modular invariance and 3d gravity alvaro v¶eliz osorio · modular invariance and 3d gravity alvaro...

Modular invariance and 3D Gravity

Alvaro Veliz Osorio

Supervisor: Dr. Stefan Vandoren

July 20, 2008

2

Abstract

We begin by discussing the basic features of pure 2+1 dimensional gravity. Then

we set the value of the cosmological constant to be negative, regime in which

the BTZ black hole and the Brown-Henneaux excitations arise. We discuss the

general geometric framework for these theories and find non-classical restrictions

on the value of the cosmological constant. Following [hep-th/07120155], we

attempt to build the modular invariant partition functions of the CFT dual of

pure 2+1 gravity. We do this by summing classical geometries over the orbits

of the group Γ∞ \ PSL(2,Z). It turns out that the result obtained cannot be

interpreted as the partition function of a conformal field theory. Subsequently

we discuss the proposal made in [hep-th/07063359] in which the CFT dual is

identified with an Extremal CFT (ECFT) with central charge c = 24k. For

the case c = 24, the resulting CFT is the so-called Monster Module M whose

partition function is known to be the J-function. Afterwards, we show how to

compute the partition functions for k ∈ Z using only the constraints imposed

by modular invariance and holomorphic factorization. Then we show that the

entropies derived from these partition functions are in good agreement with

the Bekenstein-Hawking entropy of the BTZ black hole. The result, although

appealing, is still based on rather speculative grounds, since the existence of

c > 24 ECFT’s is still uncertain.

Contents

1 Introduction 5

2 Gravity in 2+1 dimensions 11

2.1 The ADM decomposition and global charges . . . . . . . . . . . 11

2.2 The BTZ Black Hole . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Chern-Simons theory and 3D Gravity . . . . . . . . . . . . . . . 18

2.4 Brown-Henneaux excitations and the Cardy formula . . . . . . . 20

2.5 Quantization of the Cosmological Constant . . . . . . . . . . . . 22

3 AdS3 Geometry 25

3.1 Geometric structures and AdS . . . . . . . . . . . . . . . . . . . 25

3.2 Black holes as quotients . . . . . . . . . . . . . . . . . . . . . . 27

3.3 AdS/CFT Correspondence in a nutshell . . . . . . . . . . . . . . 31

4 Quantum Gravity partition functions 33

4.1 Schottky uniformization . . . . . . . . . . . . . . . . . . . . . . 34

4.2 Semiclassical approximation . . . . . . . . . . . . . . . . . . . . 36

4.3 Quantum Corrections . . . . . . . . . . . . . . . . . . . . . . . . 39

4.4 Modular invariant partition functions . . . . . . . . . . . . . . . 42

4.5 Einsenstein and Poincare sums . . . . . . . . . . . . . . . . . . . 45

4.5.1 ζ-function regularization . . . . . . . . . . . . . . . . . . 45

4.5.2 Double coset decomposition . . . . . . . . . . . . . . . . 46

4.5.3 Explicit summation . . . . . . . . . . . . . . . . . . . . . 49

3

4 CONTENTS

5 ECFT’s and Monstrous partition functions 55

5.1 3D Gravity as a Monster . . . . . . . . . . . . . . . . . . . . . . 56

5.2 Why Extremal CFT’s? . . . . . . . . . . . . . . . . . . . . . . . 60

5.3 More on holomorphic factorization . . . . . . . . . . . . . . . . 63

5.4 Hawking-Page is Lee-Yang . . . . . . . . . . . . . . . . . . . . . 67

5.5 A conundrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

6 Appendix A

Basic notions of 2D CFT 73

Chapter 1

Introduction

In this work we are concerned with a lower dimensional model for gravity, we

will be interested in three dimensional gravity. This model gives an interest-

ing setting to study Hawking radiation in a model in which the mathematics

become significatively simpler. In n dimensions, the phase space of general

relativity contains n(n − 3) degrees of freedom per space-time point. For ex-

ample in 4 dimensions we have 4 degrees of freedom which correspond to the

two gravitational wave polarizations and their time derivatives. On the other

hand, the theory we are treating here will posses no local degrees of freedom,

in other words it will have no propagating modes. However, we are still allowed

to perform global identifications which allow us to build classical solutions by

sewing patches of constant curvature 3-manifolds.

At first glance, we might think that we are dealing with a trivial theory,

nevertheless this theory hides some rather surprising properties which make

it extremely interesting. It is of special interest the case where we consider

a negative value of the cosmological constant. For this case, Achucarro and

Townsend [1], and subsequently Witten [51] showed that 3 dimensional gravity

is equivalent to a gauge theory called the Chern-Simons theory. This theories

are equivalent at classical level and furthermore, as we will see, also at a per-

turbative level. Nonetheless this is not the case non-perturbatively. This has

the interesting consequence that, perturvatibely, 3 dimensional gravity is renor-

5

6 CHAPTER 1. INTRODUCTION

malizable. Almost simultaneously, Brown and Henneaux [14] realized that a

careful analysis of the boundary conditions of the theory, implies the existence

of a Virasoro algebra living at conformal infinity. Even more interestingly, this

Virasoro algebra is centrally extended and the central charge depends only on

the value of the cosmological constant. We call the modes generated by this al-

gebra, Brown-Henneaux excitations. Few years later, Banados, Teitelboim and

Zanelli [10] found the possibility of having black hole solutions which resemble

in their thermodynamic behavior the four dimensional ones. Therefore, they

can be used to understand some properties of the black hole thermodynamics

in a more controlled scenery. These features make the study of 3 dimensional

quantum gravity enthralling. We could say that it is in some sense a theory

in the middle way between simplicity and complexity, and it may be the only

possibility to have a model with black holes that can be exactly solved at a

quantum level. In order to harvest enough information from the model we have

to ask the right questions, one of the most interesting ones is actually a puzzle

coming form the model itself:

If entropy is going to be interpreted in terms of the degeneracy of

microstates of an underlying theory, and we have a theory with no

local degrees of freedom: Where are these microstates hiding?

This is thesis is essentially about how can this question be answered.

In a first attempt to answer this question, it was realized that it actu-

ally intertwines two of the aforementioned properties, the BTZ black hole and

the existence of Brown-Henneaux excitations. The reason is that the bound-

ary Virasoro modes are the best candidates to accommodate the black hole

microstates. It was Strominger [47] who combined these two and obtained,

by invoking Cardy’s formula [15], an excellent match between the counting of

Brown-Henneaux modes and the predicted Bekenstein-Hawking entropy. Nev-

ertheless, this counting reveals nothing about the nature of the states that are

being counted i.e., gives no information about the underlying CFT. Nowadays

7

this results can be understood, at least heuristically, in terms of the conjec-

ture posed by Maldacena [39] that postulates a duality between a string theory

(gravity theory) defined on a manifold M , and a quantum field theory without

gravity defined on the conformal boundary of M . In the spirit of Maldacena’s

duality, we would like to be able to write the partition function of the dual CFT

in terms of a ”sum” over some class of bulk geometries, in this case, geome-

tries that are the possible interiors of an AdS3 asymptotic boundary. Chapter

4 is devoted to this calculation, which formally involves the computation of an

Euclidean Feynman integral over possible fillings of a torus whose modular pa-

rameter τ is a function of the ensemble parameters: temperature and angular

potential.

For every classical bulk geometry there is an associated contribution to the

partition function, this contribution can be can be calculated from the Einstein-

Hilbert action. However, we should keep in mind that we must take into account

the Brown-Henneaux excitations. This consideration leads to a 1-loop corrected

partition function for each geometry. Besides this, we should recall that tori

related by Modular transformations PSL(2,Z) should be regarded as equiva-

lent. Furthermore, we will find that the relevant group is actually Z\PSL(2,Z)

which is characterized by a couple of relatively prime integers (c, d). Because

of this equivalence we will be interested in enforcing modular invariance on the

partition on function. We can attain that by summing over the orbits of the

(quotiented) modular group. Therefore, for every fixed τ there is an infinite

number of classical geometries contributing to the partition function, one per

each group element in PSL(2,Z).

The calculation is rather involved and requires the introduction of a regular-

ization scheme, and of the introduction of certain techniques from the theory of

modular forms. The computation of this sum is performed in section 4.5.3 and

the main steps are presented in detail. The resulting partition function of the

sum turns out to be unphysical, and it is, up to now, still an open question to

explain why. It may be possible that there exist some unknown contributions to


the partition function. All sorts of possibilities are plausible, like cosmic strings

or complex saddle points. Another option is to consider the possibility that

maybe, a quantum theory of three dimensional pure gravity doesn’t exist.

In the third part of this work, we consider the possibility of adding complex

saddle points into the contributions to the partition function. We do this, by

demanding the partition function to be holomorphically factorizable. Further-

more, we discuss the proposal by Witten that the dual Conformal Field Theory

we are searching for, is a so-called Extremal Conformal Field Theory. Extremal

Conformal Field Theories were introduced by Hohn and they are defined in

terms of the structure of their partition function. If a c = 24k CFT has a mod-

ular invariant and holomorphically factorizable partition function, and the first

primary field different from the identity has a conformal dimension h = k + 1,

then it is an Extremal CFT. Up to the present date, whether this theories

exist for arbitrary c = 24k is still an open question. Notwithstanding, for

c = 24 a theory which follows this prescriptions is known and it turns out to be

the Frenkel-Lepowsky-Meurman (FLM) Monster Module M which carries the

symmetries of the Monster Group. The partition function of the the FLM con-

struction is the well-known Klein’s J(q) invariant, fact that shed light over the

mystery of the Monstrous Moonshine Conjecture. We explain how to construct

the partition function for arbitrary k and show that they can be determined

by demanding the first few energy levels to coincide with the Brown-Henneaux

partition function. If we interpret this primaries as BTZ black holes, then we

can compute the entropy associated with the degeneracy of these primary fields

and show that it is in good agreement with the predicted Bekenstein-Hawking

entropy.

The action of the Z \PSL(2,Z) group over the upper-half plane (where τ is

defined) induces a tessellation composed by fundamental regions of this group.

Every of these regions characterizes certain regime of temperatures and angular

potentials. At the classical level, each of these regions is dominated by a different

classical geometry hence the classical phase diagram is just the aforementioned

9

tessellation of the upper-half plane. Therefore we expect the occurrence of

phase transitions, in particular the phase transition from thermal AdS3 into a

BTZ black hole i.e. a Hawking-Page phase transition. This posses a puzzle,

since we know that phase transitions arise from some non-analytical behavior

of the partition function. However, holomorphic factorization seems to forbid

non-analyticities. Nevertheless, a theoretical framework introduced in the late

50’s by Lee and Yang deals with this situations and explains the occurrence of

phase transitions in terms of the condensation of zeroes in the phase boundaries

at the large volume limit. We can show that in fact holomorphically factorizable

have the required behavior.

The thesis is organized as follows:

1. The first part of the work (chapter 2 and 3) are devoted to the study of

the general aspects of 2+1 dimensional gravity. It is also introduced the

conceptual framework of holography and geometric structures.

2. The second part (chapter 4) is devoted to the computation of the partition

function as a sum over classical contributions.

3. The third part of the work is devoted to the study of the conjecture that

three dimensional gravity is dual to an Extremal Conformal Field Theory.

Chapter 2

Gravity in 2+1 dimensions

General relativity in 2+1 dimensions possesses some mysterious characteristics

that pose very interesting questions in a more simple setting than standard four

dimensional gravity. It is a theory with no local degrees of freedom i.e. it has no

propagating modes, but turns out to admit black hole solutions and even more

enigmatically, black holes with thermodynamical properties! Hence it is quite

natural to ask : What kind of microstates account for such behavior? Another

interesting (and intimately related) feature is the existence of a conformal field

theory, that arises from purely classical considerations, living at the boundary

of spacetime. This CFT can be used to compute the entropy of the three

dimensional black hole, but the deep connection between these objects remains

a mystery. In this chapter we intend to introduce the basic properties of the

aforementioned phenomena.

2.1 The ADM decomposition and global charges

We are concerned with the study of gravity in 2+1 dimensions, by this we

mean the study of the 3 dimensional geometrical configurations that extremize

the Einstein-Hilbert action:

IEH =1

16πG

∫

M

d3x√

g(R + Λ) + Icontent, (2.1)

11

12 CHAPTER 2. GRAVITY IN 2+1 DIMENSIONS

where Icontent is the action of the fields that may live in the spacetime. Possibly

one of the main features of this model is its lack of local degrees of freedom. This

manifests itself physically by the lack of propagating excitations and mathemat-

ically by the fact that the Riemann tensor can be completely determined by the

Ricci tensor, the reason is that the curvature is a map R : ∧2TpM×∧2Tp −→ R

while the Ricci form is a map Ric : T ∗p M ×T ∗

p M −→ R and in three dimensions

we can easily proof that there is an isomorphism between ∧2TpM and T ∗p M and

hence the curvature can be computed completely once we know the Ricci form.

Explicitly we have

Rµνρσ = gµρRνσ + gνσRµρ − gνρ − gµσRνρ − 1

2R(gµρgνσ − gµσgνρ). (2.2)

Hence the vacuum solutions for the Einstein’s equations

Rµν − 1

2gµνR + Λgµν = 0 (2.3)

are spacetimes with constant curvature Rµν = 2Λgµν and therefore the Riemann

tensor is

Rµνρσ = Λ(gµρgνσ − gνρgµσ), (2.4)

and any possible degree of freedom must be global or alternatively topological.

We proceed now to outline the hamiltonian approach to General Relativity,

the starting point to attain this is to perform a particular decomposition of the

metric, which involves the selection of a time direction, due to Arnowitt, Deser

and Minsner (ADM) [3]. Let’s take our spacetime manifold to be homeomorphic

to a cylinder, and foliate it with hypersurfaces Σt such that the theory turns

into the study of the evolution of these slices along timelike geodesics. If we

denote by qij(t) the intrinsic metric of each of the surfaces, the line element

takes the form:

ds2 = −N2dt2 + qij(dxi + N idt)(dxj + N jdt), (2.5)

where N and N i are called the lapse and shift functions respectively. The

other dynamical variable necessary for the canonical description is the extrinsic

2.1. THE ADM DECOMPOSITION AND GLOBAL CHARGES 13

curvature Kij which encodes the behavior of the surface’s embedding in the

full manifold, i.e we are interested in the evolution of the first and second

fundamental forms of the slices. In terms of the ADM decomposition the action

reads [16]:

IEH =

∫dt

∫d3x(πij∂τqij −NH−NiH

i), (2.6)

where

H =1√q(πijπij − π2)−√q((2)R− 2Λ), (2.7)

Hi = −2(2)∇jπij, (2.8)

and

πij =√

q(K ij − qijK). (2.9)

The momenta πij is obtained in the canonical way and therefore the phase space

is equipped with a Poisson structure:

qij(x), πkl(x′)PB = (δikδjl + δilδjk)δ(x− x′). (2.10)

By performing a Legendre transformation we obtain the Hamiltonian of the

system:

H =

∫d3x(NH + NiHi). (2.11)

Notice that the variables N and N i are non-dynamical and hence act solely

as Lagrange multipliers. By taking the variation of the action with respect to

them we see that the equations of motion are simply H = 0 and Hi = 0 and thus

the hamiltonian vanishes on shell, these equations are called hamiltonian and

momentum constraints respectively. Therefore we can see that 2+1 dimensional

gravity is constrained completely and depends only on the values of certain

global charges that arise due to boundary terms that will be introduced in

brief.

The advantage of this decomposition is that it allows us to employ the

techniques of Hamiltonian mechanics and gauge theory to gravity, carefully

though. From Dirac’s insights into the structure of constrained systems, we


know that first class constraints can be viewed as the generators of infinitesimal

gauge transformations, in this spirit we claim that the functional:

G[ξµ, q, π] :=

∫

Σ

d2xξµHµ, (2.12)

where Hµ = (H,Hi), can be identified as the generator of the gauge-like trans-

formations with parameter ξµ. These gauge transformations should be related

with the intrinsic symmetry of the theory, that as we know is diffeomorphism

invariance. In fact [48] the purely spatial part of the generator induces infinites-

imal Σ-diffeomorphisms in accordance with our intuition, on the other hand the

temporal part of the generator doesn’t fit into the gauge theory way of thinking

since it intertwines dynamics and symmetry. These generators turn out to form

an interesting Poisson algebraic structure, by using equations (2.7)-(2.12) we

obtain

G[ξµ1 ], G[ξµ

2 ]PB = G[ξµ3 ], (2.13)

with

ξ3 = ξi1∂iξ2 − ξi

2∂iξ1 := ξ1, ξ2SD, (2.14)

and

ξi3 = ξj

1∂jξi2 − ξj

2∂jξi1 + qij(ξ1∂jξ2 − ξ2∂jξ1), (2.15)

and we notice that ξ1, ξ2SD := ξ3 doesn’t constitute a Lie algebra for the pa-

rameters because of the q-dependence, and therefore the isomorphism ., .PB →., .SD is not a Lie algebra isomorphism.

In the previous analysis it is implicitly assumed that the surfaces Σt are

closed (i.e ∂Σt = ∅), in the general case some subtleties arise while performing

the surface integrals since boundary terms cannot be discarded carelessly. A

scrupulous analysis of the surface integrals, that may seem of purely mathemat-

ical motivation, will be of capital importance in the physical understanding of

the formalism.

When the slices of the space-time foliation have a non-empty boundary it

is necessary to supplement the hamiltonian with counterterms that tame the

2.2. THE BTZ BLACK HOLE 15

boundary variations because the equations of motion are simply ill defined [44].

Formally the equations of motion are:

dqij

dt=

δH

δπij, (2.16)

dπij

dt=

δH

δqij(2.17)

therefore there must exist Aij and Bij such that

δH =

∫dx2Aijδqij + Bijδπij. (2.18)

Regge and Teitelboim [44] showed that the bare hamiltonian cannot be trans-

formed into that form over an open foliation unless some counterterms are in-

troduced in order to cancel the boundary variations. In fact1 this can be viewed

in a more general way, the introduction of the surface counterterms makes the

Poisson algebra of the generators G[ξµ] well defined and we can regard the

hamiltonian as G[ξµ] with ξµ = (−N, N i), as a special case of the possible gen-

erators. These boundary terms represent the conserved quantities of the theory,

associated with the asymptotic deformations of the slices they are just the flux

integrals thought a circle at infinity, which quantities are obtained depends on

the symmetries of the system as we will see clearly in the next section.

2.2 The BTZ Black Hole

In spite the apparent simplicity of this theory there are remarkable features

that make it worthwhile of a deeper enquire. Some of these appear right away

in the classical framework, a noteworthy fact is the existence of vacuum black

hole solutions for a negative value of the cosmological constant, the first of these

was found in 1992 by Banados, Teitelboim and Zanelli (BTZ) [10]. We briefly

outline the solution and comment on some interesting properties of it.

Let’s assume that we have an axially symmetric and stationary spacetime,

i.e. we assume the existence of a pair of Killing vectors ∂φ and ∂t, then it is

1This was pointed out to me by S. Carlip


possible to choose the following ansatz for the metric:

ds2 = −N(r)2dt2 + f(r)2dr2 + r2(dφ−Nφdt)2. (2.19)

Then we plug this metric into the equations of motion, the momentum and the

hamiltonian constraints and aim for the explicit form of N(r), Nφ and f(r). The

problem melts down to find the solution of a fairly simple system of ordinary

differential equations, after rearranging the integration constants the solution

is:

ds2 = −N2dt2 + r2(dφ2 + Nφdt)2 +1

N2dr2, (2.20)

with

N2 = −M +r2

l2+

J2

4r2(2.21)

and

Nφ = − J

2r2, (2.22)

with domains −∞ < t < +∞ and 0 < φ ≤ 2π.

The BTZ black hole is characterized by its mass M and its angular mo-

mentum J , these quantities are the aforementioned asymptotic charges, M is

associated with asymptotic time translations and J with rotations at infinity.

Notice that the lapse function N has zeroes at the radial values

r2± =

l2

2M ±

√(M2 − J2

l2), (2.23)

the horizon of the BTZ black hole is located at r+ and it exists if and only if

M > 0 and |J | ≤ M . To see that r+ is indeed a bona fide event horizon is

convenient to perform the analysis in Eddington-Finkelstein coordinates ( see

[16] for details). Here we limit ourselves to present only the Carter-Penrose

diagram see figure 2.2. It can be shown in a variety of ways that the BTZ black

hole has interesting thermodynamical properties, in fact it behaves in close

analogy with the 3+1 black hole i.e. it obeys the 2+1 dimensional Hawking-

Bekenstein laws, in particular it has an entropy proportional to the perimeter

of the horizon, this feature poses a puzzle: What kind of microstates can give

2.2. THE BTZ BLACK HOLE 17

Figure 2.1: AdS3

rise to this behavior given the lack of local degrees of freedom? We will address

this question in detail in the future chapters for now we briefly summarize and

motivate these properties. A simple way to understand the emergence of these

phenomena without actually appealing to an underlying theory, is to study the

necessary counterterms for the euclidean version of the BTZ

IE = βM − 4πr+ + βNφ(r+)J (2.24)

and βIE is the free energy and we can therefore identify the entropy

SBTZ =2πr+

4~G, (2.25)

and then the Black hole radiates at a temperature

TBTZ =r2+ − r2

−2πr+

. (2.26)

More details will be provided when necessary.


2.3 Chern-Simons theory and 3D Gravity

As mentioned before, 2+1 dimensional Gravity has no local degrees of freedom

and therefore its only physical parameters are topological. This theory turns

out to be dual, in the classical level, to a well studied Topological Field Theory

(TFT) called Chern-Simons theory ([1] and [51]) with SO(2, 2) symmetry.

The symmetry group of AdS3 is SO(2, 2) (we will give more details in the

next chapter), the Lie algebra of this group is:

[Ja, Jb] = εcabJc (2.27)

[Ja, Pb] = εcabPc (2.28)

[Pa, Pb] =1

l2εcabJc (2.29)

We try to link the geometrical properties with of this Lie algebra by means

of a local frame (dreibein) and a Lie algebra valued connection. The gauge

field of three dimensional gravity can be written as a Lie algebra valued linear

combination

A = eaPa + ωaJa (2.30)

or in matrix notation

A =

ω e

l

− el

0

(2.31)

where ea is a local frame or driebein and ωa is a spin connection. The algebra

so(2, 2) can be decomposed into two commuting sectors with Lie algebra sl(2, R)

by choosing a a suitable combination of the generators

J±a =1

2(Ja ± lPa), (2.32)

and then we can split the gauge field into two so(2, 1) gauge fields with connec-

tions:

Aa± = ωa ± ea

l. (2.33)

2.3. CHERN-SIMONS THEORY AND 3D GRAVITY 19

By computing the Ricci scalar in terms of these connections we obtain, up to

boundary terms, the Einstein-Hilbert action in terms of the gauge potential

IEH =1

8πG

∫

M

ea ∧ (dωa +1

2εabcω

b ∧ ωc) +Λ

6εabce

a ∧ eb ∧ ec, (2.34)

which we identify as the sum of two Chern-Simons actions [51]

IEH = ICS[A+] + ICS[A−], (2.35)

ICS[A] =k

4π

∫

M

Tr[A ∧ dA +2

3A ∧ A ∧ A] (2.36)

where we defined k = l4G

, the so called level of the Chern-Simons theory. Given

the invertibility of the driebein the transformation rules of e and ω under local

Lorentz transformations and coordinate changes manifest themselves as gauge

transformations. This procedure can be done in dimensions other than 2+1,

but the Chern-Simons identification cannot be performed, because in three di-

mensions a Lorentz vector is equivalent to an antisymmetric tensor. We can see

that the Euler-Lagrange equations of motion are just

dA± +2

3A± ∧ A± = 0 (2.37)

which means that for classical configurations both connections should be flat

this implies some beautiful geometric consequences, in a gauge theory we can

write the gauge-invariant observables in terms of the holonomies of the theory

Hol(γ) associated with closed paths γ, but for flat connections the holonomies

respect homotopy classes and hence they are well defined over the fundamental

group. Under this approach one is tempted to see 2+1 dimensional gravity as

a gauge theory and then use the known techniques to quantize it, but there

is a subtlety that we have to take into account, even though the theories are

equivalent at tree level we might encounter difficulties on the quantum sector

of the theory, specially if one takes into account non-perturbative effects where

the dreiben may not be invertible.


2.4 Brown-Henneaux excitations and the Cardy

formula

In 1986 Brown and Henneaux [14] made a remarkable discovery while analyzing

the asymptotic symmetries of 3 dimensional gravity with a negative cosmologi-

cal constant, they found that there are two copies of the Virasoro algebra living

at infinity. By asymptotic symmetries we mean gauge transformations that pre-

serve the asymptotic behavior of the field configurations. What happens is that

the algebra of generators, if analyzed carefully, gives rise to an unexpected be-

havior at infinity. For Σ non-compact we pointed out the necessity to introduce

surface terms to regularize the functional derivatives. Under the same logic the

generators G[ξ] should also be supplemented with surface terms to make them

meaningful, so we redefine

G[ξµ] :=

∫

Σ

d2xξµHµ + J [ξµ]. (2.38)

Notice that the surface term is determined up to a constant by just enforcing

them to cancel the surface terms coming from variations on the bulk.

In order to determine the asymptotic symmetry group we should first fix

certain boundary conditions, these conditions cannot be obtained purely from

the structure of the theory, they should be motivated externally based on phys-

ical grounds. There is a physically reasonable demand that we can ask from

the boundary conditions: since the theory has no propagating modes the metric

at infinity should be the one of AdS3 up to global identifications because any

distribution (compactly supported) of matter is unable to affect distant regions

since there is no field that can propagate this signal. Nonetheless, it is possible

to weaken this demand to allow the asymptotic group to be O(2, 2) [14]. By

doing this Brown and Henneaux found the boundary conditions that make up a

good definition of asymptotically anti de Sitter space-time, then they found the

Killing vectors that generate transformations that preserve these conditions. In

light-cone coordinates x± = l−1t± φ the generating vector fields are

2.4. BROWN-HENNEAUX EXCITATIONS AND THE CARDY FORMULA21

ξ(±)t = lT± +l3

2r2∂2±T± + O(r−4), (2.39)

ξ(±)φ = lT± − l3

2r2∂2±T± + O(r−4), (2.40)

ξ(±)r = −r∂±T± + O(r−1). (2.41)

Subsequently we construct the gauge transformations generated by this set of

vector fields and compute their Poisson brackets to obtain

L±m, L±n PB = i(m− n)L±m+n +ic

12m(m2 − 1)δm+n,0 (2.42)

and

L±m, L∓n = 0 (2.43)

where L±m = G[ξ±n ] are the Fourier modes of the canonical generators, and the

central charge is

c =3l

2G. (2.44)

Notice that the generators of the center of the algebra are related with the

global charges by:

M =1

l(L0 + L0), (2.45)

and

J = L0 − L0. (2.46)

This is a truly remarkable result, the asymptotic algebra of 2+1 dimensional

gravity with Λ < 0 turns out to be the direct product of two Virasoro algebras

G ∼= diff(S1)× diff(S1), and they both have the same non-zero central charge

even at classical level. Hence we see that there is the a natural connection

between AdS3 and CFT2 that arises from purely geometrical considerations.

From this insight it follows that any consistent quantum theory of gravity on

AdS3 should be a Conformal Field Theory, nowadays this idea falls into the

broader realm of Maldacena’s AdS/CFT correspondence [39].

The fact that 3 dimensional gravity is a conformal field theory is of cap-

ital importance to address the thermodynamical properties of the BTZ black


hole. By counting Brown-Henneaux excitations on a BTZ background, Stro-

minger [47] obtained an expression for the entropy in precise agreement with

the Bekenstein-Hawking entropy for a large number of states. Strominger’s cal-

culation is to be valid in the semiclassical approximation that is l À G i.e.

the cosmological constant should be minute in Planck units. Cardy’s expres-

sion for the asymptotic density of states makes the computation rather simple,

for a 2-dimensional CFT with non-negative lowest eigenvalues ∆0 and ∆0 the

asymptotic density of states for a large conformal weight (∆, ∆) is given by [15]

ln(∆, ∆) ∼ 2π

√(c− 24∆0)∆

6+ 2π

√(c− 24∆0)∆

6, (2.47)

which in terms of the global charges is

S = π

√l(lM + J)

2G+ π

√l(lM − J)

2G, (2.48)

and coincides precisely with the Bekenstein-Hawking entropy for the BTZ Black

Hole. The power of the Cardy formula is that it is not necessary to specify the

nature of the fundamental excitations, but it is also it’s weakness since it doesn’t

tell us what kind of microstates we are counting leaving the nature of the dual

CFT veiled. There is a fundamental assumption in this derivation and it is that

the underlying 2+1 Quantum Gravity Theory actually exists, in the course of

this thesis we will see that up to now there is not an air tight argument to

assume this.

2.5 Quantization of the Cosmological Constant

Even though the Chern-Simons form of 3D Quantum Gravity cannot be used

non-perturbatively, it can be used to point some interesting directions for fur-

ther enquire. In fact we can impose restrictions in the possible values of the

cosmological constant. Equation (2.35) is a manifestation of the fact that the

Lie group SO(2, 2) is locally equivalent to SO(2, 1)×SO(2, 1). A general theory

2.5. QUANTIZATION OF THE COSMOLOGICAL CONSTANT 23

with SO(2, 1)× SO(2, 1) symmetry can be expressed in the form

I =kL

4π

∫

M

Tr[AL ∧ dAL +2

3AL ∧ AL ∧ AL]−

kR

4π

∫

M

Tr[AR ∧ dAR +2

3AR ∧ AR ∧ AR], (2.49)

with A± being so(2, 1) connections. For topological reasons, the levels k+ and k−

of the Chern-Simons theories take only integer values [53]. However, the group

SO(2, 1) is not simply connected, i.e. it has non-contractible loops, we can see

this simply by realizing that it can be contracted to the group of rotations SO(2)

and this in turn is isomorphic to U(1) which is topologically a circle therefore,

having then Z as its fundamental group. Hence it is possible to consider for

every n ∈ Z a n-fold cover of SO(2, 1) for which the restriction over the level

is k ∈ n−2Z, and for the universal covering ˜SO(2, 1) k ∈ R. But the situation

is a bit more subtle since we are not working only with a SO(2, 1) theory but

with a SO(2, 1)× SO(2, 1) one, and therefore we have to pay attention to non

independent coverings i.e. coverings that are not merely the product of two

coverings of SO(2, 1).

We relate now this result to three dimensional gravity leading to a quanti-

zation of the value of the cosmological constant. If we denote the actions of the

left and right moving sectors by

IL,R =1

4π

∫

M

Tr[AL,R ∧ dAL,R +2

3AL,R ∧ AL,R ∧ AL,R] (2.50)

we can decompose equation (2.49) into

I =kL + kR

2(IL − IR) +

kL − kR

2(IR + IL). (2.51)

The term IL − IR coincides with the Einstein-Hilbert action if we set

Λ =l

8G=

kL + kR

2. (2.52)

While the other term is a Chern-Simons interaction term, notice that if the

two levels are equal then this term doesn’t contribute this is the origin of the


terminology Chiral Gravity for a gravitational action composed by the Einstein-

Hilbert action and a Chern-Simons interaction term [37].

In section 2.4 we presented the value of the central charge of the boundary

conformal field theory that arises as the symmetry group of the boundary con-

ditions, Brown and Henneaux computed this value for a non-chiral theory (i.e

kL = kR), from the prevous discussion we can see that the level of the Chern-

Simons theory k := kL = kR is directly related to the value of the central charge,

explicitly k = l/16G and c = 3l/2G and therefore c = 24k. This result also

solves a conundrum that we overlooked, if l is regarded as a free real parameter

we would enter in conflict with Zomolodchikov’s c-theorem [? ]. According to

this theorem, the central charge associated with a continuous family of CFT’s

should be constant, this contradiction is avoided in the case c is quantized.

Chapter 3

AdS3 Geometry

The purpose of this chapter is twofold, on the one hand to give a look at some of

the previous results under a different light and foresee generalizations of them,

on the other hand to make the transition to the next few chapters smoothly

by introducing some mathematical structures of capital physical importance.

We will present a general geometric framework in which it is possible to embed

2+1 dimensional gravity, to do this we recall the key geometrical result of 2+1

gravity, the classical solutions have constant curvature. Therefore they must

be locally isometric to either of three spaces Minkowski M , de Sitter dS3 or

Anti-de Sitter AdS3 for zero, positive or negative values of Λ respectively.

3.1 Geometric structures and AdS

There is a very convenient framework to deal with global properties of a manifold

formed by gluing together patches of some know manifold. We summarize and

adapt to our notation some results provided in [49]. A geometric structure is

a triple (M, G, X) formed by a manifold M , a model space X and a Lie group

G that acts analytically on X, these ingredients are combined in the following

way to produce what is called a Geometric Structure, lets assume that:

1. M is locally diffeomorphic to X, i.e. M can be charted on X by an atlas

(Uα, ψα) where the functions ψα have images in X.

25

26 CHAPTER 3. ADS3 GEOMETRY

2. The transition functions ψαβ := ψα ψβ are contained in the action of G

on X.

Structures of this kind are particularly interesting when G is the isometry group

of X or a subgroup of it. The characteristic feature of the latter is that they

represent manifolds that are locally isometric to X, like the ones that represent

solutions of 2+1 dimensional gravity. Therefore is enough to specify the desired

model space and the isometry group to obtain a classical solution. Moreover a

manifold that admits a geometric structure modeled by X can be proven to be

diffeomorphic to some quotient of X by a discrete subgroup of ISO(X) [49].

A geometric structure is also equipped with a mechanism to discern between

the local behavior given by the charts and the global one that is hidden in the

transition functions. Let γ be a loop on M we would like to know whether this

arc can be covered by a single chart. Consider an open cover Ui with i = 1, ..., N

since the arc is a closed set, there will be always overlaps between the open sets

in particular we choose U1

⋂UN 6= ∅, now by enforcing the charts to coincide

on the intersections, by using constant group elements gi in G, hence we obtain

ψn = Hol(γ)ψ1 :=1∏

i=N

giψn (3.1)

if ψn is equal to ψ1 hence the extension of the coordinate path was successful,

but this is generally not the case which in turn implies that Hol(γ) measures the

failure of the desired extension extension of the chart. Moreover Hol(γ) respects

homotopy classes and hence it factors through the fundamental group of M and

we obtain a group homomorphism (highly non-unique), Hol(γ) : π1(M) −→ G.

Therefore if we take the universal cover M the map is trivial and hence we can

find a map Ψ : M −→ X that gives a complete representation of the universal

cover in terms of the model space.

Since we are concerned with the negative curvature solutions we embark now

to review some basic properties of AdS3 (or more precisely its universal cover

AdS3) which is going to play the role of our model space X, while M is going

3.2. BLACK HOLES AS QUOTIENTS 27

to represent space-time and G = SO(2, 2). The most straightforward way to

define AdS3 is via an embedding into a 4-dimensional space R2,2 with metric

ds2 = dx21 + dx2

2 − dx23 − dx2

4 (3.2)

and define AdS3 as

AdS3 = (x1, x2, x3, x4)|x21 + x2

2 − x23 − x2

4 = −l2. (3.3)

The space defined in such a way is a homogeneous space with isometry group

SO(2, 2) and isotropy group SO(1, 2) and it is possible to represent it as a

quotient, in the Erlangen approach. It is then easy to prove that the isometry

group of AdS3 is PSL(2,R) × PSL(2,R), where PSL(2,R) = SL(2,R)/± is

the modular group, and it is defined as the quotient of SL(2,Z) over its center.

We can see this more clearly in the particularly convenient representation:

µ : AdS3 −→ SL(2,R) (3.4)

µ : (x1, x2, x3, x4) 7−→ 1

l

x1 + x3 x2 + x4

x2 − x4 −x1 + x3

(3.5)

Notice that the characterizing embedding constraint (equation (3.3)) is con-

tained in the restriction to det(µ(X)) = 1, the isometry group PSL(2,R) ×PSL(2,R) act in both flanks of the matrix as µ(X) 7−→ ρLµ(X)ρR. Summing

up the facts just presented we can see that every solution to 2+1 dimensional

classical gravity must be and anti-de Sitter geometric structure i.e. the solu-

tions are made up of AdS3 patches weaved together with PSL(2,R)×PSL(2,R)

transition functions.

3.2 Black holes as quotients

Since the BTZ solution is just another classical solution of the Einstein-Hilbert

equations, it must be therefore possible to find a way of representing it under

the perspective of the previous section. In fact they turn out to be part of a


larger family of black hole solutions, a so called SL(2, Z) family of black holes.

First we rotate to euclidean AdS3 for matters of convenience, in this version

µ : (x1, x2, x3, x4) 7−→ 1

l

X1 + X3 X2 + iX4

X2 − iX4 −X1 + X3.

(3.6)

To understand how the family of black hole solutions arises we first study the

case of the normal BTZ solution. The key point is to find which identification of

AdS3 is necessary, the difference between the two metrics, as we can see from the

previous chapter, is the periodicity in the angular coordinate φ that we would

have to enforce by identifying φ with φ + 2π. This identification translates into

the action of two group elements [11]

ρL =

exp(π

l(r+ − r−) 0

0 exp(−πl(r+ − r−))

(3.7)

ρR =

exp(π

l(r+ + r−)) 0

0 exp(−πl(r+ + r−))

(3.8)

and to obtain the BTZ black hole we just make the identification µ(x) ∼ρLµ(x)ρR. Now we proceed to generalize this idea, for that we do the fol-

lowing. We can obtain the hyperbolic model in three dimensions by factorizing

the matrix µ(X) in the appropriate way by writing

µ(X) =1

lΩΞΩ†, (3.9)

where

Ω =

1 ω

0 1

, (3.10)

and

Ξ =

ξ 0

0 ξ−1

, (3.11)

which is a valid decomposition for h 6= 0, we choose h > 0 and see that the

metric becomes simply

ds2 =

(l

ξ

)2

(dω21 + dω2

2 + dξ2), (3.12)

3.2. BLACK HOLES AS QUOTIENTS 29

which is the pull-backed metric over the embedding map to the hyperbolic space

model. Lets consider now the action of a more general group element of the

isometry group on µ(x), let τ ∈ H be a parameter and notice that the matrix

ρ(τ) =

eiπτ 0

0 eiπτ

, (3.13)

can be applied on µ(x) as ρ(τ)µ(x)ρ†(τ), and this matrix also generates a sub-

group of PSL(2,R) isomorphic to Z we will call this the BTZ group action [22]

and we get then the space AdS3/〈(ρ(τ), ρ†(τ))〉 that would represent a black

hole solution. From the perspective of the hyperbolic model, we have to restrict

the action of the BTZ group to the region H∗ to make the action properly dis-

continuous after the identifications we get the following series of isomorphisms:

AdS3/〈(ρ(τ), ρ†(τ))〉 ∼= H∗/Z ∼= S1 ×D1. (3.14)

By choosing and appropriate homology basis we can proof that τ is the modular

parameter of the boundary (h = 0) [22]this in turn means that we have chosen

a contractible primitive cycle α(s).

It is important to recall that the modular parameter τ is defined only up to

the action of PSL(2,Z) apart from the evident integer translation invariance,

hence the procedure yields a family of black holes ”indexed” by the modular

parameter τ , but a careful revision is needed, since we are in a Wick rotated

space it is necessary to put back the distinction between space and time, this will

bring us back to a more physical discussion and to reveal the relationship with

thermal AdS. To understand the physical situation it is convenient to introduce

a different set of coordinates, let ρ ≥ 0 and u ∈ C and let ω = e2utanh(ρ) and

ξ = eu then

µ(X) =1

l

eu 0

0 e−u

cosh(ρ) sinh(ρ)

sinh(ρ) cosh(ρ)

eu∗ 0

0 e−u∗

, (3.15)

we can see that the u coordinate is defined up to the identification 2u ∼ 2u +

2πin. By performing the quotient over the BTZ group action we obtain the


identification 2u ∼ 2u + 2πτim which actually defines a non-contractible cycle.

We can define the spatial and temporal coordinates as

u =i

2(φ + itE), (3.16)

and the coordinate identification affects only φ

φ ∼ φ + 2πn. (3.17)

While the BTZ group action yields

φ ∼ φ + 2π<(τ)m

TE ∼ +2πm=(τ), (3.18)

and the resulting manifold has the φ cycle as the only contractible one and then

we can identify this manifold with thermal AdS3 (Figure 3.2). If on we choose

Figure 3.1: Thermal AdS3

the combination

u =i

2τ(φ + itE), (3.19)

for the time and space coordinates then the coordinate identification takes the

form

φ ∼ φ + 2π<(−1

τ

)n

TE ∼ +2π=(−1

τ

)n, (3.20)

3.3. ADS/CFT CORRESPONDENCE IN A NUTSHELL 31

On the other hand the BTZ group action yields

φ ∼ φ + 2πn. (3.21)

which allows us to identify the contractible cycle as the temporal one (see figure

5.33) and therefore we can identify the resulting handlebody as a black hole

solution.

Figure 3.2: AdS3

3.3 AdS/CFT Correspondence in a nutshell

The AdS/CFT has been an active research area in String Theory for more than

a decade, it was introduced by Maldacena in [39] as a realization of ’t Hooft’s

holographic principle [32] between type IIB String Theory on AdS5×S5 and and

a supersymmetric N = 4 Yang-Mills Theory, this correspondence can arise also

in other dimensions also, in the case at hand we could say that we are working on

an AdS3/CFT2 correspondence. In this section we recall some important ideas

of this duality, even though we will not use it directly for the future calculations

there is a deep conceptual link between these and the basic principles of the

correspondence. We will focus precisely in those common features mainly in


the relation between act of summing over bulk geometries and the properties

of the boundary topology. The AdS/CFT correspondence renders a duality

between a d-dimensional theory of Quantum Gravity and a d − 1 dimensional

field theory, this is remarkable since it gives a non-perturbative definition of

String Theory and does it in terms of a more familiar field theory. Up to now

there is a dictionary to translate objects from one side of the duality to the

other [2].

Given a CFT on certain manifold the gravity dual to this theory should con-

tain, at least, the sum of all the classical solutions whose conformal boundaries

coincide with the given manifold, we can see that this phenomenon is mani-

festing itself through the Brown-Henneaux excitations. Nevertheless we should,

at least in principle, include also other possible fields arising from String The-

ory solutions with the right asymptopia [13]. The statement of the AdS/CFT

correspondence is an equation relating the bulk and the boundary partition

functions

ZCFT (∂M ; γ) =

∫DΦexp(SString) '

∑

λ∈Λ

Zstring(Mλ), (3.22)

where the approximation is a saddle point (or steepest descent) one, Λ indexes

the classical solutions, γ is a particular conformal class, and the path integral

is over all the string fields that obey the boundary conditions. The rightmost

expression gives the partition function as a sum over classical solutions with

(∂M ; γ) as a conformal boundary. In this thesis we will focus in the pure

gravity contributions to the bulk side. The following chapters are devoted do

the computation of

ZCFT (∂M ; γ) '∑

λ∈Λ

Zstring(Mλ). (3.23)

for pure gravity.

Chapter 4

Quantum Gravity partition

functions

This chapter is devoted to the computation of the partition function of (min-

imal) three dimensional gravity with negative cosmological constant with the

ulterior objective of finding the spectrum of the theory or what is the same,

to find the CFT dual of 3D gravity. We consider the holographic prescription

and try to find the generating functional for the boundary CFT from the sum

over the interior geometries. On the semiclassical regime we compute the sum

over the contributions of classical geometries and then consider the one-loop

correction coming from the Brown-Henneaux excitations that we quantize after

obtaining a well defined phase space. Then we sum over the orbits of the modu-

lar group in order to enhance the symmetry and we obtain a partition function

that turns out to be ill defined, a fact that poses a series of interesting puzzles.

The information of the spectrum can be encoded in the partition function

Zk(β, θ) = Tr(exp(−βH − iθJ)) that will depend on two parameters, inverse

temperature β and angular potential θ because asymptotically AdS3 spaces have

two commuting conserved global charges H and J as was discussed in section

2.1. The value of this trace can be, at least formally, written in terms of the

Euclidean Feynman integral and it is the sum of the geometries that behave

33

34 CHAPTER 4. QUANTUM GRAVITY PARTITION FUNCTIONS

asymptotically like a genus one Riemann surface of modulus

τ = θ/2π + iβ, (4.1)

which in turn is conjectured to be equivalent to the genus 1 partition function of

a two dimensional CFT. We proceed in the usual way by taking the boundary

of AdS3 then performing a Wick rotation and then making the time coordinate

periodic, we obtain hence a torus whose size we ought to fix by choosing values

for β and θ.

4.1 Schottky uniformization

As discussed previously, every three-dimensional manifold that extremizes the

Einstein-Hilbert action with negative cosmological constant is an AdS3 geomet-

ric structure and hence it can be constructed from AdS3 by performing certain

appropriate identifications. Here we present a general recipe for discriminating

the possible identifications. Let us model AdS3 by the Poincare upper half-plane

model H3 with metric

ds2 =l2

w2(dx2 + dy2 + dw2). (4.2)

The one point compactification of the boundary i.e. w → 0 of H3 is given by the

Riemann sphere P1 which has as automorphism group Aut(P1) = PGL(2,C).

In fact acting by Mobius transformations

z 7−→ az + b

cz + d(4.3)

we can prove the chain of isomorphisms [35]:

Aut(P1) w PGL(2,C) w PSL(2,C) (4.4)

where PSL(2,C) is the isometry group of H3.

To construct the negative curvature space that we aim for, it is convenient

to focus on the action of subgroups of PSL(2,C) on the boundary, we will call

4.1. SCHOTTKY UNIFORMIZATION 35

D the closure of the stabilizer of this group action, if for certain subgroup Γ

we have D 6= P1 we will say that Γ is a Kleinan group. Kleinian groups are

interesting since they posses a region Ω = P1 − D in which they act properly

discontinuously, if we perform the quotient Σ = Ω/Γ we obtain a differentiable

manifold of constant negative curvature [49]. For every Kleinian subgroup we

can perform this construction this yields an immense family of manifolds of

negative curvature.

Figure 4.1: Schottky uniformization

We will focus on a particular subclass of Kleinian groups, the so called

Schottky groups. To construct a Schottky group we consider a paired set of


Jordan curves γi, γi with i = 1, ..., g embedded on P1 such that they have

disjoint interiors. If there is a set µi, i = 1, ..., g of Mobius transformations

that map the interior of γi to the exterior of γi then the group 〈µi〉 finitely

generated by these Mobius transformations is a Klein subgroup [42] and it is

called a Schottky group. These Mobius transformations are called Loxodromic

and they are characterized by two fixed points, one attractive and one repulsive.

Given the nature of the generators, the region exterior to all the Jordan

curves is a fundamental region of the group and hence the identifications will

be between the Jordan curves themselves see figure 4.1. After performing the

identifications we obtain a surface of genus g. It is possible to prove that the

moduli space of genus g Riemann surfaces [36] is a quotient of the parametrized

space of Schottky groups. By using this fact Koebe proved that every compact

Riemann surface can be built as a quotient of P1 by the action of some Schottky

group, this is the so called Schottky uniformization.

4.2 Semiclassical approximation

Now we embark into the calculation of the partition function for pure grav-

ity in three dimensions, in an schematic way the idea is to compute the Eu-

clidean Feynman integral over the possible geometries that follow the asymp-

totic boundary conditions

Z(β, θ) =

∫DΦexp(−IG), (4.5)

and where we demand the Euclidean time coordinate to be periodic. To carry

out this program we approximate the path integral around the saddle points of

the action

Z(β, θ) =∑

λ∈Λ

exp(−I(Mλ)), (4.6)

where Λ is a set that indexes the classical solutions, there is no a priori reason

to assume that this set is discrete, however we will soon show this fact. Intu-

itively we want to find the semiclassical g = 1 geometries that contribute to

4.2. SEMICLASSICAL APPROXIMATION 37

the partition function. From the analysis of the previous section we see that by

choosing z = 0 and z = ∞ as the fixed points of the loxodromic transformation,

we immediately see that the required Schottky group Γ is the one generated by

W (p) =

p 0

0 p−1

∈ PSL(2,C) (4.7)

Maloney and Witten proceeded in a different way (see [40]) in the original

computation, with the same conclusion. First we notice that Γ ' Z and then

the conformal boundary Σ ' Ω/Γ is diffeomorphic to a torus. The action of

Γ can be seen in the following way, a complex number is represented on the

Riemann sphere as z 7→ (z, 1) and then the action of an element of Γ takes

z to q2z. It is convenient to write this in complex polar coordinates in which

z = exp(2πiw) and w is defined modulo Z. In this coordinates the group action

yields the equivalence relation

w ∼ w +log(p)

2πi, (4.8)

which together with w ∼ w+n yields a genus one Riemann surface with modulus

τ =log(p)

2πi. (4.9)

Since such a surface is defined in terms of a lattice in C and such structure is

preserved under the action of PSL(2,Z) in the form of Mobius transformations.

In terms of the modular parameter τ ∈ H the action of this group is

τ 7→ aτ + b

cτ + d. (4.10)

Therefore we have the following equivalence between the resulting Riemann

surfaces Ω/〈W (p)〉 ∼ Ω/〈W (p)〉, where

p = exp(2πiaτ + b

cτ + d). (4.11)

Hence for every fundamental region of H we will obtain a representative of

each classical geometry. We can choose an initial fundamental region and then


attach an index to every other spacetime on the same modular orbit. Notice

that only two indices are necessary, this can be seen by recalling that modular

transformations are generated by two kinds of fundamental operations: reflec-

tion R : τ 7→ τ−1 and translation T : τ 7→ τ + 1, or we can also recall that the

the determinant should be one and that. Hence associated to every spacetime

on the fundamental region we have an indexed family Mc,d with c, d ∈ Z and

(c, d) = 1 of equivalent geometries. This is strongly related with the discussion

of black holes as quotients presented in the previous chapter we will analyze

this carefully in the near future. Notice that the group generated by

W (b) =

eb 0

0 e−b

∈ PSL(2,C) (4.12)

with β ∈ R translates points along the big cycles of the torus we can therefore

think in this subgroup as the generator of time evolution. In AdS3 polar coor-

dinates (t, ρ, φ) with range t ∈ R, ρ ∈ R+ and φ ∈ S1 in which the metric pulls

back to ds2 = cosh2(ρ)dt2 + dρ2 + sinh2(ρ)dφ, the group element W (b) acts as

t 7→ t + b

Figure 4.2: Modular orbits

Now we attempt to compute the contribution of the M0,1 space-time to

the partition function. The saddle point contribution is of the form Z0,1(τ) =

exp(−Iclass). The term Iclass requires some deeper revision by itself since at first

4.3. QUANTUM CORRECTIONS 39

glance it seems to be unbounded by below, which makes the contribution of the

partition function from this geometry infinite. This can be solved by supple-

menting the Einstein-Hilbert action with a counterterm to cancel the boundary

variations, the necessary term is the Gibbons-Hawking-York counterterm:

IGHY =1

8π

∫

∂M

d2x√

hK, (4.13)

where h and K are the first and second fundamental forms of the boundary

respectively [28]. By a long but straightforward calculation, we can compute

the contribution of M0,1 and write it in terms of the modulus

Iclass = −4πk=(τ), (4.14)

then Z0,1 ≈ exp(4πk=(τ)) which. By introducing q = exp(2πiτ), it can be

written neatly as Z0,1 ≈ |qq|−k. This result is in perfect accordance with the

Brown-Henneaux theory in which the relations H = L0 + L0 and J = L0 − L0

connect the boundary theory with the spacetimes global charges. The ground

state of the theory should be such that L0|Ω〉 = L0|Ω〉 = −k,(with 24k =

c) hence H|Ω〉 = −2k and J |Ω〉 = 0 and the ground state’s contribution to

the partition function is Tr(exp(−2π(=τ)H + 2πi(<τ)J)) = exp(−4π(=τ)) =

exp(−Iclass).

4.3 Quantum Corrections

At first glance there is no room in three dimensional gravity to accommodate

more quantum states, at least close enough to AdS3 consistently with the pertur-

bative scheme, but actually we need for every relevant quantum theory of three

dimensional gravity, states that play the role of the Brown-Henneaux modes.

We follow the general procedure [52] to find these quantum states. First we try

to describe the classical phase space associated with the theory and then we

try to build the perturbative Hilbert space, the phase space is composed of the

3-manifolds that correspond to classical solutions of Einstein’s equations below


the black hole threshold. Lets denote this space by M. However, solutions

related by rapidly decaying diffeomorphisms should be considered as equivalent

and hence the phase space and therefore we should consider the quotient space

PS w M/diff(3d). The space PS still carries a residual symmetry group

given by the diffeomorphisms that preserve boundary conditions, of course with

the rapidly decaying diffeomorphisms already collapsed to the identity (denoted

here by diff(3d)). In the case of 2+1 dimensional gravity this group is, as found

by Brown and Henneaux, G ∼= diff(S1)× diff(S1).

The space PS has a convenient geometrical structure, for small perturba-

tions every solution with the same boundary conditions is deformable by a small

diffeomorphism into AdS3, then PS consists of a single G-orbit and hence PSis a homogeneous space of G. Hence by the Erlangen prescription PS w G/H

where H is a normal subgroup of G that corresponds to the stabilizer of some

marked point on PS. For this particular case we already know the stabilizer of

this point (AdS3) and it is locally SL(2,R)× SL(2,R) and therefore

PS w diff(S1)× diff(S1)

SL(2,R)× SL(2,R). (4.15)

Now we proceed to quantize this space, we expect to obtain a Hilbert space

H such that there exists an isomorphism (diff(S1))2 → Aut(H) i.e. we want

the states to transform as representations of (diff(S1))2, furthermore this rep-

resentation is irreducible [52] and there is an eigenvector which is stable under

the action of SL(2,R)×SL(2,R). There is a familiar representation that obeys

all this requirements, the so called vacuum representation determined by a ten-

sor product of Hilbert spaces HL⊗HR each of them is characterized by a vector

|Ω〉L,R ∈ HL,R such that for n ≥ −1, Ln|Ω〉L = −kδn|Ω〉L and analogously for

the right moving modes. The representations of the factors are completely de-

termined by de value of the central charges, which we will take in as equal,

hereafter we will drop the subindices L, R. The representation is spanned by

states generated by repeated action of the Virasoro generators∞∏

n=2

Lun−n ⊗

∞∏m=2

Lvn−m|Ω〉. (4.16)

4.3. QUANTUM CORRECTIONS 41

Therefore the partition function Z0,1(τ) will get contributions from the degen-

eracies of all these Virasoro descendants, for each of the sectors

L0

∞∏n=2

Lun−n|Ω〉 = (−k +

∞∑n=2

nun)|Ω〉. (4.17)

By straightforward application of the Virasoro commutation relations we com-

pute the action of L0 and L0 on the Virasoro descendants these yield the con-

tribution to the partition function by counting degeneracies, these degeneracies

are equal to the number of partitions of integer for each level

Z0,1(τ, τ) = |qq|−k(∞∑

n=2

P (n)qn)(∞∑

m=2

P (m)qm). (4.18)

We can show that this expression equals the reciprocal of Euler’s function

∞∑n=2

P (n)qn =∞∏

n=2

1

1− qn, (4.19)

and then we arrive to the final expression for the partition function of the AdS3

thermal gas

Z0,1(τ, τ) = |qq|−k 1∏∞n=2 |1− qn|2 . (4.20)

We dwell briefly on this expression, we know that a partition function can be

written as the inverse exponential of the free energy of the system, where this

free energy can be regarded as an effective action that includes the corrections

coming from the loop diagrams,

W (τ, τ) = Iclass +∞∑

l=1

1

klWl(τ, τ) (4.21)

where Wl(τ, τ) is the connected 2-point function with l-loops. From equation

(4.27) we see that

−kW (τ, τ) = log(Z0,1(τ, τ)), (4.22)

which implies

W (τ, τ) = −log|qq| − 1

k

∞∑n=2

log(|1− qn|2), (4.23)

and then we can see that, at least in perturbation theory, we need to consider

only 1-loop corrections to the theory, such theories are called 1-loop exact.


4.4 Modular invariant partition functions

The full partition function is obtained by considering the AdS3 thermal gas

partition function and then enforcing modular invariance on it. This can be

interpreted as a sum over geometries, reflecting how modular invariance carries

the different classical geometries hidden in its orbits. Many of the results used in

this section are based on the theory of modular forms and in number theoretical

properties. The mathematical basis of the forthcoming analysis is presented

briefly, for an extensive discussion on the subject we refer the reader to [4] and

[34].

Since the symmetry group of three dimensional gravity factors through the

modular group, we want to enforce this symmetry and construct a modular

invariant expression for the partition function i.e. we expect Z(γτ) = Z(τ) for

every γ ∈ Γ. In mathematical terminology we would say that such a function is

an automorphic form of the modular group or a modular form in short, this is

precisely the subclass of functions H→ C that live naturally on H/Γ, we denote

them here by Aut(H/Γ). In general, automorphic forms can be constructed

by the method of images [34] which consists in choosing a rapidly decaying

generating function f(z) on H and then performing a so called Poincare series

φ(z) =∑γ∈Γ

f(γz) ∈ Aut(H/Γ). (4.24)

In the present case we should, in principle, perform the sum for every γ ∈PSL(2,Z) and the generating function should be Z0,1 to obtain the desired

physical results. It turns out that actually the sum must be performed over

pairs of integers c and d such that (c, d) = 1, which in turn can be interpreted

as a sum over the whole family of SL(2,Z) family of black holes introduced

in section 3.2, the reason we only sum over this pair of integers is easy to see,

we know that the complex modulus of the Riemann surface is defined up to

modular transformations which in principle depend on its four integer entries,

we discard sign changes since the transformation γ = −I can be extended on

the manifold as an orientation reversing diffeomorphism [22] so we will perform

4.4. MODULAR INVARIANT PARTITION FUNCTIONS 43

the computation for a fixed orientation and hence up to sign. The values of c

and d can be used to determine a and b up to (a, b) ∼ (a, b) + Z(c, t) by using

ac−bd = 1m this is equivalent to a choice of a contractible cycle on the manifold

(see section 3.2) and therefore we are actually summing over the SL(2,Z) family

of black holes. The objective then will be to calculate the series:

Z(τ) =∑

c,d

Z0,1(γτ). (4.25)

Before embarking in the calculation we try to localize the modular invariant part

of the expression for Z0,1(τ), first we introduce the so called Dedekind η-function

which is a function of capital importance in number theory and automorphic

forms

η(τ) := q124

∞∏n=1

(1− q)n, (4.26)

in terms of the Dedekind η-function the partition function looks:

Z0,1(τ) =|qq|−k 1

24 |1− q|2|η(τ)|2 . (4.27)

In this form, some of the (0, 1) partition function’s symmetries become evident,

in fact the term√=(τ)|η(τ)|2 is modular invariant since

Im(γτ) =det(γ)y

(cx + d)2 + c2y2, (4.28)

and [4]

η(τ) = ε(a, b, c, d)(−i(cτ + d))12 η(τ). (4.29)

Therefore the total partition function is just

Z(τ) =1√

Im(τ)|η(τ)|2∑

c,d

(√

Im(τ)|qq|−k+ 124 |1− q|2)γ, (4.30)

expanding the square we obtain

Z(τ) =1√

Im(τ)|η(τ)|2∑

c,d

(√

Im(τ)(|qq|−k+ 124 − q−k+ 1

24 q−k+ 124

+1

−q−k+ 124 q−k+ 1

24+1 + q−k+ 1

24+1q−k+ 1

24+1)γ, (4.31)


notice that in every product the exponents of q and q differ by ±1 or 0 and

therefore the series 4.31 can be decomposed into sums of the generic form

P (τ ; m,n) =∑

c,d

(√

Im(τ)q−nq−m)γ, (4.32)

the P series according to equation (4.28) transform under a Mobius transfor-

mation as

P (τ ; m,n) =√

Im(τ)∑

c,d

1

|cτ + d|γ · q−nγ · q−m, (4.33)

where γ · q = exp(2πiγτ) or more explicitly

P (τ ; m,n) =√

Im(τ)∑

c,d

1

|cτ + d|exp(−2πniγτ)exp(−2πmiγτ). (4.34)

Writing 2<(τ) = τ + τ and 2=(τ) = τ − τ yields the convenient form

P (τ ; m,n) =√

Im(τ)∑

c,d

1

|cτ + d|exp(−2πi<(γτ)(m− n) + 2πi=(γτ)(m + n)),

(4.35)

which is in terms of the variable µ := m−n that can only take values µ = 0,±1

and furthermore, these values determine also the values of κ := m + n to be:

1. for µ = 0, κ = 2k − 112

or κ = 2k + 2− 112

2. for µ = 1, κ = 2k + 1− 112

3. for µ = −1, κ = 2k + 1− 112

.

We obtain therefore four different series, one for each of the above combinations

of the form

P (τ ; µ, κ) =√

Im(τ)∑

c,d

1

|cτ + d|exp(−2πi<(γτ)µ + 2π=(γτ)κ), (4.36)

hereafter we will be concerned with the behavior of these sums.

4.5. EINSENSTEIN AND POINCARE SUMS 45

4.5 Einsenstein and Poincare sums

This section is devoted to the computation of the series 4.36 in a closed form,

this turns out to be a quite interesting computation based in many interesting

properties of the spectral analysis of automorphic forms, we will outline the

main steps of this calculation first performed by Maloney and Witten, here we

will provide further details in some relevant passages of it.

4.5.1 ζ-function regularization

Before starting the computation we should examine a very important issue

that we overlooked in the past section, to use the method of images 4.24 it is

necessary that the generating function decays rapidly on H. In fact this is not

true for the case at hand, indeed by decomposing

γ · τ =aτ + b

cτ + d= A +

B

cτ + d, (4.37)

we obtain

γ · τ =a

c+

1

c(cτ + d). (4.38)

In turn, the real and imaginary parts transform respectively as

<(γ · τ) =a

c− cx + d

c((cx + d)2 + c2y2), (4.39)

=(γ · τ) =a

c− cx + d

(cx + d)2 + c2y2. (4.40)

Now we can see that the summand doesn’t decay fast enough, notice that the

real exponent, the one depending on =(γ · τ) tends to one, leaving the sum-

mand proportional to |cτ + d|−1 and therefore the Poincare series diverges as a

harmonic series.

Hence it is necessary to select a regularization scheme for the Poincare series,

there are many ways to introduce a regulator but in this case is utterly conve-

nient to use Hawking’s ζ-function regularization [30] in a slightly modified way.


The objective of this prescription is to be able to compute the 1-loop determi-

nants of a differential operator D over the manifold, a task which is strongly

related with our objective (section 4.3). The strategy is to define a generalized

form of the familiar Riemann’s ζ-function, ζ(s) =∑

n≥1 n−s and substitute the

integers for the spectrum of the differential operator of interest

ζH(s) =∑

n

λ−sn . (4.41)

Formally the derivative of ζ(s)H is

dζH(s)

ds s=0= −

∑n

log(λn), (4.42)

and therefore we can define

det(D) = exp

(−dζH(s)

ds s=0

). (4.43)

Following this idea, we generalize our Poincare series to a more general one

P (τ ; m,n) → P (τ ; m,n, s) =∑

c,d

(√

Im(τ)q−nq−m)sγ, (4.44)

since

=(γτ)s =ys

(cτ + d)2s(4.45)

the sum will be convergent only in the region <(s) > 1, afterwards we will

perform an analytic to s = 12

and hope to get a physically sensible result.

4.5.2 Double coset decomposition

Now that we have a regularized sum at hand, we rewrite it in a convenient

way and try to find a closed form for the solution, to perform the sum it is

necessary to make a clever choice of the summation indices that we will use.

This will require to explore rather deeply the structure of the symmetry group

and perform a so called double coset decomposition. Our starting point is the

sum

P (τ, κ, µ; s) =∑

(c,d)

ys

|cτ + d|2sexp (2πκ=(γτ) + 2πiµ<(γτ)) . (4.46)


With the objective of performing the (c, d) summation we need to introduce

some interesting analytic techniques and to enlarge our repertoire on the subject

of automorphic forms, here we present a series of properties without a proof

(these can be found in [4], [34], [35]).

Let Γ ⊂ PSL(2,R) which is the group of orientation preserving isometries

on H and suppose that it acts by Mobius transformations in a stable way on

a proper connected open subset Λ of P1, we say that Γ is Fuchsian if it is a

discrete subgroup respect to the H topology, notice that every Fuchsian group

is also Kleinian. They can also be characterized, as it was proved by Poincare,

by demanding its action to be discontinuous on H. A very important property

of Fuchsian groups is that for any z ∈ P1 the stabilizer Γz is always a cyclic

subgroup [34], we say that Γ is of the first kind if every point on ∂H is a limit

point of some orbit of the group.

To be able to decompose the sum, it is necessary to know the possible

orbits generated by PSL(2,R), these motions respect the group’s conjugacy

classes, therefore we will be concerned only with the classes [γ]. In the first

possible kind of motion motion there is the identity map standing out as the

only representative of its kind, on the other hand if γ is not the identity, then

γ · τ has either one or two fixed points in H, in fact

aτ + b

cτ + d= τ, (4.47)

then the fixed points are given by

τ1,2 =(a− d)±

√(a + d)2 − 4(ad− bc)

2c. (4.48)

This result allows us to build a classification of the possible motions on H into

three main categories:

1. Parabolic: γ has one fixed point on ∂H

2. Hyperbolic: γ has two different fixed points on ∂H

3. Elliptic: γ has one fixed point in H


Groups with only hyperbolic motions perform identifications on H that yield

g ≥ 2 compact Riemann surfaces, while the ones containing parabolic motions

render non-compact ones. The way to deal with this kind of actions is to add

cusps to the quotient in order to compactify it.

As it was already mentioned the stability groups of points under a Fuchsian

action are cyclic groups Γa = 〈γa〉, in particular the ones corresponding to the

cusps. For every cusp a there exists a matrix σa ∈ PSL(2,R), denoted scaling

matrix, such that σa∞ = a and

σ−1a γaσa =

1 1

1

, (4.49)

Lets suppose that we have a Fuchsian group with cusps a1, ..., an equipped

with scaling matrices σa1 , ..., σan respectively, notice that(σ−1

akΓak

σak

)∞ = ∞and therefore these cyclic groups obey σ−1

akΓak

σak' Γ∞ i.e. they generate

integer translations in H, here we can see the first evidence of the relation with

the Maldacena-Strominger SL(2,Z) family of black holes, we will discuss this in

more detail. Consider the set σ−1ak

Γσal, lets try to split this set into double sided

lateral classes of Γ∞ it is easy to prove that the stabilizer of ∞ is non-empty if

and only if ak = al and that for any ω ∈ σ−1ak

Γσaland n,m ∈ Z, For

a ∗

c d

∈ Γ, (4.50)

with (c, d) = 1 and consider1 n

0 1

,

1 m

0 1

∈ Γ∞, (4.51)

then 1 n

0 1

a ∗

c d

1 m

0 1

=

a + nc ∗

c d + cn

, (4.52)

and hence the double cosets

Γ∞

∗ ∗

c d

Γ∞, (4.53)


fix c and determine d modulo c. Therefore we obtain the total decomposition

σ−1ak

Γσal= δk,lB ∪

⋃c>0

⋃

d(mod(c))

Γ∞

∗ ∗

c d

Γ∞, (4.54)

where the asterisks mean that the matrix depends only on c and d, we call this

particular decomposition a double coset decomposition. Now we turn back to

our Poincare series with this decomposition in mind.

4.5.3 Explicit summation

In light of the double coset decomposition, we may identify the sum over the pair

of relatively prime integers (c, d) in equation (4.46) with one over the non-trivial

double cosets γ ∈ Γ∞ \ PSL(2,Z)/Γ∞ hence we are allowed to interchange the

sums∑c>0

∑

dmod(c)

↔∑

Γ∞\PSL(2,Z)/Γ∞

(4.55)

and now we can apply the double coset decomposition, this means that we can

perform an expansion of the Poincare series around the pairs of cusps. Now we

take summand of equation (4.46) and recast it as

p(γτ) =ys

|c(τ + n) + d′|2sexp

(2πκy

|c(τ + n) + d′|2 + 2πiµ

(a

c− cx + d

c|c(τ + n) + d′|2))

,

(4.56)

where we have used equations (4.39) and (4.40). For the identity modular (i.e.

c = 0 and d = 1) we have

p(τ) := ysexp (2π(κy + iµx)) , (4.57)

and the series can be rewritten as

P (τ, κ, µ; s) = p(τ) +∑

γ∈Γ∞\PSL(2,Z)/Γ∞

Iγ(τ), (4.58)

where [34]

Iγ(τ) =∑

ξ∈Γ∞

(c(τ + n) + d)−kp(γ · ξτ), (4.59)


and more explicitly

Iγ(τ) =∑

ξ∈Γ∞

(c(τ + n) + d)−kp

(a

c− 1

c(c(z + n) + d

). (4.60)

Since Γ∞ ' Z we can apply the Poisson resumation formula to the n sum

∑

n∈ZF (n) =

∑

m∈ZF (m), (4.61)

to obtain

Iγ(τ) =∑

m∈Z

∫ ∞

∞dn(c(τ + n) + d)−kp

(a

c− 1

c(c(z + n) + d

)exp(−2πinm).

(4.62)

In our case the value of k = 0 which simplifies the expression to

Iγ(τ) =∑

m∈Z

∫ ∞

∞dnp

(a

c− 1

c(c(z + n) + d

)exp(−2πinm), (4.63)

performing the integration variable redefinition t = n + x + d/c yields

Iγ(τ) =∑

m∈Zexp

(2πi

(µa− nd

c− nx

))

∫ +∞

−∞dte2πint

(y

c2(t2 + y2)

)s

exp

(2π(ky − iµt)

c2(t2 + y2)

), (4.64)

then we perform a Taylor expansion of the exponential

exp

(2π(ky − iµt)

c2(t2 + y2)

)=

∞∑

l=0

1

l!

(2π(ky − iµt)

c2(t2 + y2)

)l

(4.65)

and subsequently we introduce the variable T = t/y to obtain

Iγ(τ) =∑

m∈Z

∞∑

l=0

exp

(2πi

(µa−md

c−mx

))c−2(l+s) (2l)

l

l!

y1−l−s

∫ +infty

−∞dTe2πimTy

(1− T 2

)−l−s(κ− iµT )l. (4.66)

Now we go back to the total sum and substitute

∑

Γ∞\PSL(2,Z)/Γ∞

→∑c>0

∑

d∈Zmod(c)

. (4.67)


Now some interesting properties of the total sum are revealed, we can see that

the integrals are independent of these sums so we can work out the series without

performing the integrations. Also recall that µ ∈ Z, this implies, due to the

periodicity of the exponential

exp

(2πi

(µa−md

c

)), (4.68)

that the expression for P (τ, κ, µ; s) depends only on a(mod(c)). Therefore we

can rearrange the total sum into the form:

P (τ, κ, µ; s) = ysexp (2π(κy + iµx)) +∑

m∈Ze−2πimxPm(τ, κ, µ; s), (4.69)

where P (τ, κ, µ; s)m is composed of two nearly independent components, one

involving the integrals and the other involving the double coset summations:

Pm(τ, κ, µ; s) =+∞∑

l=0

Il,my1−l−s∑c>0

c−c(l+s)SK(−m,µ; c), (4.70)

where

Il,m =(2π)l

l!

∫ +∞

−∞dTe2πimTy(1 + T 2)−m− s(κ− iµT )l, (4.71)

and SK(−m,µ; c) is a so called Kloosterman sum of great importance in number

theory which in fact can be performed rather easily for certain cases. Explicitly

they look:

SK(−m,µ; c) :=∑

d(mod(c))

exp

(2πi

(−md + µd∗

c

)), (4.72)

where d∗ is the inverse of d on (Zc, ·).Before executing the final cut, we recall the meaning of the modular param-

eter τ in physical terms, each τ fixes the ensemble parameters for the partition

function i.e. the (inverse) temperature β = 2π=(τ) and the angular potential

θ = 2π<(τ). We know that the action of the modular group will yield indis-

tinguishable geometries, hence we restrict ourselves to the fundamental region

and approximate the partition function semiclassically for the range of tem-

peratures and angular potentials inside this region, then we regard modularly


related temperatures and angular potentials as degenerate states. From this

procedure we expect to obtain an expression of the following form:

Z(τ) = Tr (−βH − iθJ)

=∑

m∈Ze−imθTr (−βH) . (4.73)

Equation (4.69) gives an interesting hint since we have the summation

∑

m∈Ze−2πimxPm(τ, κ, µ; s) =

∑

m∈Ze−2imθPm(τ, κ, µ; s) (4.74)

given that the eigenvalues of the angular momentum are integers, we can see

that the modes Pm(τ, κ, µ; s) should represent the traces over the hamiltonian

operator for each subspace of constant angular momentum. Now is just a matter

of setting the µ parameter and performing an analytic continuation to s = 1/2.

The integrals and the Kloosterman sums where performed explicitly by Maloney

and Witten, we just summarize briefly their results. For the zero mode we have

P0(τ, κ, µ; s) =∞∑

l=0

ωl(s, κ, µ)y1−l−s, (4.75)

where

ωl(s, κ, µ) =∑

l=0

Il,0y1−l−s

( ∞∑c=1

c−2(l+s)SK(0, µ; c)

), (4.76)

we consider now the possible cases

1. µ = 0

ωl =κl2

lπl+1/2Γ(s + l − 1/2)ζ(2(m + s)− 1)

l!Γ(l + s)ζ(2(l + s)), (4.77)

2. µ = ±1

ωl = cos

(lπ

2

)(2π)lΓ

(1+l2

)Γ

(l−12

+ s)

l!Γ(l + s)ζ(2(l + s))2F1

(l − 1

2+ s,− l

2+ s;

1

2; κ2

)

+ lκsin

(lπ

2

)(2π)lΓ

(l2

)Γ

(l−12

+ s)

l!Γ(l + s)ζ(2(l + s))2F1

(1− l

2,l

2+ s;

3

2; κ2

)

(4.78)


-1.0

-0.5

0.0

0.5

1.0

-2

-1

0

1

2

0.0

0.5

1.0

1.5

Figure 4.3: Taking s=1/2

Notice that when l > 0 both the Kloosterman sum and the integral present no

problems while taking s = 1/2, so the term that could cause a divergence is the

term l = 0, but the divergences in fact cancel. For l = 0 the equations (4.77)

and (2) transform into the prefactor

√π

Γ(s− 1/2)

Γ(s)· ζ(2s− 1)

ζ(2s)(4.79)

which is in fact regular (see figure 4.5.3) since both Γ(s− 1/2) and ζ(2s) have

simple poles at s = 1/2. Finally the θ independent part of the perturbative

corrections to the partition function is given by:

Zper(τ) =1

|η(τ)|2(−6 + f1(k)

(1

y

)+ f2(k)

(1

y

)2

+ ...

)(4.80)

where the coefficients that depend solely on the value of cosmological constant

are [40]

f1(k) =(π3 − 6π)(11 + 24k)

9ζ(3)

f2(k) =5(53π6 − 882π2) + 528(π6 − 90π2)k + 576(π6 − 90π2)k2)

2430ζ(5)(4.81)


-0.50.0

0.5

0.51.01.52.0

0

2000

4000

Figure 4.4: Perturvative corrections

Notice that the coefficients of this expansion don’t allow us to interpret this

sum as the partition function of a Conformal Field Theory. This coefficients will

multiply the terms coming from the η(τ) and these products should represent

the degeneracies of the descendants. However, they are going to be irrational

numbers, and for the case of the first term a negative one. This result calls for

further thought, since the recipe applied seems to be, at least at first sight, the

right one. It is possible that it is necessary to consider extra degrees of freedom

in the partition sums or that simply pure three dimensional quantum gravity

doesn’t exist. One of the most appealing possibilities is to extend the partition

sum to complex geometries, in such a way that the new partition function is

holomorphically factorizable. We will focus on a particular case of the former

possibility .

Chapter 5

ECFT’s and Monstrous partition

functions

This chapter deals with a collection of conjectures dealing with three dimen-

sional gravity posed by Witten in [53], and developed further in a series of

recent papers (see [25], [27], [29], [41], [55], [56], etc). The claim is that the

CFT dual to 3D Gravity is in fact a series of so-called Extremal Conformal

Field Theories and that the partition function is completely determined by de-

manding modular invariance and holomorphic factorization. This means that

for every k ∈ Z, if we take c = 24k to be the central charge associated with the

cosmological constant, then the CFT dual should be an ECFT. The existence

of these theories is not proven for k > 1 so the following material is rather spec-

ulative and based mainly on optimistic conjectures. In the special case of k = 1

a beautiful connection between quantum gravity and deep mathematical ques-

tions emerges. It turns out that the resulting Conformal Field Theory is based

on the Frenkel-Lepowsky-Meurman (FLM) construction, in which the states

transform as representatios of the Fischer-Griess monster. This fact makes us

wonder where in an apparently trivial gravitational theory such a mysterious

and complex symmetry group can be hidden.

55

56 CHAPTER 5. ECFT’S AND MONSTROUS PARTITION FUNCTIONS

5.1 3D Gravity as a Monster

To begin this section we first give a sketchy introduction to the vast theory of

the Monster Group and the Monstrous Moonshine conjecture (for extensive re-

views see [6], [7] and [45]). This theory arises as part of the quest, usually called

the Enormous Theorem, to prove that all the possible finite simple groups, i.e.

nontrivial finite groups with no nontrivial normal subgroups, fall into a fairly

well defined classification [6]. Standing out of the main classes there are 26 spo-

radic groups, these are real mathematical gems and they appeared during the

process of classification as curiosities, the largest of these is the so called Mon-

ster Group or Friendly Giant M whose existence was predicted by Fischer and

Griess and of which the former carried out the first explicit construction. Griess

visualized M as the irreducible representation of the automorphism group of a

196883-dimensional algebra, now called Griess algebra [8]. This construction

lead to a series of astonishing observations that have been an active area of

research in modern mathematics for more than a quarter of a century and has

inspired works worth of the Fields Medal and of the 2008 Abel Prize 1, one of

this observations is going to be of capital importance for us as we will see in

brief.

In a completely (apparently) different region of mathematics, the Theory of

Modular Forms and Number Theory, there is a very important object called

Klein’s j-function. The j-function is a one-to-one modular function from the

fundamental domain of PSL(2,Z) to the Riemann sphere P1. The moduli space

of Riemann surfaces of genus g = 1 is itself a Riemann surface with g = 0, and

furthermore every modular invariant function factors throughout j(τ). This

function can be characterized by its Laurent series around τ = +∞ (or q = 0)

j(τ) = q−1 + 744 + 196884q + 21493760q2 + ..., (5.1)

the constant term 744 can be absorbed by performing a SL(2,C) transforma-

tion, and therefore we can work instead with the modular function J(τ) :=

1Fields Medal: Thompson 1970, Borcherds 1998. Abel Prize: Tits and Thompson 2008.

5.1. 3D GRAVITY AS A MONSTER 57

j(τ) − 744. The size of the irreducible representations of M have been linked,

first by numerological arguments, to this expansion of the modular j-function.

From the numbers on table (5.1) we notice that curiously the order one coeffi-

cient of the expansion equals one plus the dimension of Griess’ representation

of the Monster. Although this seems to be a mere coincidence Norton, McKay

and Conway soon realized that also other coefficients can be written as lin-

ear combinations of the dimensions of almost irreducible representations of the

Monster [8].

Table 5.1: A remarkable coincidence.

Coeficients of the j-function Character degrees of M, di

1 1

196,884 196,883

21,493,760 21,296,876

864,299,970 842,609,326

20,245,856,256 18,538,750,076

For example, with the first few coefficients we can perform the combinations

196, 884 = d1 + d2

21, 493, 760 = d1 + d2 + d3

864, 299, 970 = 2d1 + 2d2 + d3 + d4. (5.2)

For this conjecture John Conway coined the term Mounstrous Moonshine using

this term with the connotation of a crazy or foolish idea [20]. However this

conjecture was proved by Borcherds, using the No-Ghost Theorem from String

Theory [9]!

It turns out that Conformal Field Theory can give a deep insight into this

remarkable phenomenon, in fact Frenkel, Lepowsky and Meurman found out


how to construct the Griess algebra from a vertex algebra [23], generally this is

considered to be the most natural construction possessing monster symmetry.

According to the FLM approach the number 196884 corresponds the number

of operators in the theory of conformal dimension ∆ = 2. These fields can

be splitted into the descendants of m dimension 1 primaries and dimension 2

primaries. For a k = 1 CFT with m primary fields the Laurent expansion of

the partition function around τ = ∞ i.e. q = 0 is of the form

Z(τ) = q−1 + m + O(q). (5.3)

The FLM point of view consists in regarding the coefficient 196884 as the

number of ∆ = 2 operators in the theory. One of them is just the stress

energy tensor while the other 196883 are ∆ = 2 primaries that transform as

representations of the Monster. So in order to be able to have enough states to

construct a representation of M we would have to be able to find a theory in

which m = 0, i.e. a theory lacking primary fields of dimension 1. For k = 1 (i.e.

c = 24) this theory exists and it is called the Monster Module [23]. This can be

achieved by compactifying a bosonic string theory in a certain particular way.

To attain this we need to introduce first an important mathematical structure,

the Leech lattice (see [45] for a didactical introduction to the subject) which

represents the densest possible packing in 24 dimensions (One dimension for

each of the transversal degrees of freedom of the Bosonic String Theory). The

points in this lattice can be regarded as the centers of balls in a 24 dimensional

space and each of them encouters 19, 560 others which is the maximum possible

in 24 dimensions, more precisely it is the lattice characterized by the following

properties: Let p ∈ L be a vector in the lattice, denote pi the basis of the

lattice and let 〈·, ·〉 be the quadratic form associated to L

1. It is an even lattice, which means that 〈p, p〉,

2. It is unimodular i.e., it has discriminant ±1, which means that det(M) =

±1 where Mi,j = 〈pi, pj〉,

5.1. 3D GRAVITY AS A MONSTER 59

3. It has rank or dimension 24 and

4. It has no vector with length squared smaller than 4.

Now label by Xi the family of 24 chiral bosonic fields and compactify this

theory over the torus R24/L, since L is unimodular this procedure yields a

holomorphic c = 24 CFT [24]. Consider a state of the form Vp(z) := Ψ =:

exp(ip ·X(z)) : i.e. a vertex operator for which the OPE is (see appendix A)

T (z)Vp(z) ∼ p2

2

Vp(ω)

(z − ω)2+

∂ωVp(ω)

z − ω(5.4)

from this we can read of that V (p) has a conformal dimension ∆ > 1. The

system still contains 24 primaries of dimension one, given by the derivatives

of the bosonic fields, these can be eliminated by orbifolding over Z2 using the

symmetry Xi → −Xi. Finally we have at hand a CFT that carries Monster

symmetry in a natural way.

Now lets go back to 3 dimensional gravity and see the consequences of this

digression into the Monsters world. There exist precisely 71 c = 24 holomorphic

CFTs [46], all but one of them carry inside also a Kac-Moody symmetry which

would imply the existence of gauge fields in the bulk 2, and we should recall

that we are interested in pure gravity. The theory that is free of Kac-Moody

symmetry turns out to be the Monster Module! Therefore it seems that k = 1

pure gravity is just another face of the monster.

Somehow this theory should be able to describe black holes inside this frame-

work, in fact in accordance with the previous discussion we can make one inter-

pretation more of the 196,883 primaries of the module, we can interpret them as

the creation operators of BTZ black holes and the descendants can be regarded

as BTZ black holes dressed with Brown-Henneaux excitations. Therefore we

have an explicit quantum degeneracy for them that yields an entropy

SM = ln(196, 8832) ∼= 8π, (5.5)

2F. Alday explained this to me by invoking the AdS/CFT correspondence


where the square comes from the fact that the actual partition function is the

product of the holomorphic and the anti-holomorphic partition functions. On

the other hand, we can compute the Bekenstein-Hawking entropy of a L0 = 1,

L0 = 1 BTZ black hole. First we rewrite equation (2.48) in terms of k, by using

equations (2.45) and (2.44) we get

SBTZ = 8π√

k. (5.6)

Hence there is a remarkable agreement between the semiclassical result and our

degeneracy counting. This is a beautiful result, basically it means that the

black holes in 3D pure quantum gravity transform as minimal representations

of the Monster group.

5.2 Why Extremal CFT’s?

We would like to be able to apply the procedure used by FLM to construct

the desired CFT for the case k > 1. Regretfully this strategy doesn’t apply in

that case (see [53]). In order to deal with the cases of higher k it is necessary

to introduce the so-called Extremal Conformal Field Theories. Before trying

to motivate rhe use of Extremal Conformal Field Theories (ECFT) we need to

define them in a precise way and to discuss some of the basic properties that

they possess. In a CFT with c = 24k the lowest primary field different from

the identity cannot have a conformal dimension bigger than k + 1. Extremal

Conformal Field Theories are holomorphic CFT’s where the lowest dimension

primary is of conformal dimension exactly k + 1. These kind of CFT’s was

introduced by Hohn in [33] and are constructed with the objective of avoiding

the presence of extra Kac-Moody symmetries which under the AdS/CFT cor-

respondence point of view will imply the existence of non-gravitational fields in

the bulk. Intuitively we would like to demand that there are no low dimensional

primary fields but the identity, at c = 24k the lowest dimension a primary dif-

ferent from the identity cannot surpass c, in a ECFT we demand the equality

5.2. WHY EXTREMAL CFT’S? 61

to hold. The existence of this kind of theories is uncertain for k > 1, but in case

these exist they are an interesting candidate for the CFT dual of 3D gravity.

We could regard the k + 1 primaries as the creation operators of BTZ black

holes since this dimension is in good agreement with the minimum BTZ mass.

Now we will approach the construction of the modular invariant partition

function in a different way from the one used in chapter 4. Classical black holes

don’t arise for L0 < 0 and furthermore the L0 = 0 ones do not contribute to the

entropy, hence we would expect the corrections to the partition function 4.27

to be of order q.

Z(q) = Z0,1(q) + O(q), (5.7)

where Z0,1(q) is given by equation (4.27). Now we go back to study the modular

J-function and use Hohn’s result to get a series of modular invariant partition

functions. If we want the partition function to be modular invariant this should

be a function of the J-function, in fact we can write

Z(q) =k∑

n=0

cnJn. (5.8)

the reason we can do this is that the only pole of J , located at q = 0, coincides

with the order k pole of the partition function. The cn coefficients are obtained

by matching order by order equation (5.8) with equation (5.7).

For example for k=2 we proceed in the following way. Take

q−2 1∏∞n=2(1− qn)

= (1− q)q−2 1∏∞n=1(1− qn)

, (5.9)

then

Z0,1(2) = (1− q)q−2 1∏∞n=1(1− qn)

= q−2

∞∑n=0

P (n)(qn − qn+1), (5.10)

and

Z2(q) = c0 + c1J1 + c2J

2. (5.11)

Then we substitute the expansion (5.8) and match the coefficients for q−2, q−1


and q0

c0 + 2c2(196884) = P (2)− P (1)

(c1 + c2)1

q= 0

c21

q2= P (0)

1

q2, (5.12)

(5.13)

therefore c0 = −3973767, c1 = 0 and c2 = 1 and hence

Z2 = −3973767 + J2(τ) = q−2 + 1 + 42987520q + 40491909396q2 + ... (5.14)

This can be done in a completely analogous way for other values of k. The first

few orders are

Z1(q) = J(q)

Z2(q) = q−2 + 1 + 42987520q + 40491909396q2 + ...

Z3(q) = q−3 + q−1 + 1 + 2593096794q + 12756091394048q2 + ... (5.15)

Z4(q) = q−4 + q−2 + q−1 + 2 + 8102669428q + 160467192452452276q2 + ...

Notice that all of them are partition functions corresponding to ECFTs and that

Z1(q) corresponds to the Monster Module of the previous section. Summing up,

we put together equations (2.45), (2.48), the result of section 2.5 and equation

(5.15) in order to compare the degeneracy of the microstates and the value of

the Bekenstein-Hawking Entropy. The agreement is remarkable and it improves

as we increase the level k, in accordance with the fact that the Bekenstein-

Hawking entropy is computed semiclassically and the semiclassical limit of our

theory is at large k. We should note that this results should be taken with a bit

of scepticism, since arguments have been given which may show that ECFT’s

don’t exist for k > 1 [25] and also that in case they exist they don’t have

monster symmetry in fact, D. Gaiotto already killed the two-headed monster

(k = 2) in [26].

5.3. MORE ON HOLOMORPHIC FACTORIZATION 63

Table 5.2: Comparison of entropies (holomorphic sector).

Level k SECFT SBH

1 12.19 12.57

2 17.58 17.77

3 21.68 21.77

4 25.12 25.13

5.3 More on holomorphic factorization

This section intends to build a bridge between the ECFT computations of the

previous section and the approach used in chapter 4 to compute the partition

function. This will help us also to visualize the partition sums 5.15 as actual

sums over geometries, regretfully geometries of a rather puzzling nature. As

we discussed in section 4.5.3 there exists a huge amount of possible approaches

to heal the partition functions of chapter 4. In spite of our ignorance about

the existence of k > 1 ECFT’s, the partition functions found in the previous

section give an interesting hint of the possible route to follow: To demand also

holomorphic factorization. But, what would that imply physically? To achieve

this connection we try to explain how are the calculations of chapter 4 and of

section 5.2 related. The key lies again in the coefficients of expansion of the

Klein’s j-function. If we write

j(q) = q−1 + 744 ++∞∑n=1

cnqn, (5.16)

then the coefficients are given by [43]

cn =2π√n

+∞∑m=1

1

mSK(−m,n)I1

(4π√

n

m

), (5.17)

where SK(−m,n) is a Kloosterman sum defined in equation (4.72) and Iν(z) is

just a Bessel function

Iν =

(12z)

ν

2πi

∫ c+i∞

c−i∞tν−1et+ z2

4t dt, (5.18)


with ν ∈ H and c ∈ R+. After applying the prescription (4.55), the complete

Rademacher expansion [43] for J is

J(τ) = −12 +∑c>0

∑

d∈Zmod(c)

exp

(2πi

(−aτ + b

cτ + d

))− exp

(−2πi

a

c

)(1− δc,0),

(5.19)

and from equations (5.17), (5.18) and (5.19) we can see the connection with

the approach of the previous chapter i.e. we can regard the partition function

as an actual sum over geometries. The second term in the right hand side of

5.19 consists of (see chapter 4) in a sum over the Maldacena-Stromiger family

of black holes, or equivalently a sum over choices of contractible cycles. The

other two terms don’t have an immediate physical interpretation. However to

partially tackle this issue Manschot [41] relied on the so called Farey transform

technique from [22], we briefly outline this procedure. The Farey transform of

a modular form of weight w is defined as

DF (f) =

(q

∂

∂q

)1−w

f. (5.20)

The inverse Farey transform of DF (J(τ)) is equal to J(τ) up to a constant.

Klein’s function is a w = 1 modular form [4] and its Poincare series is [41]

DF J(τ) = −1

2

∑

γ∈Γ∞\PSL(2,Z)

exp (−2πiγ · τ)

(cτ + d)2. (5.21)

The interesting thing about this sum is that it can be actually understood as a

sum over the geometries as we can see from the index of summation and we can

recall that J(τ) is the partition function of the k = 1 theory. Now we simply

repeat the generalization to higher k performed in section 5.1 i.e. we expand

the partition function as a polynomial in J and then we fit the coefficients order

by order and obtain

DF Z(τ) = −1

2

∑

γ∈Γ∞\PSL(2,Z)

cnexp (−2nπiγ · τ)

(cτ + d)2. (5.22)

Therefore in some restricted sense we can picture the ECFT partition functions

as sums over geometries up to a Farey transform.

5.3. MORE ON HOLOMORPHIC FACTORIZATION 65

Now we discuss some generalities of holomorphic factorization and see their

consequences in this scheme. In chapter 4 we constructed a partition function

of the form

Z(β, τ) =∑

γ∈Γ∞PSL(2,Z)

Z0,1(γ · τ), (5.23)

where Z0,1 was computed in section 4.2 and is the partition function of AdS3

thermal gas. Notice that Z0,1 is holomorphically factorizable

Z0,1(q, q) = Zh0,1(q)Z

a0,1(q), (5.24)

where the superscripts stand for holomorphic and anti-holomorphic sectors.

Perturbatively this property is kept and a good physical motivation for that,

comes from the fact that at this level three dimensional gravity can be modelled

by a Chern-Simons theory (see section 2.3). This theory has a symmetry group

SL(2,R)× SL(2,R) composed of two decoupled factors, which would manifest

themselves as two independent sectors on the boundary CFT and therefore this

implies the holomorphic factorization of the partition function. In general, for

every γ ∈ Γ∞ \ PSL(2,Z) labelled by c, d ∈ Z we have

Zγ(q) = Zhγ (q)Za

γ (q), (5.25)

which is also holomorphically factorizable. Then we follow the prescription of

chapter 4 in order to enforce modular invariance on Z, we sum over the orbits

of the Γ∞ \ PSL(2,Z)

Z(q) =∑

γ∈Γ∞\PSL(2,Z)

Zhγ (q)Za

γ (q). (5.26)

Notice that this partition function is no longer holomorphically factorizable.

The reason is that the sum over the topologies is common for both right and

left moving modes. In order to heal this problem, we can enlarge the sum to

one over independent topologies

Z(q) :=∑

γ,γ′∈Γ∞\PSL(2,Z)

Z ′hγ (q)Za

γ (q), (5.27)


which is indeed holomorphically factorizable

Zγ(q) =

∑

γ∈Γ∞\PSL(2,Z)

Z ′hγ (q)

∑

γ′∈Γ∞\PSL(2,Z)

Zaγ (q)

. (5.28)

This new sum will include a great amount of geometries that were not considered

in the previous partition function. The nature of this geometries is rather

puzzling. In general γ 6= γ′, so the action associated with the geometry Mγ,γ′

is not real and therefore it is not a classical solution of Einstein’s equations.

When we demand holomorphic factorization, this is reflected in the physical

spectrum in a curious way. Suppose |Ψh〉 and |Ψa〉 correspond to the ground

states of the holomorphic and anti-holomorphic sectors of the theory respec-

tively, and that analogously |Φh〉 and |Φa〉 correspond to primary descendants.

We know that the tensor product|Φh〉 ⊗ |Φa〉 and the states that emerge due

to the repeated application of the Virasoro generator represent AdS3 geometric

structures dressed with Brown-Henneaux excitations. But the states that de-

scend from |Φh〉 ⊗ |Ψa〉 have no trivial classical interpretation, they would be

states that resemble AdS3 for left movers and a BTZ black hole for the right

ones. This states are needed in order to have holomorphic factorization in our

theory, hence we need to find at least some arguments that favor their inclusion

in the partition sums.

There is no precise argument to demand the doubled sum over the geome-

tries, but we can at least show that the sum has the right classical limit i.e.

one that makes the non-classical geometries irrelevant. To estimate the classical

limit we fix the temperature and the angular potential so that τ is constant, and

then we consider de limit k → ∞. At this limit and inside each fundamental

region, Z(τ) [53] is dominated by a unique γ, this will give rise to a very inter-

esting phase diagram that we will discuss in section 5.4. But, what happens in

the case Z(q)? We aim for the minimum of the real part of the Zγ(τ) partition

5.4. HAWKING-PAGE IS LEE-YANG 67

functions

<(Iγγ′) = <(2πiγτ) + <(−2πiγτ) (5.29)

= <(2πiγ′τ) + <(−2πiγ′τ),

and therefore

<(Iγγ′) =1

2(Iγ(τ) + I ′γ(τ), (5.30)

which then implies that the minimum appears when γ = γ′. These are great

news, since it means that the semiclassical limit will be controlled by actual

solutions to the Einstein’s equations.

5.4 Hawking-Page is Lee-Yang

With the aim of studying the physical properties of the phase space of three

dimensional gravity, we introduce some interesting thermodynamical properties

of black holes in space-times with negative cosmological constant. First we

introduce an interesting physical property of four dimensional black holes that

will facilitate the understanding of the three dimensional phenomena. One

interesting property of asymptotically AdS space-times is that they have no

characteristic temperature (just like Minkowski and differently from dS), this

is manifest from the fact that it’s ground state has no periodicity in imaginary

time. Then we can build thermal states of arbitrary temperature by imposing

a periodicity β = T−1 in the imaginary time coordinate. The fundamental

difference between black holes in flat space-time and black holes on AdS is that

even though the former can be in thermal equilibrium with radiation, this state

is unstable at fixed temperature [31]. This implies that a canonical ensemble

cannot be defined for black holes in asymptotically flat space-time. The physical

picture is simple, if the black hole increases its mass slightly it will cool down and

this would imply an increase in the absorption rate of the black hole and hence

it will keep growing steadily. On the other hand AdS allows us to build stable

black holes in equilibrium with thermal radiation. In an asymptotically Anti-de


Sitter space-time we know that at large distances the gravitational potential

increases, because of the bending caused by the negative cosmological constant.

As a consequence, it is unnecessary to confine the radiation in a box since

the AdS radius gives already a confinement mechanism. In four dimensions,

Hawking and Page [31] found that at a temperature

THP =1

2π

√−Λ, (5.31)

there is a phase separation, at T < THP thermal radiation is stable and there

is no black hole formation, above it it is possible the formation of black holes

and these are stable for a particular value of temperature, this is a so called

Hawking-Page phase transition.

In the case of three dimensional gravity, the thermodynamic behavior is anal-

ogous to the four dimensional one, in our case we are only concerned with AdS

black holes, since they are the only ones that arise in pure 3D gravity. Semiclas-

sically we expect a Hawking-Page phase transition to occur from thermal AdS3

at inverse temperature β = =(τ) into a BTZ black hole. After performing a

Wick rotation the conformal boundary is a torus of modulus τ and in the semi-

classical regime, the partition function is dominated by a solid torus that fills

in this boundary. Defining this handlebody involves the choice of a contractible

cycle γ (see section 3.2). Geometrically a Hawking-Page phase transition con-

sists in exchanging the contractible cycles in the homology basis with thermal

AdS3 having a spatial cycle and the BTZ a temporal one as contractible.

The (Euclidean) action for thermal AdS3 is given by equation (4.14)

IAdS3 = −4πk=(τ), (5.32)

the action of the BTZ Black hole is related by a transformation τ → −τ−1

IBTZ =4πk

=(τ). (5.33)

As we discussed before, the action of Γ∞\PSL(2,Z) will generate the degenerate

geometries that we sum in the path integral. On the other hand, we know that

5.4. HAWKING-PAGE IS LEE-YANG 69

this group is indexed by a pair of relatively prime integers c, d ∈ Z which

in turn denote a choice of primitive cycles. In fact we expect that when we

cross the boundary of the different fundamental regions, we will obtain different

semiclassical physical scenarios. Notice that the tessellation of the upper-half

plain to consider is just a sub-tessellation (see figure 5.4) of the familiar one

since the group we consider is Γ∞ \ PSL(2,Z).

Figure 5.1: Tesselation of H under the action of Γ∞ \ PSL(2,Z)

The partition function Z(τ) is computed on a canonical ensemble at a fixed

(inverse) temperature =(τ) and a fixed angular potential <(τ). For every fixed

value of τ , the aforementioned infinite number of geometries will contribute

to the partition function. To determine the dominant one in the semiclassical

regime i.e. k → ∞ we just have to extremize Iclass in terms of the group

element γ ∈ Γ∞ \ PSL(2,Z). In the standard fundamental domain (or any

integer translation of it)

F = τ ∈ H||τ | > 1, |<(τ)| ≤ 1

2 (5.34)

the dominant geometry is M0,1 [22], which it nothing but Euclidean thermal

AdS3. If τ is in another domain then we simply have to find the Mobius trans-

formation that takes τ into the standard fundamental domain.


There seems to be a fundamental flaw in this argumentation, we are assum-

ing that our partition function is holomorphically factorizable, but in order to

obtain a phase transition it is necessary to find some non-analytic behavior for

Z over the phase boundaries. The solution to this problem was tackled in a

different context by Lee and Yang [54]. In a nutshell the idea is the following:

systems with finite volume cannot undergo a phase transition, however it’s par-

tition function can have zeroes over the complex plane. In the limit when we

take the volume of the system to infinity, these zeroes condense along the phase

boundaries giving rise to phase transitions. The analog of the infinite volume

limit for our partition function is k →∞ because as we know l = 16Gk implies

that k directly proportional to the AdS3 radius. In fact the ECFT partition

function has the predicted Lee-Yang behavior over the phase boundary as it was

showed in [40] and [5], here we just give a heuristic example that motivates this

fact at tree level but that shows the most relevant features of the phenomenom

. Disregarding the one loop correction, we have Za(τ) = Zh(τ) hence we can

just focus on the holomorphic part. Lets suppose that we are restricted to the

region dominated by either AdS3 or the BTZ black hole, therefore

Zh(τ) ≈ exp(−IAdS3) + exp(−IBTZ). (5.35)

By using equations 5.32 and 5.33 we obtain the approximation

Zh(τ) ≈ exp(−2πikτ) + exp(2πik

τ). (5.36)

Now we notice that the boundary between the regions dominated by the thermal

AdS3 and the BTZ black hole can be parametrized in polar coordinates as

B = τ ∈ H|τ = exp(iφ), φ ∈ [π/3, 2π/3]. (5.37)

After a simple calculation, we find that that the zeroes are characterized by

cos(φ) =2m + 1

4k. (5.38)

The curve B can also be characterized by cos(φ) ∈ [0, 0.5] therefore, which in

turn allows us to show that there are k different zeroes over B. Hence in the

5.5. A CONUNDRUM 71

large k limit, this zeroes should become a dense subset of B giving rise to a

Lee-Yang condensation which represents a Hawking-Page phase transition.

5.5 A conundrum

Besides all the interesting features that the ECFT partition functions have, it

is still an open problem whether this theories exist or not. Some consistency

checks have been performed in [55],[56] and [53].In these works it was proved

that for k = 2 the CFT partition function can be computed for any hyperelliptic

Riemann surface and that the genus 2 partiotion function can be computed

exactly for k = 3. However, it is not possible to assure the existence of the

ECFT’s based in these proofs. On the other hand, there has been work done

in order to show the non-existence of these kind of theories. Gaberdiel in [25]

argued that from k = 42, these theories are inconsistent withe the axioms of

CFT. However, Gaiotto in [26] showed a loop-hole in Gaberdiel’s argument and

on the other hand he showed that, even though ECFT’s may exist, they do not

carry monster symmetry. The question is still open and the main complication

is that we can only rely on the axiomatics of CFT to attempt to construct

ECFT’s.

Chapter 6

Appendix A

Basic notions of 2D CFT

In this appendix we introduce some basic notions and terminology about two

dimensional conformal field theory, an extensive introduction to the subject can

be found in [24]. A conformal field theory (CFT) is a quantum field theory that

is invariant under conformal transformations. In two dimensions there is an

infinite-dimensional group of local conformal transformations, described by the

holomorphic functions.

The Fourier modes Ln, of the energy-momentum tensor T (z), called Virasoro

generators, generate local comformal transformations on the Hilbert space. In

particular, the hamiltionian should be proportional to L0+L0. These generators

obey a commutation algebra called Virasoro algebra

[Ln, Lm] = (n−m)Ln+m +c

12n(n2 − 1)δn+m

[Ln, Lm

]= (n−m)Ln+m +

c

12n(n2 − 1)δn+m

[Ln, Lm

]= 0, (6.1)

where c is called the central charge.

The Hilbert space of a CFT can be constructed as follows. We can define

73

74 CHAPTER 6. APPENDIX A BASIC NOTIONS OF 2D CFT

vacuum state |0〉 such that for n ≥ −1

Ln|0〉 = 0

Ln|0〉 = 0, (6.2)

which in particular implies the invariance of |0〉 under the subalgebra L−1, L0, L1.We define now a class of states that are of capital importance in a CFT. A

primary state of conformal weight (∆, ∆) is a state such that

L0|∆, ∆〉 = ∆|∆, ∆〉L0|∆, ∆〉 = ∆|∆, ∆〉, (6.3)

and for n > 0

Ln|∆, ∆〉 = 0

Ln|∆, ∆〉 = 0〉. (6.4)

Notice that primary states are eigenvalues of the Hamiltonian. From the Vira-

soro algebra, we can show that for n > 0,

[L0, L−n] = nL−n, (6.5)

hence we can increase the conformal dimension of a state by applying L−n to

it. Therefore we can construct exited states by the repeated application of the

negatively indexed Virasoro modes, we call the resulting states descendants

L−n1L−n2 ...L−nk|∆, ∆〉, (6.6)

where the indices are ordered according to 1 ≤ n1 ≤ ... ≤ nk. From equation

(6.5) we can see that the conformal weight of the state (6.6) is

∆′ = ∆ +k∑

l=1

nk := ∆ + N. (6.7)

We call N the level of the descendant. Since we can apply the Virasoro genera-

tors in arbitrary different orders, there is a degeneracy for each level of descen-

dants. For each level we have a degeneracy equal to the number of partitions

75

of integer P (N). The generating function of P (N) is

∞∑n=0

P (n)qn =∞∏

n=1

1

1− qn(6.8)

A tool of great importance in the subject of CFT’s is the so-called operator

product expansion (OPE), which is based in the holomorphic form of the Ward

identities. For a primary field φ of dimension (∆, ∆) the OPE is of the form

(the expressions are similar for the anti-holomorphic sector)

T (z)φ(ω, ω) ∼ ∆

(z − ω)2+

1

z − ω∂ωφ(ω, ω), (6.9)

where ∼ means equality up to regular terms. A very important OPE is the one

of the energy momemtum tensor itself

T (z)T (ω) ∼ c/2

(z − ω)4+

2T (ω)

(z − ω)2+

∂ωT (ω)

z − ω, (6.10)

where c is the central charge of the theory. Another operator whose OPE is

commonly used is the so-called vertex operator Vα(ω) :=: exp(iαφ(ω)):

T (z)V(ω) ∼ α2

2

Vα(ω)

(z − ω)2+

∂ωVα(ω)

z − ω. (6.11)

These results will be used in chapters 4 and 5.

76 CHAPTER 6. APPENDIX A BASIC NOTIONS OF 2D CFT

Bibliography

[1] A. Achucarro, P.K. Towsend (1986) A Chern-Simons action for three-

dimensional anti-de Sitter supergravity theories Phys. Rev. Lett. B180

89

[2] O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri, Y. Oz (1999), Large

N Field Theories, String Theory and Gravity, hep-th/9905111.

[3] Arnowitt, R. Deser, S., Misner C.W. (1962), The dynamics of General

Relavity, in Gravitation: An introduction to current research, ed. L. Wit-

ten.

[4] T. Apostol (1990), Modular Functions and Dirichlet series in Number

Theory, Springer Verlag.

[5] T. Asai, M. Kaneko, H. Ninoyima (1997), Zeroes of Certain Modular

Functions and Applications, Commen. Math. Univ. Sancti Pauli, Vol46-1

93-101.

[6] M. Aschbacher (2004), The Status of the Classification of Finite Simple

Groups, Notices of the AMS.

[7] M. Aschbacher (2000), Finite Groups, Cambridge Studies in Advanced

Mathematics.

[8] R. Borcherds (2002), What is the Monster?, Notices of the AMS, Volume

49, No. 9.

77

78 BIBLIOGRAPHY

[9] R. Borcherds (1992), Monstrous Moonshine and Monstrous Lie Superal-

gebras, Invent. Math. 109, 405444.

[10] M. Banados, C. Teitelboim, J. Zanelli (1992), Black hole in three-

dimensional spacetime, Phys. Rev. Lett.69 1894.

[11] M. Banados, M. Henneaux, C. Teitelboim, J. Zanelli. (1993), Geometry

of the 2+1 Black Hole, gr-qc/9302012.

[12] M. Banados (1995), Global Charges in Chern-Simons Theory and the 2+1

black hole, hep-th/9405171.

[13] J. de Boer, L. Maoz, A. Naqvi (2004), Some aspects of the AdS/CFT

correspondence, hep-th/0407212v3.

[14] J. Brown, M. Henneaux (1986), Central charges in the canonical real-

ization of asymptotic symmetries: An example from three dimensional

gravity, Commun. Math. Phys. 104 207-226.

[15] J.L. Cardy (1986), Operator content of two-dimensional conformally in-

variant theories, Nucl. Phys. B 270.

[16] S. Carlip (1998), Quantum Gravity in 2+1 dimensions, Cambridge Uni-

versity Press.

[17] S. Carlip (2001), Quantum Gravity: A progress report, gr-qc/0108040.

[18] S. Carlip (2005), Conformal Field Theory, (2+1)-Dimensional Gravity,

and the BTZ Black Hole gr-qc/050302v4.

[19] S. Carlip, C. Teitelboim (1994), Aspects of Black Hole Quantum Mechan-

ics and Thermodynamics in 2+1 dimensions, gr-qc/9405070v1.

[20] J. Conway, S. Norton (1979), Monstruos Moonshine, Bull. London Math.

Soc. 11, 308339.

BIBLIOGRAPHY 79

[21] S. Deser, R. Jackiw, G. ’t Hooft (1984), Three Dimensional Einstein Grav-

ity: The Dynamics of Flat spacetime, Annals, Phys. 152 405.

[22] R. Dijkgraaf, J. Maldacena, G. Moore, E. Verlinde (2007), A Black Hole

Farey tail, hep-th/0005003v3.

[23] I. Frenkel, J. Lepowsky and A. Meurman (1981), A Natural Represen-

tation of the Fischer-Griess Monster with the Modular function J as a

character, Proc. Natl. Aad. Sci. 81.

[24] P. Di Francesco, P. Mathieu, D. Senechal (1997), Conformal Field Theory,

Springer Verlag.

[25] M. Gaberdiel (2007), Constraints on Extremal Self-Dual CFTs, hep-

th/0707.4073.

[26] D. Gaiotto (2008) Monster symmetry and Extremal CFTs, hep-

th/0801.0988.

[27] D. Gaiotto, X. Yin(2007), Genus two partition functions of Extremal Con-

formal Field Theories, hep-th/0707.3437v1.

[28] G. Gibbons, S. Hawking, (1977) Action Integrals and Partition Functions

in Quantum Gravity Phys. Rev. D15 2752.

[29] S. Giombi, A. Maloney, X. Yin(2008) One-loop Partition Functions of 3D

Gravity hep-th/0804.1773v1.

[30] S. Hawking (1977) Zeta Function Regularisation of Path Integrals in

Curved Spacetime Commun. Math. Phys. 55 133.

[31] S. Hawking, D. Page (1983) Thermodynamics of Black Holes in Anti-de

Sitter Space Commun. Math. Phys. 87 577.

[32] G. ’t Hooft (1993), Dimensional Reduction in Quantum Gravity, gr-

qc/9310026v1.

80 BIBLIOGRAPHY

[33] G. Hohn (1995) Selbstduale Vertexoperatorsuperalgebren un das Babymon-

ster (Universitat Bonn PhD Thesis) ArXiv:07060236.

[34] H. Iwaniec (1997), Topics in classical automorphic forms, American Math-

ematical Society.

[35] J. Jost (2006), Compact Riemann Surfaces: An Introduction to Contem-

porary Mathematics, Springer Verlag.

[36] K. Krasnov (2000), Holography and Riemann surfaces, hep-th/0005106v2.

[37] W. Li, W. Song, A. Strominger (2008),Chiral Gravity in Three Dimen-

sions, hep-th/08014566.

[38] J. Maldacena, A. Strominger (1998), AdS3 Black Holes and a stringy

exclusion principle, hep-th/9804085v3.

[39] J. Maldacena (1997), The Large N Limit of Superconformal Field Theories

and Supergravity, hep-th/9711200.

[40] A. Maloney, E. Witten (2007), Quantum Gravity Partition Functions in

Three Dimensions, hep-th/0712.01553.

[41] J. Manschot (2007), AdS3 Partition Functions Reconstructed, hep-

th/0707.1159.

[42] D. Mumford, C. Series, D. Wright, D.J. Wright (2002), Indras Pearls:

The Vision on Felix Klein, Cambridge University Press.

[43] H. Rademacher (1938), The Fourier Coeficients of the Modular Invariant

j(τ), American Journal of Mathematics 60 501-512.

[44] Regge, T. Teitelboim C. Role of surface integrals in the Hamiltonian for-

mulation of General Relativity, Ann, Phys 88, 286-318.

[45] M. Ronan (2006), Symmetry and the Monster: One of the Greatest Quests

of Mathematics, Oxford University Press.

BIBLIOGRAPHY 81

[46] A.N. Schellekens (1993), The Seventy Relatives of the Monster Module,

hep-th/9205072.

[47] A. Strominger (1998), Black Hole Entropy from Near-Horizon Mi-

crostates, hep-th/9712251.

[48] C. Teitelboim (1973), How commutators of constraints reflect the space-

time structure, Ann. Phys. 79, 542-557.

[49] W. Thurston, S. Levy (1997), Three-dimensional Geometry and Topology

v1, Princeton University Press.

[50] M. Tuite (2000), Genus two meromorphic conformal field theory,

math/9910136v2.

[51] E. Witten (1988), (2+1)-Dimensional Gravity as an Exactly Soluble Sys-

tem, Nucl. Phys. 46.

[52] E. Witten (1988), Coadjoint orbits of the Virasoro group, Commun. Math.

Phys. 114 1-53.

[53] E. Witten (2007), Three dimensional gravity reconsidered, hep-

th/0706.3359v1.

[54] C.N. Yang, T.D. Lee (1952), Statistical Theory of Equations of State and

Phase Transitions I. Theory of Condensation, Phys. Rev. Lett. 87 404-9.

[55] X. Yin (2007a), Partition functions for Three-Dimensional pure Quantum

Gravity, hep-th/0710.2129v1.

[56] X. Yin (2007b), On Non-handlebody Instantons in 3D Gravity, hep-

th/0711.2803v1.

[57] A.B. Zomolodchikov. (1986), ”Irreversibility of the Flux of the Renormal-

ization Group Flow in a 2D Field Theory, Pis’ma Zh. Eksp. Theor. Fiz.

43 No. 12.

modular invariance and 3d gravity alvaro v¶eliz osorio · modular invariance and 3d gravity alvaro...

Documents