The Kerr/CFT correspondence

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The Kerr/CFT correspondence

Teresa Bautista Thesis for the

Master's in Theoretical Physics Supervised by

Georey Compère, Erik Verlinde

University of Amsterdam ITFA

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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.

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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

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Chapter 1: Introduction

true for non-spherical spinning black holes.

• First Law: a balance equation is satised, namely dM = κ

8πGdA+µdQ+ΩdJ, where

Mis 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 = ~κ

2π.

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.

1_{As 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.

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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/CF T2

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

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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 AdS3spacetimes, 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.

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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 → x0 that leave the metric invariant up to a rescaling factor

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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}0µ _{= x}µ₊

µ and imposes that the relation (2.1) is satised. In this way, one nds the following dierential equation for the vectors µ

Lηµν(x) =

2

d(∂ · )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, µ _{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 1

2(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 µ _{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_{= dzd¯}_z1_.

Assuming that the innitesimal transformations (z), ¯(¯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

1_{This 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.

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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] = 2¯l0

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∂}

rand −∂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) = 1

2(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) = Lτ σ

l0+ ¯l0 = −τ ∂τ − σ∂σ = −iD

l1+ ¯l1 = −2τ σ∂τ− (τ2+ σ2)∂σ = −iKσ l1− ¯l1= −(τ2+ σ2)∂τ− 2τ σ∂σ = iKτ

These generators obey the following brackets

[D, Pµ] = iPµ [D, Kµ] = −iKµ

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

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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}

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

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2.2 Conformal Field Theory

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) → Φ0(z, ¯z) = ∂f ∂z h ∂ ¯f ∂ ¯z ¯h Φ(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)dzh_d¯_zh¯ _{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 h N Y i=1 Φi(zi, ¯zi)i = N Y i=1 ∂f ∂z hi z=zi ∂ ¯f ∂ ¯z ¯hi ¯ z=¯zi h N Y j=1 Φj(f (zj), ¯f (¯zj))i (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

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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}−2πi¯ζ/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_2πLe2πinζ/L∂ζ, ¯ln= −i_2πLe−2πin¯ζ/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 δΦ = [Q, Φ]. The stress-energy

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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 Tz ¯z = 0and the divergenceless property implies Tzz =

T (z), Tz ¯¯z= ¯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 ₌

Tµνν. 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_{= T}µ

νxν). One can also

think of it in the following way: because T (z) is conserved, then T (z)(z) is also conserved, for every holomorphic function (z). Therefore, as we will see, the theory acquires an innite set of conserved charges Qn≡ Ln(associated to every vector n= −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

Q = 1 2πi I dz(z)T (z), Q¯ = 1 2πi I d¯z¯(¯z) ¯T (¯z). (2.10)

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

commutator. δ,¯Φ(w, ¯w) = [Q+ Q¯, Φ(w, ¯w)] = 1 2πi[ I dz(z)T (z), Φ(w, ¯w)] + 1 2πi[ I d¯z¯(¯z) ¯T (¯z), Φ(w, ¯w)]

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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

δΦ(w, ¯w) = 1 2πi( I |z|>|w| − I |z|<|w| )(dz(z)R(T (z)Φ(w, ¯w))) = 1 2πi I w dz(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∂(w)Φ(w, ¯w) + (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∂w¯Φ(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 n = −zn+1, as we have briey

mentioned before. These charges, Ln= 1 2πi I dzzn+1T (z), L¯n= 1 2πi I d¯z ¯zn+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

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operators and are called mode expansions. For the stress-energy tensor T (z) =X n∈Z z−n−2Ln, T (¯¯ z) = X n∈Z ¯ z−n−2L¯n. (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) =X

n∈Z

φnz−n−h. (2.16)

When going to the cylinder, using the transformation rule (2.8), the mode expansion becomes Φcyl(ζ) = X n∈Z φne−niζ = X 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) δ,¯Φ(w, ¯w) = (h∂(w) +

(w)∂)Φ(w, ¯w) with (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 2π_L.

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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 T T 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 form

On

(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] = ( I dz 2πi I dw 2πi− I dw 2πi I dz 2πi)z n+1_{T (z)w}m+1_{T (w) =} = I 0 dw 2πi I w dz 2πiz n+1_wm+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,

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also for the antiholomorphic part, is an innite dimensional algebra, called the Virasoro algebra [Ln, Lm] = (n − m)Ln+m+ c 12n(n 2_{− 1)δ} n+m,0 [ ¯Ln, ¯Lm] = (n − m) ¯Ln+m+ ¯ c 12n(n 2_{− 1)δ} n+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

δT (z) = 1 2πi I z dw(w)T (w)T (z) = c 12∂ 3 z(z) + 2∂z(z)T (z) + (z)∂zT (z) (2.20)

This transformation can be integrated to a nite one as T (z) = (∂z 0 ∂z) 2_T0_(z0_{) +} c 12{z 0_{, z},} _where _{{f (z), z} =} f000 f0 − 3 2( f00 f0) 2 _(2.21) where T0 _{≡ T}

z0_z0 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 hT (z)T (w)i = c/2(z − w)4_.

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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}−2πi¯ζ/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

Tcyl(ζ) = (2π L)

2_[−z2_Tplane_{(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 = Z dσTτ τ = − Z dσ(Tζζ+ ¯T_{ζ ¯}¯_ζ). (2.23)

Supposing that the ground state energy vanishes when the theory is dened on the plane, i.e. hTplane_{i = 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.

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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 hT r(T )i 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

T r(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

|ψi that is an eigenstate of L0 and ¯L0 as

L0|ψi = h|ψi, L¯0|ψi = ¯h|ψi.

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|ψi = (LnL0− nLn)|ψi = (h − n)Ln|ψi.

This tells us that Lnare 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

4_{This 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.

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initial primary state |ψi, the tower starts |ψi L−1|ψi

L2₋₁|ψi, L−2|ψi

L3₋₁|ψi, L−1L−2|ψi, L−3|ψi

...

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 |0i 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) =

Z 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] = Z Dφi Z 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

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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] =

Z φ_f=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 |Oi 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|Oi = I _dz 2πiz n+1_{T (z)O(z = 0) =} I _dz 2πiz n+1₍hO z2 + ∂O z + ...),

where the path integral is implicit, we can see the eect of various generators on the state |Oi:

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

the state |Oi is a primary. From here the bijective correspondence.

• L₀|Oi = h|Oi, which establishes that the conformal weight of the operator coincides with the energy eigenvalue of the corresponding primary state.

• L−1|Oi = |∂Oi. 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 = Tτ τ = −(T (ζ) + ¯T ( ¯ζ)) = ( 2π L) 2X n (Lne−2πinζ/L+ ¯Lne−2πin¯ζ/L).

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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|ψi|2 = hψ|L+1L−1|ψi =

hψ|[L+1, L−1]|ψi = 2hhψ|ψi ≥ 0. The only state with h = 0 is the vacuum.

• c > 0. This follows from the norm |L−n|0i|2= h0|[LnL−n]|0i = ₁₂cn(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_{= dzd¯}_z_.

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+ τ σ2, z = σ¯ 1 + ¯τ σ2 the identications

become σ1 ∼ σ1+ 1, σ2∼ σ2+ 1. The line element becomes ds2 = |dσ1+ τ dσ2|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 → τ

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2.3 The Cardy formula

of these two transformations generate the whole group of modular transformations. A common set of generators is given by T and T−1_XT−1 _{≡ S : τ → −1/τ}_{. The most}

general transformation can be parametrized by τ0 = 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 = P SL(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 = T r[e−βH] = Z dqhq, β|q, 0iE = Z [dq]Pe−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 = T r[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}

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

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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 = 2π

L(L cyl 0 + ¯L cyl 0 )and P = 2π L(L cyl 0 − ¯L cyl 0 ), 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)

Tcyl(ζ) = (2π L) 2_[−z2_{T (z) +} c 24] = −( 2π L) 2_[X 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), Tcyl_{(ζ) = −(}2π

L) 2P

n∈ZL cyl

n e−2πinζ/L and it follows that6

Lcyl_n = Ln−

c 24δn,0. The partition function then becomes

Z(τ, ¯τ ) = T r[e−βHeiσP] = T r[e2πiτ Lcyl0 _e−2πi¯τ ¯L cyl

0 _{] = T r[q}L0−c/24_q_¯L¯0−¯c/24_] (2.28)

where q ≡ e2πiτ_{, ¯}_{q ≡ e}−2πi¯τ_.

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 Z0(τ, ¯τ ) = T r[qL0_q_¯L¯0_{] =} ∞ X nR,nL=0 ρ(nR, nL)qnR+hq¯nL+¯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 Z0_{(τ, ¯}_{τ )} _{by contour integration}

ρ(nR, nL) = 1 (2πi)2 I _dq qnR+h+1 d¯q ¯ qnL+¯h+1 Z0(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) =

6_{The 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.

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ρ(nR) ¯ρ(nL) which translates into the factorization of the partition function Z0(τ, ¯τ ) =

Z0(τ ) ¯Z0(¯τ ). Therefore, we can write ρ(nR) = 1 2πi I _dq qnR+h+1Z 0_{(τ ),} _ρ(n_¯ L) = 1 2πi I _d¯_q ¯ qnL+¯h+1 ¯ Z0(¯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 Z0_{, ¯}_Z0 _{are not modular invariant. However, one can}

notice that Z0_{(τ ) = q}c/24_{Z(τ )} _{and the same for the antichiral sector. We can now use}

modular invariance under the transformation S to write

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and introduce it to (2.30) ρ(n) =

I

dτ e−2πiτ (n+h)e2πicτ /24e2πic/24τZ0(−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 H 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 f0_(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∗) − |f00(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 ) = Z0(−1/τ ) =Xρ(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∗ ≈ ip_24nc ≡ i. Substituting this

into g(τ)

g(i) = Z0(i/) =Xρ(n)e−2π(n+h)/.

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

˜

Z0(τ ) = q−hZ0(τ ) =Xρ(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 ) = ˜Z0(−1/τ ) =Xρ(n)e−2πin/τ.

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

q

(c−24h)n

6 . The degeneracy of states results in

ρ(n) = ρ(0)e2π

q

(c−24h)n 6 _.

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)−−−→ en→∞ 2π q (c−24h)n 6 _{= e}2π q ceff n 6 (2.32)

and the same follows for the antichiral degeneracy. The notation of cef f 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(r cef fnR

6 +

r ¯cef fnL

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.

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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/T7_{. 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−1and ˜σ ∈ [0, 1). For convenience we redene the partition function Z(β) = T r[e−βH_{] = T r[e}−L ˜βH_{] ≡ 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 = −π2_c/6L _{(restricting to the chiral sector). The partition function at low temperatures}

becomes

Z( ˜β) ≈ eπ2c ˜β/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

˜ σ0 = 1 ˜ βσ,˜ ˜ t0 = 1 ˜ β ˜ t.

The timelike coordinate acquires periodicity ˜σ0 _{∈ [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) ≈ e

π2c

6 ˜β _. (2.38)

7_{We 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

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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 ≈ − π 2_c 6L ˜β2 = − π2c 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 = π

2

3 c ˜T .

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 cef f 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

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then as 1/TL= β + σand 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 = π

2

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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 = −dX₀2+ n−1 X i=1 dX_i2− dX_n2 (3.1)

by the hyperboloid equation

− X2 0 + n−1 X i=1 X_i2− X2 n= −l2 (3.2)

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

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Chapter 3: Anti-de Sitter spacetime coordinates (t, u, xi_{), dened by} X0 = u 2[1 + 1 u2(l 2_{− t}2₊ n−2 X i=1 (xi)2)] Xn = lt u Xi = lxi u , i = 1, ..., n − 2 Xn−1 = u 2[1 − 1 u2(l 2_{+ t}2₋ n−2 X i=1 (xi)2)]

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

ds2 = l

2

u2(−dt

2_{+ du}2_{+ d~}_x2_). _(3.3)

A related metric that we will encounter later on is the one obtained doing r = 1/u ds2 = l2(dr

2

r2 + r

2_(−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; X i Ω2_i = 1 (3.5)

which lead to the metric

ds2 = l2(− cosh2ρdτ2+ dρ2+ sinh2ρdΩ2_n−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 _{' l}2_(−dτ2 _{+ dρ}2 _{+ ρ}2_dΩ2₎_{, 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+ dy

2

(1 + y2₎ + y 2_dΩ2

The Kerr/CFT correspondence