..., d− 1.

117

Appendix A: Hamiltonian formulation of General Relativity

A hypersurface can be written in two equivalent ways: as a constraint Φ(xµ) = ctt or as

a set of parametric equations xµ = xµ(ya). The constraint is often used to get the unit

vector nµ normal to the hypersurface, whereas the parametric relations give the vectors

eµa tangent to the hypersurface

nµ = N∂µΦ(xµ), eµa =∂xµ

∂ya.

The constant N , called the lapse function, is obtained from the normalization condition

gµνnµnν = −1 and is N = |gµν∂µΦ∂νΦ|−1/2. The tangent vectors do not have unit

length in general and are always orthogonal to the normal vector, nµeµa = 0. The tangent

vectors can be used to project, pull-back1, a (d+1)-tensor onto the hypersurface, turning

it into a d-tensor. The most basic example of this is the induced metric qab, which is the

pull-back of the metric itself onto the hypersurface qab = eµaeνbgµν . The two metrics are

related through the completeness relation

gµν = qabeµaeνb − nµnν .

All quantities intrinsic to the hypersurface (like Christoel symbols, Riemann tensor,

Ricci tensor, etc.), can be obtained from qab in a way that is similar to how we calculate

quantities from the full spacetime metric. To relate the intrinsic quantities to the full-

spacetime ones we need information on how the hypersurface is embedded into the full

spacetime. This is described by the extrinsic curvature, which amounts for the change in

qab as one moves in the normal direction and is given by the Lie derivative of the metric

pulled back onto the hypersurface

Kab =1

2eµae

νbLngµν = eµae

νb∇(µnν).

The extrinsic curvature becomes important when writing down the action from which

the Einstein equations are derived through the variational principle, which is

S = SEH + SGH + Smatter =1

2κ

∫Mdd+1x

√g(R− 2Λ) +

1

κ

∮∂M

ddy√qK + Smatter.

where κ = 8πG. The Bulk term corresponds to the Einstein-Hilbert action SEH . This

term is enough to derive the eld equations of general relativity when the spacetime

1Consider two manifolds M and N , with coordinate systems xµ and yα respectively, a map φ : M →N ,a function f : N → R and a vector V (p) of TpM . Naively, the pullback of f by φ, φ∗f , is simply

the composition φ∗f = f φ and the pushforward vector φ∗V at the point φ(p) on N is given by its

action (φ∗V )(f) = V (φ∗f). Rewriting the vectors in the last expression in the coordinate basis of the

tangent spaces of the two manifolds and using the chain rule, the pushforward operator can be expressed

as (φ∗)αµ = ∂yα

∂xµ. The pullback operator then becomes (φ∗)µ

α = ∂yα

∂xµ, which generalizes to covariant

tensors as (φ∗T )µ1...µl = ∂yα1

∂xµ1... ∂y

αl

∂xµlTα1...αl .

118

manifold doesn't have a boundary. If the manifold has a boundary ∂M , then the action

has to be supplemented with a boundary term so that the variational principle is well-

dened. The boundary integral SGH is called the Gibbons-Hawking term and contains

the trace of the extrinsic curvature K = qabKab. This term allows the integrand to be a

well dened Lagrangian density because the geometry of the boundary encoded in the

induced metric qab is taken xed when performing the variational principle. Then, the

variation of this term cancels the boundary terms arising from the variation of the bulk

piece.

Temporal foliation

In the ADM picture, spacetime M is foliated into a stack of constant-time hypersurfaces

Σt, in other words M = R× Σt, see gure A. We can dene a scalar function Φ(xµ) = t

which describes every hypersurface through Φ = const. Here t becomes the foliation

parameter and the parametric relations xµ = xµ(t, ya) depend now on d+ 1 parameters

instead of d. Besides the normal and tangent vectors (which are now dened at t xed)

to the hypersurface Σt, we can dene another vector, known as ow vector

tµ =∂xµ

∂t|yafixed,

which points in the direction of increasing time and is normalized according to tµ∂µΦ = 1

(notice the use of t for both the foliation parameter and the ow vector t = tµ∂µ. It is

important to realize that tµ does not necessarily point in the same direction as the unit

normal nµ, we may write

tµ = Nnµ +Naeµa .

The tangent piece Na is called the shift function. This separation of the normal and

tangent components is called the ADM decomposition.

We now present the notation we will use in the following for the dierent embeddings

and their quantities, which can be seen in gure A.

ADM foliation manifold metric intr. cuv. extr. curv. normal tangent

full spacetime M gµν Rµνκλboundary Σ qab Kab nµ eµaspatial hypersurface Σt qab Rabcd Kab nµ eµahypersurf. boundary ∂Σt γij kij ra eai

119


Figure A.1 Temporal foliation of the n-dimensional spacetime (n − 3 dimensions

are suppressed in the gures). The relevant hypersurfaces and their normal vectors are

represented. (Figure extracted from [49].)

The gravitational Lagrangian

Now, we will try to write the Einstein-Hilbert action in a way that the Lagrangian

can be read o depending only on Afterwards, we will compute the Hamiltonian by

doing a Legendre transform on the Lagrangian, which replaces the generalized velocity

by the canonical momentum. First then, we must specify the canonical variables in

the gravitational theory. The gravitational equivalent of a point-particle position is the

induced metric qab on the constant-time hypersurface Σt. We dene the generalized

velocity qab as the Lie derivative along the ow vector t as

qab := Ltqab = eµaeνbLtgµν .

The last equality holds because the tangent vectors can be parallel-transported along tµ,

so they commute with the Lie derivative. Notice then that qab resembles the extrinsic

curvature Kab = 12eµaeνbLngµν , with the derivative taken along t instead of n. However,

these two are related through t = Nn+Naea, so we can write

Ltgµν = ∇µtν +∇νtµ = (∇µN)nν +N(∇µnν) +∇µ(Naeaν) + (µ↔ ν).

Now, using that the covariant derivative ∇a on the hypersurface Σt is the pullback of

the full-spacetime covariant derivative and using the orthogonality between the normal

and tangent vectors, we can write

qab = 2NKab +∇aNb +∇bNa. (A.1)

120

We can now proceed to write the Einstein-Hilbert action. Since we now know the exact

relation between the generalized velocity and the extrinsic curvature, it will be enough

the express the action in terms of the latter. In the Einstein-Hilbert action, the Riemann

tensor contains both canonical quantities qab and Kab. It would be necessary then to

express the integrand of this action explicitly in terms ofKab so that later we can perform

the Legendre transform. For this, we will use the so-called Gauss-Codazzi equations,

which relate the intrinsic Riemann tensor to the full spacetime one, splitting it into

intrinsic and extrinsic quantities. We will not derive this equations here, more about

them can be found in [48]. The (contracted) Gauss-Codazzi equation reads

R = R− (K2 −KabKab)− 2∇κ(nλ∇λnκ − nκ∇λnλ).

We introduce this expression in the Einstein-Hilbert action to obtain

2κSEH =

∫M

dd+1x√g(R− 2Λ)

=

∫M

dd+1x√g(R+K2 −KabKab − 2Λ) + 2

∮∂M

(nλ∇λnκ − nκ∇λnλ)dσκ.

The boundary ∂M can be split into the three pieces shown in gure A as ∂M = Σ−∞ ∪Σ∞ ∪ Σ. For each of these pieces, the surface elements become

dσκ = −nκ√qddy on Σ−∞

dσκ = nκ√qddy on Σ∞

dσκ = nκ√qddy on Σ. (A.2)

Now, we use the following tricks: 2nκ∇λnκ = ∇λ(nκnκ) = 0, K = qabkab = ∇λnλ and

nκnκ = 0. Introducing this into the boundary term

2

∮∂M

(nλ∇λnκ − nκ∇λnλ)dσκ = (

∫Σ−∞

−∫

Σ∞

)ddy√q2K − 2

∫Σ

ddy√qnκnλ∇λnκ. (A.3)

where we have integrated by parts in the last term. Now, all these surface terms should

be related to the Gibbons-Hawking term, which we must also split into the three pieces

of the boundary and use the same surface elements (A.2). For the term at t = −∞, we

have to take into account that the normal vector is dened inwards. An extra minus sign

results since K = ∇λnλoutward = −∇λnλ. The Gibbons-Hawking term becomes

2κSGH = (−∫

Σ−∞

+

∫Σ∞

)ddy√q2K +

∫Σ

ddy√q2K.

121


where Kab = eµa eνb∇µnν . Adding this Gibbons-Hawking term to the boundary term (A.3)

of the Einstein-Hilbert action makes all of the unhatted terms cancel. The integrand of

the remaining Σ integral is K − nµnν∇µnν = k2.

In this way, we obtain the action

Sgrav = SEH + SGH =1

2κ

∫M

dd+1x√g(R+K2 −KabKab − 2Λ) +

1

κ

∫Σ

ddy√qk.

The Lagrangian density obtained from this integrand is still well-dened despite the

boundary term because the embedded extrinsic curvature k is not a canonical variable.

Then, the gravitational Lagrangian is dened through writing the action as an integral

over the foliation parameter t Sgrav =∫∞−∞ dtLgrav

Lgrav =1

2κ

∫Σt

ddyN√q(R+K2 −KabKab − 2Λ) +

1

κ

∮∂Σt

dd−1θN√γk.

where we used√g = N

√q and

√q = N

√γ. Having now the Lagrangian written in

canonical form, we can perform the Legendre transform.

The gravitational Hamiltonian

The Legendre transformation in the gravitational case generalizes directly from the point-

particle's case

Hgrav(q, p) =

∫Σt

ddy√qpabqab − Lgrav(q, q), with pab =

1√q

δLgravδqab

.

To compute pab, the derivative has to be with respect to the extrinsic curvature on which

the Lagrangian depends explicitly. From (A.1), we can write

δ

δqab=

1

2N

δ

δKab

with which the canonical momentum results

pab =1

2κ(Kab − qabK).

2The intermediate steps: K − nµnν∇µnν = (qabeµa eνb − nµnν)∇µnν = (gµν − nµnν − nµnν)∇µnν =

(qabeµaeνb − nµnν)∇µnν = qabeµae

νb∇µnν = qab∇arb = (γijeai e

bj + rarb)∇arb = γijeai e

bj∇arb = k. In the

5th step we have used that nµ = eµara.

122

Next we can see that the gravitational Hamiltonian will also separate into a bulk term

and a boundary term. Substituting the expression of the momentum, it reads

Hgrav =

∫Σt

ddy√q(NH+NaHa) +

∮∂Σt

dd−1θ√γ(NHbdy +NaHbdya ).

The so-called Hamilton and momentum constraint functions are respectively

H = −R− 2Λ

2κ+ 2κ(pabpab −

1

d− 1p2) and Ha = −2∇bpab,

and the boundary terms are

Hbdy =k

κand Hbdya = 2rbpab. (A.4)

The equations of motion consist of the Hamilton constraint H = 0 and the momentum

constraints Ha = 0 together with the Hamilton evolution equations qab = δHgrav/δpab

and pab = −δHgrav/δqab.

The variation of the above boundary terms cancel all boundary terms coming from the

variation of the bulk piece when one assumes δN = δNa = δqab = 0. In section 4.1, we

actually relax this condition to allow for surface charges.

123

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Documents

The Kerr/CFT correspondenceChapter 1: Introduction hole with rotation, i.e. the Kerr black hole, and a conformal eld theory. The main motivation to study such a correspondence is that