AdS-CFT correspondence
Bilal Khalid
Senior Year Project Report
Department of Physics
Syed Babar Ali School of Science and Engineering
Lahore University of Management Sciences
Supervisor: Dr. Babar Qureshi
May 22, 2018
Contents
Preface 5
1 An Introduction to String Theory 7
1.1 Action . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.2 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.1 Classical Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.2.2 Quantization and Spectrum . . . . . . . . . . . . . . . . . . . . . . . . 12
1.3 String Interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.5 Superstring theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2 The AdS-CFT Correspondence 23
2.1 N coincident D-branes in type IIB superstring theory . . . . . . . . . . . . . 23
2.2 D-branes as spacetime geometry . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3 The correspondence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 IR/UV connection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3 Black Hole Phase Transition 27
3.1 Schwarzschild black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.2 AdS black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3 The Hawking-Page phase transition . . . . . . . . . . . . . . . . . . . . . . . . 30
3.4 The confinement-deconfinement phase transition . . . . . . . . . . . . . . . . . 32
A The path integral and field theory at a finite temperature 35
3
B Spinors 39
C Conformal field theory 45
D Anti-de Sitter Spacetime 51
E Schwarzschild Geometry 55
Bibliography 59
4
Preface
Holography is the equivalence of a theory of gravity in an (n+ 1)-dimensional spacetime with
a theory, which does not involve gravity, on its n-dimensional boundary. In this thesis, we
derive a realization of holography in the framework of string theory. Specifically, we will relate
type IIB superstring theory (a theory of gravity) in AdS5 × S5 spacetime with N = 4 Super
Yang-Mills theory on the boundary of AdS5 i.e. M4. Since N = 4 SYM is a conformal field
theory, this duality is called the AdS − CFT correspondence. We also study a black hole
phase transition in AdS5 on the gravity side. An attempt to figure out the corresponding phase
transition on the field theory side shows that the black hole phase transition can be interpreted as
a confinement-deconfinement phase transition on the field theory side, which provides further
evidence for the correspondence. Moreover, since confinement-deconfinement transition is also
an important feature of Quantum Chromodynamics (QCD), this interpretation of the black hole
phase transition is also an evidence for the AdS − QCD correspodence, which hasn’t been
established yet.
The plan of the thesis goes like this. In chapter 1, we give an introduction to string theory.
More specifically, we quantize the bosonic string theory action and derive the spectrum. We
study D-branes which are higher dimensional dynamical objects that arise in string theory.
To incorporate fermions into our model, we develop superstring theory. The spectrum of the
superstring theory tells us the dimensionality of the D-branes that are stable. Then we employ
3 dimensional D-branes to motivate the AdS − CFT correspondence in chapter 3. In chapter
4, we study the Hawking-Page phase transition, which takes place in the bulk and makes black
holes the preferred state of the AdS universe over thermal radiation, above a certain critical
temperature. Then we connect this phase transition with the confinement-deconfinement phase
transition on the boundary theory through AdS − CFT .
Appendices on a variety of topics have been added in the end to make the thesis as self-
5
contained as possible. The prerequisites for understanding the thesis are introductory courses in
quantum field theory and general relativity.
6
Chapter 1
An Introduction to String Theory
In order to derive the AdS −CFT correspondence, one needs to look at objects called D-branes
from two different perspectives. First from the point of view of string theory and second through
the picture of supergravity. D-branes in string theory are multi dimensional objects on which
gauge fields live. In the low energy limit, these objects decouple from the bulk and then the
physics of D-branes is described by gauge theories. Whereas, in supergravity, D-branes are
objects that gravitate the spacetime around them. In this framework, when we work out the
solution with D-branes, the geometry comes out to be AdS. Since the string theory picture
and the supergravity picture are just two different ways of describing the same physics, we
equate these two pictures of D-branes. And the result is the AdS −CFT correspondence, which
equates a gauge theory with string theory in AdS space. In order to derive and understand the
correspondence in detail, it is imperative to understand what D-branes are and how they arise in
string theory. That is exactly the purpose of this chapter. We’ll first write the string theory action
and quantize it. This chapter is based on ref. [4, 1, 2, 3].
1.1 Action
We know that a particle is a 0-dimensional object that traces out a 1-dimensional worldline in
spacetime. As a natural extension of this concept we can have higher dimensional objects with
higher dimensional trajectories. Let’s call the 1-dimensional objects strings. Their trajectories
will be described by 2-dimensional worldsheets. Generally, a p-dimensional object, called a Dp-
brane (a particle is a D0-brane, a string is a D1-brane), has a (p+ 1)-dimensional worldvolume.
We’ll write the string theory action by considering only 1 dimensional objects i.e. strings.
7
However, later we’ll see that implementation of certain boundary conditions gives rise to higher
dimensional objects as well.
In string theory, every (quantum) vibration mode of the string gives us a particle. Different
modes correspond to different particles. In this section we’ll write the classical action for strings
and in the next section, we’ll quantize it to see what kind of particles do these oscillations of
the strings give us. We’ll write the action of strings in analogy to that of particles. We know
that particles follow worldlines that have the minimum spacetime distance and the action is
thus proportional to the worldline distance. Likewise, we define string worldsheets to have
minimum area. So, string action is proportional to the worldsheet area. This action is called the
Nambu-Goto action and it is written as:
SNG = T
∫Σ
dA =1
2πα′
∫Σ
dA, (1.1)
where Σ represents the worldsheet, and in the last expression we have re-written T as 1/2πα′. T
is called the tension of the string. Since the mass dimension of dA is −2 and that of the action
is 0, so T must have mass dimension 2. Similarly, α′ has mass dimension −2. Here we have
introduced T just as a parameter to get the dimensions right.
Before we proceed to quantization, we’ll re-write the action in a form that is easier to
quantize,
SP =1
4πα′
∫Σ
d2σ√−hhabgµν∂aXµ∂bX
ν . (1.2)
Here Xµ are the co-ordinates that the target spacetime (the spacetime in which the strings live)
assigns to the worldsheet and g is the metric of the target spacetime. Since, the worlsheet is
2-dimensional, we parameterize it using two parameters, τ and σ. Later, we’ll assign a temporal
nature to τ and a spatial one to σ. For convenience, to refer to indices of the worldsheet co-
ordinates we’ll use English letters a, b, c etc. and for the target spacetime we’ll use the Greek
symbols µ, ν, γ etc. a and b can take the values 0 or 1. σ0 refers to τ and σ1 refers to σ. In
eq. (1.2), h refers to the metric in the worldsheet co-ordinates. As, gµν∂aXµ∂bXν is the metric
inherited by the worldsheet from the target spacetime, it must be equal to hab, which contracts
with hab to give 2, and we get the Nambu-Goto action back since d2σ√−h is the area element.
From now on, we’ll work with Minkowskian target spacetime i.e. gµν = ηµν .
8
1.2 Quantization
1.2.1 Classical Solutions
The target spacetime co-ordinatesXµ are fields living on the worldsheet, we are going to quantize
them. We’ll follow the usual canonical quantization procedure. First we need to find the classical
equations of motion. The equations of motion we get after varying the action are,
hab : ∂aXµ∂bXµ −
1
2habh
cd∂cXµ∂dXµ = 0, (1.3)
Xµ : ∂a(√−hhab∂bXµ) = 0. (1.4)
There are two kinds of strings: closed and open. Closed strings are those that loop back to the
starting point. For such strings, we choose the spatial parameterization σ ∈ [0, 2π] with a period
of 2π. On the other hand, for open strings σ ∈ [0, π]. On varying the action, in addition to getting
the equations of motion, we find that the open strings also have to satisfy the following boundary
conditions,
(δXµ)hσb∂bXµ
∣∣∣∣σ=0,π
= 0. (1.5)
So, there are two possible choices of boundary conditions at each σ = 0, π: δXµ = 0 (Dirichlet)
or hσb∂bXµ = 0 (Neumann). For the time being, we’ll use Neumann boundary conditions for all
Xµ. Later, we’ll also see the implications of Dirichlet b.c., which give rise to D-branes.
We need to remove gauge redundancies in the degrees of freedom before quantization. To do
that, we first notice that the Polyakov action has the following symmetries:
• Poincare symmetry of the target spacetime,
• Re-parameterization of the worldsheet,
• Weyl scaling: hab → e2ω(τ,σ).hab.
Using a co-ordinate transformation on the worldsheet we can always get a metric of the form
hab = e2ω(τ,σ)ηab and after Weyl scaling we obtain hab = ηab. By making this choice we have
almost fixed the gauge. We say almost, because there is still some gauge redundancy. To see it,
let’s go to the light cone co-ordinates: σ± = 1√2(τ ± σ). The light cone metric is,
ds2 = −2dσ+dσ−. (1.6)
9
Under the transformation σ+ → σ+ = f(σ+), σ− → σ− = g(σ−):
ds2 → ds2 = −2dσ+dσ− = −2f ′(σ+)g′(σ−)dσ+dσ−. (1.7)
The pre-factor can be removed by Weyl scaling which gives us the metric we started with. The
gauge redundancy in terms of τ is,
τ =1√2
(σ+ + σ−) =1√2
(f(τ + σ) + g(τ − σ)
). (1.8)
To fix the gauge, we need to fix the functions f and g. And there is a natural choice for these
functions. To see that, let’s use hab = ηab to simplify eq. (1.4) to,
∂20X
µ − ∂21X
µ = 0. (1.9)
This is a wave equation. Generally, all solutions of Xµ will have a left moving and a right
moving part, and these solutions fit the bill for f and g. We choose v+τ = 1√2(X0 +X1) where
v+ is a constant. This choice is called the light cone gauge.
The most general solution of eq. (1.9) is,
Xµ(τ, σ) = xµ + vµaσa +Xµ
R(τ − σ) +XµL(τ + σ). (1.10)
For closed strings, Xµ(τ, σ) = Xµ(τ, σ + 2π),
xµ + vµ0 τ +XµR(τ −σ)+Xµ
L(τ +σ) = xµ + vµ0 τ +2πvµ1 +XµR(τ −σ−2π)+Xµ
L(τ +σ+2π).
Thus, vµ1 = 0 and XR and XL are independent periodic functions with period 2π. For open
strings we have to satisfy the Neumann b.c. i.e. ∂σXµ∣∣σ=0,π
= 0. At σ = 0, we get vµ1 = 0 and
X ′R(τ) = X ′L(τ). At σ = π, X ′R(τ − π) = X ′L(τ + π). So, we conclude that for an open string
the left and right moving parts are equal and periodic in 2π. Since, vµ1 = 0 for both closed and
open strings, we’ll use vµ in place of vµ0 for convenience.
In the target spacetime, we transform to the co-ordinates X± = 1√2(X0 ± X1) with the
new metric ds2 = −2dX+dX− + dX idX i where i = 2, 3, ..., (D − 1) and there is an implicit
summation over i. In the light cone gauge, X+ = v+τ . As, X+ is fully determined in the light
cone gauge, it no longer remains a degree of freedom.
The equations of motion for hab given in eq. (1.3) are called the Virasoro constraints. If we
plug in the possible combinations of (a, b) we get only two unique equations:
∂0Xµ∂0Xµ + ∂1X
µ∂1Xµ = 0, (1.11)
10
∂0Xµ∂1Xµ = 0. (1.12)
In the light cone gauge, eq. (1.11) implies,
(∂0Xi)2 + (∂1X
i)2 = 2v+∂0X−. (1.13)
Similarly, eq. (1.12) gives,
∂0Xi∂1X
i = v+∂1X−. (1.14)
We can fully determine X− given all X i from eq. (1.13) and (1.14). This means that in the light
cone gauge, even X− isn’t a true degree of freedom. The only degrees of freedom we have are
X i.
We can expand the left and right moving parts of the general solution given in eq. (1.10) in
terms of their Fourier modes. For closed strings,
Xµ(τ, σ) = xµ + vµτ + ι
√α′
2
∑n6=0
1
n
(αµne
−ιn(τ+σ) + αµne−ιn(τ−σ)
). (1.15)
For open strings, the left and right moving parts are equal i.e. αµn = αµn, so
Xµ(τ, σ) = xµ + vµτ + ι√
2α′∑n6=0
1
nαµne
−ιnτ cos(nσ). (1.16)
Substituting these expansions in eq. (1.13) and comparing the 0th mode on both sides of the
equations gives:
Closed: 2v+v− = (vi)2 + α′∑n6=0
(αi−nα
in + αi−nα
in
), (1.17)
Open: 2v+v− = (vi)2 +α′
2
∑n6=0
αi−nαin. (1.18)
If we plug in closed string mode expansion given in eq. (1.15) to eq. (1.14) and again compare
0th oscillation modes on both sides of the equation, we get what is called the level matching
condition for closed strings, ∑n6=1
αi−nαin =
∑n 6=1
αi−nαin. (1.19)
The canonical momenta current Πµa correspoding to translational symmetry in Xµ are:
Πµa =
∂L∂(∂aXµ)
=1
2πα′∂aX
µ. (1.20)
Πµ0 is the canonical momentum density. We integrate Πµ
0 over the spatial co-ordinate to get the
canonical momentum P µ. The oscillatory modes integrate out to zero. The only contribution we
get is,
P µ =
∫ l
0
dσΠµ0 =
l
2πα′vµ, (1.21)
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where l = 2π for closed and l = π for open strings. So,
P µ =
vµ/α′ closed
vµ/2α′ open.
(1.22)
Dividing eq. (1.17) by (α′)2 and noting that P µ = vµ/α′ for closed strings,
2P+P− − (P i)2 =2
α′
∞∑n=1
(αi−nαin + αi−nα
in). (1.23)
The left hand side of this equation is −P µPµ which is equal to M2 where M is the mass of the
string. So,
M2 =2
α′
∞∑n=1
(αi−nαin + αi−nα
in). (1.24)
Similarly for open strings using eq. (1.18) and noting that P µ = vµ/2α′ gives
M2 =1
α′
∞∑n=1
αi−nαin. (1.25)
Eq. (1.24) and (1.25) are called the mass-shell conditions for closed and open strings. These
equations determine mass of the string from its oscillation modes.
1.2.2 Quantization and Spectrum
We impose the commutation relations:[X i(τ, σ), Xj(τ, σ′)
]= 0, (1.26)[
Πi(τ, σ),Πj(τ, σ′)]
= 0, (1.27)[X i(τ, σ),Πj(τ, σ′)
]= ιδijδ(σ − σ′). (1.28)
Given the above commutation relations, the commutation relations for Fourier modes are:[αin, α
jm
]=[αin, α
jm
]= nδijδm+n,0. (1.29)
For open strings we’ll have only one Fourier mode algebra. For m > 0, defining aim = αim/√m
and aim† = αi−m/
√m. Similarly, aim = αim/
√m and aim
† = αi−m/√m. Then,
[ain, ain†] = [ain, a
in†] = 1, (1.30)
which is the same as the algebra of creation and annihilation operators of harmonic oscillator. For
open strings, we only have one such algebra, whereas for closed strings, we have two identical
copies of this algebra. This algebra can be used to construct quantum states of the string.
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After quantization, the mass-shell condition given in eq. (1.24) becomes,
M2 =2
α′
∞∑n=1
[n(ain†ain + ain
†ain
)+ n]
=2
α′
∞∑n=1
(nN i
n + nN in + n
)(1.31)
where the extra n comes from the ordering corrections. Each ith mode must contribute i/2 per
oscillator. But since there are two oscillators per mode, ith mode contributes i in total. The
ground state has M2 = 2/α′(1 + 2 + 3 + ...) and using a famous result we can renormalize
(1 + 2 + 3 + ...) to −1/12 which gives M2 = −(D − 2)/6α′ for the ground state. We get a
(D − 2) because there was an implicit summation over i as well. So, for closed strings,
M2 =2
α′
∞∑n=1
(nN i
n + nN in
)− (D − 2)
6α′. (1.32)
Similarly, for open strings,
M2 =1
α′
∞∑n=1
(nN i
n
)− (D − 2)
24α′. (1.33)
Every excitation of a string gives us a particle with a different mass. If m > 0, the open string
ground state |0; pµ〉 will be given by αim |0; pµ〉 = 0. The first excited state is αi−1 |0; pµ〉 and
it has M2 = (26 −D)/24α′. Here, i is a polarization label. Since there are (D − 2) possible
polarizations, we conclude that this state must be that of a photon. So, for this state to be
massless, we must have D = 26. All the higher excited states will be massive. For the ground
state, M2 = −1/α′. A particle with negative mass-squared is called a tachyon. So, the open
string has a tachyonic ground state. Tachyons imply instability of the vacuum.
After quantization, the level matching condition for closed strings given in eq. (1.19) be-
comes,∞∑n=1
nN in =
∞∑n=1
nN in. (1.34)
This condition tells us that the excitations of the left and right movers are not independent
but equal for closed strings. The ground state |0; pµ〉 of a closed string is again tachyonic
with M2 = −4/α′. The first excited state can’t be αi−1 |0; pµ〉 because of eq. (1.34). Instead,
the first excited state is αi−1αj−1 |0; pµ〉 and it is massless. In fact, most generally, the first
excited state is Rijαi−1α
j−1 |0; pµ〉. We can always write Rij = Sij + Aij + S ′δij , where Sij is
the symmetric traceless part, Aij is the anti-symmetric part and R′δij is the trace part. Then,
Sijαi−1α
j−1 |0; pµ〉 gives the graviton, Aijαi−1α
j−1 |0; pµ〉 an anti-symmetric massless spin-2 field,
and S ′αi−1αi−1 |0; pµ〉 gives a massless scalar particle called the dilaton. The emergence of
graviton in the string spectrum makes string theory a candidate for the theory of quantum gravity.
Higher excited states are all massive.
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1.3 String Interactions
We can extend the string action given in eq. (1.1) to include other terms consistent with its
symmetries (Poincare, reparameterization, Weyl scaling). For instance, another action consistent
with these symmetries is,
Sχ =λ
4π
∫d2σ√−hR = λχ (1.35)
whereR = Ricci scalar of the worldsheet and χ = 2− 2h. h is called the genus of a surface and
it specifies its topology. Given the action S = SNG + Sχ, the amplitude of any process, in the
Euclidean signature, with given initial and final string states can be found as,∑all configurations
e−S =∞∑
h=0
e−λχ∑
all surfaces with a given topology
e−SNG . (1.36)
This shows us that in string theory, summing over topologies of surfaces gives us interactions of
strings. Let’s call gs = eλ. For a spherical surface (h = 0), the amplitude is proportional to 1/g2s .
This surface can be interpreted as the nucleation of a closed string from the south pole and after
travelling along the surface of the sphere, its disappearance into the north pole. Similarly, a torus
(h = 1) represents the nucleation of a closed string, its splitting into two closed strings, then the
joining back of these two strings into one and ultimately the disappearance of this closed string.
Its amplitude goes as g0s .
So far we have seen only those processes in which the strings originate from vacuum and
then ultimately disappear. What about the processes in which some initial string configuration
leads to some final configuration? In that case, the surfaces have some boundaries (or external
arms). In the presence of n boundaries (the number of external strings), χ becomes equal to
2− 2h− n. The amplitude of a configuration in which a string splits into two or two strings join
to form one is proportional to gs as h = 0 and n = 3. So, we say that gs is the coupling strength
parameter for the closed strings.
To find the coupling parameter for the open strings go in terms of gs, consider a band. This
surface represents an open string moving through time. Now, if we add a circle (a boundary)
inside this band, it shows the splitting and joining back of open strings. Or in other words, the
amplitude is multiplied by g2o . On the other hand, as we saw before, adding a boundary brings in
an extra factor of gs in the amplitude. So we conclude that g2o = gs.
Now we’re going to find Newton’s gravitational constant GN in terms of gs. Conisder the
process: G + G → G + G, where G represents a graviton. The amplitude of this process
should be proportional to GN . From the string perspective, since we are in the weak coupling
14
limit (gs << 1), only the lowest order term i.e. g2s in the amplitude will survive. So, GN is
proportional to g2s . Since gs is dimensionless, dimensional consistency dictates that GN ∼ g2
sα′4.
1.4 D-branes
We saw before that for open strings, we had two possible choices for boundary conditions:
δXµ∣∣σ=0,π
= 0 (Dirichlet) or ∂σXµ∣∣σ=0,π
= 0 (Neumann). So far we have only used Neumann
b.c. for all Xµ. Now, we are going to see the consequences of using the Dirichlet b.c. So, on
Xα = (X0, X1, ..., Xp) we’ll use Newmann b.c. like before and on XA = (Xp+1, ..., XD−1)
we’ll use Dirichlet b.c. i.e. δXA∣∣σ=0,π
= 0. This means that the endpoints of the open string stay
fixated at a constant value, say bA, of XA. XA = bA gives a hypersurface in the target spacetime.
Since Xα = (X0, X1, ..., Xp) are free to vary on this hypersurface, it is a (p+ 1)-dimensional
hypersurface. Right now it’s just a surface on which open strings live. But when we quantize
these strings, we’ll see that these surfaces can be interpreted as dynamical objects. They are
called Dp-branes where p indicates the number of spatial dimensions along these surfaces.
For Xα we’ll get the same solutions as before. For XA we pick the most general solution
given in eq. (1.10) and use the Dirichlet b.c. We get vAa = 0,XAR (τ) = −XA
L (τ) andXAR (τ−π) =
−XAL (τ + π) i.e. the right and left moving parts are related by a negative sign and both are
periodic in 2π. For Xα the mode expansions are the same as given in eq. (1.16). On the other
hand for XA we get,
Xa(τ, σ) = ba + ι√
2α′∑n 6=0
1
nαane
−ιnτ sin(nσ). (1.37)
Again we go to the light cone gauge and eliminate two target spacetime co-ordinates normal to
the brane and label X i = (X2, X3, ..., Xp). Using exactly the same procedure as before, we find
the mass shell condition,
M2 =1
α′
∞∑n=1
(αi−nα
in + αA−nα
An
). (1.38)
We then impose exactly the same commutation relations.
The ground state |0; pα〉 defined by αim |0; pα〉 = 0 (m > 0) is tachyonic, which means that
the brane is unstable. Note that, for someone living on the Dp-brane, the SO(1, 25) symmetry of
the target spacetime breaks down into the local SO(1, p) symmetry and the global SO(25− p)
symmetry. This has the consequence that the indices for directions normal to the brane no longer
act as Lorentz indices. So, αi−1 |0; pα〉 is the state of a photon living on the Dp-brane with (p− 1)
15
possible polarizations and αA−1 |0; pα〉 gives us (25−p) massless scalar particles since the indices
A are no longer Lorentz indices, rather just counting labels. There is a massless scalar for every
direction normal to the brane. The physical interpretation of these massless scalars is that they
describe the fluctuations of the brane in the perpendicular directions (hence a scalar per normal
direction). We introduced branes simply as boundary conditions for open strings but now they
can be interpreted as dynamical objects.
1.5 Superstring theories
The version of string theory we have discussed so far is the bosonic string theory. However it
is unsatisfactory for two reasons. Firstly, the vacuum is tachyonic. Tachyons imply instability
of the vacuum. Secondly, the spectrum of the bosonic string theory does not contain fermions.
Any model aiming to describe nature must contain fermions. So, we would like to incorporate
fermions in the framework of string theory. The string theories with fermionic degrees of freedom
are called superstring theories. Here we’ll see only one particular version of superstring theory
i.e. the Ramond-Neveu-Schwarz (RNS) formalism.
In this framework, we proceed by adding fermionic degrees of freedom to the Polyakov
action given in eq. (1.2) in the following way,
S =1
4πα′
∫d2σ∂aXµ∂aXµ + ιψµγa∂aψµ. (1.39)
The new fields ψµ(τ, σ) are two-component spinors1 on the worldsheet and ψµ(τ, σ) is their
Dirac adjoint defined as ψµ = ψµ†γ0. γa are two dimensional Dirac matrices obeying the Dirac
algebra: γa, γb = 2ηab. We choose the following representation for γa,
γo =
0 −1
1 0
, γ1 =
0 1
1 0
. (1.40)
ψµ has two components,
ψ =
ψµ−ψµ+
. (1.41)
Since, the Dirac matrices are real, we impose the Majorana condition by simply demanding that
(ψµ−)∗ = ψµ− and (ψµ+)∗ = ψµ+. The fermionic part of the action in eq. (1.39) can be rewritten as,
Sf =ι
2πα′
∫d2σψ+µ∂−ψ
µ+ + ψ−µ∂+ψ
µ−, (1.42)
1for a detailed discussion on spinors, see Appendix B
16
where ∂± = 12(∂0 ± ∂1). The equations of motion and boundary conditions for the bosonic part
are the same. For the fermionic part the equations of motion are,
∂−ψµ+ = 0, ∂+ψ
µ− = 0. (1.43)
So, ψµ+ is a left moving solution and a function of (τ + σ) i.e. ψµ+(τ + σ). On the other hand, ψµ−
is right moving, hence a function of (τ − σ) i.e. ψµ−(τ − σ). These solutions must satisfy the
boundary conditions that we obtain by varying the action. The boundary terms are,
δS ∼∫dτψ+µ(δψµ+)− ψ−µ(δψµ−)
∣∣∣σ=l− ψ+µ(δψµ+)− ψ−µ(δψµ−)
∣∣∣σ=0
, (1.44)
where l = π for open strings and l = 2π for closed strings. For open strings, the terms in
eq. (1.44) for different ends of the string must be put separately equal to zero . So, at each end of
the string we have two possible choices: ψµ+ = ±ψµ−. The overall sign of ψµ+ and ψµ− doesn’t
matter so, by convention we choose ψµ+ = ψµ− at σ = 0. Then we have two choices at σ = π:
• Ramond boundary conditions: ψµ+∣∣∣σ=π
= ψµ−
∣∣∣σ=π
Given these boundary conditions, the mode expansions for ψµ+(τ + σ) and ψµ−(τ − σ) are:
ψµ+ =√α′∑n∈Z
dµne−ιn(τ+σ), ψµ− =
√α′∑n∈Z
dµne−ιn(τ−σ). (1.45)
• Neveu-Schwarz boundary conditions: ψµ+∣∣∣σ=π
= −ψµ−∣∣∣σ=π
The mode expansions for ψµ+(τ + σ) and ψµ−(τ − σ) are:
ψµ+ =√α′∑r∈Z+ 1
2
bµr e−ιr(τ+σ), ψµ− =
√α′∑r∈Z+ 1
2
bµr e−ιr(τ−σ). (1.46)
For closed strings, we have to impose conditions on ψµ+ and ψµ− separately. There are two
possible choices for each, ψµ±(τ) = ±ψµ±(τ + 2π) i.e. periodicity (R) or antiperiodicity (NS).
For right movers, the mode expansion in each case is,
R : ψµ− =√α′∑n∈Z
dµne−ιn(τ−σ), (1.47)
NS : ψµ− =√α′∑r∈Z+ 1
2
bµr e−ιr(τ−σ). (1.48)
17
Similarly for left movers,
R : ψµ− =√α′∑n∈Z
dµne−ιn(τ+σ), (1.49)
NS : ψµ− =√α′∑r∈Z+ 1
2
bµr e−ιr(τ+σ). (1.50)
Since the choice of boundary conditions for the left and right movers is independent (i.e. R or
NS), there are four sectors for closed strings: R-R, R-NS, NS-R, NS-NS. Now we are going
to impose the quantization conditions on the fields. For the bosonic fields, the quantization
procedure is exactly the same as that in the bosonic string theory. For fermionic fields, we impose
the anti-commutation relations:
ψµA(σ, τ), ψµB(σ′, τ) = 2πα′ηµνδABδ(σ − σ′), (1.51)
where A, B can take values + or −. This implies that the Fourier coefficients for open strings
satisfy the following anti-commutation relations,
dµm, dνn = ηµνδm+n,0, bµr , bνs = ηµνδr+s,0. (1.52)
The ground states in the two sectors are defined as,
R : αµm |0〉R = dµm |0〉R = 0 (m > 0), (1.53)
NS : αµm |0〉NS = bµr |0〉NS = 0 (m, r > 0). (1.54)
Now, let’s see the excited states in the NS sector. Before we do that, we’ll write down the mass
shell condition in this sector including the ordering corrections:
α′M2 =∞∑n=1
αi−nαin +
∞∑r= 1
2
rbi−rbir −
(D − 2)
16. (1.55)
The first excited state will be bi−1/2 |0〉NS with M2 = (10−D)/16α′. Using the same argument
we used in the bosonic string theory, this state must be massless since it represents a photon i.e.
D = 10. So, for this version of superstring theory to be consistent with the world, D must be
equal to 10. With this choice of D, the ground state is tachyonic i.e. M2 = −1/2α′. Higher
excited states are all massive. All the states in the NS sector are spacetime bosons. On the other
hand, the R sector has a special operator dµ0 which satisfies the algebra, dµ0 , dν0 = ηµν . This
operator takes a ground state to another ground state (the ground state is degenerate). And the
algebra this operator satisfies is identical to Dirac algebra apart from the factor of 2. So, the R
18
sector ground state must be a representation of the Dirac algebra. The mass shell condition for
the R sector after including the ordering corrections is,
α′M2 =∞∑n=1
αi−nαin +
∞∑n=1
rdi−ndin. (1.56)
We conclude that, the R sector ground state is a massless spacetime spinor. Since all the excited
states are constructed by acting on the ground state with objects that have vector indices, all the
excited states are also spacetime fermions and they are all massive.
In spite of having found a way to incorporate fermions in the string spectrum, there is still a
problem. The ground state in the NS sector is still tachyonic and unphysical. This problem can
be dealt with by removing the troublesome part of the spectrum using a procedure called the
GSO projection. For that, we first define the G-parity operator in the NS sector as,
G = (−1)∑∞r=1
2bi−rb
ir+1
. (1.57)
The prescription of the projection is to keep only those states that have a positive G-parity and
eliminate all the states with a negative G-parity. As a result, the tachyonic ground state from the
spectrum is eliminated because it has a negative G-parity and the massless vector becomes the
new ground state in the NS sector.
The R sector ground state is a spinor. This ground state must be a 32-component spinor,
because the Dirac matrix is 32× 32 in 10 dimensions. However, we can reduce the number of
components by imposing the Weyl and the Majorana conditions. The Weyl condition gives a
spinor with a definite chirality and the number of components is reduced to 16. The Majorana
condition further reduces the number of components by half to give 8 complex valued components
(or 16 real valued components). Thus, in this spacetime, a Majorana-Weyl spinor has 8 physical
degrees of freedom (half of the dimension of the phase space which is 16 dimensional). On
the other hand, we just saw that after GSO projection the ground state in the NS sector is a
massless vector which also has 8 degrees of freedom. Therefore, at the massless level, there are
equal number of fermionic and bosonic degrees of freedom. In fact, there are equal number of
fermionic and bosonic degrees of freedom for each value of mass after the GSO projection. This
is a significant evidence for the presence of spacetime supersymmetry2 in this formalism. And
indeed it can be proved that supersymmetry is there.
Now let’s analyze the closed string spectrum. As we mentioned before there are four sectors
for closed strings: R-R, R-NS, NS-R, NS-NS. Since the ground states in the R sectors are2supersymmetry is a symmetry under the exchange of bosons and fermions
19
Majorana-Weyl spinors, we can get two different theories (IIB or IIA) depending on the
chirality of the ground states of the left and right moving R sectors. In IIB, the ground states
of both the R sectors have the same chirality and in IIA, they have the opposite chirality. The
massless spectrum for the IIB theory contains,
|+〉R ⊗ |+〉R , (1.58)
bi− 12|0〉NS ⊗ b
j
− 12
|0〉NS , (1.59)
bi− 12|0〉NS ⊗ |+〉R , (1.60)
|+〉R ⊗ bi− 1
2|0〉NS . (1.61)
The massless spectrum for type IIA theory is,
|−〉R ⊗ |+〉R , (1.62)
bi− 12|0〉NS ⊗ b
j
− 12
|0〉NS , (1.63)
bi− 12|0〉NS ⊗ |+〉R , (1.64)
|−〉R ⊗ bi− 1
2|0〉NS . (1.65)
Now we give the description of the massless states in all the sectors.
• R-R: Since the states are a tensor product of spinors, they are bosonic. For the IIA theory,
the spinors have opposite chirality, so the spectrum contains a vector and a three form
gauge field, whereas for the IIB case, the spectrum contains a scalar, a two form field and
a four form field.
• NS-NS: For both IIA and IIB, this sector is identical. It contains the dilaton (scalar), a
two form and the graviton.
• NS-R and R-NS: Both IIA and IIB contain a spin-3/2 particle (gravitino) and a spin-1/2
particle (dilatino) in each of these sectors. The only difference is that in the IIA case, the
two gravitinos have opposite chirality unlike the IIB case.
An n-form gauge field is a generalization of the Maxwell field and is given by,
An =1
n!Aµ1µ2...µndx
µ1 ∧ dxµ2 ∧ ... ∧ dxµn . (1.66)
20
The n = 1 case corresponds to the Maxwell field. The field strength corresponding to An is an
(n+ 1)-form Fn+1 given by Fn+1 = dAn, where d represents the exterior derivative. A charged
particle (D0-brane) couples to the Maxwell gauge field Aµ in the following way,
S =
∫C
AµdXµ
dτdτ, (1.67)
where C represents the worldline of the particle. Similarly, a Dp-brane couples electrically to a
(p+ 1) form gauge field,
S =
∫Σ
Aµ1...µp+1
∂Xµ1
∂σ0
...∂Xµp+1
∂σpdp+1σ. (1.68)
Here Σ is the worldvolume of the brane. In 4 dimensions, a particle also couples magnetically
to Aµ. Or in other words it couples electrically to ∗F where F = dA and ∗ is the Hodge star
operator. Similarly, a D(6− p)-brane couples magnetically to a (p+ 1) form gauge field in 10
dimensions since its field strength is a (p+ 2) form which means that its Hodge dual is an (8− p)
form and this (8 − p) form field strength will have a (7 − p) form gauge field (that couples
electrically to a D(6 − p)-brane). In short, a Dp-brane is charged electrically under a (p + 1)
form gauge field and charged magnetically under a (7− p) form gauge field.
In type IIA superstring theory, one form and three form gauge fields are present in the
massless spectrum, so there must be stable D-branes in the theory that carry the corresponding
charges. Thus, IIA theory should have stable D0, D2, D4, D6 branes. Similarly, the IIB theory
has zero form, two form and four form gauge fields, so it should have stable D(−1), D1, D3, D5,
D7 branes.
Now that we have seen the existence of stable D3-branes in type IIB superstring theory, in
the next chapter, we’ll consider a configuration of N coincident D3-branes. Since, the branes lie
on top of each other, the endpoints of the open strings can be tied to any of the N D-branes. So,
the string states will be symmetric under the exchange of endpoints from one brane to another. In
the low energy limit, this will give rise to an SU(N) gauge theory on the branes. This is where
the gauge theory side of the AdS − CFT correspondence comes from.
21
Chapter 2
The AdS-CFT Correspondence
As we discussed briefly in chapter 1, to derive the AdS − CFT correspondence, we need to
look at D-branes from two different point of views: the superstring picture and the supergravity
picture. To be able to develop the superstring picture for D-branes, we explored string theory in
detail in chapter 1. Now that we have expounded on D-branes in string theory, we are ready to
formulate the superstring picture. That is what we’re going to do in the next section. This will
give rise to a gauge theory on the D-branes, as we mentioned at the end of chapter 1. However,
the supergravity picture won’t be developed with the same detail. Rather we’ll just assume the
solution for D-branes and move on from there. This chapter is based on ref. [4, 5, 6, 7, 8].
2.1 N coincident D-branes in type IIB superstring
theory
We have already seen at the end of chapter 1 that type IIB superstring theory must have
stable D3-branes. Consider N such branes lying on top of each other. Since, an open string
can have its end points attached to any of these branes, so, there are N2 possible open string
configurations that have exactly the same spectrum. The massless spectrum contains a one form
gauge field Aα where α labels the spacetime co-ordinates along the brane, a scalar φA for each
direction orthogonal to the brane and their supersymmetric partners. The open string state will be
represented by |ψ; I, J〉 with I, J labelling the branes the end points of the string are attached to
and I, J = 1, 2, ..., N . In other words, the state can be represented by an N ×N matrix and the
fields can be written as (Aα)IJ and (φA)IJ . Since the branes are indistinguishable from each other
we have the freedom to reshuffle the indices. It turns out that this has to be done by an element of
23
U(N). So, from the worldsheet point of view there is a global U(N) symmetry. But in the point
of view of the worldvolume of the branes, this must be a gauge symmetry. Next we take the low
energy limit which means that the squared of the energies have to be much less than 1/α′. Or
alternatively we can take α′ → 0. In this limit, the open and closed strings will live in massless
modes only and the open strings living on the branes will decouple from the bulk. In fact, U(1)
of the U(N) gauge group can also be decoupled from SU(N) in the low energy limit. So, in the
low energy limit, the branes realize an SU(N) gauge theory. Moreover, the spectrum contains
a vector field, 6 scalar fields and their superpartners. This description can be identified with
N = 4 Super Yang-Mills theory, which is a conformal field theory1, with g2YM ∼ g2
o = gs. The
description of the bulk is given by the low energy limit of type IIB superstring theory, which is
called type IIB supergravity. Thus, a stack of D3-branes in type IIB superstring theory in the
low energy limit is described by N = 4 SYM inM4 and type IIB SUGRA inM10.
2.2 D-branes as spacetime geometry
Einstein’s gravity is a gauge theory with the gauge group being the Poincare group. One may
ask if we can add internal symmetries to the Poincare group to extend the algebra. The answer
is no. According to the Coleman-Mandula theorem, we cannot combine Poincare and internal
symmetries otherwise the S-matrices for all the processes will be zero. However, there is a hidden
assumption in this theorem. The theorem works only when the final algebra is a Lie algebra.
However, we can evade the theorem by extending the concept of a Lie algebra to a graded Lie
algebra. In a graded Lie algebra, some generators satisfy an anti-commutation rule instead of a
commutation rule. The theory we get under this new gauge group is called supergravity. There is
also another way to obtain supergravity, that is by taking the low energy limit of the superstring
theory.
In supergravity, D-branes gravitate the spacetime around them. So, in this picture, we’ll look
at how D3-branes affect the 10 dimenional spacetime around them in the low energy limit. The
metric in the supergravity solution involving N D3-branes is,
ds2 = H−12 (r)(−dt2 + d~x2) +H
12 (r)(dr2 + r2dΩ2
5), (2.1)
where H(r) = 1 + R4/r4 and R4 = N4πgs(α′)2. Also, ~x = (x1, x2, x3) represents spatial
dimensions along the branes.1see Appendix C for a comprehensive discussion on conformal field theory
24
In the limit r →∞, H(r)→ 1, so the spacetime becomesM10. On the other hand, in the
limit r → 0, H(r) ≈ R4/r4, and we get,
ds2 =r2
R2(−dt2 + d~x2) +
R2
r2dr2 +R2dΩ2
5, (2.2)
which is the metric for AdS5 × S5. Actually, this metric represents only the Poincare patch of
AdS52.
Now we would like to take the low energy limit. We have to be careful regarding the observer
with respect to which we are taking the low energy limit. In the previous section, we took the
limit with respect to t given in eq. (2.1), which is the time for the observer at r =∞. The local
proper time at a general value of r is dτ = H−1/4dt, which means that energy is given locally by
Eloc = H1/4E∞. As r → 0, Eloc ≈ (R/r)E∞. Taking the low energy limit means E2∞α′ → 0.
So, at r =∞, only the massless modes will exist in the low energy limit and since the spacetime
isM10 at r =∞, the description of physics here is given by type IIB SUGRA inM10. The low
energy limit can also be written as E2loc(r
2/R2)α′ → 0. This means that for r → 0, we can have
any value of Eloc. Thus, close to the branes, any string excitation is allowed. The spacetime here
is AdS5× S5. The sectors at r = 0 and r =∞ decouple from each other in the low energy limit.
So, N coincident D3-branes in the low energy limit can be described by type IIB superstring
theory in AdS5 × S5 and type IIB SUGRA inM10.
2.3 The correspondence
We have given two alternative descriptions for coincident D3-branes in the previous two sections.
Since the two descriptions must be equivalent and both the descriptions involve type IIB SUGRA
inM10, we conclude that, N = 4 SYM inM4 must be equivalent to type IIB superstring
theory in AdS5 × S5. This is the AdS5 − CFT4 correspondence. The boundary of the Poincare
patch of AdS5 isM4, so we can say that SYM lives on the boundary of the gravity theory.
More evidence for the correspondence is provided by the matching of symmetries on both
sides. N = 4 SYM has conformal symmetry SO(4, 2), while the conformal group SO(4, 2) is
also the isometry group of AdS5. SYM has SO(6) global symmetry, whereas the isometry group
of S5 is also SO(6). Moreover, both sides of the correspondence have the same supersymmetry.
The definition of R is R4 = N4πgs(α′)2. But we also have g2
YM ∼ gs, so, R4 ∼
N4πg2YM(α′)2. Using GN ∼ g2
s(α′)4 gives GN/R
8 ∼ 1/N2. This shows that the dual gravity2for a detailed discussion on AdS space, see Appendix D
25
theory for the gauge theory in the t’Hooft limit i.e. N → ∞, is weakly coupled. It can be
shown that the large N limit in the gauge theory implies strong coupling. Since, we cannot
study strongly coupled gauge theories by perturbation theory, AdS − CFT provides a more
approachable way by the study of a weakly coupled gravity theory.
2.4 IR/UV connection
From the context of holography, it is clear that the field theory describing the gravity theory lives
in one dimension lower. However, from the perspective of field theory, we may ask what does
the extra dimension on the gravity side represent? To answer the question let’s start with the
AdS5 metric,
ds2 =r2
R2(−dt2 + d~x2) +
R2
r2dr2. (2.3)
Making the co-ordinate transformation z = R2/r gives,
ds2 =R2
z2(−dt2 + dz2 + d~x2). (2.4)
The boundary of AdS5 lies at r =∞ or z = 0, which is where SYM lives. Local proper time at
a fixed z is,
dτ =R
zdt. (2.5)
So, the energy scales of the bulk and the boundary theory are related by,
EYM =R
zEloc. (2.6)
The same value of Eloc at two different values of z correspond to different values of EYM . So,
the extra dimension in the bulk can be interpreted as the energy scale of the boundary theory. In
particular, the UV processes in SYM (EYM →∞) correspond to z → 0 and the IR processes
(EYM → 0) correspond to z →∞. This relation of the energy scale of the boundary theory to
the depth of the corresponding process inside the bulk is called the IR/UV connection.
In the next chapter, we’ll discuss a phase transition in the gravity theory. Having derived
the correspondence, we’ll then use the AdS − CFT toolbox to guess the corresponding phase
transition in the boundary theory. We’ll see that the resemblance between the bulk and the
boundary phase transition will be so nice, that it will further reinforce our confidence in the
correspondence.
26
Chapter 3
Black Hole Phase Transition
In AdS spacetime, above a critical temperature, the preferred state of the universe switches from
radiation to black holes. This phase transition is called the Hawking-Page phase transition. Since
the gravity theory in the bulk is related to to a field theory on the boundary, a phase transition in
the bulk must also have a dual in the boundary theory. As we’ll see, AdS−CFT correspondence
will tell us that the entropy jumps fromO(N0) toO(N2) upon the Hawking-Page phase transition,
where N is the number of colors of the boundary theory. This has a very nice resemblance with
the confinement-deconfinement phase transition in gauge theories in which the confining phase
with entropy O(N0) makes a phase transition to the deconfining phase with entropy O(N2). To
derive the Hawking-Page phase transition, we need to know about black holes in AdS space. A
better appreciation for such black holes can only be obtained by first seeing the problems with
black holes in asymptotically flat spacetime, which are thermodynamically unstable. We start
with black holes in flat spacetime in the next section. This chapter is based on ref. [10, 11, 12].
3.1 Schwarzschild black holes
Black hole in an asymptotically flat geometry,M4, is described by the Schwarzschild metric1,
ds2 = −f(r)dt2 + f(r)−1dr2 + r2dΩ2, (3.1)
where f(r) = 1− 2GNM/r and M is the mass of the black hole. The event horizon lies at the
Schwarzschild radius rs = 2GNM . The temperature that the quanta feel in this geometry can be
determined by computing the period of the Euclidean time of the geometry2. So, introducing
1for a detailed account on Schwarzschild geometry, see Appendix E2see Appendix A
27
the proper distance from the horizon ρ as dρ = dr/√f(r). We can perform a Taylor series
expansion of f(r) around rs,
f(r) = f(rs) + f ′(rs)(r − rs) + .... (3.2)
But here f(rs) = 0, and when we are very close to the horizon f(r) ≈ f ′(rs)(r − rs). So,
f(r) = f ′(rs)(r − rs) = ρ2f ′(rs)2/4. Let’s call K = f ′(rs)/2. Then, f(r) = K2ρ2 close to the
horizon. We can write the metric in this limit as,
ds2 = −K2ρ2dt2 + dρ2 + r2sdΩ2. (3.3)
Now, going to the Euclidean time t→ −ιτ , and further introducing θ = Kτ , we get,
ds2E = (ρ2dθ2 + dρ2) + r2
sdΩ2. (3.4)
The terms in the bracket resemble the metric for polar co-ordinates. This metric has a conical
singularity at ρ = 0 unless θ is periodic in 2π. So, τ must be periodic in 2π/K. Thus,
β = 2π/~K and then,
TBH =~
8πGNM. (3.5)
And to obtain black hole entropy,
SBH =
∫ M
0
1
TBH(m)dm =
4πr2s
4GN~=
AH4GN~
. (3.6)
This is the famous result of Bekenstein and Hawking.
Since TBH ∼ 1/M , so such a black hole has negative specific heat i.e. ∂M/∂TBH , and that
implies instability. Consider the black hole to be in equilibrium with an infinite heat reservoir.
If we raise the temperature of the black hole by an infinitesimal amount, it radiates some mass
away which further raises its temperature and the black hole keeps radiating until it evaporates.
Similarly, a small reduction in the black hole temperature makes it absorb radiation which further
lowers its temperature and the black hole grows indefinitely. So, a Schwarzschild black hole is
thermodynamically unstable.
To find a stable black hole solution, let’s place the black hole in a finite box, filled with
radiation. The total energy of the box will be E = M + Erad, where M is the mass of the black
hole. We want to maximize the entropy of the box, S = SBH + Srad. So,
∂S
∂M=∂SBH∂M
+∂Srad∂M
=∂SBH∂M
− ∂Srad∂Erad
= 0. (3.7)
28
This is essentially the statement that TBH = Trad. Now, computing the second derivative,
∂2S
∂M2=∂2SBH∂M2
+∂2Srad∂E2
rad
< 0 =⇒ − 1
T 2BH
∂TBH∂M
− 1
T 2rad
∂Trad∂Erad
< 0. (3.8)
Since, Erad ∼ T 4rad, so, ∂Trad/∂Erad = Trad/(4Erad). Also, TBH ∼ 1/M . Plugging this in
eq. (3.8) gives,
Erad <1
4M. (3.9)
So, by placing the black hole in a box, although unphysical, we can get a stable black hole
solution (as long as Erad < M/4).
3.2 AdS black holes
Now we’ll put the black hole in a more natural box. Our spacetime will no longer be asymp-
totically flat. Rather, we’ll put the black hole in AdS spacetime because AdS behaves like a
confining box. Consider the 5 dimensional metric,
ds2 = −V (r)dt2 + V (r)−1dr2 + r2dΩ2, (3.10)
where V (r) = 1 − 2M/r2 + r2/R2. For small r, this metric behaves like the Schwarzschild
metric while for large r, it behaves like AdS. This metric is called the Schwarzschild-AdS
metric.
The event horizon of the black hole lies at r = r+, where r+ is a root of V (r) = 1−2M/r2 +
r2/R2. Now consider we are very close to the event horizon. Let r = r+ + ρ2, where ρ << 1.
Then,
V (ρ) = 1− 2M
(r+ + ρ2)2+
(r+ + ρ2)2
R2
≈R2r2
+ρ2 + 2r+ρ
2R2 − 2MR2 + r4+ + 4r3
+ρ2
R2(r+ + 2r+ρ2)
≈2(R2 + 2r2
+)
R2r+
ρ2. (3.11)
In the second line, the fact that R2r2+ − 2MR2 + r4
+ = 0 (because r+ is a root of V (r)) was
used. And we neglected all the powers of ρ higher than 2. Also, dr2 = 4ρ2dρ2. So, close to the
horizon the metric in eq. (3.10) becomes,
ds2 ≈ −(2(R2 + 2r2
+)
R2r+
ρ2)dt2 + 4
( R2r+
2(R2 + 2r2+)
)dρ2 + r2
+dΩ2. (3.12)
29
Going to Euclidean time t→ −ιdτ , and further defining θ = 2(R2 + 2r2+)τ/(2R2r+), we get,
ds2 ≈ 2( R2r+
R2 + 2r2+
)[ρ2dθ2 + dρ2] + r2
+dΩ2. (3.13)
For the part of the metric in the square brackets to not have a conical singularity at ρ = 0, θ must
be periodic in 2π. That means that the period of Euclidean time must be β = 2π(R2r+)/(R2 +
2r2+). So the temperature of the black hole is,
TBH =1
β=R2 + 2r2
+
2πR2r+
. (3.14)
Since r+ ∝M approximately, we can plot TBH with the mass of the black hole.
Figure 3.1: Comparison of the dependence of TBH on M for the Schwarzschild and the Schwarzschild-AdS black hole. (a)TBH versus M for Schwarzschild black hole. (b) For Schwarzschild-AdS black hole.
Fig. 3.1 shows that, unlike the case for Schwarzschild black hole, the temperature of
Schwarzschild-AdS black hole does not decrease monotonically with mass. Rather, it attains a
minimum and starts increasing again. The value of r+ at which temperature attains its minimum
is ro = R/√
2. This gives To =√
2/(πR). For small black holes (r < ro), the heat capacity is
negative and they are thermodynamically unstable. On the other hand larger black holes (r > ro)
have positive heat capacity and are stable.
3.3 The Hawking-Page phase transition
Although we have found that stable black holes can exist above some temperature, we don’t
know yet if they are the preferred state of the universe over radiation or not. To find the answer
to this question, we’ll have to compare the free energy of the Schwarzschild-AdS and the AdS
metric. And that will lead us to Hawking-Page phase transition.
The partition function of a theory can be found by the path integral3,
Z =
∫Dφe−I[φ], (3.15)
3see Appendix A
30
where the integral is done over all fields φ(~x, t) periodic in imaginary time with period β and
I[φ] denotes the action. The minima of the action, where δI = 0, provides the most dominant
contribution to the path integral. So, we can approximate Z ≈ e−I . The free energy is given by
F = −T logZ = TI .
The action for gravity is the Einstein-Hilbert action,
I = − 1
16π
∫d4x√−g(R− 2Λ), (3.16)
where R is the Ricci curvature scalar and Λ is the cosmological constant. Both AdS and
Schwarzschild-AdS metrics satisfy vacuum Einstein’s equations, soR = 2dΛ/(d− 2), where d
is the dimension of the spacetime. Since we have considered d = 5, I = −(Λ/12π)∫d4x√−g.
Let’s compute the integral for AdS first. The integral needs to be regularized because it is infinite.
So, we need to integrate the radial variable to some cutoff value r = K,
I1 = − Λ
12πR3
∫ β1
0
dτ
∫ K
0
r3dr
∫S3
dΩ = − Ω3
12πR3
Λβ1K4
4∼ β1K
4
R5, (3.17)
where Ω3 = (2π)3/2/Γ(3/2). Similarly, for Schwarzschild-AdS metric,
I0 = − Λ
12πR3
∫ β0
0
dτ
∫ K
r+
r3dr
∫S3
dΩ = − Ω3
12πR3
Λβ0(K4 − r+4)
4∼ β0(K4 − r+
4)
R5.
(3.18)
Where the constants common to both I1 and I0 have been ignored. We have already found β0 in
the previous section. To find β1, we require that the observer at r = K in both the spacetimes
observes the same temperature at his location. The metrics at r = K are,
ds2AdS = −
(1 +
K2
R2
)dt2 +K2dΩ2, ds2
BH−AdS = −(
1− 2M
K+K2
R2
)dt2 +K2dΩ2.
Since t in AdS is periodic in β1 and that in Schwarzschild-AdS is periodic in β0, the matching
of proper time of the observer at r = K in the two metrics gives,
β1
√1 +
K2
R2= β0
√1− 2M
K+K2
R2. (3.19)
Thus,
I = I0 − I1 ∼β0
R5
((K4 − r4
+)− β1
β0
K4
)
=β0
R5
((K4 − r4
+)−
√1− 2MR2
K(R2 +K2)K4
).
31
In the limit of large K,
I ≈ β0
R5
((K4 − r4
+)−
(1− MR2
K(R2 +K2)
)K4
)≈ − β0
R5(r4
+ −MR2).
Using MR2 = r2+(R2 + r2
+)/2, we get,
I =r2
+β0
2R3
(1−
r2+
R2
). (3.20)
Note that I = 0 at r+ = R. This corresponds to TBH ≡ T1 = 3/(2πR). Recall from
thermodynamics that the preference between two same temperature states, in equilibrium with
the environment, is determined by the free energy. Specifically, the state with the lower free
energy is preferred. So, for T < T1, we have r+ < R, and eq. (3.20) implies I1 < I0 which
means that the preferred state will be thermal radiation. Similarly, for T > T1, I1 > I0, making
black holes the preferred state instead of radiation. So, a phase transiton occurs at T = T1
that switches the preferred state of the universe from radiation to black holes. This phase
transition is called the Hawking phase transition. In summary, for T < To, black holes are
thermodynamically unstable and the universe is filled with radiation, for To < T < T1, black
holes are thermodynamically stable but the preferred state of the universe is still radiation, and
for T > T1, black holes become the preferred state of the universe.
3.4 The confinement-deconfinement phase transition
The easiest way to understand confinement-deconfinement transition is to consider Quantum
chromodynamics (QCD) which is an SU(3) Yang-Mills theory. In QCD, when quarks are far
apart the coupling constant is large but when they are close together the coupling constant is
small. As a result, when quarks are distant, they attract each other to form a bound state called a
hadron. And after they achieve the hadronic state, they behave like free particles. Quarks are said
to be in confinement when they are in such bounded states. However, quarks do not stay in this
state when the temperature becomes very large. Instead, quarks and gluons mix with each other
to form a quark-gluon plasma (QGP ). Quarks in such a state are said to be in deconfinement. So,
as the temperature increases a phase transition switches the state of the quarks from confinement
to deconfinement.
32
Now, we’re interested in finding out the phase transition in the boundary theory that is dual
to the Hawking-Page phase transition. As we have mentioned previously, the phase transition
dual to the Hawking-Page phase transition is the confinement-deconfinement phase transition.
To show that, we’ll compute the entropy of the gravity system before and after the phase
transition. When that is done, we’ll see that the expressions for entropy will resemble that of
confinement-deconfinement transition.
First, we’ll compute the entropy for the AdS black hole solution. The action I0 is divergent.
But the divergence can be removed if we consider thermal AdS as the reference background.
Then the relevant quantity will be I = I0 − I1 given in eq. (3.20). Uptill now, we have ignored
G5 i.e. the 5 dimensional Newton’s constant, however, now in the context of AdS −CFT it will
become important. So, rewriting eq. (3.20) with G5,
I ∼r2
+β
2R3G5
(1−
r2+
R2
). (3.21)
By definition, S = βE − βF . But βF = I . So, S = βE − I . The energy can be computed as,
E = − ∂
∂β(logZ) ∼ ∂I
∂β
=r2
+
2R3G5
(1−
r2+
R2
)+
β
2R3G5
(2r+ −
4r3+
R2
)∂r+
∂β.
From eq. (3.14), we can compute ∂r+/∂β. So, we finally get,
E ∼r2
+
2R3G5
(1−
r2+
R2
)+
r2+
R3G5
(1 +
2r2+
R2
)=
3r2+
2R3G5
(1 +
r2+
R2
). (3.22)
Thus, the entropy S = βE − I is,
S ∼r3
+
R3G5
. (3.23)
For a black hole at a given temperature, r+ can be found from eq. (3.14). This means that
r+ ∼ R2 at a given temperature and eq. (3.23) becomes,
S ∼ R3
G5
. (3.24)
Now, we will use AdS − CFT to determine the dependence of this entropy on N (number of
colors parameter) of the boundary theory. First note that for AdS5 × S5 which is the spacetime
on the gravity side of the duality,
1
16πG10
∫d5xd5Ω
√−g10R10 =
V5
16πG10
∫d5x√−g5R5, (3.25)
33
where V5 is the volume of S5. We can see from the equation above that G5 = G10/V5 ∼ G10/R5
(recall from the metric in eq. (2.2) that the radius for the S5 part was R). And G5R5 ∼ G10 ∼
g2sα′4. Also, recall from chapter 2, that the scalar R for AdS5 is determined by the parameters of
the boundary theory. The relation was R4 ∼ Ngsα′2. So, the entropy becomes,
S ∼ R8
G5R5∼ (Ngsα
′2)2
g2sα′4 = N2. (3.26)
On the other hand, thermal AdS has zero entropy because we have chosen it to be the reference
background. So, in the thermal AdS, entropy goes as O(N0). When the phase transition occurs,
the entropy jumps to O(N2). This sounds like confinement. In the confining phase, the entropy
is S ∼ O(N0), whereas, in the deconfining phase, the entropy is S ∼ O(N2). Thus our result
for the jump of entropy due to Hawking-Page phase transition agrees nicely with the jump due to
the confinement-deconfinement phase transition. Moreover, the confining phase in the boundary
theory corresponds to the thermal AdS in the bulk and the deconfining phase corresponds to the
black hole solution.
This matching of the behavior of phase transitions on the bulk and boundary theory pro-
vides further evidence for the AdS − CFT correspondence. Moreover, since confinement-
deconfinement is also a feature ofQCD, this leads us to suspect the existence of theAdS−QCD
correspondence. This correspondence is different from AdS − CFT because QCD isn’t a
conformal field theory. The AdS − QCD correspondence hasn’t been established yet, but
Hawking-Page phase transition might have pointed us in the right direction.
34
Appendix A
The path integral and field theory at a
finite temperature
In this appendix, we’ll be focussed on deriving the result that the periodicity of the Euclidean
(imaginary) time of a theory gives us the inverse of the temperature felt by the quanta in that
theory. The appendix is based on ref. [13].
In quantum mechanics, the probability amplitude for a particle to go from an inital state |qI〉
to the final state |qF 〉 in time T is given by 〈qF | e−ιHT |qI〉 (where q denotes the configuration
space co-ordinates of the particle). We can divide T into N segments, where for each segment
δt = T/N . We can re-write the amplitude as,
〈qF | e−ιHT |qI〉 = 〈qF | e−ιHδte−ιHδt · · · e−ιHδt |qI〉 . (A.1)
Using the fact that |q〉 forms a complete set of basis states so that∫dq |q〉 〈q|, we can insert a
completeness relation between every adjacent propagators in the above expression,
〈qF | e−ιHT |qI〉 =
∫ (N−1∏i=1
dqi
)〈qF | e−ιHδt |qN−1〉 〈qN−1| e−ιHδt |qN−1〉 · ··
· · · 〈q2| e−ιHδt |q1〉 〈q1| e−ιHδt |qI〉 . (A.2)
Now we’ll compute each individual factor 〈qi+1| e−ιHδt |qi〉. For a particle, in general, H =
p2/2m+ V (q), so,
〈qi+1| e−ιHδt |qi〉 = e−ιV (qi)δt 〈qi+1| e−ιδt(p2/2m) |qi〉
= e−ιV (qi)δt
∫dp 〈qi+1| e−ιδt(p
2/2m) |p〉 〈p|qi〉
= e−ιV (qi)δt
∫dp
2πe−ιδt(p
2/2m)eιp(qi+1−qi).
35
The integral over p is a Gaussian integral and after evaluating it we get,
〈qi+1| e−ιHδt |qi〉 =
(−ι2πmδt
) 12
eιδt(m/2)[(qi+1−qi)/δt]2e−ιδtV (qi). (A.3)
Putting this into eq. (A.2),
〈qF | e−ιHT |qI〉 =
(−ι2πmδt
)N2N−1∏i=1
∫dqie
ιδt∑N−1j=0 (m/2)[(qj+1−qj)/δt]2−V (qj), (A.4)
where j = 0 corresponds to qI and j = N to qF . Going to the continuum limit i.e. δt→ 0, we
can replace (qi+1− qi)/δt by q and δt∑N−1
j=0 by∫ T
0dt. Further, we define the integral over paths
as ∫Dq(t) = lim
N→∞
(−ι2πmδt
)N2N−1∏i=1
∫dqi. (A.5)
With this definition, we can write the probability amplitude as
〈qF | e−ιHT |qI〉 =
∫Dq(t)eι
∫ T0 dt[ 1
2mq2−V (q)]. (A.6)
Identifying the integral in the exponent as the action, we obtain the path integral
〈qF | e−ιHT |qI〉 =
∫Dq(t)eιS(q). (A.7)
So, to compute the amplitude for the particle to go from |qI〉 to |qF 〉 in time T , take all the possible
paths that the particle can take between these two states, assign each path a corresponding weight
eιS , and sum them all. Note that we have to impose the boundary conditions on q(t): q(0) = qI
and q(t) = qF . Although we have derived the path integral for a single particle system, but this
principle is true for all quantum systems. Given any quantum system, the amplitude of a process
is given by the sum over all paths weighed by eιS .
If instead of taking the initial and final states to be the configuration space eigenstates, we
take any general states |I〉 and |F 〉, the probability amplitude can be found in the following way,
〈F | e−ιHT |I〉 =
∫dqF
∫dqI 〈F |qF 〉 〈qF | e−ιHT |qI〉 〈qI |I〉
=
∫dqF
∫dqIψF (qF )∗ 〈qF | e−ιHT |qI〉ψI(qI).
In most cases, we are interested in taking the initial and final states to be the ground state.
This amplitude is denoted by Z . So, Z = 〈0| e−ιHT |0〉. To compute the path integral, it is
somewhat more rigorous to perform a Wick rotation to Euclidean time by replacing t→ −ιτ
36
and rotating the integration contour in the complex plane. Under this transformation the path
integral becomes,
Z =
∫Dq(t)e−
∫ T0 dτ [ 1
2m(dq/dτ)2+V (q)]. (A.8)
This integral is called the Euclidean path integral. The Euclidean path integral provides an
integral representation for the partition function. The partition function from quantum statistical
mechanics is
Z = Tr e−βH =∑n
〈n| e−βH |n〉 . (A.9)
Clearly, we have an integral representation for the partition function if we replace T by −ιβ in
〈F | e−ιHT |I〉. We get,
Z = Tr e−βH =
∫PBC
Dq(t)e−∫ β0 dτL(q). (A.10)
Where L(q) = 12m(dq/dτ)2 + V (q). The PBC subscript reminds us that we are taking a trace
so the functional integral should be done over all the paths q(τ) such that q(0) = q(β).
We can extend this observation to field theory. For instance, consider a scalar field theory in
(D + 1) dimensions. The partition function will be,
Z = Tr e−βH =
∫PBC
Dφe−∫ β0 dτ
∫dDxL(φ). (A.11)
The integral is evaluated over all φ(~x, t) such that φ(~x, 0) = φ(~x, β). So, to study a field theory
at a finite temperature, we only need to perform a Wick rotation and then impose the periodic
boundary conditions. Conversely, if upon Wick rotation, we find that the theory is periodic in
Euclidean time with period β, we would say that the quanta of the theory feel a temperature of
1/β.
37
Appendix B
Spinors
One learns in a standard introduction to quantum field theory that scalar field φ gives rise to
spin-0 particles and the one form gauge field Aµ produces spin-1 particles. Both of these fields
produce bosons. However, to have a complete model of the world, one must also incorporate
fermions. The fields that describe fermions are called spinor fields. The quanta of spinor fields
carry a spin-1/2. The aim of this appendix is to introduce spinors. Although, we’ll only work in
4 dimensions in this appendix, the generalization to any finite dimension is straightforward. This
appendix is based on ref. [14].
B.1 Lorentz algebra
In special relativity, the Lorentz transformation Λµν must satisfy the condition Λµ
αΛνβη
αβ = ηµν ,
where ηµν is the Minkowskian metric. If we write an infinitesimal transformation Λµν = δµν + ωµν ,
this condition becomes the requirement that ωµν is anti-symmetric in indices i.e. ωµν + ωνµ = 0.
In 4 dimensions, the number of independent components for an anti-symmetric matrix are
(4× 4− 4)/2 = 6. We are going to introduce a basis for these anti-symmetric matrices. Let’s
denote these basis elements by (Mργ)µν , where µ and ν give us an element of the matrix and ρ
and γ tell us what matrix we’re dealing with. In addition to being anti-symmetric in µ and ν,
these matrices are also anti-symmetric in ρ and γ so that eventually we have only 6 independent
matrices. We can make the following choice,
(Mργ)µν = ηρµηγν − ηγµηρν . (B.1)
Or after lowering the last index,
(Mργ)µν = ηρµδγν − ηγµδρν . (B.2)
39
In this basis, M01 generates boosts in the x1 direction, M12 is the generator of rotations in the
(x1, x2)-plane and so on. Any general ωµν is given by a linear combination of Mργ of the form,
ωµν =1
2Ωργ(M
ργ)µν . (B.3)
Here Ωργ is also anti-symmetric in indices. These generators Mργ of the Lorentz transformations
obey the following commutation relations,
[Mργ,M τν ] = ηγτMρν − ηρτMγν + ηρνMγτ − ηγνMρτ . (B.4)
These commutation relations define the Lorentz Lie algebra. An arbitrary Lorentz transformation
can be written as,
Λ = exp(1
2ΩργM
ργ). (B.5)
B.2 The spinor representation
We are interested in finding other representations of the Lorentz algebra. To construct the spinor
representation, we are going to need Clifford algebra defined by,
γµ, γν = 2ηµν1. (B.6)
We need to find four matrices that satisfy γµγν = −γνγµ when µ 6= ν and (γ0)2 = 1, (γi)2 = −1
for i = 1, 2, 3. One possible representation is,
γ0 =
0 1
1 0
, γi =
0 σi
−σi 0
, (B.7)
where every element in these matrices is itself a 2×2 matrix and σ1, σ2, σ3 are the Pauli matrices.
This representation is called the Weyl or chiral representation. To see what Clifford algebra has
got to do with the Lorentz group, consider,
Sρν =1
4[γρ, γν ] =
1
2γργν − 1
2ηρν . (B.8)
Then,
[Sµν , γρ] =1
2[γµγν , γρ],
=1
2γµγνγρ − 1
2γργµγν ,
=1
2γµγν , γρ − 1
2γµγργν − 1
2γρ, γµγν +
1
2γµγργν ,
= γµηνρ − γνηρµ. (B.9)
40
Further,
[Sµν , Sρα] =1
2[Sµν , γργα],
=1
2[Sµν , γρ]γα +
1
2γρ[Sµν , γα].
Using the result we’ve just derived for [Sµν,γρ],
[Sµν , Sρα] =1
2γµγαηνρ − 1
2γνγαηρµ +
1
2γργµηνα − 1
2γργνηαµ.
Noting that γµγα = 2Sµα + ηµα, we get,
[Sµν , Sρα] = Sµαηνρ − Sναηρµ + Sρµηνα − Sρνηαµ, (B.10)
i.e. the matrices Sµν form a representation of the Lorentz algebra given in eq. (B.4). We’ll label
the rows and columns of these matrices using α, β = 1, 2, 3, 4. Now, we need to construct a field
for this representation to act upon. So, we introduce the field with four complex components
ψα(x) and we’ll call it the Dirac spinor field. This field transforms under Lorentz transformations
as, ψα(x)→ S[Λ]αβψβ(Λ−1x), where
Λ = exp(1
2ΩρσM
ρσ), (B.11)
S[Λ] = exp(1
2ΩρσS
ρσ). (B.12)
Now we’ll see in detail how do these fields transform under rotations and boosts. Before doing
that, it is important to note that,
Sij =1
2
0 σi
−σi 0
0 σj
−σj 0
= − ι2εijk
σk 0
0 σk
, (B.13)
S0i =1
2
0 1
1 0
0 σi
−σi 0
=1
2
−σi 0
0 σi
. (B.14)
To find the explicit form of the rotation matrix, we write the rotation parameter as Ωij = −εijkϕk,
then under rotations,
S[Λ] = exp(1
2ΩρσS
ρσ)
=
exp(ι ~ϕ.~σ
2
)0
0 exp(ι ~ϕ.~σ
2
) . (B.15)
Consider a rotation about x3-axis by 2π i.e. ~ϕ = (0, 0, 2π). The transformation matrix for this
rotation is,
S[Λ] =
exp(ιπσ3) 0
0 exp(ιπσ3)
= −1. (B.16)
41
Therefore under a 2π rotation ψα(x)→ −ψα(x). This isn’t what happens to a vector under a 2π
rotation. This shows that S[Λ] is indeed a different representation from Λµν . Under boosts, the
transformation matrix becomes,
S[Λ] = exp(1
2ΩρσS
ρσ)
=
exp(~χ.~σ2
)0
0 exp(− ~χ.~σ
2
) , (B.17)
where we have represented the boost parameter as Ωi0 = −Ω0i = χi.
B.3 The Dirac action
Now, we would like to construct a Lorentz invariant action for the spinor fields. Let’s start by
making some guess. Consider the term ψ†ψ. Although this term won’t fit the bill, but it will tell
us how to proceed. Under Lorentz transformation,
ψ(x)→ S[Λ]ψ(Λ−1x),
ψ†(x)→ ψ†(Λ−1x)S[Λ]†.
So, ψ†ψ → ψ†S[Λ]†S[Λ]ψ. However, S[Λ]†S[Λ] 6= 1 in general. Because for S[Λ] to be unitary,
we can see from eq. (B.12), Sµν should be anti-hermitian i.e. (Sµν)† = −Sµν . And since
Sµν ∼ [γµ, γν ], this would require all γµ to be either hermitian or anti-hermitian. However, this
is not possible because from eq. (B.6), (γ0)2 = 1 (real eigenvalues) and (γi)2 = −1 (imaginary
eigenvalues). So, γ0 is hermitian but then γi can’t be hermitian and vice versa. So, S[Λ] isn’t
unitary. In fact, the Lorentz group doesn’t have any finite dimensional unitary representation.
In chiral representation (γ0)† = γ0 and (γi)† = −γi. Or we can simply write (γµ)† =
γ0γµγ0. This means (Sµν)† ∼ [(γν)†, (γµ)†] ∼ −γ0Sµνγ0. And S[Λ]† = exp(
12Ωρσ(Sρσ)†
)=
γ0S[Λ]−1γ0. We define the Dirac adjoint, ψ = ψ†γ0. Now consider the term, ψψ. Then under
Lorentz transformation,
ψ(x)ψ(x) = ψ†(x)γ0ψ(x)
→ ψ†(Λ−1x)S[Λ]†γ0S[Λ]ψ(Λ−1x)
= ψ†(Λ−1x)γ0(γ0S[Λ]†γ0)S[Λ]ψ(Λ−1x)
= ψ†(Λ−1x)γ0ψ(Λ−1x)
= ¯ψ(Λ−1x)ψ(Λ−1x).
42
So, ψψ is a Lorentz scalar. Further, ψγµψ transforms as a Lorentz vector,
ψγµψ → ψS[Λ]−1γµS[Λ]ψ.
For this to transform as a vector we must have S[Λ]−1γµS[Λ] = Λµνγ
ν . For an infinitesi-
mal Lorentz transformation, we have Λ ≈ 1 + ΩρσMρσ/2 and S[Λ] ≈ 1 + ΩρσS
ρσ/2. So,
the requirement above reduces to −[Sρσ, γµ] = (Mρσ)µνγν . But we showed in eq. (B.9) that
[Sρσ, γµ] = γρησµ − γσηµρ. And also, (Mρσ)µνγν = (ηρµδσν − ησµδρν)γν = ηρµγσ − ησµγρ. So
indeed, ψγµψ transforms as a vector under Lorentz transformations. Using these two Lorentz
covariant terms, we can construct the action,
S =
∫d4xψ(x)(ιγµ∂µ −m)ψ(x), (B.18)
called the Dirac action. After quantization, this action describes particles of spin-1/2 and mass
m. Varying the action with respect to ψ we get the Dirac equation,
(ι/∂ −m)ψ(x) = 0. (B.19)
Here, /∂ = γµ∂µ.
At first sight, it looks like the Dirac spinor has 8 degrees of freedom because it has 4 complex
components. But it actually has only 4 degrees of freedom. The conjugate momentum to ψ is
πψ = ∂L/∂(∂0ψ) = ιψ†. Unlike the case of a scalar field, the conjugate momentum doesn’t
depend on the time derivative of the field. So, the phase space of the Dirac spinor has 8 real
dimensions. And the number of degrees of freedom is half the dimension of the phase space i.e.
4.
For the quantization of spinor fields, instead of using commutation relations, we have to
use anti-commutation relations. This is a consequence of the spin-statistics theorem. It states
that integer spin fields should be quantized as bosons whereas half integer spin fields should be
quantized as fermions, otherwise we’ll encounter inconsistencies. So, spinors describe fermions.
B.4 Weyl spinors
We can decompose the Dirac spinor representation into two irreducible representations acting on
the two component spinors u±. We decompose ψ as,
ψ =
u+
u−
. (B.20)
43
Then under rotations both u+ and u− transform in the same way i.e. u± → exp(ι ~ϕ.~σ
2
)u±, while
under boosts they transform oppositely, u± → exp(± ~χ.~σ
2
)u±. The two component spinors u±
are called Weyl spinors. u+ and u− are said to have opposite chirality.
B.5 Majorana spinors
The components of a spinor ψ are complex. If we try to make them real by imposing ψ∗ = ψ,
then they wouldn’t stay real after we make a Lorentz transformation because S[Λ] is also complex.
However, if we choose a real representation for the Sµν , then we can work with real spinors by
imposing the condition ψ∗ = ψ. Such spinors are called Majorana spinors. For instance, this can
be done by choosing γµ to be purely imaginary so that Sµν ∼ [γµ, γν ] are real. However, we can
generalize the definition of Majorana spinors even to the case when the representation is not real.
But still we’ll assume the γµ to satisfy (γ0)† = γ0 and (γi)† = −γi. We define a 4× 4 matrix C
satisfying
C†C = 1, C†γµC = −(γµ)∗. (B.21)
Then using this matrix, we define the charge conjugate of a Dirac spinor as,
ψ(C) = Cψ∗. (B.22)
ψ(C) transforms under a Lorentz transformation as,
ψ(C) → CS[Λ]∗ψ∗ = S[Λ]Cψ∗ = S[Λ]ψ(C).
Now we impose the representation independent Majorana condition ψ(C) = ψ to obtain a
Majorana spinor. For the case when γµ is purely imaginary, clearly C = 1 will work and we get
the simple condition ψ∗ = ψ as expected.
44
Appendix C
Conformal field theory
Conformal field theory is a field theory possessing conformal symmetry i.e. invariance under
conformal transformations. Conformal transformations are those that preserve angles between
vectors e.g. translations, scaling etc. A conformal field theory is so restrictive that sometimes
we’re able to solve the theory without even writing an action. Nevertheless, the form of the
two point and the three point functions of fields involved are completely fixed due to conformal
invariance. This appendix is based on ref. [15, 16, 5].
C.1 Conformal Killing equation
A conformal transformation is a differentiable map between two manifolds that scales the dot
product of vectors locally by a positive factor i.e.
ηρσdxρdxσ = Λ(x)ηµνdx
µdxν , (C.1)
where ηµν is the Minkowskian metric because we are assuming both the manifolds to be the
Minkowskian space and Λ(x) is positive. The above condition implies,
∂xρ∂xµ
∂xσ∂xν
ηρσ = Λ(x)ηµν . (C.2)
Under a conformal transformation, the vectors are scaled uniformly, hence the angles are
preserved. Now consider an infinitesimal transformation xµ → xµ = xµ + εµ(x) and plug this
into eq. (C.2),
∂xρ∂xµ
∂xσ∂xν
ηρσ = ηρσ
(δρµ +
∂ερ
∂xµ
)(δσν +
∂εσ
∂xν
)= ηµν +
(∂εµ∂xν
+∂εν∂xµ
).
45
For a conformal transformation this should be equal to Λ(x)ηµν . So we conclude that,
∂µεν + ∂νεµ = K(x)ηµν , (C.3)
where Λ(x) = 1 +K(x). K(x) can be determined in terms of εµ by taking the trace of the above
equation,
ηµν(∂µεν + ∂νεµ) = K(x)ηµνηµν
2(∂.ε) = K(x)d.
Here d is the dimension of the spacetime. So, replacingK(x) in eq. (C.3) by the above expression
gives,
∂µεν + ∂νεµ =2
dηµν(∂.ε). (C.4)
This is the conformal killing equation.
C.2 Conformal group in d ≥ 3
Let’s derive some other important equations from eq. (C.4) that’ll be helpful in finding out the
conformal group. First, applying ∂ν on both sides of eq. (C.4) gives,
∂µ(∂.ε) + εµ =2
d∂µ(∂.ε) (C.5)
where = ∂ν∂ν . Further taking ∂ν of this equation,
∂ν∂µ(∂.ε) + ∂νεµ =2
d∂ν∂µ(∂.ε). (C.6)
Now interchange µ and ν in this equation, and add the new equation obtained to this equation
and then use eq. (C.5) to get,
2∂ν∂µ(∂.ε) +
(2
d(∂.ε)
)ηµν =
4
d∂ν∂µ(∂.ε),
=⇒(ηµν + (d− 2)∂µ∂ν
)(∂.ε) = 0.
Contracting this equation with ηµν ,
(d− 1)(∂.ε) = 0 (C.7)
46
Another expression we need is obtained by first applying ∂ρ on eq. (C.4) and permuting indices,
∂ρ∂µεν + ∂ρ∂νεµ =2
dηµν∂ρ(∂.ε),
∂ν∂ρεµ + ∂µ∂ρεν =2
dηρµ∂ν(∂.ε),
∂µ∂νερ + ∂ν∂µερ =2
dηνρ∂µ(∂.ε).
Subtracting the first equation from the sum of the last two gives us the desired equation,
2∂µ∂νερ =2
d(−ηµν∂ρ + ηρµ∂ν + ηνρ∂µ)(∂.ε). (C.8)
Eq. (C.7) shows that for d 6= 1, εµ can be quadratic at most. So, εµ = aµ + bµνxν + cµνρx
νxρ.
Now we’ll examine each of these terms separately.
• εµ = aµ corresponds to translations and the generator is Pµ = −ι∂µ.
• If we insert the general expression for εµ in eq. (C.4) we find,
bµν + bνµ =2
d(ηρσbρσ)ηµν (C.9)
which shows that the symmetric part of the bµν is equal to a constant times ηµν . We can
separate bµν into a symmetric and an anti-symmetric part, bµν = αηµν +mµν where mµν =
−mνµ. The first part of this transformation is scaling and the generator is D = −ixµ∂µ.
The second part corresponds to the Lorentz transformations with the generators being
Lµν = ι(xµ∂ν − xν∂µ).
• For the last term let’s put the general form of εµ in eq. (C.8) to get ∂.ε = bµµ + 2cµµρxρ and
further taking ∂ν of this equation gives ∂ν(∂.ε) = 2cµµν . From this equation, we find that
cµνρ = ηµρdν + ηµνdρ − ηνρdµ,
where dµ = 1dcρρµ. The resulting transformations are called special conformal transforma-
tions (SCT). The generator is Kµ = −i(2xµxν∂ν − (x.x)∂µ).
47
It is straightforward to show that these generators satisfy the following commutation relations,
[D,Pµ] = ιPµ,[D,Kµ] = −ιKµ,[Kµ, Pν ] = 2ι(ηµνD − Lµν),[Kρ, Lµν ] = ι(ηρµKν − ηρνKµ),[Pρ, Lµν ] = ι(ηρµPν − ηρνPµ),[Lµν , Lρσ] = ι(ηνρLµσ + ηµσLνρ − ηµρLνσ − ηνσLµρ).
We can actually redefine the generators in a way that compresses all these commutation relations
into one. Redefining, Jµν = Lµν , J−1,0 = D, J−1,µ = 12(Pµ −Kµ) and J0µ = 1
2(Pµ +Kµ). And
also that Jmn = −Jnm where m,n = −1, 0, 1, ...., d. Then all the commutation relations above
are contained in the following commutation relation,
[Jmn, Jrs] = i(ηmsJnr + ηnrJms − ηmrJns − ηnsJmr), (C.10)
where ηms has the signature diag(−1,−1, 1, · · ·, 1). So, the conformal Lie algebra is contained
in this commutation relation. But this is the same commutation relation that would define the
Lie algebra of SO(d, 2). We thus conclude that, in d ≥ 3 dimensions, the conformal group is
SO(d, 2).
C.3 Conformal group in d = 2
Conformal group in d = 2 is very special, it’s Lie algebra is infinite dimensional. We could ask
why doesn’t the treatment for d ≥ 3 doesn’t also apply to the case of d = 2. As we’ll soon see
that the conformal group for d = 2 is infinitely large and the previous treatment did not make
that obvious. Putting d = 2 in eq. (C.4) gives us two equations,
∂0ε0 = ∂1ε1, ∂0ε1 = −∂1ε0. (C.11)
These equations are the Cauchy-Riemann equations from complex analysis. Thus, we can
replace x0 and x1 by z = x0 + ιx1 and then any continuous transformation will be given by
z → z′ = z+ ε(z) where ε(z) = ε0 + ιε1 and the requirement of conformality now becomes the
48
requirement of analyticity of ε(z). In general, ε(z) can be written as a Laurent series expansion,
z → z′ = z + ε(z) = z +∑n∈Z
εn(−zn+1),
z → z′ = z + ε(z) = z +∑n∈Z
εn(−zn+1).
The transformation generators are,
ln = −zn+1∂z, ln = −zn+1∂z, (C.12)
where ∂z = 12(∂0 − ι∂1) and ∂z = 1
2(∂0 + ι∂1). The commutators of these generators are,
[lm, ln] = zm+1∂z(zn+1∂z)− zn+1∂z(z
m+1∂z)
= (n+ 1)zn+m+1∂z − (m+ 1)zn+m+1∂z
= (m− n)lm+n, (C.13)
[lm, ln] = (m− n)lm+n, (C.14)
[lm, ln] = 0. (C.15)
The algebra defined by the first commutation relation is called the Witt algebra. The algebra
defined by the second commutation relation is exactly the same as the first and the last commuta-
tion relation shows that the two algebras are independent. In summary, the conformal algebra in
d = 2 is infinite dimensional.
The conformal transformations we found out in the previous subsection (i.e. translations,
scaling, Lorentz, SCT) correspond to l−1, l0 and l1 in the d = 2 case.
• l−1 generates z → z′ = z − ε−1, which are just translations.
• l0 = −z∂z generates transformations of the form z → az where a is some complex number.
The multiplication by |a| represents scalings and the multiplcation with the phase factor of
a are just rotations (i.e. boosts since one of the two dimensions has temporal nature).
• l1 generates translations of ω = −1/z, which, can be checked, are just the special
conformal transformations.
If a 2D quantum field theory is conformally invariant, the infinite extent of the conformal
group restraints the theory so much that sometimes we are able to solve the theory without
needing to explicitly write an action.
49
C.4 Two and Three-Point Functions of Chiral
quasi-Primary Fields
Chiral fields are those which only depend on z (i.e. independent of z). Primary fields are those
that transform under a conformal transformation z → f(z) in the following fashion,
φ(z, z)→ φ′(z, z) = (∂zf)h(∂zf)hφ(f(z), f(z)).
Such a field is said to be a primary field of conformal dimension (h, h) (h is not the complex
conjugate of h, both of them are real numbers and they may or may not be the same). If this
transformation law holds true globally, then the field is said to be quasi-primary.
If a theory is conformally invariant, the form of two and three-point functions of chiral
quasi-primary fields is completely fixed by virtue of this symmetry. Let’s consider the two-point
function of two chiral quasi-primary fields, 〈φ1(z)φ2(ω)〉 = g(z, ω). Translational invariance
requires g(z, ω) = g(z − ω). Under the scaling transformation z → λz,
〈φ1(z)φ2(ω)〉 → 〈Λh1φ1(λz)Λh2φ2(λω)〉 = λh1+h2g(λ(z − ω)
).
Since this should be equal to g(z − ω), we conclude
g(z − ω) =d12
(z − ω)h1+h2, (C.16)
where d12 is some constant. Finally, under SCTs which essentially imply invariance under
z → −1/z,
〈φ1(z)φ2(ω)〉 →⟨ 1
z2h1
1
ω2h2φ1
(− 1
z
)φ2
(− 1
ω
)⟩=
1
z2h1ω2h2
d12(− 1
z+ 1
ω
)h1+h2.
And this will be equal to d12/(z − ω)h1+h2 only if h1 = h2. So the final form of the two point
function is,
〈φ1(z)φ2(ω)〉 =d12δh1,h2
(z − ω)2h1. (C.17)
By using similar arguments we can also work out the three point function which comes out to be,
〈φ1(z1)φ2(z3)φ3(z3)〉 =C123
(z1 − z2)h1+h2−h3(z2 − z3)h2+h3−h1(z1 − z3)h1+h3−h2. (C.18)
50
Appendix D
Anti-de Sitter Spacetime
Anti-de Sitter space is a maximally symmetric space with negative curvature scalar. We introduce
the notion of a maximally symmetric space more formally in the next section, but the meaning is
evident from the name. Physically, AdS space is like a confining box because it takes a light ray
finite amount of time to reach the boundary. This appendix is based on ref. [4, 17].
D.1 Maximally symmetric space
As the name suggests, a maximally symmetric space has maximum possible symmetry. But what
is the maximum possible symmetry? For a better grasp of the concept, let’s move forward with
an example that perfectly captures the notion, Rd with flat Euclidean metric. The isometries of
this space are translations and rotations. Pick a point p. We can make a translation in any one
of the d directions from this point. So, there are d total translations. A rotation can be thought
of as moving one of the axes into another. Since there are d axes, and d − 1 axes into which
an axis can be rotated, we expect that there are d(d− 1) rotations. But the rotation of the first
axis to the second should be counted the same as the rotation of the second into the first. So,
there are d(d− 1)/2 rotations. Therefore, there are d+ d(d− 1)/2 = d(d+ 1)/2 independent
isometries of Rd and thus it has d(d+ 1)/2 Killing vectors. We define a space with d(d+ 1)/2
Killing vectors to be maximally symmetric.
The curvature scalar of a maximally symmetric space is the same everywhere. So, the
curvature scalar is used to classify maximally symmetric spaces. It is also possible to construct
the Riemann tensor from the curvature scalar for a maximally symmetric space. To see that, let’s
choose locally inertial co-ordinates at some point p so that gµν = ηµν . We can perform a Lorentz
51
transformation at p without changing the metric ηµν . Since the space is maximally symmetric,
the Riemann curvature tensor should also be Lorentz invariant at this point. That is because
there is no preferred direction. But there are only a few Lorentz invariant tensors: the metric, the
Kronecker delta, and the Levi-Civita tensor. So, the curvature tensor at p in these locally inertial
co-ordinates must be constructed from these Lorentz invariant tensors. So, using these tensors
we have to construct a tensor with the symmetries of the Riemann tensor and it turns out to be,
Rµνρσ ∝ gµρgνσ − gµσgνρ. (D.1)
Since this is a tensorial relation, this must be true in all co-ordinate systems. Taking the trace on
both sides fixes the proportionality constant and we get,
Rµνρσ =R
d(d− 1)(gµρgνσ − gµσgνρ). (D.2)
So, we have found the curvature tensor for a maximally symmetric space. Thus, the Ricci scalar
completely specifies a maximally symmetric space locally.
D.2 Construction of AdS
As mentioned before, AdS is a maximally symmetric space with negative curvature scalar. It
satisfies vacuum Einstein’s equations with negative cosmological constant. The source free
Einstein’s field equations are,
Rµν −1
2Rgµν + Λgµν = 0. (D.3)
Taking trace on both sides of this equation gives,
Λ =d− 2
2dR. (D.4)
And sinceR < 0, Λ < 0 for d > 2.
Now we’ll construct the metric for AdS. Consider a flat Lorentz spacetime of signature
(2,D). The metric will be,
ds2 = −dx2−1 − dx2
0 + d~x2, (D.5)
where ~x = (x1, x2, ..., xD). Now take a surface in this space given by,
− x2−1 − x2
0 + ~x2 = −R2. (D.6)
52
Since, the isometry group of R2,D is SO(2, D), so SO(2, D) must also be the isometry group of
the surface we’ve constructed. But SO(2, D) has (D + 1)(D + 2)/2 generators and since the
surface is (D + 1)-dimensional, it is thus a maximally symmetric space. One can check that
R = −1/R2. So, this embedded surface is the AdS space.
With an eye on eq. (D.6), we can parameterize,
x0 = R√
1 + r2 cos(t), x−1 = R√
1 + r2 sin(t), ~x2 = R2r2, (D.7)
where t ∈ [0, π] and r ∈ [0,∞). Then, from eq. (D.5),
ds2 = R2
(− (1 + r2)dt2 +
dr2
1 + r2+ r2dΩD−1
). (D.8)
Define r = tan ρ where ρ ∈ [0, π/2),
ds2 =R2
cos2(ρ)(−dt2 + dρ2 + sin2(ρ)dΩ2
D−1). (D.9)
This is a cylinder like geometry. Imagine t running along the axis of the cylinder from 0 to π and
ρ along the radial direction going from 0 to π/2. The boundary of this geometry lies at ρ = π/2
and the metric at the boundary is,
ds2boundary ∼ −dt2 + dΩ2
D−1. (D.10)
So, the boundary of AdSD+1 is R× SD−1 with the metric of signature (−+).
For a light ray, we’ll have ds2 = 0 and that gives −dt2 + dρ2 = 0. So it takes a light ray a
finite amount of time t = π/2 to reach the boundary. Thus, the AdS space is like a confining
box.
D.3 The Poincare co-ordinates
Another choice of parameterization to satisfy eq. (D.6) is,
x−1 =R2
2r
(1 +
r2
R4(R2 + ~y2 − t2)
), (D.11)
xD =R2
2r
(1− r2
R4(R2 − ~y2 − t2)
), (D.12)
x0 =r
Rt, (D.13)
xi =r
Ryi. (D.14)
53
where r > 0. We can see that x−1 + xD = r, but r > 0 means that this co-ordinate choice
doesn’t correspond to the full AdS space, rather only to a patch of it called the Poincare patch.
The metric in this co-ordinate system is,
ds2 =r2
R2(−dt2 + d~y2) +
R2
r2dr2. (D.15)
The boundary of Poincare patch lies at r =∞ and isMD.
54
Appendix E
Schwarzschild Geometry
Schwarzschild solution is the most general static spherically symmetric solution to the source
free Einstein’s equations with zero cosmological constant. In this section we’ll only work with
the Schwarzschild solution in 4 dimensions, but higher dimension solutions also look similar.
This appendix is based on ref. [17].
E.1 The Schwarzschild metric
To find the general spherically symmetric and static solution to the source free Einstein’s field
equations, we begin by guessing the metric. First consider the Minkowskian metric,
ds2Minkowski = −dt2 + dr2 + r2dΩ2. (E.1)
We want to preserve the spherical symmetry of this metric. Thus generally,
ds2 = −e2α(r)dt2 + e2β(r)dr2 + e2γ(r)r2dΩ2. (E.2)
Defining r = eγ(r)r. This means that dr = eγ(1 + rdγ/dr)dr and hence,
ds2 = −e2α(r)dt2 +(
1 + rdγ
dr
)−2
e2β(r)−2γ(r)dr2 + r2dΩ2. (E.3)
Let’s relabel r → r and (1 + rdγ/dr)−2e2β(r)−2γ(r) → e2β ,
ds2 = −e2α(r)dt2 + e2β(r)dr2 + r2dΩ2. (E.4)
α(r) and β(r) can be determined using Einstein’s field equations. First compute Rµν and R.
Then we would like to set R = 0. Further, use source free Einstein’s field equations with
55
zero cosmological constant, Rµν = 0 to determine α(r) and β(r). We won’t be doing all this
computation here, rather we’ll just state the final result,
e2α = 1− Rs
r, e2β =
(1− Rs
r
)−1
, (E.5)
where Rs is some undetermined constant. The metric is,
ds2 = −
(1− Rs
r
)dt2 +
(1− Rs
r
)−1
dr2 + r2dΩ2. (E.6)
This is the Schwarzschild metric. Recall that in the weak field limit of GR we have, gtt =
−(1 − 2GM/r) where M was the mass of the gravitating object. Since, the Schwarzschild
metric should reduce to the weak field metric when r becomes large, we must have Rs = 2GM .
Clearly, the metric is problematic at r = 0 and r = 2GM . But we don’t know yet whether these
singularities are honest or just a fault in our co-ordinate system. r = 0 is indeed a co-ordinate
independent singularity as it can be checked by computing the curvature scalars. One can show
that,
RµνρσRµνρσ =48G2M2
r6. (E.7)
This blows up at r = 0. On the other hand, r = 2GM is a co-ordinate singularity and it can be
completely removed in a better co-ordinate system as we’ll show in the next section.
E.2 Schwarzschild black holes
To understand the causal structure of the spacetime, we find the radial null curves. So,
ds2 = 0 = −
(1− 2GM
r
)dt2 +
(1− 2GM
r
)−1
. (E.8)
This gives,
dt
dr= ±
(1− 2GM
r
)−1
. (E.9)
For large r, this is equal to ±1, like it is for flat spacetime. As r → 2GM , dt/dr → ±∞, so,
light gets closer and closer to r = 2GM but never reaches it in this co-ordinate system. Or we
can say that the light cones close up as we approach 2GM .
This inability to reach r = 2GM is an illusion. In fact, light doesn’t have any trouble
reaching r = 2GM but an observer from the outside would never see it reaching r = 2GM .
Solving eq. (E.9) gives,
t = ±r∗ + constant, (E.10)
56
where r∗ is called the tortoise co-ordinate and is given by,
r∗ = r + 2GM ln( r
2GM− 1). (E.11)
This definition makes sense only for r ≥ 2GM . But our current co-ordinates aren’t very good
beyond this anyway. The tortoise co-ordinate for r = 2GM is r∗ = −∞. The Schwarzschild
metric can be re-written in terms of the tortoise co-ordinate as,
ds2 =
(1− 2GM
r
)(−dt2 + dr∗2) + r2dΩ2. (E.12)
where r is a function of r∗. Null geodesics are now given by t = ±r∗. The light cones now don’t
close up. Define new co-ordinates,
v = t+ r∗, u = t− r∗. (E.13)
The infalling radial null geodesic is characterized by v = constant and an outgoing one by
u = constant. Now we go back to the original co-ordinate r, but replace t by the new co-ordinate
v. These are the Eddington-Finkelstein co-ordinates and the metric becomes,
ds2 = −
(1− 2GM
r
)dv2 + (dvdr + drdv) + r2dΩ2. (E.14)
This metric is perfectly regular at r = 2GM . Even though gvv vanishes at r = 2GM , but the
metric is invertible since the determinant is −r4 sin2 θ. In these co-ordinates, the condition for
radial null curves is,
infalling:dv
dr= 0, (E.15)
outgoing:dv
dr= 2
(1− 2GM
r
)−1
. (E.16)
These co-ordinates behave perfectly well at r = 2GM and beyond it. We can see that something
interesting is happening beyond r = 2GM . For r < 2GM , dv/dr < 0. This means that the
light cone tilts over inside r = 2GM and all the future paths are directed towards the decreasing
direction of r. Thus r = 2GM functions as a point of no return. Once a particle goes past it,
it can never come out. Not even light can escape it. The surface specified by r = 2GM is,
thus, called the event horizon. The event horizon is a null surface because at r = 2GM , a null
geodesic is given simply by dr/dv = 0, which is true if r is fixed at 2GM . So, it is really the
causal structure of this spacetime that makes it impossible to cross the horizon in an outward
going direction. Since nothing can cross the horizon, we cannot see anything inside, thus the
region of spacetime inside the horizon is called the black hole.
57
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