macroscopic and microscopic structure of black ... - Stefan Kowalczyk

AUGUST 2006
MACROSCOPIC AND MICROSCOPIC
STRUCTURE OF BLACK HOLES
Master’s Thesis
SZCZEPAN WOJCIECH KOWALCZYK 1
SUPERVISOR: PROF. DR. ERIK P. VERLINDE
1 email: stefan@kowalczyk.nl
Keywords: Black Holes, Hawking Radiation,
Supersymmetry, D-branes, Elliptic Genus

ABSTRACT
This thesis is an introduction to classical, quantum, and stringy black holes with special
focus on black hole entropy.
We start with a brief review of the geometry of classical black holes in a variety of
different coordinate systems. The curious analogy between thermodynamics and the
laws of black hole mechanics is demystified. We try to give an answer to the question
whether general relativity can be interpreted as a statistical theory by studying a
derivation of the Einstein equations from thermodynamical principles. Using quantum
field theory in Rindler space we give an explicit derivation of the Hawking effect. In
the main part of this thesis we try to understand the microscopic origin of the entropy
of certain extremal, BPS-saturated black holes by making use of D-brane technology.
As a prelude to this famous entropy matching due to Strominger and Vafa, we examine
the so-called correspondence principle between strings and black holes. We also explain
why extremal black holes are sometimes called supersymmetric solitons. In order
to illustrate some standard counting techniques, we count 1/2-BPS heterotic states on
T 4 × T 2 . Finally, we focus on modular forms, elliptic genera and other mathematical
tools, which we use to review the counting of dyons in N = 4 string theory.

ACKNOWLEDGEMENTS
I would like to thank my advisor, Prof. dr. Erik Verlinde, for motivating me to dive
into the physics of black holes and explaining many things to me ever since we met at
the student initiated seminar.
I am also very grateful to Prof. dr. Jan de Boer, Prof. dr. Klaas Landsman, and
Prof. dr. Erik Verlinde for writing recommendation letters for me, which helped me
getting into the State University of New York at Stony Brook.
Last but not least, I would like to thank my girlfriend, Rebecca Hegeman, and my
family for always being there for me.

CONTENTS
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Black Holes in Einstein Gravity . . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Einstein gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 The Schwarzschild black hole . . . . . . . . . . . . . . . . . . . . . 7
2.3 Charged black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3.1 The extreme Reissner-Nordström solution . . . . . . . . . . . 12
3. Motivation for Black Hole Thermodynamics . . . . . . . . . . . . . . . . 14
3.1 Two derivations of the first black hole law . . . . . . . . . . . . . . . 15
3.2 Black hole entropy and the generalized second law . . . . . . . . . . 18
3.3 The holographic principle . . . . . . . . . . . . . . . . . . . . . . . . 19
3.4 Derivation of Einstein equations from thermodynamics . . . . . . . . 20
3.5 Wald entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4. Quantum Black Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.1 Rindler space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
4.2 Field theory in Rindler space . . . . . . . . . . . . . . . . . . . . . . 28
4.2.1 Derivation of the Hawking effect . . . . . . . . . . . . . . . . 30
4.3 Path integral approach . . . . . . . . . . . . . . . . . . . . . . . . . 33
5. Extremal Black Holes in String Theory . . . . . . . . . . . . . . . . . . . 38
5.1 Perturbative microstates in string theory . . . . . . . . . . . . . . . . 38
5.2 Correspondence principle between strings and black holes . . . . . . 40
5.3 Extremal black holes and extended supersymmetry . . . . . . . . . . 41
5.4 Type II supergravity . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
5.5 Form fields and D-branes . . . . . . . . . . . . . . . . . . . . . . . . 49
5.6 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.7 Counting of 1/2 BPS heterotic states on T 4 × T 2 . . . . . . . . . . . 52
5.8 D=5 Reissner-Nordström black holes . . . . . . . . . . . . . . . . . . 55
5.8.1 Derivation of macroscopic entropy . . . . . . . . . . . . . . . 56
5.8.2 Derivation of microscopic entropy . . . . . . . . . . . . . . . 61

5.9 Link with the Strominger-Vafa result . . . . . . . . . . . . . . . . . . 63
6. Mathematical Tools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.1 Modular forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
6.2 (Weak) Jacobi forms . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.3 Elliptic genus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
6.3.1 Definition and properties . . . . . . . . . . . . . . . . . . . . 69
6.3.2 A geometrical description . . . . . . . . . . . . . . . . . . . 71
7. Counting N = 4 Dyons via Automorphic Forms . . . . . . . . . . . . . . 74
7.1 Electric-magnetic duality, S-duality and dyons of charge eθ/2π . . . . 74
7.2 Making use of the attractor mechanism . . . . . . . . . . . . . . . . . 76
7.3 The Dijkgraaf-Verlinde-Verlinde degeneracy formula . . . . . . . . . 77
7.3.1 Consistency checks . . . . . . . . . . . . . . . . . . . . . . . 79
7.4 State counting and the elliptic genus . . . . . . . . . . . . . . . . . . 80
8. Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
Appendix A: Geodesic Deviation and the Raychaudhuri Equation . . . . . . . . 90
Appendix B: Samenvatting in het Nederlands . . . . . . . . . . . . . . . . . . 90
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
1

1. INTRODUCTION
Theoretical physics is founded on two pillars: quantum theory and general relativity.
The former is extremely successful in describing physics at microscopic scales, while
the latter is equally successful in describing physics at cosmological scales. However,
the two are very different. For example general relativity is deterministic, whereas
quantum theory is not. Attempts to unify them into a quantum theory of gravity stum-
ble upon the problem of the non-renormalizability of the theory. Yet one could ask:
is it really necessary to have a unified theory of quantum gravity? The answer is yes.
In the case of purely classical interactions, one could, without such a unified theory,
obtain exact position and velocity of particles, thus violating Heisenberg’s uncertainty
principle. Both quantum theory and general theory are supposed to apply everywhere
and there cannot be two separate everywhere’s. Nature works and does not care about
our renormalization problems. So our mission is clear. We have to find a theory of
quantum gravity. With this in mind, black holes are very interesting objects to study.
For a complete understanding of black holes we need to apply both the idea of dynam-
ical spacetime geometry and of quantum physics simultaneously.
Classically, black holes are completely black. Objects inside their event horizon
are eternally trapped. Their gravitational pull does not allow even light rays to escape.
In the seventies it was found that there was a very close analogy between the laws that
govern black hole physics and the four laws of thermodynamics [1]. The laws of black
hole mechanics become the laws of thermodynamics if one replaces the surface gravity
κ of the black hole by the temperature T of a body in thermal equilibrium, the mass M
of the black hole by the energy of the system E, etc. At a classical level this seems to
be just a formal analogy with no physical meaning, since classically black holes have
strictly zero temperature. However, Hawking showed, under very general assumptions,

1. Introduction 3
that quantum mechanics implies that black holes emit particles [2]. In his so-called
semi-classical approximation 1 , the black hole radiation is exactly thermal and contains
no information about the state of the black hole. This leads to the famous “information
loss paradox”, since infalling particles can carry information, while outgoing ones
cannot [3]. As Hawking argued, this would lead to non-unitary evolution, so that the
basic principles of quantum mechanics would have to be modified. Intuitively, it is
clear that information should not be lost in reality (and thus that the semi-classical
approximation breaks down at some point), but should somehow be returned in subtle
corrections of the outgoing radiation.
The above mentioned analogy between the laws of black holes and the laws of
thermodynamics, combined with the Hawking effect, suggests to assign to a black
hole of area A a ‘macroscopic’ or ‘thermodynamic’ entropy 2
Smacro = A
. (1.1)
4
This ‘macroscopic’ entropy can be measured far away from the black hole and is de-
termined by its mass M, angular momentum J and charge Q. A quantum theory of
gravity should also be able to specify the microstates of the black hole which give rise
to this macrostate. If there are N states corresponding to such a black hole, then the
associated ‘microscopic’ or ‘statistical’ entropy is
Smicro = log N. (1.2)
These two entropies are expected to agree. The microscopic origin of black hole en-
tropy was not understood until the discovery of so-called “D-branes”, dynamical ob-
jects in string theory.
In [4, 5] it was proposed to take the fundamental strings themselves as candidates
for the black hole microstates. This string-black hole correspondence principle pre-
dicts the black hole entropy in terms of string states correctly up to a constant of order
1 In the semi-classical approximation the gravitational field is treated classically (it is a solution of the
Einstein equation) while the matter field is treated quantum mechanically in this classical background
(one replaces the flat Minkowski metric in the wave equation by the curved metric that solves the
Einstein equation).
2 For the moment we set c = � = GN = kB = 1.

1. Introduction 4
unity and it also explains the final state of a Schwarzschild black hole as a highly
excited single string state.
For charged extremal black holes (i.e., black holes with Q = M) one can apply
even more quantitative tests. These turn out to be “BPS solitons” when embedded into
theories with extended supersymmetry [6]. The essential properties of such objects are
determined by supersymmetry, and because of this it is possible to do calculations in
the weak coupling perturbation regime and then to increase the string coupling, thereby
moving to a regime where the backreaction onto spacetime has led to the formation of
a black hole, see figure 1.1.
Fig. 1.1: It turns out that one can study (the microscopic structure of) black holes at strong
string coupling by doing ‘easy’ calculations in the weak string coupling regime.
This weak ↔ strong coupling became a very powerful tool after it was discovered
that charged black holes have a dual, perturbative description in terms of D-branes [7].
Strominger and Vafa showed, by making use of these D-branes, how one can derive
the Bekenstein-Hawking area-entropy relation for a class of five-dimensional extremal
black holes in string theory by counting the degeneracy of BPS soliton bound states
[8]. At the end of this brief introduction, and at the beginning of the real work, let us
state that black holes are very interesting objects to study and may well turn out to be
the ‘Hydrogen atom’ of quantum gravity.

2. BLACK HOLES IN EINSTEIN GRAVITY
In this chapter we examine the geometry of classical black holes in a variety of differ-
ent coordinate systems. Each coordinate system turns out to have its own advantages
and disadvantages.
2.1 Einstein gravity
The crucial idea of Einstein gravity is that spacetime is dynamical: the geometry of
spacetime is determined by the distribution of matter, whereas the motion of matter is
determined by the spacetime geometry.
Mathematically speaking, spacetime is a four dimensional (pseudo-) Riemannian
manifold with metric gµν. We choose our signature to be (− + + +). Once we re-
strict the action to be at most quadratic in derivatives and ignore the possibility of a
cosmological constant, the unique gravitational action is the Einstein-Hilbert action
SEH = 1
2κ2 �
d 4 x √ −gR. (2.1)
Here κ is related to Newton’s constant by κ 2 = 8πGN, g is the trace of the metric, and
R is the Ricci scalar. The coupling of matter to gravity is determined by the principle
of minimal coupling. This principle basically amounts to taking a law of physics
valid in inertial coordinates in flat spacetime, writing it in a coordinate-invariant form
(usually this boils down to replacing partial derivatives by covariant ones), and, finally,
asserting that the resulting law remains true in curved spacetime. We define an energy-
momentum tensor of matter by
where SM is the action for matter.
Tµν = −2 δSM
√ , (2.2)
−g δg µν

2. Black Holes in Einstein Gravity 7
The Einstein equations, which are the Euler-Lagrange equations that one gets by
varying the combined action SEH + SM with respect to the metric,
Rµν − 1
2 gµνR = κ 2 Tµν, (2.3)
where Rµν is the Ricci tensor, form the a central result of Einstein gravity
The left-hand side is a reflection of the geometry of spacetime, while the right-hand
side contains all information about the location and motion of the matter.
2.2 The Schwarzschild black hole
First let’s take a look at the simplest black hole that general relativity predicts: a spher-
ically symmetric static uncharged black hole. It is described by the Schwarzschild
metric
ds 2 �
= − 1 − 2GM
�
dt
r
2 �
+ 1 − 2GM
�−1 dr
r
2 + r 2 dΩ 2 , (2.4)
where dΩ 2 ≡ dθ 2 + sin 2 θdφ 2 is the metric on the unit two-sphere.
According to Birkhoff’s theorem the Schwarzschild solution is the unique spheri-
cally symmetric vacuum solution.
In vacuum Tµν = 0. By taking the trace of equation (2.3) this implies R = 0,
which in turn implies Rµν = 0. So the vacuum solutions to Einstein’s equations are
precisely the Ricci-flat spacetimes.
The coefficients of the Schwarzschild metric in the form (2.4) become infinite at
r = 0 and r = 2GM. It is well known that r = 0 represents an honest singularity
(as may be verified by calculation of the invariant RµνλψR µνλψ ), whereas r = 2GM is
just a coordinate singularity which disappears when going to appropriate coordinates.
An observer in free fall at r = 2GM would not notice anything unusual. Globally,
however, the horizon is a very special place, since to a distant observer it represents
the boundary of the world, or, more precisely, the boundary of the part of the world
which can influence his detectors.
One way in which we can study a spacetime is by looking at its causal structure.
This can be done by studying the behavior of light cones. Setting ds 2 = 0 in equation

2. Black Holes in Einstein Gravity 8
(2.4) gives us for radial null curves (for constant angles θ and φ)
�
0 = − 1 − 2GM
�
dt
r
2 �
+ 1 − 2GM
�−1 dr
r
2 , (2.5)
thus
dt
dr
�
= ± 1 − 2GM
�−1 . (2.6)
r
Clearly, in Schwarzschild coordinates, light cones appear to close up as we approach
the Schwarzschild radius r = 2GM. Does this mean that a light ray that approaches
r = 2GM never gets there? No! This is just an illusion, caused by the fact that our
coordinates are not well behaved at the horizon. Therefor we would like to find a
coordinate system that is better behaved at the horizon.
We can explicitly solve the above equation (2.6) to obtain
�
� ��
r
t = ± r + 2GM ln − 1 + const. (2.7)
2GM
The expression in square brackets is known as the tortoise coordinate r ∗ . Since beyond
r < 2GM our coordinates are not good anyway, we don’t have to be worried about the
logarithm. We have that
dr ∗ �
�
r
= d r + 2GM ln
2GM
�� �
− 1 = 1 − 2GM
�−1 dr. (2.8)
r
Plugging this into the Schwarzschild metric gives
ds 2 �
= 1 − 2GM
��
− dt
r
2 + dr ∗2
�
+ r 2 dΩ 2 . (2.9)
Here r should be thought of as a function of r ∗ . And indeed, in these coordinates we
see that none of the metric coefficients is infinite at the horizon. However, we also see
that in these coordinates the horizon has been pushed to r ∗ = −∞, which is not so
nice. In a moment we will introduce new coordinates to repair this awkward behavior.
Notice that the lines v ≡ t + r ∗ = constant (at constant angles θ and φ) are the
geodesics of infalling light rays. This can immediately be read of from equation (2.7).
The outgoing ones satisfy u ≡ t − r ∗ = constant. In the (v, r) coordinates we can
follow future-directed timelike paths across the event horizon r = 2GM (thereby
arriving in region III in figure 2.1).

2. Black Holes in Einstein Gravity 9
Note that for infalling null curves we have dv/dr = 0, whereas outgoing null
curves satisfy
dv
dr = d(t − r∗ + 2r∗ )
dr
= d(2r∗ )
dr
�
= 2 1 − 2GM
�−1 . (2.10)
r
So we see that this time the light cones don’t close up but tilt over. In the region
0 < r < 2GM the local future light cone points entirely toward the singularity at
r = 0.
At first sight it may look as if we were done. After all, our spacetime includes
r ≤ 2GM. However, we have no guarantee that there are no other directions in which
to extend our manifold. In fact there are other directions. For example, we could have
taken (u, r) as our new coordinates, making it possible to cross the horizon across past-
directed curves (thereby arriving in region IV in figure 2.1). In this way one arrives
at a different place than by crossing the horizon along future-directed curves in (v, r)
coordinates.
Let us now introduce new coordinates that are good all over the place:
¯v = e v/4GM , ū = −e −u/4GM
(2.11)
In these new coordinates (¯v, ū, θ, φ), known as the Kruskal coordinates 1 , the metric
becomes
ds 2 = − 32G3M 3
e
r
−r/2GM dūd¯v + r 2 dΩ 2 . (2.12)
Since the metric components are nonsingular at r = 2M we can analytically continue
the solution to the whole region −∞ < ū, ¯v < ∞. This is the extended Schwarzschild
solution, shown in figure 2.1.
For every r, t we have two solutions for ū and ¯v. Therefore the region r < 2GM
occurs twice in (ū,¯v) space; these two regions are indicated by III and IV. Region III
is the black hole. Once an observer has entered region III, he or she will not be able
to escape from it, and will fall into the singularity at r = 0 within finite proper time.
1 The related coordinates (T, R, θ, φ), where T ≡ (¯v + ū)/2 and R ≡ (¯v − ū)/2, are sometimes also
called Kruskal coordinates. The nice thing about taking (T, R, θ, φ) is that in that case one coordinate
is timelike while the remaining ones are spacelike.

2. Black Holes in Einstein Gravity 10
u
I I
r=0
r=0
T
III
IV
I
v
r=2GM
R
r=2GM
Fig. 2.1: The Schwarzschild solution in Kruskal coordinates.
Region IV is simply the time-reverse of region III. It can be thought of as a white hole.
The central region, |ū| ≪ 1, |¯v| ≪ 1, will turn out to be very important in our study
of the quantum mechanics of the black hole. If we ignore the curvature in this central
region, and replace it by flat space, we have the so-called Rindler space.
2.3 Charged black holes
In this section we will briefly describe some main facts about electrically charged black
holes. For compactness we set G = 1 from now on. The energy-momentum tensor for
electromagnetism is given by
Tµν = FµρF ρ ν − 1
4 gµνFρσF ρσ , (2.13)
where Fµν is the electromagnetic field strength tensor. Thus the gravitational field
equations are
Rµν − 1
2 gµνR = κ 2
�
FµρF ρ ν − 1
ρσ
gµνFρσF
4
�
. (2.14)
A generalized version of Birkhoff’s theorem now states that the unique spherically
symmetric solution of equation (2.14) is the Reissner-Nordström solution
ds 2 = −C(r)dt 2 + dr2
C(r) + r2 dΩ 2 , (2.15)

where
2. Black Holes in Einstein Gravity 11
C(r) = 1 − 2M
r
Q2
+ . (2.16)
r2 In this expression Q is the charge of the black hole. By definition
Qe = 1
�
4π
⋆
F, (2.17)
where S 2 ∞ is an asymptotic spacelike sphere at infinity and ⋆ F is the dual of the field
strength two-form F = 1
2 Fµνdx µ ∧ dx ν . Here µ, ν denote the indices t, r, φ, θ.
Although isolated magnetic charges have never been observed in nature, we may
write in general Q = � Q 2 e + Q 2 m where Qe is the electric and Qm the magnetic charge
(which is defined like the electric charge (2.17) but without the star). In the electrically
charged case the radial component of the electric field is Er = Frt = Qe/r 2 and in
the magnetically charged case Br = Fθφ = Qm/r 2 . (The electric field is curl free,
whereas the magnetic field is divergence free.)
Clearly for a charged black hole the horizon structure is more complicated than for
the Schwarzschild black hole. By solving
1 − 2M
r
S 2 ∞
Q2
+ = 0 (2.18)
r2 we get two horizons: an inner one (r−) and an outer one (r+) located at
� �
r± = M 1 ± 1 − Q2
M 2
�
. (2.19)
There are three cases to distinguish: Q 2 > M 2 , Q 2 < M 2 , and Q 2 = M 2 .
In the case Q 2 > M 2 the metric is completely regular all the way down to r = 0.
At r = 0 we have a singularity: a naked one, which according to the cosmic censorship
conjecture is not physical. It is however good to note that objects satisfying Q 2 > M 2
do exist in nature (an electron is an example of such an object), but just cannot be
described by classical general relativity.
The case M 2 > Q 2 is called the non-extreme Reissner-Nordström black hole and
is certainly the most realistic case, but we will not discuss it here.
Our main point of focus will be on the somewhat less realistic, though mathemat-
ically easier to study, so-called extreme Reissner-Nordström black holes, for which

2. Black Holes in Einstein Gravity 12
the charge is equal to the mass, Q 2 = M 2 . In this case the two horizons coincide at
r+ = r− = M.
2.3.1 The extreme Reissner-Nordström solution
Plugging Q2 = M 2 into equation (2.16) gives
�
C(r) = 1 − Q
�2 . (2.20)
r
The metric thus takes the form
ds 2 � �2 r − Q
= − dt
r
2 � �2 r
+ dr
r − Q
2 + r 2 dΩ 2 . (2.21)
It is convenient to define a shifted radial coordinate
ρ ≡ r − Q. (2.22)
We can write the metric as
ds 2 � �2 ρ
= − dt
ρ + Q
2 � �2 ρ + Q
+ d(ρ + Q)
ρ
2 + (ρ + Q) 2 dΩ 2
(2.23)
�
= − 1 + Q
�−2 dt
ρ
2 �
+ 1 + Q
�2 [dρ
ρ
2 + ρ 2 dΩ 2 ]. (2.24)
And since dρ 2 + ρ 2 dΩ 2 is just the flat metric in three spatial dimensions, we can write
where
ds 2 = −H −2 (�x)dt 2 + H 2 (�x)d�x 2 , (2.25)
H = 1 + Q
. (2.26)
|�x|
For N black holes with charges Qi (or equivalently multiple masses Mi) sitting at
positions �xi we have [9]
H = 1 +
N�
i=1
Qi
. (2.27)
|�x − �xi|
Such a stationary configuration of N extremal black holes is possible because of the
exact cancelation of the electric repulsion and gravitational attraction. This is called
the no-force property.

2. Black Holes in Einstein Gravity 13
We can find an additional special property of the extremal black hole geometry as
follows. Near the horizon ρ = 0 the metric (see (2.23) and (2.24)) behaves like
Defining another new coordinate
so that dz/dr = Q 2 /r 2 = z/r, we find
ds 2 → − ρ2
Q 2 dt2 + Q2
ρ 2 dρ2 + Q 2 dΩ 2 2
z ≡ Q2
ρ
(2.28)
(2.29)
ds 2 → Q2
z 2 (−dt2 + dz 2 ) + Q 2 dΩ 2 2. (2.30)
This is AdS2×S 2 , the direct product of an Anti-de-Sitter spacetime ((SO(2, 1)/SO(1, 1))
with a sphere. This is a maximally symmetric space. It is interesting to note that the
Reissner-Nordström spacetime is also maximally symmetric and asymptotically flat.
The extreme Reissner-Nordström black hole will turn out to possess a certain super-
symmetry which makes it an interesting and relatively easy object to study.

3. MOTIVATION FOR BLACK HOLE THERMODYNAMICS
An amazing result from black hole physics is that one can derive a certain set of laws
for black holes that has the same structure as the laws of thermodynamics. They are
called the laws of black hole mechanics [1]. One can think of these “black hole laws”
as theorems within the theory of general relativity. In this section we will investigate
the question whether this curious analogy between thermodynamics and black hole
laws is more than a formal coincidence.
Let us first introduce some terminology. We assume that the reader is familiar with
Killing vectors (see for example [10] or [11]). A vector ξ is called a Killing vector if
it satisfies the Killing equation
∇µξν + ∇νξµ = 0, (3.1)
where ∇µ is the covariant derivative. Remember that if a spacetime has a Killing
vector, we can find a coordinate system in which the metric is independent of one of
the coordinates, and the quantity pµξ µ will be constant along geodesics with tangent
vector p µ . If a Killing vector field ξ µ is null along some null hypersurface Σ, we say
that Σ is a Killing horizon 1 of ξ µ .
Now let K be a Killing horizon with normal Killing field ξ µ . The Killing horizon
is defined by the equation ξ ν ξν = 0. Therefore the gradient ∇ µ (ξ ν ξν) is normal to the
horizon. By definition ξ µ is also normal to the horizon. So the two must be proportional
to each other. In other words, there exist a proportionality function κS on K. κS is
defined by the equation
∇ µ (ξ ν ξν) = −2κSξ µ
(3.2)
1 It can be proven that in Einstein gravity event horizons of stationary black holes are always Killing
horizons (i.e., one can always find an appropriate Killing vector field). See [12].

3. Motivation for Black Hole Thermodynamics 15
and is called the surface gravity 2 . It turns out that this so-called surface gravity is
actually a constant function on the horizon,
κS = const. (3.3)
This is the case whenever Einstein’s equations hold, with the energy-momentum tensor
satisfying the dominant energy condition (which basically means that the energy and
pressure always remain positive), see [1].
This constancy of the surface gravity for a stationary black hole is called the zeroth
law of black hole mechanics. It is reminiscent of the zeroth law of thermodynam-
ics, which states that the temperature is uniform everywhere in a system in thermal
equilibrium. This suggests that surface gravity is analogous to temperature.
The first law of black hole mechanics is a law of energy conservation. It gives a
relation between the change in mass (δM), horizon area (δA), charge (δQ), and angular
momentum (δJ) of a black hole when it is perturbed, as follows
δM = κS
δA + µδQ + ΩδJ. (3.4)
8π
Obviously, it has the same mathematical form as the first law of thermodynamics
dE = T dS − P dV + µdN. (3.5)
Note that the µ in equation (3.5) stands for the chemical potential, but in equation (3.4)
is does not. (We will soon find out where it does stand for.) Above we concluded that
surface gravity is analogous to temperature. We can now guess that area plays the role
of entropy.
3.1 Two derivations of the first black hole law
We will now present a short derivation of equation (3.4). Obviously, we will not find
the ΩδJ term, since the black hole we study is not spinning. Starting from equation
2 One can think of the surface gravity κS as the limiting force which would have to be exerted at
infinity to keep an object stationary at the horizon. Another - basically the same - way of thinking about
it, is that surface gravity is the acceleration of a static particle near the horizon as measured at spatial
infinity.

(2.16) we have
So
3. Motivation for Black Hole Thermodynamics 16
C ′ (r+)dr+ = dC(r+) = − 2dM
r+
1
2 C′ (r+)r+dr+ = −dM + Q
+ 2Q
r2 dQ. (3.6)
+
r+
dQ. (3.7)
Now we can plug in the area of the (outer) horizon into this equation using A = 4πr 2 +,
so dA = 8πr+dr+ and we get
dM = −1
16π C′ (r+)dA + µdQ, (3.8)
where we put µ ≡ Q/r+. We see that this indeed looks like equation (3.4), but without
the ΩδJ term. This is as expected, since our black hole is not spinning.
In order to find this spinning term one has to consider a rotating black hole. We
will now give another derivation of equation (3.4), but this time for a rotating black
hole. The metric for a charged rotating black hole is the Kerr-Newman metric, [13]
[14]. We will use it in the form
ds 2 � �
2 2 ∆ − a sin θ
= −
dt
Σ
2 − 2a sin2 θ(r2 + a2 − ∆)
dtdφ +
Σ
� �
2 2 2 2 2 (r + a ) − ∆a sin θ
+
sin
Σ
2 θdφ 2 + Σ
∆ dr2 + Σdθ 2 , (3.9)
where � Σ = r 2 + a 2 cos 2 θ,
∆ = r 2 + a 2 + Q 2 − 2Mr.
Here a is the angular momentum per unit mass, a = J/M. Notice that for a = 0 the
Kerr-Newman metric reduces to the Reissner-Nordström metric.
Now we will study the variation of the area of the event horizon of a Kerr black
hole (A) with respect to M and J. Let us first write down an explicit equation for A:
A =
=
�
�
r=r+
r=r+
dθdφ √ gθθgφφ
�
dθdφ Σ (r2 + + a2 ) 2 sin2 Σ
= 4π(2Mr+ − Q 2 ), (3.10)

3. Motivation for Black Hole Thermodynamics 17
where the rotating version of equation (2.19) was used, namely
r+ = M + � M 2 − a 2 − Q 2 . (3.11)
We are now ready to calculate the variation of A with respect to M and J ≡ Ma.
Ignoring the δQ contribution for the moment we have from equation (3.10) that
Let’s work this out:
So, since
we get
1
8π δA = r+δM + Mδr+. (3.12)
δr+ = δ(M + � M 2 − a2 − Q2 )
= δM + 1
2 (M 2 − a 2 − Q 2 1
−
) 2 2(MδM − aδa)
= δM + (r+ − M) −1 (MδM − aδa). (3.13)
δa = δ(J/M) = 1 a
δJ − δM (3.14)
M M
1
8π = r+δM + MδM + (r+ − M) −1 {(M 2 + a 2 )δM − aδJ}. (3.15)
Rewriting this gives
δM = r+ − M
8π(r2 + + a2 a
δA +
) r2 δJ. (3.16)
+ + a2 We see that κS should equal (r+ − M)/(r 2 + + a 2 ) and ΩH = a/(r 2 + + a 2 ) in order to
get the desired
δM = κS
8π δA + ΩHδJ. (3.17)
Note that if we would also have varied with respect to Q we would have found an
additional µδQ term in the above equation, as one can immediately see from equation
(3.10).

3. Motivation for Black Hole Thermodynamics 18
3.2 Black hole entropy and the generalized second law
So what do we learn from this first law of black hole mechanics? By comparing
equations (3.4) and (3.5) it seems that black holes have entropy (and that their entropy
is proportional to their area). This was already anticipated by Bekenstein, who noted
that we can violate the second law of thermodynamics by throwing matter into a black
hole. It is clear that the second law must be reformulated in the presence of black
holes. The way this is done is by associating a certain entropy with each black hole
and reformulating the second law to a generalized second law (GSL) which states that
the sum of the entropy outside the black hole and the entropy of the black hole itself
will never decrease [15].
It is not obvious at all that this should work, and actually at a purely classical level
it simply appears to be false. Perhaps the easiest way to see why, is by considering a
box containing entropy in the form of radiation which can be lowered to the horizon
of a black hole and be dropped in. Now, in principle, for an ideal infinitesimal box all
of its energy can be extracted at infinity, so when the box falls into the black hole, no
mass is added to the black hole. Therefore its area and thus its entropy is unchanged.
The entropy of the exterior, however, has decreased. It thus appears that the GSL is
simply not true at a purely classical level.
Another problem was that if black holes really have entropy, they must have a
temperature because of the thermodynamic identity 1/T = ∂S/∂E. Since E ∼ M
and S ∼ A ∼ M 2 we have that ∂S/∂M ∼ M ∼ T −1 and our guess would thus be that
black holes have a temperature T ∼ M −1 . Indeed, Hawking’s calculation [2] shows
that a black hole of mass M indeed has a temperature
and we get that
T =
�
, (3.18)
8πGM
S = A
. (3.19)
4G�
We will look at a derivation of this using the path integral approach in section 4.3.
This solves yet another mystery, because it looked like the units did not match, since

3. Motivation for Black Hole Thermodynamics 19
entropy is dimensionless, whereas horizon area is length squared. But now we see that
indeed S = A/4 where the area is measured in units of the Planck length.
There are also second and third laws of black hole mechanics, which we will not
discuss here, see for example [1].
3.3 The holographic principle
At this point it is worthwhile to pause for a moment and wonder about the strange fact
that black hole entropy scales with its area and not with its volume, as we are used
for thermodynamic systems. (Consider for example a simple thermodynamic system
of volume V divided in little unit cubes which can take either value 0 or 1. The total
number of states for such a system is 2 V . To get the entropy we have to take the
logarithm of this quantity, so we indeed see that S ∝ V .)
Imagine the following. Suppose we have a volume V somewhere in the universe;
for simplicity let’s take it to be a sphere with radius R. It seems logical that the amount
of information (or entropy) in this volume depends on the volume V ∝ R 3 . Our claim,
and also the claim of the holographic principle which is due to ’t Hooft [16], is that
this is not the case! The amount of information we can store in a sphere with radius R
is proportional to the area A ∝ R 2 . So, so to speak, if we can put a maximum of 100
Gigabyte into a spherical laptop with radius R, we can put only 400 (and not 800!)
Gigabyte into a laptop of radius 2R.
We can understand this as follows. Our volume V can be turned into a black hole
in the same region by throwing matter in. Its entropy will than be equal to the area
SBH ∼ A surrounding V , in suitable units. Thus, we learn that it is impossible for
the entropy of our volume to scale with the volume, since we always can make a black
hole with entropy proportional to area out of it.
So in short, the holographic principle states that information (entropy) scales with
area and not with volume, as one would expect. More precisely, it states that a field
theory on a two-dimensional closed surface suffices to describe all processes that take
place in the three-dimensional volume within this surface. Further evidence for holog-

3. Motivation for Black Hole Thermodynamics 20
raphy is given by the AdS/CFT correspondence [17], which basically is a dictionary
encoding the equivalence between string theory on five dimensional Anti-de-Sitter
space and conformal fields on its four dimensional boundary.
3.4 Derivation of Einstein equations from thermodynamics
Let us turn the logic around and show that not only does general relativity give ther-
modynamic kind of identities, but that one actually can derive the Einstein equations
from thermodynamics principles.
In this section we will study Jacobson’s derivation [18] of the Einstein equations
from thermodynamical principles in some detail. In turns out that locally the Einstein
equations can be derived from the proportionality of entropy and horizon area together
with the fundamental relation δQ = T dS. If this is indeed the right way to do things,
gravity should not be quantized but rather seen as a many particle effect (just like, for
example, sound waves). For some general relativity background (geodesic deviation,
Raychaudhuri equation, etc.), see Appendix A.
Jacobson considered locally static horizons and created locally accelerated ob-
servers in order to study information entering the spacetime through the local past
horizon of the local Rindler space. Assuming local equilibrium he then derived the
Einstein equations. He was thus led to the conclusion that the Einstein equations give
an effective field description of some more fundamental fields, and actually do not
describe a fundamental force: the Einstein equation is an equation of state.
So what exactly is meant here? It is clear what we mean by heat in thermodynam-
ics: it is energy that flows between degrees of freedom that are not macroscopically
observable. We will define heat to be energy that flows across some causal horizon
(not necessarily a black hole event horizon). Next, we have to say what we mean with
the temperature of the system into which the heat is flowing. We define it to be the
Unruh temperature [19] associated with a uniformly accelerated observer hovering just
inside the horizon. Since different accelerated observers will measure different things,
the same observer should be used to measure the heat flow (which is defined by the

3. Motivation for Black Hole Thermodynamics 21
energy flux through the horizon). Another quantity appearing in the relation Q = T dS
that needs our attention is the entropy, S. We use a comoving volume element which
moves along with the energy towards the horizon to define the entropy. As the volume
element falls towards the horizon it is flattened (as seen by an outside observer) and the
entropy of the volume element scales as the surface area of the local horizon. Having
identified all the relevant quantities, we will now derive the (local) Einstein equations.
Let p be some point in the spacetime. According to the equivalence principle we
can find local coordinates in which the metric is Minkowski with vanishing first deriva-
tives at p. A local horizon can be created by considering Lorentz boosted observers
near p, and is called a “local Rindler horizon”. In Jacobson’s terminology, we have
a small spacelike 2-surface element P through p, whose past directed null normal
congruence to one side (the “inside”) has vanishing expansion and shear at p (see Ap-
pendix A). It is always possible to choose this P such that both the expansion and
shear vanish close to p. He calls the past horizon of such a p the “local Rindler horizon
of P”.
Let χ µ be a (local) Killing field generating boosts orthogonal to P and vanishing
at P. Note that our Rindler spacetime is local, and thus posses only local symmetries,
which means that χ µ has the Killing property only in a spacelike neighborhood of p. (It
exists also in the future and past of p, but does not satisfy Killing’s equation there.) The
heat flow is defined to be Tµνχ µ , where Tµν is the matter energy-momentum tensor.
Now let us consider any such local Rindler horizon through a spacetime point p
(see figure 3.1). We assume that all the heat flux across the horizon is energy carried
by matter. Since we take the thermodynamic limit we can assume all fluctuations in
Tµν to be negligible. The heat flux to the past of P can now be obtained by integrating
the boost-energy current of the matter, Tµνχ µ , over a pencil of generators of the “in-
side” past horizon H of P. The χ µ can be reparameterised by using vectors k µ , which
are tangent to the horizon generators for an affine parameter λ, which vanishes at a
neighborhood of point p and is negative to the past of P, as follows: χ µ = −κλk µ ,
where κ is the acceleration of the Killing orbit on which the norm of χ is one. The
volume dV over which we integrate is the volume of the spacelike hypersurface gen-

3. Motivation for Black Hole Thermodynamics 22
¢¡
a
Q
Fig. 3.1: Spacetime diagram showing the heat flux δQ across the local Rindler horizon H of a
2-surface element P. Each point in the diagram represents a two dimensional spacelike
surface. The hyperbola is a uniformly accelerated worldline, and χ a is the approximate
boost Killing vector on H. Adapted from [18].
erated by the timelike geodesic vector field k µ . This is the natural normal vector to the
volume. Putting all this in one equation, we get
�
δQ = −κ
λTµνk
H
µ k ν dλdA, (3.20)
where we substituted dV = dλdA (where A is the area element on a cross section of
the horizon) since near the horizon the volume becomes flatter and flatter, as seen by
the observer on χ µ . This gives us, on the one hand, from dS = δQ/T , an expression
for dS. On the other hand, as we have seen, entropy seems to be proportional with
area and we can assume that the entropy variation associated with a piece of horizon
satisfies dS = ηδA, where δA is the area variation of a cross section of a pencil of
generators of H and η is a constant (equal to 1/4). So by working out δA we can find
some condition. It will turn out that this condition is precisely the Einstein equation.
One can show that the change of the horizon area is
dA
dλ
= θA, (3.21)
with θ the expansion of the congruence of null geodesics k µ , which generates the
horizon. Integrating (3.21) the horizon area element at affine parameter λ is
A(λ) = A0e R λ
0 dλ′ θ . (3.22)

3. Motivation for Black Hole Thermodynamics 23
where A0 is the surface element of the horizon at λ = 0. Expanding the exponent we
find
A(λ) ≈ A0(1 + λθ(0) + λ2
2
dθ
(0)). (3.23)
dλ
Noting that, due to the fact that the accelerated observer observes a local heat bath
which is homogeneous, there is no contribution from the first order term and that we
can use the Raychaudhuri equation (8.11) to work out dθ/dλ. We get
A(λ) ≈ A0
�
1 + λ2
2 Rµνk µ k ν
�
. (3.24)
Here we used the fact that the local Rindler horizon was chosen to be instantaneously
stationary at P, so that θ and σ (the shear) vanish at P. We can thus neglect the con-
tributions from θ 2 and σ 2 = σµνσ µν when integrating to find θ near P. Summarizing
all this,
�
δA = − λRabk
H
µ k ν dλdA. (3.25)
As announced, comparing (3.20) and (3.25) we conclude that
or, for λ �= 0,
− 2πλTµνk µ k ν dλdA = ηλRµνk µ k ν dλdA, (3.26)
− 2π
η Tµνk µ k ν = Rµνk µ k ν . (3.27)
Because the k µ are null vector fields we can add a term fgµνk µ k ν proportional to the
metric to the right hand side (3.27). As we will see in a moment, this corresponds to
the freedom one has in adding an integration constant in the Einstein equation. There
is a close relation between f and Λ, the cosmological constant. We have
− 2π
η Tµνk µ k ν = Rµνk µ k ν + fgµνk µ k ν . (3.28)
Since δQ = T dS holds for any family of geodesics, the above equation holds for all
null vector fields on the horizon and we have
2π
η Tµν + Rµν + fgµν = 0. (3.29)

3. Motivation for Black Hole Thermodynamics 24
We would like to find out what f is. A way to find this is out, is to take the diver-
gence of equation (3.29). The divergence of the energy momentum tensor vanishes
(by definition) and using the Bianchi identities
we find
∇µ(Rµν − R
2 gµν) = 0 (3.30)
f = − R
2
with C some constant. Plugging this f into equation 3.29 gives
Rµν − 1
2 gµνR + Cgµν = − 2π
η Tµν
+ C (3.31)
(3.32)
which is precisely the Einstein equation if we identify C with (minus) the cosmological
constant, C = Λ, and with η = 1/4.
A perhaps somewhat preliminary conclusion of all this would be that the general
theory of relativity can be interpreted as a statistical theory. The equivalence principle
corresponds to the statement that the neighborhood of each point in spacetime is in
local thermodynamic equilibrium.
3.5 Wald entropy
We saw that for the Einstein-Hilbert action (2.1) we get the Bekenstein-Hawking equa-
tion S = AH/4. In general (in string theory) we can get a more complicated action I
with corrections involving the Riemann tensor and other fields. This will modify the
laws of black hole thermodynamics. However, Wald [20] has a ‘simple’ solution for
this: he gives the following formal expression for the entropy
�
ˆS
δI
:= 2π
S2 ɛ
δRµναβ
µν ɛ ρσ√ hd 2 Ω. (3.33)
Here Rµναβ is regarded as formally independent of the metric.
As an example let us calculate the Wald entropy for the Einstein-Hilbert action
L = 1
16π Rµναβg µα g νβ . (3.34)

We get
Thus
3. Motivation for Black Hole Thermodynamics 25
∂L
∂Rµναβ
= 1 1
16π 2 (gµαg νβ − g να g µβ ). (3.35)
ˆS = 1
�
8 S2 1
2 (gµαg νβ − g να g µβ √
2
)ɛµνɛαβ hd σ)). (3.36)
Only the tt and rr terms in the metric survive. So we get
ˆS = 1
�
8
For the Schwarzschild black hole
so we get
g tt =
g rr =
which is indeed what we hoped to get.
S2 2g tt g rr√ hd 2 σ. (3.37)
�
1 − 2GM
�
, (3.38)
r
�
1 − 2GM
�−1 , (3.39)
r
ˆS = 1
�
4 S2 √
2
hd σ = A/4, (3.40)
Wald’s redefinition of entropy is needed (besides the proper inclusion of higher
curvature terms on the supergravity side) to find agreement between geometric entropy
computed in four-dimensional N = 2 supergravity and the statistical entropy found by
state counting [21].

4. QUANTUM BLACK HOLES
First, we shall describe the standard picture of Rindler space arising from the near-
horizon approximation of the Schwarzschild metric. Then we derive the Hawking
effect following the treatment by ’t Hooft in [22]. Some attention will be paid to
the gravitational back reaction. We also comment on the information loss paradox, or
better said, on information recovery. Finally, we discuss an alternative approach which
makes use of Feynman’s path integral method.
4.1 Rindler space
Let us start with motivating the introduction of the Rindler geometry. It is not difficult
to see that the (t − r)-part of the Schwarzschild metric (2.4) is very well approximated
by flat space near the horizon. Introducing the positive coordinate x defined by
r = 2M + x2
8M
(4.1)
near the horizon r ≈ 2M (or r ≈ 0) we get for the Schwarzschild metric the approxi-
mation
ds 2 ≈ −(κx) 2 dt 2 + dx 2
+ (2M)
� �� �
2−dim Rindler space
2 dΩ 2
,
� �� �
(4.2)
2−sphere
where κ = 1/4M is the surface gravity of the Schwarzschild metric and where we
used the near horizon expansion
1 − 2M
r
= 1 −
2M
2M + x2
8M
= 1 + x2
16M 2 − 1
1 + x2
16M 2
= x2 x4
+ O( ). (4.3)
16M 2 M 4
We can easily check that x = 0 is just a coordinate singularity and that Rindler space-
time is just (a part of) Minkowski space in unusual coordinates by introducing Kruskal-
type coordinates
U = −xe −κt , V = xe κt . (4.4)

One indeed checks that
dUdV =
4. Quantum Black Holes 27
�
∂U ∂U
dx +
∂x ∂t dt
��
∂V ∂V
dx +
∂x ∂t dt
�
= (−e −κt dx + κxe −κt dt)(e κt dx + κxe κt dt)
= (κx) 2 dt 2 − dx 2 = −ds 2 , (4.5)
which is Minkowski space in null coordinates 1 . So the Rindler coordinates with x > 0
cover precisely the U < 0, V > 0 region of Minkowski space.
For the study of uniformly accelerated observers (simply called ‘Rindler observers’;
we encountered them already in section 3.4) it is convenient to introduce new coordi-
nates (ξ, η) on the right wedge of two-dimensional Minkowski space (x > |t|), given
by
t = a −1 e aξ sinh aη, (4.6)
x = a −1 e aξ cosh aη, (4.7)
where a is a positive constant. As one can easily check, the near horizon two-dimensional
Rindler metric becomes
Lines of constant ξ are hyperbolae
ds 2 = e 2aξ (dη 2 − dξ 2 ). (4.8)
x 2 − t 2 = a −2 e 2aξ = constant, (4.9)
and represent the world lines of uniformly accelerated observers, while lines of con-
stant η are straight (from (4.6) and (4.7) we see that x ∝ t). This is indicated in figure
4.1.
We can also define coordinates in the region II by
t = −a −1 e aξ sinh aη, (4.10)
x = −a −1 e aξ cosh aη. (4.11)
We will use the coordinates (ξ, η) for both region I and II although their ranges are not
the same in each region. We will simply indicate to which region we are referring. The
1 Setting U = T − X and V = T + X we get the standard form ds 2 = −dT 2 + dX 2 .

4. Quantum Black Holes 28
t
constant
z
constant
Fig. 4.1: Rindler spacetime is a wedge of Minkowski spacetime. Lines of constant ξ represent
the world lines of uniformly accelerated observers.
advantage is that we will be able to use the same metric, namely the one in equation
(4.8), for both region I and II.
The important point to keep in mind is that Rindler space is a model for a gravita-
tional field, although there are no real gravitational forces present.
4.2 Field theory in Rindler space
For the above discussion we could safely ignore two spatial directions, but now we
want to study four and not two dimensional Minkowski space, so let us repair this:
we consider Minkowski space described by the coordinates t, z and ˜x = (x, y) and
introduce new coordinates ξ, η and ˜x, given by
z = a −1 e aξ cosh aη,
t = a −1 e aξ sinh aη, (4.12)
˜x = ˜x.
Note that the coordinate z is playing the role of the coordinate x in section 4.1.
In this section we will consider a quantum field theory, possibly with interacting
particles, in a flat Minkowski background. The Hamiltonian, HM, of such a theory can
usually be written as
HM =
�
HM(�x)d 3 �x, (4.13)

4. Quantum Black Holes 29
where HM is the Hamiltonian density. A time boost in Rindler space is generated by
the operator
HR =
�
(H(�x)z − Pz(�x)t)d 3 �x. (4.14)
This can be understood by analogy by looking at how the boost matrix
�
cosh α
�
sinh α
sinh α cosh α
works on a vector (z, t). In the infinitesimal case we get
� � � � � � � � �
z 1 α z z + αt z
→
=
=
t α 1 t αz + t t
We have that
�
�
t
+ α
z
(4.15)
�
. (4.16)
δϕ(z, t) = tδzϕ − zδtϕ. (4.17)
Usually one considers the operator HR at t = 0,
�
HR = HM(�x)zd 3 �x. (4.18)
In contrast with (4.13), this is not bounded from below. We would like to write this as
with
H1 =
HR = H1 − H2, (4.19)
�
�
z>0
H(�x)zd 3 �x, (4.20)
H2 = H(�x)|z|d
z

4. Quantum Black Holes 30
In the limit ξ → −∞ the last two terms do not contribute and we get plane waves.
There is, however, a problem in that there is an asymptotic region at ξ → −∞ into
which wave packets may disappear at η → ∞ or may come from at η → −∞. We
need a boundary condition at ξ = −∞ in order to formulate scattering against the
Rindler horizon. We do not know this boundary condition and would like to find it.
The explanation for this weird situation is that there seems to be thermal (Hawking)
radiation emerging from the region ξ = −∞. We will now consider the derivation
of this phenomenon. Good references are Gerard ’t Hooft’s article in [22] and the
excellent book by Birrell and Davies [23].
4.2.1 Derivation of the Hawking effect
Let us consider only non-interacting scalar particles. Other cases are quite similar. For
a scalar field Φ in Minkowski space we can make an expansion (see [24] for a nice
introduction to quantum field theory)
�
Φ(�r, t) =
d 3� k
�
2k0( � k)(2π) 3
Things get nicer if we work in lightcone coordinates
We define new annihilation operators a1 by
�
a( �k)e i�k�r−ik0t †
+ a ( � −i
k)e � �
k�r+ik0t
. (4.24)
u = (t − z)/2, v = (t + z)/2, (4.25)
k+ = k0 + k3, k− = k0 − k3. (4.26)
a( � k) � k0 =: a1( ˜ k, k+) � k+. (4.27)
It turns out to be convenient to Fourier transform a1 with respect to the logarithm of
k+ (see [22]). We will call the new annihilation operator obtained in this way a2, so
a1( ˜ k, k+) � k+ = (2π) −1/2
� ∞
where µ 2 = ˜ k 2 + m 2 = k+k−.
−∞
dωa2( ˜ k, ω)e −iω ln(k+/µ) , (4.28)

From the Minkowski Hamiltonian
we get the Rindler Hamiltonian
4. Quantum Black Holes 31
HM(r, t) = 1
2 ((� ∂ϕ) 2 + (∂tϕ) 2 + m 2 ϕ 2 ) (4.29)
HR =
�
zHM(�r, 0)d 3 �r, (4.30)
and one can check by plugging a bit modified version of (4.24) and (4.28) into (4.30)
that
HR =
�
d 2˜
� ∞
k dωa
−∞
†
2( ˜ k, ω)a2( ˜ k, ω). (4.31)
So indeed, our claim that a2 is an annihilation operator was well grounded; it corre-
sponds to a Rindler energy ω.
Nevertheless a2 is not the annihilation operator we want to work with. As is well
motivated in [22] we are interested in an operator aI( ˜ k, ω) that is defined as follows:
and similarly
aI( ˜ k, ω) √ 1 − e 2πω = a2( ˜ k, ω) + e −πω a †
2(− ˜ k, ω), (4.32)
aII( ˜ k, ω) √ 1 − e 2πω = a2( ˜ k, −ω) + e −πω a †
2(− ˜ k, ω). (4.33)
These transformations are called Bogolyubov transformations. One can check that the
value of Φ in region I depends only on aI and its Hermitean conjugate, and similarly
for aII. The above definitions are motivated by the fact that we can expand Φ(�r, t) as
� ∞
d
Φ(�r, t) = dω
2˜ k
�
4π(2π) 3 ei˜ k˜r
where
0
�
K(−ω, µu, µv) � a2( ˜ k, ω) + e −πω a †
2(− ˜ k, −ω) � +
K ∗ (−ω, µu, µv) � a †
2(− ˜ k, ω) + e −πω a2( ˜ k, −ω) ��
, (4.34)
� ∞
K(ω, α, β) =
0
dx
x xiω e −ixα−iβ/x
(4.35)
is an integration kernel; see [22] for some properties of K. From the commutation
relations
[aI( ˜ k, ω), a †
I (˜ k ′ , ω ′ )] = δ 2 ( ˜ k − ˜ k ′ )δ(ω − ω ′ ) (4.36)

(and similarly for [aII, a †
II ]) and
4. Quantum Black Holes 32
[aI, aI] = [aII, aII] = [aI, aII] = [aII, aI] = 0 (4.37)
we see that all observables in region II commute with all aI, a †
I . Our conclusion is
that not a2 and a †
2, but aI and a †
I are the proper annihilation and creation operators for
region I (and aII and a †
II for region II).
Let us now study the vacuum Ω as defined by an observer in Minkowski space:
a |Ω〉 = a1 |Ω〉 = a2 |Ω〉 = 0 , for all k, ω. (4.38)
In our Rindler basis this turns, using the Bogolyubov transformations (4.32) and (4.33),
into
aI( ˜ k, ω) |Ω〉 − e −πω a †
II (−˜ k, ω) |Ω〉 = 0, (4.39)
aII( ˜ k, ω) |Ω〉 − e −πω a †
I (−˜ k, ω) |Ω〉 = 0, = 0. (4.40)
So we learn that the vacuum only consists of states with equal number operators nI =
nII, since the above implies that when acting on the vacuum Ω we have
nI := a †
I (˜ k, ω)aI( ˜ kω) = e −πω a †
Ia† II = e−πωa †
IIa† I = a†
IIaII =: nII. (4.41)
We thus find by plugging
into (4.39) that
�
n
fn
Ω = �
fn |n, n〉 (4.42)
√ n |n − 1, n〉 = e −πω �
and we thus see that the coefficients fn satisfy
n
n
√
fn n + 1 |n, n + 1〉, (4.43)
fn+1 = e −πω fn. (4.44)
Consequently, we can write the Minkowski vacuum in the Rindler basis as
|Ω〉 = � √
1 − e−2πω ˜k,ω
∞�
e −πnω |n, n〉. (4.45)
n=0

4. Quantum Black Holes 33
The strange conclusion we can make now, is that the Rindler observer detects par-
ticles radiating in all directions at some temperature TR! To see this we write down the
probability that a Rindler observer in region I, while looking at the states Ω, observes
nI particles with energy ω and transverse momentum ˜ k in region I. From (4.42) and
(4.45) we see that this is
PnI
= �
nII
We can write this as
〈Ω|nI, nII〉〈nI, nII|Ω〉 = |fnI |2 = (1 − e −2πω ) −1 e −2πnIω . (4.46)
e −β(E−F ) , (4.47)
where E = nω is the energy, β = 1/T the inverse temperature, and F the free energy.
The Rindler temperature can be read of to be TR = 1/2π.
Now, for the distant observer the central region of the Kruskal space of a black hole
is just Rindler space. All the particles he sees are ingoing particles. There seem to be
no particles coming out (in the local Minkowski frame). We thus need the state |Ω〉 to
describe these. So the reason why an observer in the Kruskal region I will see thermal
radiation with a temperature T is that he thinks he sees the state |Ω〉. Finally, notice
that when Rindler space is replaced by Kruskal space we must insert an extra factor
of 4M in the unit of time. Therefore the temperature of the expected radiation from a
black hole is
which is the Hawking temperature [2].
TH = 1/8πM, (4.48)
4.3 Path integral approach
There is also an elegant derivation of the Hawking temperature which makes use of
path integrals. The evaluation of the generating function W [J] for the Green functions
of the quantum field theory (at J = 0) gives us the partition function, which can
be compared to the partition function ln Z = −βF of a thermodynamic system at
temperature T = β −1 . Let us now take a brief look at how this goes.

4. Quantum Black Holes 34
As is well know, the generating function W [J] is a functional integral over the
exponent of the action:
where
�
W [J] =
�
S[φ, J] =
Dφe iS[φ,J] , (4.49)
d 4 x[L(φ, ∂µφ) + Jφ] (4.50)
is the action; L is the Lagrangian density, and J is a source current. One could try to
take the Einstein-Hilbert action and just try to do the integral. But this gives an infinite
result. What one should do, instead, is to consider a finite region and add a boundary
term to the action. The action looks then like
S[g, 0] = 1
�
�
√ D 1
−gRd x +
16π
8π
M
∂M
√ −hBd D−1 ξ (4.51)
where hµν is the induced metric on the boundary ∂M of a D-dimensional spacetime,
ξ µ are the coordinates on this boundary, and B is essentially the trace of the extrinsic
curvature (the second fundamental form). We will not treat this in detail, but by doing
this, and going to imaginary time, one can evaluate this action and find it to be
for the Schwarzschild action, and
S[g(S), 0] = iπ
M (4.52)
κ
S[g(KN), 0] = iπ
κ (M − ΦHQ) (4.53)
for the Kerr-Newman metrics (3.9). So how do we get our partition function? The
dominant contribution to the path integral (4.49) comes from the fields (in our case
metrics) that minimize the action (thus solutions of the equations of motion). So in the
case of rotating charged black holes we find
ln W [0] = ln Z � iS[g(KN), 0]. (4.54)
On the other hand we know that the partition function of a grand canonical ensemble at
inverse temperature β with chemical potentials µi associated with conserved charges
Ni is given (by definition) by
ln Z = −β(E − T S − �
µiNi). (4.55)
i

4. Quantum Black Holes 35
For the case we are considering β equals 2π/κ (for a motivation, see the end of this
section) and we find combining equations (4.53), (4.54), (4.55) that
1
1
M = T S +
2 2 ΦHQ + ΩHJ, (4.56)
where ΦH and ΩH are our ‘conserved charges’ at the horizon: respectively the co-
rotating electrostatic potential, and the angular velocity. Comparing this with the fol-
lowing equation for M, due to Smarr [25],
1 κ
M =
2 8π AH + 1
2 ΦHQ + ΩHJ, (4.57)
and restoring all constants that we have not been writing we indeed find
T =
and also the Bekenstein-Hawking formula
�c 3
8πGNMkB
(4.58)
S = c3AH . (4.59)
4GD�
Let us give a concrete example — which is at the same time a shortcut to find the
Hawking temperature — of how this imaginary time trick is being used in practice.
We will take the Schwarzschild metric for simplicity.
In the (T, R, θ, φ) Kruskal coordinates from chapter 2 the Schwarzschild metric
takes the form
with
ds 2 = −
32M 3
r
T 2 − X 2 =
e −r/2GM (dT 2 − dX 2 ) + r 2 dΩ 2
(4.60)
�
1 − r
�
e
2M
r/2M , (4.61)
X + T
X − T = et/2M , (4.62)
(cf. equations (2.11) and (2.12)). From equation (4.61) we see that the singularity
at r = 0 corresponds to the surface −T 2 + X 2 = −1. A smart way of avoiding this
singularity is by defining a new coordinate ˜ T = iT , so that r = 0 is being avoided since
it corresponds to ˜ T 2 + X 2 = −1, which has no real solutions. The metric becomes
ds 2 =
32M 3
r
e −r/2GM (d ˜ T 2 + dX 2 ) + r 2 dΩ 2
(4.63)

with
4. Quantum Black Holes 36
˜T 2 + X 2 =
� r
2M
�
− 1 e r/2M , (4.64)
X − i ˜ T
X + i ˜ T = et/2M . (4.65)
Note that on the contour where ˜ T and X are real, r is real and larger than 2M. As
promised, we define an imaginary time coordinate τ by
τ = it = i2M ln X − i ˜ T
X + i ˜ T
(4.66)
The logarithm of a complex number can be easily found: it is not defined on the
negative real axis, and every other point the complex number (a + ib) can be written as
re iφ (with r > 0, −π < φ < π) so we see that ln(a + ib) = ln|a 2 + b 2 | + iArg(a + ib).
Thus from 4.66 we have
τ = i2M[ln(X − i ˜ T ) − ln(X + i ˜ T )] (4.67)
= i2Mi(Arg(X − i ˜ T ) − Arg(X + i ˜ T )) = 4MArg(X + i ˜ T ) (4.68)
This is an angular coordinate with period β = 4M ·2π = 8πM. The functional integral
defining the generating function should therefore be over matter fields and metrics with
this periodicity in τ. But this is just the partition function Z for a canonical ensemble
of the fields at temperature T = β −1 = (8πM) −1 . For a Schwarzschild black hole
the surface gravity is known to be κ = (4M) −1 , so that the temperature is indeed
T = κ/2π in accordance with the above.

5. EXTREMAL BLACK HOLES IN STRING THEORY
There is only one candidate theory, named string theory, that can provide us with an
answer to the question what the microstates of a black hole are. In this chapter we will
examine the macroscopic and microscopic structure of black holes in string theory.
Some good references for this chapter are [26, 27, 28, 29, 30, 31, 32].
5.1 Perturbative microstates in string theory
As we have seen in the previous chapters, black holes should be thought of as thermo-
dynamical systems. Since we know that thermodynamics is only an approximation of
a more fundamental description in terms of microstates of the system, we would like
to find out what these microstates of a black hole are. Roughly speaking, a system
with an entropy S has e S microstates. As we have seen before, for the Schwarzschild
black hole Sbh ∼ A ∼ M 2 M 2
. The associated number of states thus grows like e . At
this point it is interesting to think about whether this agrees or not with the number of
perturbative string states at mass level M, as this is what one would perhaps expect
(or hope for?). One could object that associating perturbative string states with a black
hole is absurd. In the end, perturbative string states are obtained by quantizing a string
in a flat background spacetime, so how can they be equivalent with a black hole? This
is certainly a legitimate question. To jump ahead a bit, let us say something about this.
As we will see, as the string coupling strength gS increases, the mass M of the pertur-
bative state is unaltered both in the classical (independence of gS) as in the quantum
world (no corrections due to supersymmetry). Now the gravitational field of the state
is determined by GNM (in four dimensions) and since GN is proportional to g2 S , we
find that as gS increases, the gravitational field increases. There is thus a back reaction
on the perturbative state and, eventually, it may be described by a curved spacetime

5. Extremal Black Holes in String Theory 39
with large curvature. Thus, at least in principle, it might be possible to associate cer-
tain black-hole spacetimes with perturbative states. We will see that this is the case for
extreme (BPS) black holes.
Consider the open bosonic string, with mode expansion
X µ = x µ + l 2 sp µ � 1
τ + ils
n
n�=0
αµ n cos(nσ). (5.1)
Remember that in this expression the oscillator coefficients α µ n are creation operators
for n ≤ 0 and annihilation operators for n ≥ 0. We will take units in which the string
length scale ls = 1/ √ πT (where T is the string tension) is 1. Then one can easily
show that the mass eigenvalues are given by the eigenvalues of
where
1
2 M 2 = N − 1, (5.2)
N =
∞�
α i −nα i n, (5.3)
n=1
is the number operator (where there is an implied sum over i, the 24 transverse dimen-
sions of the bosonic string). Our goal is to estimate the number dn of states at level n.
It is convenient to define a generating function
G(w) ≡ tr w N =
∞�
dnw n , (5.4)
with |w| < 1. We will now examine the behavior of G(w) and use Cauchy’s equation
to get an approximation of dn
dn = 1
�
2πi C
n=0
G(w)
dw. (5.5)
wn+1 where C is a closed loop around w = 0. Plugging in the expression for the number
operator N, we have (using the geometric series to get rid of the sum over i) that
tr w N ∞�
= tr w αi −nαi ∞�
�
1
n =
1 − wm �24 . (5.6)
Let us try to simplify
m=1
f(w) ≡
m=1
∞�
(1 − w m ) (5.7)
m=1

5. Extremal Black Holes in String Theory 40
by using (once again) the geometric series and a Taylor expansion for the logarithm:
ln f(w) =
∞�
ln(1 − w m ) = −
m=1
∞�
m,p=1
wmp p =
∞�
p=1
wp p(1 − wp . (5.8)
)
Close to w = 1 we can Taylor expand w p as w p ∼ = 1 + p(w − 1) + . . .. We get
and thus
when w ∼ 1.
From (5.5) we get that
ln f(w) ∼ = −1
1 − w
∞�
p=1
p −2 = − π2
6(1 − w)
� � 2 4π
G(w) ∼ exp
1 − w
dn ∝ e 4π√ n
(5.9)
(5.10)
as n → ∞. (5.11)
Therefore, from (5.2), the number of states of a free string increases as
ρ(M) ∝ e √ 2πM . (5.12)
That is, with an M in the exponent. Note that this is very different from the black hole
requirement that the number of states increases as
which has an M 2 in the exponent.
ρbh(M) ∝ e 4πGN M 2
, (5.13)
5.2 Correspondence principle between strings and black holes
In [4] Susskind speculates that all black hole states are in one to one correspondence
with single string states. Susskind’s proposition is based on the following. By in-
creasing the string coupling, the size of a highly excited string becomes less than its
Schwarzschild radius. We should thus think of it as a black hole. Decreasing the
coupling, on the other hand, generically makes the size of a black hole smaller, and
eventually even smaller than the string scale. In such a case, the metric becomes ill-
defined and we can no longer interpret the black hole as being a black hole. Susskind

5. Extremal Black Holes in String Theory 41
suggests that we should describe this configuration in terms of some string state. At
weak coupling the black hole can be thought of as a small number of (or to leading
order just one) highly excited strings. There seems to be a problem, however. As we
have shown in section 5.1, the entropy of a free string is proportional to the mass of
the string state, whereas the Bekenstein-Hawking entropy is proportional to the square
of the mass of the black hole. As a resolution to this discrepancy, Susskind’s suggests
that in the above M versus M 2 comparison the effect of a large gravitational redshift
is not taken into account.
In [5] Horowitz and Polchinski give a more precise, although still somewhat qual-
itative, formulation of the relation between strings and black holes. Based on this fact
that for black holes in string theory generically the Schwarzschild radius in string units
decreases when the string coupling is reduced, they formulate a correspondence prin-
ciple, which states that (i) when the size of the horizon drops below the size of a string,
the typical black hole states becomes a typical state of strings (and so-called D-branes)
with the same charges, and (ii) the mass does not change abruptly during the transi-
tion. Their approach does not give right numerical coefficients, but it does give the
correct dependence on mass and charge for a large variety of cases. For the exact right
numerical coefficients we have to introduce some more concepts, which we will do in
the following sections.
5.3 Extremal black holes and extended supersymmetry
Let us next review the supersymmetry algebra and its representations. We can view
extremal Reissner-Nordström black holes as ‘supersymmetric solitons’ because they
behave very much like BPS objects where M ≥ Q is the BPS bound. The reason why
we can associate BPS states with extremal black holes is that they have the crucial
property that they cannot receive quantum corrections, which is similar to the fact that
extremal black holes have vanishing temperature and thus don’t radiate. In this section
we will clarify all this.
Supersymmetry is a symmetry between fermions and bosons and supersymmetric

5. Extremal Black Holes in String Theory 42
theories are theories with conserved spinorial currents. For each such current we get
four conserved charges. We can organize these charges into a Weyl spinor, or, in gen-
eral, into N Weyl spinors Q A α if we have N spinorial currents (and thus 4N conserved
charges). Here A = 1, ..., N counts the supersymmetries and α = 1, 2 is a Weyl spinor
index. A theorem of Haag, Lopuzanski and Sohnius states that the most general form
of the supersymmetry algebra 1 (in four spacetime dimensions), is given by
{Q A α, ¯ Q B ˙ β } = 2δ A Bσ µ
α ˙ β Pµ, (5.14)
{Q A α, Q B β } = 2ɛαβZ AB . (5.15)
The Q bars are the hermitean conjugates and have opposite chirality (this is indicated
by the dot), so ¯ Q B ˙ β = (Q B β )† ; the 2 × 2 matrices σ µ with µ = 1, 2, 3 are the Pauli
matrices, and σ 0 is the identity matrix; ɛαβ is antisymmetric with ɛ12 = +1; the matrix
Z AB is also antisymmetric and the operators in it commute with all the operators in
the super Poincaré algebra which explains why they are called central charges; by
definition we set Z 12 ≡ Z, which we may choose without loss of generality to be non-
negative. Let us for concreteness take N = 2 and start with massive representations,
M 2 > 0. We may work in the rest frame Pµ = (M,�0). By forming two appropriate
linear combinations aα = Q 1 α + ɛαβQ 2 β and bα = Q 1 α − ɛαβQ 2 β
can bring the algebra to the form
{aα, a †
β } = 4(M + Z)δαβ,
{bα, b †
β } = 4(M − Z)δαβ,
of the generators, we
{aα, b †
β } = {aα, bβ} = 0. (5.16)
Note that (up to a non-relevant normalization factor) these are precisely the anticom-
mutation relations satisfied by two independent sets of fermion annihilation and cre-
ation operators. Starting with some state |ψ〉 that is annihilated by the annihilation
operators aα and bα we can construct a total of 16 states using the creation operators
a † α and b † α. Since quantum mechanics is a unitary theory we require the absence of
1 This is is an extension of the usual Poincaré algebra (the Lie algebra of the 10-dimensional Poincaré
Lie group of isometries of Minkowski space).

5. Extremal Black Holes in String Theory 43
negative norm states, i.e., we have 〈φ|{aα, a † α}|φ〉 ≥ 0 for any state |φ〉 (and similarly
for bα). This implies that the mass is bounded by the central charge. This is called the
BPS-bound:
M ≥ |Z|. (5.17)
In the case M = Z we have to be careful, since in this case the representation contains
null states. These have to be divided out in order to get unitary representations, which
corresponds to |ψ〉 being annihilated by one set of creation operators (b † α if Z > 0).
Such ‘BPS’-states are thus constructed using only the a †
β
creation operators. As a
consequence they are invariant under half of the supersymmetry algebra. They are
so-called ‘short’ massive representations of the supersymmetry algebra.
An interesting feature of BPS states is that they cannot receive quantum correc-
tions, assuming that the full theory is supersymmetric. This is the case because the
relation M = Z is a consequence of the supersymmetry algebra. The analogy with ex-
tremal Reissner-Nordström black holes, which have exactly vanishing Hawking tem-
perature, is clear.
The extreme Reissner-Nordström black hole is sometimes called a supersymmetric
soliton. In the remainder of this section we will explain this terminology.
First let us say what we mean by a soliton. A soliton is a stationary, regular, and
stable finite energy solution to the equations of motion. Some of these features are
immediately clear for the extreme Reissner-Nordström black hole: it is static and thus
in particular it is stationary; it has a finite mass M and thus finite energy; it is regular in
the sense that it has no naked singularity. It is also stable, at least when considered as
a solution of N = 2 supergravity into which it can be embedded, as we will see below.
The next question is: what do we mean by a ‘supersymmetric’ soliton? A solution is
called supersymmetric if it is invariant under a rigid transformation, which can be seen
as an analogue of an isometry. By ‘being invariant’ we mean here
δɛ(x)Φ |Φ0= 0. (5.18)
That is to say, a field configuration Φ0 is supersymmetric if there exists a choice ɛ(x) of
the supersymmetry transformation parameters such that the configuration is invariant.

5. Extremal Black Holes in String Theory 44
The notation should speak for itself: one should perform a supersymmetry variation
of all the fields Φ and then evaluate it at Φ0; this should be zero. It is instructive
to note that the transformation parameters ɛ(x) are fermionic analogues of Killing
vectors. (They are called Killing spinors and equation (5.18) is the Killing spinor
equation.) Killing vectors are the generators of spacetime symmetries. The trivial
vacuum (Minkowski space) has ten Killing vectors. A generic spacetime has none,
whereas more symmetric spacetimes can have some. For example the RN black hole
has 4 Killing vectors (1 for time, 3 for rotation invariance). Similarly, we have ‘Killing
spinors’ ɛ(x) which are supersymmetry transformation parameters.
As mentioned above, the extreme RN black hole can be embedded into N = 2
supergravity. This can be done by adding two gravitini ψ A µ . The extreme RN black
hole is not only a solution of Einstein-Maxwell theory, but also of the extended theory,
with ψ A µ = 0. It also possess Killing spinors and is thus a supersymmetric solution in
the above sense. Actually the only condition comes from the gravitino variation
δɛψµA = ∇µɛA − 1 −
Fab 4 γaγ b γµεABɛ B = 0, (5.19)
where we suppressed the spinor indices and introduced some new notation. We use
four-dimensional Majorana spinors, but project onto one chirality. This is encoded
it the placement of the index A = 1, 2: letting γ5 work on ɛ A gives ɛ A , whereas
for a lower index A we pick up a minus sign. Half of the components of ɛA, ɛ A are
independentFµν is the graviphoton field strength and the minus superscript indicates
its antiselfdual part:
F ± µν = 1
2 (Fµν ± i ⋆ Fµν). (5.20)
One can now check (we will not do this) that the extreme RN black hole has Killing
spinors which can be split in a part h(�x) which is completely fixed in terms of the
coefficients in the metric, and a part at infinity
where the part at infinity satisfies an additional equation
ɛA(h(�x)) = h(�x)ɛA(∞), (5.21)
ɛA(∞) + iγ 0 Z
|Z| ɛABɛ B (∞) = 0. (5.22)

5. Extremal Black Holes in String Theory 45
Consequently, half of the parameters are fixed in terms of others, which means that
we half of the maximum, which is 8/2=4, Killing spinors. Furthermore, it can be
shown that the extreme RN black hole is part of a hypermultiplet. The factor Z/|Z|
is the central charge. In local supersymmetric theories central charge transformations
are local U(1) transformation. The corresponding gauge field is a graviphoton. The
central charge is a linear combination of the electric and magnetic charge of this U(1):
Z = 1
� �
− 2F = p − iq. So |Z| = p2 + q2 = Q = M gives us our relation between
4π
the extreme RN black hole and the supersymmetric BPS limit. The extreme RN black
hole has thus all the features we expect from a BPS state:
• It is invariant under half of the supersymmetry transformations.
• It sits in a short multiplet.
• It saturates the supersymmetric mass bound.
The strange no-force property that admits that a number of extremal black holes can
sit next to each other without feeling each others presence (see chapter 2) can now be
understood as a consequence of the additional supersymmetry.
Our next goal will be to try to find a microscopic description of the Reissner-
Nordström black hole. It turns out that the simplest example one can consider in
string theory is its five dimensional analogue. We will need to consider low energy
effective field theory describing string theory. This is a generalization of Einstein-
Maxwell theory. The number of spacetime dimensions involved is ten. There are also
more fields involved: they are precisely the massless modes that arise in (perturbative)
string theory. Higher massive modes decouple and do not appear in the low energy
effective theory. We will now study type II supergravity which is the field theory that
describes such massless modes. As the name already implies, supergravity is the union
of general relativity (a theory of gravity) and supersymmetry.

5. Extremal Black Holes in String Theory 46
5.4 Type II supergravity
Supersymmetry is a symmetry between bosons and fermions and the massless states
in (super)string theory include both bosons and fermions. However, since black holes
are solutions of the classical bosonic field equations, we will only be interested in the
bosonic degrees of freedom. Closed string theories have massless (bosonic) modes
associated with a symmetric, traceless, rank-2 tensor (the graviton field Gµν), an an-
tisymmetric rank-2 tensor (the Kalb-Ramond field Bµν), and the dilaton 2 field φ. In
type II string theories, these states arise in the NS-NS sector. Before we continue, let
us remind ourselves of some string theory basics.
In string theory a two-dimensional non-linear sigma-model with action
�
SW S = 1
4πα ′
d
Σ
2 σ √ −hh αβ (σ)∂αX µ ∂βX ν Gµν(X) (5.23)
is being used to model the motion of a string in a curved spacetime background with
metric Gµν(X). The integration is over the world-sheet Σ. There are two coordinates
(σ 0 , σ 1 ) = σ and hαβ(σ) is the world-sheet metric. The coordinates of the string in
spacetime are X µ (σ). The parameter α ′ is related to the string tension T and the string
length scale lS via α ′ = 1
2πT
1 = 2l2 S and is the only independent dimensionful parameter
in string theory. One can impose different conditions on the boundaries of the-world
sheet. One of them is
∂1X µ |∂Σ= 0. (5.24)
This is the Neumann boundary condition and it describes open strings. One can also
fix the endpoints of the string. This is the Dirichlet boundary condition
∂0X µ |∂Σ= 0. (5.25)
Since the momentum at the end of the string is not conserved, Dirichlet boundary con-
ditions require the string to be coupled to another dynamical object. Such objects are
called D-branes. They are non-perturbative objects since they have a tension propor-
tional to 1/gS. D-branes are very important in understanding the microscopic origin
2 The expectation value of the dilaton fixes the string coupling constant via gS = 〈e φ 〉.

5. Extremal Black Holes in String Theory 47
of black hole entropy. At this point let us just state that one can form so-called Dp-
branes by imposing Neumann conditions along time and p directions, and Dirichlet
conditions among the remaining D − p − 1 directions. (Here D is the dimension of the
spacetime.) We will come back to this in the next section.
Type II string theory consists of closed oriented strings and the world-sheet action
is extended to a (1,1) supersymmetric action by adding world-sheet fermions ψ µ (σ).
For these fermions one can choose the boundary conditions independently to be either
periodic (Ramond) or anti-periodic (Neveu-Schwarz). We thus have four scenarios:
NS-NS, NS-R, R-NS, R-R. The states in the NS-NS and R-R sectors are bosonic, the
ones in the R-NS and NS-R sectors are fermionic.
As stated above, the massless spectrum of the NS-NS sector for both type IIA
and IIB string theory consists of Gµν, Bµν, φ. They are constructed using the Neveu-
Schwarz (half-integer moded) world-sheet fermion creation operators for both left-
and right-movers. In the R-R sector we also have massless bosonic states. They are
constructed using the Ramond (integer moded) world-sheet fermion creation operators
for both left- and right-movers. For the R-R sector we have to distinguish between the
IIA and IIB theories. Let us look at IIA for the moment. We can do this without loss
of generality, since one can relate the (chiral) IIB to the (non-chiral) IIA theory via
T-duality. The R-R states may be represented by form fields C(1) and C(3) (for type IIB
these are C(0), C(2) and C(4)), where
C(n) ≡ 1
n! Cµ1µ2...µndx µ1 ∧ dx µ2 ∧ ... ∧ dx µn . (5.26)
We could as well have represented these fields by C(7) and C(5). They are related by
the 8-dimensional epsilon tensor. It is a known fact [26] that the condition for the tree-
level Weyl invariance of the string world-sheet action is preserved in the (quantum)
string theory boils down to the vanishing of the associated (renormalization-group)
beta functions for these fields. As a result, one gets the following effective action
(which is a good approximation in the low energy limit α ′ → 0) in the so-called string

frame:
SIIA =
5. Extremal Black Holes in String Theory 48
1
2κ 2 10
�
+
� �
d 10 x(−G) 1/2 e −2φ R(G)
[e −2φ (4dφ ∧ ∗ dφ − 1
2 H(3) ∧ ∗ H(3)) − 1
2 F(2) ∧ ∗ F(2)
− 1
2 ˜ F(4) ∧ ∗ F(4) − 1
2 B(2) ∧ F(4) ∧ F(4)]
�
. (5.27)
We see that all terms from the NS-NS sector are coupled to the dilaton, while terms
from the R-R sector are not. In this expression G is the determinant of Gµν (the
metric appearing in the non-linear sigma model (5.23)). κ10 has the dimensions of
a ten-dimensional gravitational coupling, but can not be be directly identified with the
physical gravitational coupling. The reason for this is simply that a rescaling of κ
can be compensated (as can be seen from equation (5.27)) by a shift of the dilaton’s
vacuum expectation value. It is clear what the relation between κ10, α ′ and gS should
be, namely κ10 = (α ′ ) 2 gS · const. In fact,
2κ 2 10 = (2π) 7 (α ′ ) 4 g 2 S. (5.28)
Furthermore we have the dilaton 1-form dφ, which is simply ∂µφdx µ . The field-
strength forms are related to the potentials by
F(2) = dC(1), F(4) = dC(3) and H(3) = dB(2)
where B(2) is the 2-form associated with Bµν, and
(5.29)
˜F(4) = F(4) + C(1) ∧ H(3). (5.30)
Finally, Hodge duals in D dimensions (so in our case, D = 10) are defined by
∗ C(p) = (−G)1/2
p!(D − p)! ɛν1ν2...νD−pµ1µ2...µpC µ1µ2...µp dx ν1 ∧ dx ν2 ∧ ... ∧ dx νD−p . (5.31)
It is easy to transform this ‘string frame’ action into the ‘Einstein frame’ action, in
which there is no dilaton factor in the curvature term, by plugging in the field redefini-
tion
Gµν = e φ/2 gµν. (5.32)

This gives
SIIA =
1
2κ 2 10
− 1
2
5. Extremal Black Holes in String Theory 49
�
� �
d 10 x(−g) 1/2 R(g)
[dφ ∧ ∗ −φ 1
dφ + e
2 H(3) ∧ ∗ H(3) + e 3φ/2 F(2) ∧ ∗ F(2)
+e φ/2 ˜ F(4) ∧ ∗ ˜ F(4) + B(2) ∧ F(4) ∧ F(4)]
�
(5.33)
for the effective bosonic action in the Einstein frame. Another possibility would be to
absorb the dilaton vacuum expectation value in the prefactor of the action, see [30].
We see that type II string theory effective action involves various field strengths. In
the same fashion as an electron is coupled by its charge to the Maxwell gauge potential
Aµ, one would expect that there should be some objects in the underlying string theory
that couple to the associated gauge form fields. Indeed, such extended objects exist:
they are called p-branes, where p stands for the number of spatial dimensions. In fact,
p-branes are higher dimensional analogues of the extremal Reissner-Nordström black
hole. Before we come back to this, let us study our form fields a bit more.
5.5 Form fields and D-branes
Starting with the gauge field Aµ, which is coupled naturally to the world line X µ (τ)
of some charged particle by a term
�
Aµ
dX µ
dτ (5.34)
dτ
in the action, and whose vector potential 1-form Aµdx µ ≡ A(1) transforms as A(1) →
A(1) + dΛ0 (where Λ(0) is a function) we get the field strength 2-form F(2) = dA(1).
The Maxwell equations in four dimensions are
dF(2) = 0 and d ∗ F(2) = ∗ J(1), (5.35)
where J(1) = Jµdx µ is the current one form. Since we are in four dimensions, ∗J(1) is
a 3-form and we can find the electric charge by integrating over some spatial volume
V3 which is enclosed by some surface S2. Using Gauss’ theorem we get
� �
∗
Q = J(1) =
V 3
S 2
∗ F(2). (5.36)

5. Extremal Black Holes in String Theory 50
Now let us look at an analogous thing in string theory.
Analogously to the gauge field/world line coupling above, one has the coupling of
the Kalb-Ramond field to the string world-sheet by
�
S ∼
Bµνɛ αβ ∂αX µ ∂βX ν d 2 σ. (5.37)
We have dH(3) = ddB(2) = 0. The electric charge Q1 of the string in 10 dimensions
is given by
Q1 =
�
S 7
∗H(3). (5.38)
This can be generalized for higher dimensional charges. Obviously, the generalization
of (5.34) and (5.37) for a general antisymmteric (p + 1) dimensional tensor gauge field
is a term in the action of the form
�
Cµ1µ2...µp+1ɛ α0α1...αp ∂α0X µ1 ∂α1X µ2 ...∂αpX µp+1 d p+1 σ. (5.39)
So what this describes, is the coupling of the C(p+1) gauge field to the (p + 1) dimen-
sional world volume of some extended object. This object has p spatial dimensions and
is called a p-brane. The electric charge of a p-brane that is enclosed by a hypersurface
S8−p is given by
� 8−p
∗
Qp = F(p+2), (5.40)
S
where F(p+2) = dC(p+1) is the field strength, which is invariant under gauge transfor-
mations C(p+1) → C(p+1) + dΛ(p).
In the effective action for type II superstrings (5.27) we see both the field strengths
H3 and Fn appear. There is quite a difference between these. The H3 comes from
the NS-NS form field B(2) which is coupled electrically to the string world sheet or its
excitations. The Fn, with n even, are derived from the R-R fields C(n−1) and do not
couple electrically. This suggests that R-R field strengths F(n) are naturally associated
with branes having p = n − 2 dimensions.
Since perturbative string states are neutral with respect to the charges associated
with the R-R fields, we have to add non-perturbative dynamical objects that do have R-
R charges to type II string theory. (By R-R charges we mean electric charges associated

5. Extremal Black Holes in String Theory 51
with the R-R gauge form fields.) These objects are called Dp-branes. The D stands
for Dirichlet: there has also to be an open string sector where open strings end on
p-dimensional planes [7]. Thus, the open string world sheet has Dirichlet boundary
conditions in the directions perpendicular to the branes. These Dp-branes do couple
electrically to the associated R-R fields. (Just as the fundamental string is coupled to
the NS-NS field Bµν.) It is precisely this enlargement of string theory that lead to the
understanding of black hole entropy in terms of associated microstates.
But before we construct explicit black hole solutions of the type II supergravity
field equations (whose entropy we shall be able to explain in terms of Dp-branes) let
us briefly discuss dualities in string theory and give an example to get to know some
of the techniques for doing microscopic entropy calculations.
5.6 Dualities
By a duality we mean two different ways to look at the same physics. The first duality
symmetry we will consider, T-duality, is a symmetry that relates compactifications on
a manifold of (large) volume V to ones with (small) volume 1/V . Obviously, the
simplest case is compactification on a circle with radius R. It turns out that type IIA
compactified on a circle with radius R is equivalent to type IIB compactified on a circle
with radius 1/R, and likewise for heterotic E8 × E8 and heterotic SO(32).
Next we consider S-duality. This is a (conjectured) non-perturbative duality. (Non-
perturbative because it acts on the dilaton as gS → 1/gS.) S-duality is a weak ↔ strong
duality: it relates the weak coupling regime of one theory to the strong coupling regime
of another one. In particular, IIB theory is self-dual under S-duality and Heterotic
SO(32) is S-dual to Type I theory. Type IIB string theory is believed to have an exact
non-perturbative SL(2, Z) symmetry. We will have more to say about S-duality in
chapter 7.

5. Extremal Black Holes in String Theory 52
5.7 Counting of 1/2 BPS heterotic states on T 4 × T 2
In this section we will count 1/2 BPS states as example to illustrate some standard
counting techniques. We will consider heterotic string theory on T 4 × T 2 with the
following coordinates:
T 4
����
6789
× ˜ S 1
����
5
× S 1
����
4
For convenience we will set α ′ = 1 in this section.
× M 4
����
0123
. (5.41)
We consider a string around the ”5” direction, winding it ω times and with momen-
tum n. The question is how many such states there are, subject to the usual Virasoro
constraints that the left moving Hamiltonian is
and the other constraint is
where
2 M
−
2 M
−
4 + NR + p2R 2
4 + NL − 1 + p2L 2
pL,R =
= 0, (5.42)
= 0, (5.43)
� � �
1 n
∓ wR . (5.44)
2 R
We look at the right moving ground state, so NR = 0 an thus, from (5.42) we get that
M = √ 2pR, (5.45)
which looks like a BPS bound. Furthermore, the right moving superstring has a 16
dimensional representation (8v for the NS sector, and 8s for the R sector) and this is
thus a short representations (a long one would be 256 dimensional). From these two
facts we see that this is BPS state. The question we want to consider is: how many
such BPS states are there?
The second equation says we can have arbitrary left-moving oscillations. The left-
moving string is a 26 dimensional bosonic string. We choose one direction for the
string to carry momentum and one for time, so we have 24 left moving oscillator.
We have spacetime bosons X i , and internal bosons Y I which parametrize the torus

5. Extremal Black Holes in String Theory 53
of E8 × E8. Let us denote the oscillators of X i as α i n and of Y I as β I n. Then the
left-moving oscillations can be written as
NL = �
n>0
(nα i −nα i n + nβ I −nβ I n), (5.46)
where i = 1, ..., 8 because there are 8 transverse dimensions in 10 dimensions and
I = 1, ..., 16 because the rank of E8 × E8 is 16. To take care of the zero-point energy
we introduce N = NL − 1 and from (5.43) we get
N = NL − 1 = p2 R − p2 L
2
= nω. (5.47)
We are interested in Ω(N), the number of states. Basically this is the harmonic oscil-
lator problem with 24 oscillators. This Ω(N) appears in the partition function on the
world-sheet,
Z(β) = Tre −βN = � Ω(N)e −βN , (5.48)
and we can us this to find Ω(N) by doing an inverse Laplace transform
Ω(N) = 1
�
2πi
e βN Z(β)dβ. (5.49)
So our problem is reduced to finding this inverse Laplace transform. We would like
to know the asymptotics of Ω(N) at large N, or in other words, at large temperature.
Plugging in the expansions for N into (5.48), we get
Z(β) = Tre −β(P (nα i −n αi n+nβ I −n βI n)−1 . (5.50)
By introducing q = e −β and using the geometric series for 1 + q n + q 2n + ... we get
Z(β) =
1
q �
n (1 − qn . (5.51)
) 24
The trick is to use the fact that this is a modular form of weight 12 (it also appears in
the one loop partition function of the bosonic string), namely
Z(β) =
� �12 β
Z
2π
� 4π 2
β
�
. (5.52)

5. Extremal Black Holes in String Theory 54
We can relate high temperature with low temperature partition function by this equa-
tion. For convenience, let’s introduce
So at high temperature we get
and thus
˜β ≡ 4π2
β , ˜q ≡ e− ˜ β
(5.53)
β → 0 ⇒ ˜ β → ∞ ⇒ ˜q → 1. (5.54)
Z( ˜ β) =
1
˜q � (1 − ˜q n 1
�
) 24 ˜q
= e 4π2
β (5.55)
for ˜q → 1 The above implies the following behavior at large N, or, at high temperature:
Ω(N) = 1
�
2πi
e βN e 4π2
� �12 β
β dβ
2π
(5.56)
In our case we have a central charge c = 24. The more general expression is
Ω(N) = 1
�
2πi
e βN e 4π2
� �12 c β
β 24 dβ
2π
(5.57)
We now want to do a saddle point approximation, so let us spend a few words on
this. Basically, we want to find where the exponent has its peak. Let us do this a bit
more general: suppose (and this is indeed what we want) that we want to calculate an
integral
� ∞
A = dx e −ξf(x)
−∞
(5.58)
in the limit of large ξ. Assuming f(x) has a global minimum at some x0 we can Taylor
expand f(x) and get to quadratic order
f(x) ≈ f(x0) + 1
2 f ′′ (x0)(x − x0) 2 . (5.59)
Assuming that our global minimum is sufficiently separated from other local minima
and whose value is sufficiently smaller than the value of those, the dominant contri-
bution to the integral comes from a small region around x0. Our integral turns into a
standard Gaussian integral and the result for ξ goes to infinity is simply
A ≈ e −ξf(x0)
�
2π
ξf ′′ �1/2 . (5.60)
(x0)

5. Extremal Black Holes in String Theory 55
Let us now use this method, called the saddle point approximation, for (5.57). We
assume the second derivative (which is a 1/β 3 term) to be small. Then we can easily
find the maximum,
Plugging this value of β in give
f(β) = βN + 4π2
β
f ′ (β) = 0 at β = β0 = 2π
√ N
(5.61)
(5.62)
f(β0) = 4π √ N = 4π √ nω (5.63)
So now we simply apply the above saddle point approximation and get (in our case
c = 24)
Ω(N) ∼ e 4π√ N = e 4π √ nω . (5.64)
Clearly, for some general central charge c, we get what is called Cardy’s formula,
�
cN
f(β0) = 2π
(5.65)
6
and the number of states is
Ω(N) ∼ e 2π
√
cN
6 . (5.66)
This kind of manipulations were already studies by Ramanujan and Cardy because
what we have been doing is basically equivalent to counting the number of ways to
partition an integer N using integers of 24 colors. Physically, what we have been
solving is the problem of distributing integer energy among 24 oscillators.
5.8 D=5 Reissner-Nordström black holes
In this important section we will investigate the simplest example of a black hole in
string theory for which a microscopic description can be found. For this we will need
the five dimensional analogue of the charged black hole solution (2.15) that we already
studies in chapter 2:
ds 2 �
= 1 − 2M Q2
+
r2 r4 �
dt 2 �
− 1 − 2M Q2
+
r2 r4 �−1 dr 2 − r 2 dΩ 2 3. (5.67)

5. Extremal Black Holes in String Theory 56
The gauge potential 1-form has an 1/r 2 dependence because we are in four spatial
dimensions
A = Q
dt. (5.68)
r2 Analogously to the four dimensional case, this may, in the extremal case M = Q, be
cast into an isotropic form by introducing a new coordinate ρ ≡ � r 2 − Q. We get
ds 2 = H −2 dt 2 − H(dρ 2 + ρdΩ 2 3), (5.69)
A = (1 − H −1 )dt, H = 1 + Q
. (5.70)
ρ2 We will now try to connect this 5D black hole with string theory via the effective field
theory describing string theory in the low-energy limit.
5.8.1 Derivation of macroscopic entropy
From our discussion in sections 5.4 and 5.5 it follows that a solution of the field equa-
tions that one gets from the effective action (5.33) for the metric gµν, with non-zero
field strength F(n), gives the gravitational field associated with an (n − 2)-brane hav-
ing the associated NS-NS or R-R charge. This motivates us to spilt our coordinates
into two groups. One for the n − 1 = p + 1 dimensions of the p-brane world volume
with Poincaré invariance (‘longitudinal’) and one for the rotational invariance of the
remaining 10−(p+1) = 9−p directions (‘transversal’). So let us split our coordinates
x µ as x a , with a = 0, 1, ..., p for the longitudinal ones, and y i , with i = (p + 1), ..., 9
for the transversal ones. We would like to know the metric. It turns out that the line
element of this configuration in the string frame is of the form
ds 2 = H(r) −1/2
�
(dx 0 ) 2 −
p�
a=1
(dx a ) 2
�
− H(r) 1/2
Let us say briefly where this expression comes from.
9�
(dy i ) 2 . (5.71)
i=p+1

5. Extremal Black Holes in String Theory 57
Ansatz
A logical ansatz would be something like the 5D Reissner-Nordström black hole, so
we assume the metric to have the form
ds 2 = e 2A(r) ηabdx a dx b − e 2B(r) δijdy i dy j , (5.72)
where r is the radial coordinate, r = � y i y i . For the dilaton and antisymmetric tensor,
our ansatz is
φ = f(r), (5.73)
C012...p = e C(r) − 1, (5.74)
Now the claim is that there is a solution for the classical field equations following from
(5.33) in which all of the functions A, B, C, f are determined by a single harmonic
function H(r) [33], as follows
ds 2 = Hp(r) −(7−p)/8 ηabdx a dx b − Hp(r) (p+1)/8 δijdy i dy j , (5.75)
e φ = Hp(r) (3−p)/4 , (5.76)
C(p+1) = (Hp(r) −1 − 1)dx 0 ∧ dx 1 ∧ ... ∧ dx p . (5.77)
where the harmonic function Hp(r) is given by
Hp(r) = 1 + L7−p p
, (5.78)
r7−p with
L 7−p
p = 2κ10
√
πqp
(2π
(7 − p)Ω8−p
√ α ′ ) 3−p , (5.79)
where qp is an integer and Ωn is the volume of the unit n-sphere. This looks so com-
plicated because it is for general p. For example for p = 5 this reduces (using (5.28))
to
L 2 5 = q5gsα ′ . (5.80)
By transforming (5.75) using (5.32) and (5.76) we indeed get a line element that in the
string frame looks like
ds 2 = Hp(r) −1/2 ηabdx a dx b − Hp(r) 1/2 δijdy i dy j . (5.81)

5. Extremal Black Holes in String Theory 58
Do any of these solutions give the metric of a black hole? Well, maybe. At this point
it is clear that there are two problems:
• The warp factors Hp(r) are wrong.
• The dilaton is singular at the event horizon.
Extremality and supersymmetry
The above solutions (5.81) are extreme solitons, as one can check comparing the mass
that can be read of from Hp(r), and the charge associated with the C(p+1) form gauge
potential that one gets by integrating ∗ F(p+2), namely
2π
Mp = Qp = qp
gs
�
2π √ α ′
�−(1+p) . (5.82)
(Note that this diverges as gs → 0, as expected for non-perturbative objects.) It is
important to note that the solitons have qp units of some fundamental Dp-brane charge.
We would like to get our 5D Reissner-Nordtröm metric back, and by looking at
(5.78) this seems to have the best chances if we compactify the x 1 , ..., x 5 coordinates
on a five torus, and take p = 5.
Indeed, this is a good choice in the light of a result that follows from supersym-
metry considerations. The point is that the solution is invariant under 16 supercharges
Qα + P ˜ Qα, where Q acts on the rightmovers, ˜ Q on the leftmovers, and P represents
the operator that reflects the directions y i transverse to the brane. By looking at two
reflection operators P1 and P2 we find that the preserved supersymmetry charges sat-
isfy
Qα + P1 ˜ Qα = Qα + P2 ˜ Qα = Qα + P1(P −1 P2) ˜ Qα. (5.83)
This can be shown to imply that the number of directions that are transverse one brane
and parallel to the other is a multiple of four [34]. So because of the above extremality
and supersymmetry argument we may combine the p = 5 solutions with the p = 1
solutions. We call the coordinates on the p = 1 soliton x 0 and x 1 . The remaining four
coordinates x a are transverse to the 1-brane but parallel to the 5-brane.

5. Extremal Black Holes in String Theory 59
Compactification and ‘smearing’
The directions x 2 , ..., x 5 are compactified on T 4 ⊂ T 5 of volume V4 which leads to a
smearing of the D1-branes. This is analogous to the situation of the ‘method of images’
used in Kaluza-Klein theory. Let us for example look only at the compactification in
the x 2 direction on a circle of radius R. The D1-branes are then replaced by an infinite
array of parallel branes a distance 2πR apart. This gives an effective harmonic function
˜H1 for the array. For r ≫ R this function varies as
� ˜r 2 − (x 2 ) 2 � −5/2 . (5.84)
It is as if the D1-branes also wrap x2 √
. The effective number of D2-branes is q1 α ′ /R.
This case is easily generalized to the case where all coordinates are compactified on
T 4 . The effective number ˜q5 of D5-branes is given by
˜q5 = (2π√α ′ ) 4
q1. (5.85)
This is the number that should be plugged into (5.79) which then gives ˜ H1 from (5.78)
with p = 5. Now by the ‘harmonic function sum rule’ [35] we can simply take products
of the warp factors and get for the combined system
ds 2 =H −1/4
5
− H 3/4
5
V4
˜H −3/4
1 [(dx 0 ) 2 − (dx 1 ) 2 ] − H −1/4 ˜H 5
1/4
1 [(dz 2 ) 2 + ... + (dx 5 ) 2 ]
˜H 1/4
1 [(dy 6 ) 2 + ... + (dy 9 ) 2 ], (5.86)
e φ = H −1/2
5
˜H 1/2
1 , (5.87)
C(5) = [H −1
5 − 1]dx 0 ∧ dx 1 ∧ ... ∧ dx 5 , C(1) = [ ˜ H −1
1 − 1]dx 0 ∧ dx 1 . (5.88)
This solves one of our problems. The dilaton is now finite at the event horizon r = 0.
Our final mission is to repair the warp factor.

5. Extremal Black Holes in String Theory 60
Adding momentum in the x 1 -direction
The solution is to add momentum along the x 1 -direction. This leads [35] to the final
form of the metric
ds 2 =H −1/4
5
− H −1/4
5
˜H −3/4
1 [(dx 0 ) 2 − (dx 1 ) 2 + HP (dx 0 − dx 1 ) 2 ]
˜H 1/4
1 [(dx 2 ) 2 + ... + (dx 5 ) 2 ] − H 3/4 ˜H 5
1/4
1 [dr 2 + r 2 dΩ 2 3]. (5.89)
The harmonic functions may be written in the form
Using (5.28), (5.79) and (5.85),
˜H1 = 1 + r2 1
r 2 , H5 = 1 + L2 5
r2 , HP = r2 P
r
2 . (5.90)
r 2 1 = q1gsα ′ 2π√α ′ ) 4
, L
V4
2 5 = q5gsα ′ , r 2 P = ng 2 sα ′ 2π√α ′ ) 4 α
V4
′
, (5.91)
R2 with R the radius of the circle upon which the x 1 coordinate is compactified and n an
integer specifying the momentum P = n/R on the circle.
The entropy of this solution
Taking the (eight-dimensional) volume of the time slice at the horizon, we get the
‘area’ AH of the event horizon, which is, taking the limit r → 0 (from above) equal to
AH = � (r −3/4
1
L −1/4
rP )2πR �� (r 1/4
5
1 L −1/4
5 ) 4 V4
�� (r 1/4
1 L 3/4
5 )2π 2�
= 4π 3 RV4r1L5rP . (5.92)
The surface gravity is zero. Thus the temperature of this extreme black hole is zero.
This is not surprising since this in an extreme black hole. It does however have an
entropy. The entropy associated with this black hole simplifies to
Sbh = AH
4G10
= 2π √ q1q5n (5.93)
and is thus independent of both the string coupling gS and the size of the compact
dimension. It only depends on the integers q1, q5 and n.
Next, we will show how one gets this formula on the microscopic side.

5. Extremal Black Holes in String Theory 61
5.8.2 Derivation of microscopic entropy
The above constructed black hole solution can be interpreted as a bound state of q5
D5-branes and q1 D1-branes with some momentum n/R. The reason behind this is
that, as we have shown in (5.82), the soliton solution has qp units of p-brane charge
(we also argued that Dp-branes are sources of R-R charges).
lows
Now, our configuration (5.89) breaks the 10-dimensional Lorentz symmetry as fol-
SO(1, 9) → SO(1, 1) × SO(4)� × SO(4)⊥. (5.94)
The SO(1, 1) acts on the D1-brane world sheet (x 0 , x 1 ). The SO(4)� acts on the
rest of the D5-brane world volume (x 2 , ..., x 5 ). The final factor SO(4)⊥ acts on the
remaining dimensions (y 6 , ..., y 9 ) that are transverse to both. Since the rigid branes
don’t carry linear nor angular momentum, it is an important question to ask, what
degrees of freedom do carry the momentum n/R. Well, an obvious possibility is the
these degrees are the massless states of open strings that begin and end on D-branes.
(The massive excitations of the D-branes themselves do not play any role when the
coupling is weak, since they have masses proportional to g −1
S .) Now there are four
possibilities: 1-1 states, 5-5 states, 1-5 states and 5-1 states (by this notation we mean
that in an n-m state the open string begins on a Dn-brane and ends on a Dm-brane).
For the 1-1 and 5-5 state we are taken away from the maximal degeneracy state, which
is the black hole state, by the separation of the individual D1- and D5-branes from
each other. We thus only consider the 1-5 and 5-1 states. These give a total of 4q1q5
bosons and an equal number of fermions.
What we want to do, is to distribute the momentum n/R among the 4q1q5 bosons
and fermions. This is basically the same problem as we have seen before, in section
5.1 where we estimated the number of bosonic string states at mass level n. There is,
however, a small complication, because now also have fermions.
As before, we solve the problem by using a generating function (this is the partition
function)
Z(w) ≡ tr w N =
∞�
dnw n , (5.95)
n=0

5. Extremal Black Holes in String Theory 62
where N = P R and dn is the number of states with eigenvalue n of N (so it has
momentum n/R). Doing analogous steps to the one in section 5.1 we get
tr w N =
∞�
� m 1 + w
1 − wn �4q1q5 . (5.96)
m=1
The fermionic contribution in the numerator is even simpler than the bosonic. In the
one dimensional bosonic case we have many multiparticle states, α−n|0〉, (α−n) 2 |0〉,
etc., which, as we have seen lead to
tr w P ∞
n=1 α−nαn =
∞�
tr w α−nαn =
n=1
∞�
∞�
n=1 i=1
(w n ) i =
∞�
n=1
(1 − w n ) −1 . (5.97)
In the case that our modes were fermions, ψ−n, we would, by Pauli’s exclusion
principle, not be able to make multiparticle states (ψ−n) 2 |0〉, so our partition function
is simply
result
or
tr w P ∞
n=1 ψ−nψn =
∞�
tr w ψ−nψn =
n=1
∞�
(1 + w n ). (5.98)
Finally, we may estimate dn for large values of n, as we did in section 5.1 with the
n=1
dn ∼ e 2π√ q1q5n , (5.99)
Sbh = 2π √ q1q5n (5.100)
Basically, this is the Strominger-Vafa result [8]. There are however some differences,
which we explain in the next section. Before we do that let us note that we can permute
the three charges among themselves in all possible ways. Note that by doing four T-
dualities along the T 4 we can exchange the D1 and D5 branes with each other, leaving
the momentum P unchanged. Another set of dualities can map D1-D5-P to P-D1-D5.
Via S-duality this then can be mapped to P-NS1-NS5. (Therefore, keeping only the
first two charges the D1-D5 bound state is dual to the P-NS1 bound state, which is
simply an elementary string carrying vibrations.) For more details see section 2.5 of
[36].

5. Extremal Black Holes in String Theory 63
5.9 Link with the Strominger-Vafa result
The case treated by Strominger and Vafa in their break-through article [8] was only a
little bit different. They took (type IIB string theory) on the K3 manifold instead of
our T 4 . So in their case there are
Q5 D5−branes wrapping K3 × S 1 , and
Q1 D1−branes wrapping S 1 .
Again, the D1-brane can move in four directions transverse direction inside the D5-
brane, and we have 4Q1Q5 bosons and as many fermions. Now bosons have central
charge 1, whereas the fermions have central charge 1/2. This gives a central charge of
the corresponding conformal field theory for this situation
The Cardy formula, equation (5.66), gives us the answer
(1 + 1/2)4Q1Q5 = 6Q1Q5. (5.101)
Ω(N) = e 2π
√
cN
6 = e 2πQ1Q5N
, (5.102)
where N is the momentum. In fact, Strominger and Vafa use axion charge QH and
electric charge QF of the fundamental string. By the axion charge QH we mean the flux
of the H-field; it stands for the charge of the NS5-brane. By applying S-duality QH
and QF become equal to the Q1 and Q5 charges of the D1- and D5-branes above. So
this is the link between what we did in the previous section, and what Strominger and
Vafa do in their paper. They get for the degeneracy of the BPS solitons (or equivalently
the growth of the so-called elliptic genus, which they introduce in their paper) for large
charge QH
S = 2π
�
QH( 1
2 Q2 F + 1), (5.103)
which agrees to leading order with the Bekenstein-Hawking entropy for large Q 2 F ,
�
QHQ
SBH = 2π
2 F
. (5.104)
2

5. Extremal Black Holes in String Theory 64
Strominger and Vafa speculate that the elliptic genus is a more relevant physical
quantity than the actual number of BPS states, which may depend on the moduli of the
K3. The point is that the elliptic genus is an appropriately weighted sum (with weights
±1) and is moduli independent. However, either quantity will give the same leading
degeneracy as a function of the charges, and therefore this subtlety is unimportant
for their paper. Nevertheless, for many other purposes, like our counting of dyons in
N = 4 string theory in chapter 7, the elliptic genus turned out to be very important.

6. MATHEMATICAL TOOLS
6.1 Modular forms
A function of rational character on the plane can have two (or more?) complex periods.
A nice example of such a function is Jacobi’s sinus amplitudinus [37]. Let f be such a
(nonconstant, single-valued) function with periods ω: f(x + ω) = f(x). These form a
module L over Z. (Any integer combination of periods n1ω1 + n2ω2 is also a period.)
For ‘most’ functions L is trivial (most functions are not periodic), for some functions
(like the sinus) it is of rank one (L = 2πZ), and for some (like this so-called sinus
amplitudinus) L is of rank two. Jacobi [37] proved that there are no other possibilities.
In the third case, we call L a period lattice, and f an elliptic function.
The question one can ask now is: What are all the possible pairs of primitive peri-
ods for a fixed lattice L = ω1Z ⊕ ω2Z? (For example, ω1 + 6ω2 and ω2 will do just as
well as ω1 and ω2.) One can write one pair in terms of another, like
ω ′ 2 = aω2 + bω1, (6.1)
ω ′ 1 = cω2 + dω1, (6.2)
where a, b, c, d ∈ Z. It is easily seen that the determinant of the transformation matrix
should be −1 or +1. Without further comment, we choose it (we can do this 1 ) to be
+1. So what we learn from this, is that two pairs of primitive periods are related by a
linear substitution � a b
c d
�
∈ SL(2, Z). (6.3)
Conversely, any such substitution produces new primitive periods from old ones. The
corresponding period ratios ω = ω2/ω1 are related by the associated fractional linear
1 This can be done by imposing the condition that the period ratio ω = ω2/ω1 should have positive
imaginary part.

substitution by
6. Mathematical Tools 67
ω ↦→
aω + b
cω + d
(6.4)
from the projective group P SL(2, Z) = SL(2, Z)/(±1). This is called the modular
group of first level Γ1. Two tori (tori can be written as C/L) are (conformally) equiv-
alent if and only if their periods are related by a substitution of this kind.
tions
One easily checks that an element from P SL(2, Z) is generated by the transforma-
T : ω → ω + 1, S : ω → −1/ω. (6.5)
The change ω → ω ′ = (aω + b)/(cω + d) is called a modular transformation. The
generators T and S satisfy S 2 = (ST ) 3 .
A modular form of weight 2n ∈ N is an analytic function f on the open half-plane
H that transforms under the modular group Γ1 according to
� �
aω + b
f = (cω + d)
cω + d
2n f(ω). (6.6)
Such a form is of period 1 and so can be expanded in whole powers of the parameter
q = e 2πiω in a neighborhood of q = 0 (so at ω = i∞ so to speak). It is furthermore
required that this expansion contains no negative powers of q. Also, f is to be pole-free
at ∞. When f vanishes at infinity, we call it a cusp form. If we do not require f to be
pole-free at infinity, f is said to be a modular function. It has a Fourier expansion of
the form
f(ω) = �
n≥−N
anq n
(6.7)
which converges for 0 < q < ɛ ≤ 1 for some ɛ. (If N > 0 then f is said to have a pole
of order N at infinity. (So one speaks of a modular form precisely when f converges
for all |q| < 1, which implies N ≤ 0; a0 is it’s value an infinity.)
A typical example of a modular function is the Eisenstein series. A typical example
of a modular form is the Dedekind eta-function. Or perhaps one should say that raised
to a certain power this becomes a real modular form.

6. Mathematical Tools 68
6.2 (Weak) Jacobi forms
The definition of a Jacobi function that we give here is a generalization of the one
in [38]. We include half integer index, in order to incorporate elliptic genera of odd-
dimensional Calabi-Yau Manifolds. Let 0 ≤ k ∈ Z and 0 ≤ m ∈ Z/2. We call a
function φ from the upper half-plane H × C → C a Jacobi function of weight k and
index r if it transforms like
�
aω + b
φ
cω + d ,
�
z
= (cω + d)
cω + d
k rcz2
2πi
e cω+d φ(ω, z), (6.8)
φ(ω, z + λω + µ) = (−1) 2r(λ+µ) e −2πir(λ2 ω+2λz) φ(ω, z), (6.9)
where � a b
c d
and which has Fourier expansion of the form
φ(ω, z) =
�
∈ SL(2, Z), λ, µ ∈ Z (6.10)
�
m≥−M,l∈Z+r
c(n, l)q m y l
(6.11)
(where q = e 2πiω , y = e 2πiz , and M ∈ Z), which converges for all 0 < |q| < ɛ ≤ 1
and z ∈ C. One easily checks that the product of two Jacobi functions with weights
k, k ′ and indices r, r ′ is a Jacobi function of weight k + k ′ and index r + r ′ , so the class
just defined is closed under multiplication. We see that a modular function is a special
kind of Jacobi function: namely a Jacobi function of index r = 0. (φ(ω, 0) transforms
as a modular function of weight k. Also, for r = 0 we have c(m, l) = 0 if l �= 0.)
An important fact, which is essentially proven in [38], is that the coefficients c(m, l)
depend only on 4rm − l 2 and on l mod 2r. A weak Jacobi form is a special kind of
Jacobi function, namely one with a Fourier expansion of the form
φ(ω, z) = �
m≥0,l∈Z+r
c(m, l)q m y l , (6.12)
which converges for all |q| < 1. The reason why we are interested in this weak Jacobi
form, is that the elliptic genus of a Calabi-Yau manifold of complex dimensions d is
given by a weak Jacobi form of weight zero and index m = d/2, [39].

6. Mathematical Tools 69
In physics it seems we must use weak Jacobi forms and not Jacobi forms because
L0 − c/24 > 0 in the Ramond sector of a unitary theory. Let us now give the definition
of the elliptic genus, and study some of its properties.
6.3 Elliptic genus
Let us now give a proper definition of the elliptic genus for a Kähler manifold M of
complex dimensions d, study some properties of its properties and give a geometrical
description [40, 41, 42].
6.3.1 Definition and properties
Lets start with an elliptic curve E with modulus τ and a line bundle labeled by z ∈
Jac(E) ∼ = E. Recall that a an elliptic curve is basically a type of a cubic curve whose
solutions are confined to a region of space that is topologically equivalent to a torus.
In fact, an elliptic curve or (complex) torus, provided with a complex structure, can be
defined by the quotient
Eτ = C/(Z + τZ). (6.13)
The complex parameter τ determines the complex structure of the torus, and is called
the modulus of the torus. One can find the modulus of a torus by considering so-called
periods. Each torus has, up to scalings, a unique holomorphic one-form λ = adx,
where a is an arbitrary scaling factor, and x is a global complex coordinate. Now
the first homology of the torus is spanned by two fundamental 1-cycles, which can be
chosen to be the paths joining the origin with the points τ, and 1 (all in C). These are
closed curves on the torus and we call them γ1 and γ2 respectively. The modulus can
than be found from the quotient of the periods
�
γ1
τ =
λ
�
γ2 λ.
(6.14)
Different fundamental cycles give values for τ which are related by modular transfor-
mations.

6. Mathematical Tools 70
Next let us introduce q = e 2πiτ , y = e 2πiz . The elliptic genus is then defined as
χ(M; q, y) = TrH(M)(−1) F y FL q L0− d
8 ¯q ¯ L0− d
8 , (6.15)
where F = FL + FR and H(M) is the Hilbert space of the N = 2 supersymmetric
field theory with (d-dimensional) target space M. Physically, the elliptic genus count
the number of string states with ¯ L0 = 0. It is a genus because it satisfies the following
relations. It is additive under disjoint union,
χ(M ⊔ M ′ ; q, y) = χ(M; q, y) + χ(M ′ ; q, y). (6.16)
For any product space M × M ′ we have
And, finally,
in the sense of a complex bordism.
χ(M × M ′ ; q, y) = χ(M; q, y) · χ(M ′ ; q, y). (6.17)
χ(M = ∂N; q, y) = 0, (6.18)
For q = 0 the elliptic genus reduces to a weighted sum over the Hodge numbers,
which is essentially the Hirzeburch χy-genus,
χ(M; 0, y) = �
(−1) p+q y
p,q
p− d
2 h p,q (M). (6.19)
Recall that the Hodge numbers are the dimensions of the Hodge cohomology,
h p,q = dim H p,q (M) = dim ker ¯ ∂p,q
im ¯ , (6.20)
∂p,q−1
where ¯ ∂p,q indicates the anti-holomorphic differential ¯ ∂ (which maps a (p, q)-form to
a (p, q + 1)-form) restricted to (p, q)-forms.
For y = 1 the elliptic genus simply reduces the Euler number of M,
χ(M; q, 1) = χ(M) = �
(−1) n bn, (6.21)
where bn are the Betti numbers. The second equality is the Euler-Poincaré theorem.
For any topological space, the n-th Betti number bn is defined as the rank of the n-th
homology group. The relation between Betti numbers and Hodge numbers is
bn = �
h p,q . (6.22)
p+q=n
n

6. Mathematical Tools 71
6.3.2 A geometrical description
First, let us recall the notions of a Chern character and a Todd class. By the splitting
principle and the Whitney product formula, we can write the Chern class of a vector
bundle E, denoted by c(E), as [43]
c(E) =
r�
(1 + xi), (6.23)
i=1
where xi is the first Chern class of the complex line bundle Li in the Whitney sum; xi
is called a Chern root. Let x1, ..., xn be Chern roots of a rank r vector bundle E → M.
Then the Chern character of the vector bundle is defined as
r�
ch(E) = e xi . (6.24)
Here e xi is to be regarded as a formal power series. (And in fact it is a polynomial,
since the Chern roots are expressed in symmetric functions which are cohomology
classes that vanish for degree > r.) The Todd class is defined as
i=1
td(E) = � xi 1
i = 1r = 1 + −xi 1 − e 2 c1 + 1
12 (c21 + c2) + 1
24 c1c2 + ... (6.25)
If we take as bundle E the (holomorphic) tangent bundle of the some variety X then
we just write td(X).
We defined these two concepts in order to introduce the Hirzebruch-Riemann-Roch
formula, which states that
n�
i=0
(−1) i dim H i (X, E)) = deg((ch(E) · td(X))n), (6.26)
with X a non-singular, projective variety of dimension n, and E a holomorphic vector
bundle over X. On the right hand side of (6.26), ch(E) · td(X) is to be calculated in
the so-called Chow ring CH(X), and then the n-subscript indicates that we have to
take the part of the result which lies in CH n (X). We will not do anything with Chow
rings here, so we will not give the definition; this can be found in Appendix A of [44].
For smooth manifolds M, this makes an alternative geometrical definition of the
elliptic genus possible. For any vector bundle V define the formal sums
�
V = �
q k
k�
V, SqV = �
q
k≥0
k≥0
q k S k V, (6.27)

6. Mathematical Tools 72
where ∧ k and S k denote the k’th exterior and symmetric product respectively. Now
let TM and ¯ TM be holomorphic and anti-holomorphic tangent bundles of M. Then one
can define the bundle
�
�
�
d
−
Eq,y = y 2
n≥1
−q n−1 y
TM ⊗ �
−q n y −1
¯TM ⊗ Sq nTM ⊗ Sq n ¯ TM
�
. (6.28)
The elliptic genus can then we defined via the Hirzebruch-Riemann-Roch formula as
�
χ(M; q, y) = ch(Eq,y)td(M). (6.29)
M

7. COUNTING N = 4 DYONS VIA AUTOMORPHIC FORMS
In this final chapter we will focus on the dyonic spectrum of N = 4 string theory in
four dimensions. Dyonic states are states that are both electric and magnetic at the
same time. These preserve only 1/4 of the supersymmetries and are quite mysterious
since they can not be treated perturbatively in any description. This in contrast with
purely electric (or magnetic) states which preserve 1/2 of the supercharges and whose
degeneracies are easily determined (they correspond to the heterotic string states that
are in the right-moving ground state). We will study a proposal of Dijkgraaf, Verlinde
and Verlinde [45] for the exact degeneracy of these dyonic BPS states. Recently, their
proposal was confirmed [46].
Let us do some preparatory work first.
7.1 Electric-magnetic duality, S-duality and dyons of charge eθ/2π
S-duality is a (conjectured) equivalence between weakly coupled and strongly cou-
pled string theory; it relates the perturbative expansion around zero to an expansion
around infinite coupling. It is thus not an equivalence that holds order-by-order in
string perturbation theory. S-duality is a far-reaching equivalence that involves non-
trivial relations among all orders of perturbation theory.
Let us begin with considering the free Maxwell equations (so we take J = 0 in
(5.35)), dF = 0 and d(∗F ) = 0, or, written out with indices
where F µν
d
∂µF µν = 0, (7.1)
∂µF µν
d
= 0, (7.2)
= 1
2 ɛµναβ Fαβ. One observes that these stay perfectly valid under the trans-

7. Counting N = 4 Dyons via Automorphic Forms 75
formation F ↔ ∗F , or equivalently,
F µν ↔ F µν
d , (7.3)
which is called electric-magnetic duality (or “abelian S-duality”) since it interchanges
the E and B field. Note that the above Maxwell equations are different in nature: the
former is an equation of motion, derived from the action, while the latter is a Bianchi
identity.
This can be extended to the quantum theory provided the Dirac quantization con-
dition is satisfied [47]. This means that if a particle of electric and magnetic charge
(Qe, Qm) exists, and another of charges (Q ′ e, Q ′ m), then
QeQ ′ m − QmQ ′ e ∈ 2πZ. (7.4)
The simplest solution is that the electric charge is a multiple of e, and the magnetic
charge is a multiple of 2π/e, so we get a lattice
Qe = en1, Qm = 2π
e n2, (7.5)
with n1, n2 integers. Note that at this moment something of the strong ↔ weak cou-
pling (or S-duality) becomes apparent, since a theory with an electrically charged field
(e, 0) and a magnetically charged field (0, 2π/e) is invariant under (7.3) if we also take
e → 2π/e, interchanging strong and weak coupling.
We can get a general solution to the Dirac quantization condition by including a
θ-parameter for the gauge field; the electric charges shift by an amount proportional to
the magnetic charge [48],
At this point it is useful to form the combination
θ
Qe = en1 + en2
2π , Qm = 2π
e n2. (7.6)
τ = θ
2π
i
+ . (7.7)
e2 The solution of the Dirac quantization condition (7.6) is invariant under θ → θ + 2π,
with n1 → n1 − n2, which can be rephrased as
τ → τ + 1, n1 → n1 − n2. (7.8)

7. Counting N = 4 Dyons via Automorphic Forms 76
Furthermore, one can check that under electric-magnetic duality and θ → θ + 2π we
have
τ → − 1
τ , n1 ↔ n2. (7.9)
So these generate the familiar SL(2, Z).
We will use electromagnetism with an angle θ, as above, and the special combina-
tion which we called τ to show how one gets the entropy formula [49]
SBH = π � q 2 eq 2 m − (qe · qm) 2 , (7.10)
for the Bekenstein-Hawking entropy for 1/4-BPS black holes that we will encounter in
the following.
7.2 Making use of the attractor mechanism
From (7.5) the ratio of qe and qm is equal to e 2 up to some constant and we can write
(in this section we will write p and q for qe and qm respectively)
√ pq ∼ |ep + iq
e |2 = e 2 |p + iq
e 2 |2 . (7.11)
We can write this in a more general form by using τ as in the previous section
τ = θ
2π
i
+ . (7.12)
e2 The angle θ can vary and for θ = 0 we get our equation (7.11) back; the generalized
expression is
1
|�p + τ�q| 2 . (7.13)
τ2
Here τ2 is the imaginary part of τ and one can check that for θ = 0 we indeed get back
(7.11) for θ = 0, since (7.13) is just
e 2 |�p + �qθ i�q
+
2π e2 |2 . (7.14)
The thing we would like to do is to minimize this function (7.13) of τ. The so-
called attractor mechanism [50] explains why this we should minimize this at the hori-
zon of the black hole. So let us minimize
1
|�p + τ�q| 2 = 1
�
|�p| 2 + (τ 2 1 + τ 2 2 )|�q| 2 + 2τ1(�p · �q) 2
�
τ2
τ2
(7.15)

7. Counting N = 4 Dyons via Automorphic Forms 77
first with respect to τ1 and plug the result into the minimization with respect to τ2.
Minimizing with respect to τ1 gives
Minimizing with respect to τ2 gives
− |�p|2
τ 2 2
τ1 = −
(�p · �q)
. (7.16)
|�q| 2
− τ 2 1
τ 2 |�q|
2
2 + |�q| 2 − 2τ1(�p · �q)
τ 2 2
By plugging in the τ1 that we found in (7.16) we get
So
and we get out desired formula
− 1
τ 2 � 2 (�p · �q)
|�p| +
2
2
|�q| 2 − |�q|2 τ 2 2 −
= 0. (7.17)
2(�p · �q)2
|�q| 2
�
= 0 (7.18)
(�p · �q) 2 − |�p| 2 |�q| 2 = −|�q| 4 τ 2 2 , (7.19)
|�q| 2 τ2 = � |�p| 2 |�q| 2 − (�p · �q) 2 . (7.20)
7.3 The Dijkgraaf-Verlinde-Verlinde degeneracy formula
In this section, we shall recall an interesting conjecture, due to R. Dijkgraaf, E. Ver-
linde and H. Verlinde (DVV), which relates the exact degeneracies of 1/4 BPS states
in N = 4 string theory, to Fourier coefficients of certain automorphic forms [45].
N = 4 string theory has two perturbative formulations, one in terms of the heterotic
string compactified on T 6 , and a dual one in terms of type II strings compactified on
K3 × T 2 . The moduli space factorizes into
SL(2, R)
U(1)
× SO(6, 22, R)
SO(6) × SO(22)
and points related by an action of the duality group
(7.21)
Γ = SL(2, R) × SO(6, 22, R) (7.22)
are conjectured to be equivalent under non-perturbative dualities. In the point of view
of the heterotic string we have 28 electric charges qe and 28 magnetic charges qm which

7. Counting N = 4 Dyons via Automorphic Forms 78
both lie on the Γ 22,6 lattice. We see that the Bekenstein-Hawking entropy for 1/4-BPS
black holes, which is given by 1
SBH = π � q 2 eq 2 m − (qe · qm) 2 (7.23)
is invariant under the duality group (7.22). This entropy equation stays invariant un-
der qe ↔ (−)qm which, of course, is related to the first factor of (7.22), being an
electromagnetic S-duality which acts on electric charges qeΛ and magnetic charges q Λ m,
Λ = 0, ..., 27 transforming in the 28 of the second factor.
DVV proposed that the exact degeneracies, or, to be more precise, the number
of bosonic minus the number of fermionic BPS-multiplets for a given electric and
magnetic charge, should be given by
�
d(qe, qm) =
dρdσdν eiπ(q2 mρ+q2 eσ+2qe·qmν) . (7.24)
Φ(ρ, σ, ν)
The integral is over the contour 0 ≤ ρ, ν, σ ≤ 2π. The function Ω(ρ, ν, σ) is manifestly
SL(2, Z) invariant, which suggest that it can be written as a modular form in
� �
ρ ν
Ω =
ν σ
(7.25)
It is natural to identify the matrix Ω with the period matrix of a genus two Riemann
surface. This period matrix Ω transforms under Sp(2, Z), which can be written as a
4 × 4 matrix decomposed onto four real 2 × 2 blocks, A, B, C, D,
� �
A B
C D
with
Then Ω transforms as follows under Sp(2, Z),
(7.26)
A T D − C T B = DA T − CB T = 12, (7.27)
A T C = C T A, B T D = D T B. (7.28)
Ω → Ω ′ = (AΩ + B)(CΩ + D) −1
1 See section 7.2 for a quantitative explanation of where this expression comes from.
(7.29)

7. Counting N = 4 Dyons via Automorphic Forms 79
In fact, Ω is the unique automorphic form of weight 10 of the modular group Sp(2, Z).
Combining the electric and magnetic charges into a vector
� �
qm
q =
the degeneracy formula takes the form
�
d(qe, qm) =
qe
(7.30)
dΩ eiπq·Ω·q
. (7.31)
Φ(Ω)
This can be seen as a ‘logical’ generalization of the 1/2-BPS purely electric case, in
which case the number of multiplets is given by
�
d(qe) =
dσ eiπσq2 e
, (7.32)
η(σ) 24
where the integral over σ is from 0 to 1 and η(σ) is the Dedekind η-function.
DVV carried out several consistency checks, some of which we will briefly discuss.
7.3.1 Consistency checks
One consistency check one can do, that was done by DVV, is to extract the large
charge behavior of d(qe, qm) by computing the contour integral in (7.24) by residues.
Of course, what one expects to find, is agreement with (7.23), and indeed it can be
shown that in the large charge limit
d(qe, qm) ∼ e π
√
q2 eq2 m−(qe·qm)2 . (7.33)
This check can be done by picking the residue at the divisor D = ρσ − ν 2 + ν and
using
where
See [45], [51] and [52] for some more details.
Φ = D 2 η 24 (ρ ′ )η 24 (σ ′ )/det 12 (Ω), (7.34)
ρ ′ = − σ
ρσ − ν2 , σ′ = − ρ
. (7.35)
ρσ − ν2 As another check, and continuing our discussion of Sp(2, Z) above, let us note that
the SL(2, Z) duality transformations are identified with the subgroup of Sp(2, Z) that

7. Counting N = 4 Dyons via Automorphic Forms 80
leave the genus two modular form Φ(Ω) invariant, which leads us to the conclusion
that the presented degeneracies are manifestly duality symmetric. (Using the modular
invariance of Φ, one can cancel the action of SL(2, Z) by a change of the integration
contour γ → γ ′ , and deform γ ′ back to γ while avoiding singularities.)
7.4 State counting and the elliptic genus
The way in which the elliptic genus makes its entrance in the counting of N = 4 dyons
is via the cusp form Φ, which was an infinite product representation
Φ(ρ, σ, ν) = e 2πi(ρ+σ+ν)
�
�
1 − e 2πi(kρ+lσ+mν)
�c(4kl−m2 )
, (7.36)
(k,l,m)>0
where (k, l, m) > 0 means that k, l ≥ 0 and m ∈ Z, m < 0 for k = l = 0. The
coefficients c(k) are the Fourier coefficients of the elliptic genus of K3,
χ(K3; ρ, ν) = �
c(4h − m 2 )e 2πi(hρ+mz) . (7.37)
h≥0,m∈Z
It is known [41] that the elliptic genus of K3, which is defined by
χ(K3, τ, z) = Tr(−1) F c
2πi(τ(L0−
e 24 )+zFL)
(7.38)
plays an import role in the counting of certain string states. In the above expression
the trace is over the RR-sector of a superconformal sigma-model and F = FL + FR is
the sum of the left- and right- moving fermion number. One can think of the elliptic
genus as a partition function. An interesting identity for the elliptic genus that we will
just quote (a proof can be found in [42]) is a relation between orbifold elliptic genera
of the symmetric product manifolds in terms of that of a Kähler manifold M is
�
p N χ(S N M; q, y) = �
(7.39)
N=0
n>0,m≥0,l
where the coefficients c(m, l) are defined via the expansion
χ(M; q, y) = �
m≥0,l
1
(1 − p n q m y l ) c(nm,l)
c(m, l)q m y l . (7.40)
There is a nice physical interpretation of this identity in terms of second quantized
string theory [42].

8. OUTLOOK
At the end of this thesis we come to the starting point of recent research work on black
holes in the context of string theory. Much can be said at this point, but let us only
mention briefly (quite arbitratily): 1. Near-extremal black holes, 2. Mathur’s fuzzball
proposal, and 3. Topological strings and the OSV conjecture. All three could be a
subject for a thesis of its own, so we will just scratch the surfaces.
Near-extremal black holes
The Strominger-Vafa result can be generalized to near-extremal black holes, as was
soon realized by Callan and Maldacena [53]. In the near-extremal case the entropy
becomes a function of the mass of the black hole as well as its four charges. Probably
the most prominent application of going away from the BPS limit and studying near-
BPS states is the derivation of Hawking radiation from the D-brane perspective. By
adding a small amount of right-moving momentum to a purely left-moving BPS state
one gets a near-BPS state. Intuitively one can imagine that what happens in the D-
brane picture is that left- and right-moving open strings interact to form closed string
states and leave the brane. This is indicated in picture 8.1. A nice review of various
aspects of near extremal black holes is Maldacena’s PhD thesis [27].
Note that extremal black holes do not Hawking radiate (as we already discussed).
This is clear from the picture: you get Hawking radiation only when a left mover
attaches to a right mover and they leave the brane as a closed string (this is the near-
extremal case).

8. Outlook 82
Fig. 8.1: In the D-brane picture left- and right-moving open strings can interact, form closed
string states and leave the brane as Hawking radiation.
Mathur’s fuzzball proposal
Mathur et al, see e.g. [36], set up a whole program in which the aim is to give all mi-
croscopic states of a black hole an own gravitational solution. This so-called ‘fuzzball’
approach looks on the one hand very promising, but is on the other hand quite con-
troversial. For example, in the recent article on the Farey tail expansion for N = 2
attractor black holes [54], quite the contrary seems to be the case. The major part of
the black hole entropy is carried by one particular black hole background, with no one
to one correspondence between microstates and gravitational backgrounds.
Topological strings and the OSV conjecture
The claim of the OSV conjecture [55] is that there is a direct relation between black
hole entropy and the topological string partition function, |Ztop| 2 = ZBH. An excellent
review is [52].

APPENDIX
Appendix A: Geodesic Deviation and the Raychaudhuri Equation
As is well known, the defining property of flat geometry is the postulate that initially
parallel lines always remain parallel. In curved space this is not true (think for example
of a sphere). To study this ‘curved behavior’ we consider a family of non-crossing
geodesics, γs(λ). Here for each s ∈ R, γs(λ) is a geodesic parameterized by the affine
parameter λ. On the two-dimensional surfaces that these geodesics define we pick
coordinates s and t. We can think of our surface as a set of points x µ (s, t). It is natural
to define vector fields T µ (tangent vectors) and S µ (deviation vectors) as indicated in
figure A.1, namely by
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Geodesics
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Deviation vector
s
(8.1)
Fig. 8.2: A set of geodesics with tangent vectors and deviation vectors. The vector field consisting
of the deviation vectors measures the deviation between nearby geodesics.
We want to look at how S µ changes in the T direction, so we define the “relative

velocity of geodesics” as
8. Outlook 84
V µ = (∇T S) µ = T ρ ∇ρS µ
and also the “relative acceleration of geodesics”
(8.2)
A µ = (∇T V ) µ = T ρ ∇ρV µ . (8.3)
A straightforward calculation (see [10], page 146) shows that the relative acceleration
between two neighboring geodesics is proportional to the curvature:
A µ = R µ νρσT ν T ρ S σ
This equation is called the geodesic deviation equation.
(8.4)
Instead of looking at a one-parameter family of geodesics we will now consider an
entire congruence of geodesics. A congruence in O ⊂ M, where O is an open subset
of a manifold M, is a family of curves such that through each point p ∈ O there passes
exactly one curve in this family. Clearly, we get a vector field for each congruence and
the other way around. A congruence is said to be smooth if the corresponding vector
field is smooth.
Our goal in the rest of this section will be to define the expansion, shear, and twist
of timelike geodesic congruences in a spacetime (M, gµnu) and derive an equation (the
Raychaudhuri equation) for their rate of change as one moves along the curves in the
congruence. One can imagine setting up a set of three normal vectors orthogonal to our
timelike geodesic and looking at their failure to be parallel transported as we follow
their evolution along the geodesics.
cated.
We will also briefly discuss the null geodesic case which is slightly more compli-
Let T µ = dx µ /dτ be the tangent vector field to a four-dimensional timelike geodesic
congruence; T µ is normalized, TµT µ = −1, and obeys the geodesic equation T λ ∇λT µ =
0. We define a tensor field Bµν by
Bµν = ∇νTµ
(8.5)

8. Outlook 85
Physically, Bµν measures the failure of S µ to be parallel transported along the congru-
ence. Next we define a “spatial metric” 1
The expansion θ is defined by
hµν = gµν + TµTν
θ = B µν hµν
(8.6)
(8.7)
and it describes the change of a volume of a sphere of test particles centered at some
point when moving along their central geodesic. Next we define the shear σµν, which
represents a distortion in the shape of our collection of test particles from a sphere into
an ellipsoid, by
Finally, we define the twist or rotation
Thus, we can decompose Bµν as
σµν = B(µν) − 1
3 θhµν
(8.8)
ωµν = B[µν]. (8.9)
Bµν = 1
3 θhµν + σµν + ωµν. (8.10)
Now the geodesic deviation equation (8.4) yields equations for the rate of change
of θ, σµν, and ωµν:
dθ
dτ
= −1
3 θ2 − σµνσ µν + ωµνω µν − RµνT µ T ν
(8.11)
This is the timelike version of the Raychaudhuri equation. It gives a relation be-
tween the rate of change of the expansion, and the shear and twist of timelike geodesics
as one moves along the curves in the congruence.
One can also consider null geodesics. The derivation of the null version of Ray-
chaudhuri’s equation is slightly more complicated (see for example [10] Appendix F,
or [11] chapter 9.2)] and we will not treat it here. The main difference is that for null
1 One can think of this as some kind of projection: any vector from the tangent space TpM can
be projected onto a subspace corresponding to vectors normal to T µ by the ‘projection tensor’ h µ ν =
g µρ hρν = δ µ ν + T µ Tν.

8. Outlook 86
geodesics studying the evolution of vectors in a three-dimensional subspace normal to
the tangent field doesn’t make sense, since tangent vectors to null curves are normal
to themselves. Instead, we can study a two-dimensional subspace of spatial vectors
normal to the null tangent vector field k µ = dx µ /dλ. The difficulty is that observers in
different Lorentz frames will have different notions of what constitutes a spatial vec-
tor, so there is no unique way of defining this subspace. For more details we refer the
reader to Appendix F of [10]. This difference between the three and two dimensional
subspaces in respectively the timelike and null geodesic case results in a pre-factor of
1/2 instead of 1/3 in the null Raychauhuri equation, which reads
where
dθ
dλ
= −1
2 θ2 − ˆσµν ˆσ µν + ˆωµν ˆω µν − Rµνk µ k ν , (8.12)
θ = ˆ h µν Bµν
ˆ (8.13)
ˆσµν = ˆ B(µν) − 1
2 θˆ hµν
(8.14)
ˆωµν = ˆ B[µν] (8.15)
are again to be interpreted as the expansion, shear, and twist of the congruence. The
hatted tensor field ˆ Bµν is given by
ˆBµν = 1
2 θˆ hµν + ˆσµν + ˆωµν, (8.16)
and the hatted metric ˆ h µν is the metric on our two-dimensional subspace. Let us be a
little bit more precise about what is meant by this. The tangent vectors in the tangent
space Vp at a point p ∈ M which are orthogonal to k µ form a three-dimensional
subspace ˜ Vp. Now define ˆ Vp to be the vector subspace of equivalence classes of vectors
in ˜ Vp, with the equivalence relation
x µ ∼ y µ ⇐⇒ x µ − y µ = ck µ
(8.17)
for some c ∈ R. The spacetime metric gµν gives now rise to a ‘hatted metric’ on
ˆVp. This hatted metric is denoted by ˆ hµν. The reason why we put no hat on θ is that
ˆh µν ˆ Bµν = h µν Bµν as one can easily verify.

8. Outlook 87
Appendix B: Samenvatting in het Nederlands
Dit stukje is — in gewijzigde vorm — verschenen op http://www.natuurkunde.nl/
Van zwarte gaten heeft iedereen wel eens gehoord. “Een zwart gat is een bijzonder
soort object in het heelal, waar als gevolg van het sterke zwaartekrachtsveld geen licht
of materie kan ontsnappen”, is wat Wikipedia ons vertelt. Maar dat is nog niet het hele
verhaal. Zwarte gaten zijn namelijk zo zwart nog niet. Dat komt door een vreemde
eigenschap van het vacuüm. Je zou denken dat vacuüm ‘niks’ is. Maar dat is niet zo:
uit het ‘niks’ worden aan één stuk door deeltjes en antideeltjes gevormd die vervolgens
weer snel verdwijnen, althans, tenzij er ééntje in een zwart gat valt.
Zwarte gaten
In dit stukje zal ik ervan uitgaan dat u in ieder geval een vaag beeld hebt van wat een
zwart gat is, maar meer dan dat hoeft ook niet. Dat zwarte gaten zwart zijn klinkt
ergens wel logisch. Maar als u zich er wat meer in verdiept blijkt al snel dat zwarte
gaten eigenlijk ‘grijs’ zijn. Ze slokken wel alles op, dat is waar, maar er komen ook
deeltjes uit! Wel altijd deeltjes van ‘hetzelfde soort’. Het maakt niet uit wat u in
het zwarte gat hebt gegooid, of het nou een fles was waar nog statiegeld op zat (dat
statiegeld kunt u dan trouwens wel vergeten) of een tafel of een stoel, er komt altijd
hetzelfde uit.
Tja, zult u zich afvragen, hoe kan dat eigenlijk? Zwarte gaten zijn toch van die
zware sterren waar de ontsnappingssnelheid groter is dan de lichtsnelheid. En niks
kan sneller dan het licht. Dus niks kan ontsnappen, zelfs licht niet. Hoe kunnen die
deeltjes dan wèl ontsnappen?
Vacuüm is niet niks
Helemaal niks bestaat niet in de kwantum mechanica. De onzekerheidsrelaties van
Heisenberg vertellen ons dat dat niet kan. In de kwantum mechanica is namelijk niks
zeker, en als het vacuüm echt helemaal leeg zou zijn, zou u zeker weten dat het exact

8. Outlook 88
leeg is, maar aangezien niks zeker is, kan vacuüm dus simpelweg niet helemaal leeg
zijn. Volgt u het nog?
De natuur lost dit probleem op een wonderbaarlijke manier op. Er worden constant
paren deeltjes en antideeltjes uit het niks gevormd die snel weer verdwijnen door tegen
elkaar aan te botsen (zie figuur 8.3). Zulke deeltjes heten ‘virtuele deeltjes’.
Fig. 8.3: Het vacuüm is niet niks. Er worden constant paren deeltjes en antideeltjes uit het
niks gevormd die snel weer verdwijnen door tegen elkaar aan te botsen. Deze deeltjes
noemen we virtuele deeltjes.
Maar om op het verhaal van de zwarte gaten terug te komen: stelt u zich eens voor
dat er zo een deeltje-antideeltje paar wordt gevormd net naast een zwart gat. Het kan
dan gebeuren dat het antideeltje het zwarte gat invalt, terwijl het deeltje de andere kant
op gaat. Ze kunnen dan niet meer tegen elkaar aanbotsen en verdwijnen. Voor ons lijkt
het er dan op of er een deeltje door het zwarte gat is uitgezonden (zie figuur 8.4).
De energie van het zwarte gat neemt op deze manier af. Denk maar aan energiebe-
houd. Het deeltje wat onze kant op is gevlogen heeft een zekere energie en omdat
energie behouden is moet het zwarte gat wel een zelfde hoeveelheid energie zijn ver-
loren. Nu is energie via E = mc 2 equivalent is aan massa. We zien dus dat een zwart
gat op deze manier massa kwijtraakt. Dit is het Hawking effect.

8. Outlook 89
Fig. 8.4: Virtuele deeltjes in de buurt van een zwart gat.
Kwantum zwarte gaten
Waarom is het eigenlijk interessant om zwarte gaten te bestuderen? Daar zijn meerdere
redenen voor. De belangrijkste is dat zwarte gaten zowel erg zwaar als erg klein zijn.
Daarom kan men natuurkundige theorieën die gebruikt worden om zwarte gaten te
beschrijven ook gebruiken om de Big Bang te bestuderen. Toen was immers ook alle
materie heel dicht op elkaar gepakt.
Voor de meeste verschijnselen in de natuur volstaat het om ofwel kwantum me-
chanica ofwel relativiteitstheorie te gebruiken. Maar voor het bestuderen van een zwart
gat zijn allebei de theorieën tegelijk nodig en dat brengt een lastig probleem met zich
mee. Het punt is dat kwantum mechanica en relativiteitstheorie elkaar niet zo mogen.
Zodra men berekeningen gaat doen waarbij beide theorieën tegelijk gebruikt worden,
komt er steeds een vervelend ‘oneindig’ uit. Er is dus een nieuwe theorie - zeg maar
een theorie van ‘kwantum gravitatie’ - nodig om verder te komen. Momenteel is de
enige serieuze kandidaat voor zo een theorie de snaartheorie. Deze scriptie gaat voor
een groot gedeelte over hoe men gebruik makend van snaartheorie een goed begrip van
zwarte gaten kan krijgen. Aan de andere kant kan een goed begrip van zwarte gaten de
snaartheorie, die nog steeds een theorie in wording is, wellicht een stap verder brengen.

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