introduction to superstring theory1 introduction at the moment, string theory is the most promising...
TRANSCRIPT
Introduction to
Superstring Theory
Lecture Notes – Spring 2005
Esko Keski-Vakkuri
S. Fawad Hassan
1
Contents
1 Introduction 5
2 The bosonic string 5
2.1 The Nambu-Goto action . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.1.1 Strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 The Polyakov action . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.3 Classical symmetries of the Polyakov action . . . . . . . . . . . . . . 11
2.3.1 Gauge fixing . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.4 Conformal invariance: . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.5 Equations of motion and boundary conditions . . . . . . . . . . . . . 14
2.6 Mode expansion and quantization . . . . . . . . . . . . . . . . . . . . 16
2.7 Constraints . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Low-lying string states. . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.9 The light-cone gauge . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.10 Lowest lying states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.11 Open strings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.12 Path integral quantization of the bosonic string . . . . . . . . . . . . 30
2.13 Conformal field theory (CFT) . . . . . . . . . . . . . . . . . . . . . . 35
2.13.1 Commutators in CFT and Radial Ordering . . . . . . . . . . . 45
2.13.2 Operator Product Expansions . . . . . . . . . . . . . . . . . . 45
2.13.3 Correlation Functions . . . . . . . . . . . . . . . . . . . . . . . 47
2.13.4 Wick’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 48
2.13.5 Operator-state Correspondence . . . . . . . . . . . . . . . . . 50
2.14 Tree-level Bosonic String Interactions . . . . . . . . . . . . . . . . . . 55
2.14.1 Scattering of Open String Tachyons . . . . . . . . . . . . . . . 58
2.14.2 Tree-level Scattering of Closed String Tachyons . . . . . . . . 64
2.15 Strings in Background Fields . . . . . . . . . . . . . . . . . . . . . . . 66
2.16 Weyl Invariance and the Weyl Anomaly . . . . . . . . . . . . . . . . . 70
2.17 The Bosonic String Beta Functions and the Effective Action . . . . . 71
2.18 An Example of a One-loop Amplitude: the Vacuum-to-vacuum Am-
plitude, i.e., the Partition Function . . . . . . . . . . . . . . . . . . . 74
2.18.1 The Vacuum Energy . . . . . . . . . . . . . . . . . . . . . . . 79
3 Superstrings (“Where It Begins Again”) 81
3.1 The Superstring Action . . . . . . . . . . . . . . . . . . . . . . . . . . 82
3.2 Equations of Motion and Boundary Conditions . . . . . . . . . . . . . 84
3.3 Mode Expansions and Quantization . . . . . . . . . . . . . . . . . . . 85
3.4 Constraints on Physical States . . . . . . . . . . . . . . . . . . . . . . 88
3.5 Emergence of Spacetime Spinors . . . . . . . . . . . . . . . . . . . . . 90
2
3.5.1 Chirality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
3.6 The Spin Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
3.7 Lowest Lying Excitations of Closed Superstrings . . . . . . . . . . . . 93
3.7.1 NS-NS Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.7.2 R-NS Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
3.7.3 NS-R Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.7.4 R-R Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
3.7.5 Problems with the Spectrum . . . . . . . . . . . . . . . . . . . 94
3.8 The GSO projection (GSO=Gliozzi-Scherk-Olive) . . . . . . . . . . . 95
3.8.1 NS Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.8.2 R Sector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
3.9 Type IIA and Type IIB Superstrings . . . . . . . . . . . . . . . . . . 96
3.9.1 Gamma Matrix Conventions . . . . . . . . . . . . . . . . . . . 97
3.9.2 R-R Ground States . . . . . . . . . . . . . . . . . . . . . . . . 98
3.10 Type IIA and IIB Supergravity . . . . . . . . . . . . . . . . . . . . . 101
3.11 Toroidal Compactification and T-duality . . . . . . . . . . . . . . . . 104
3.11.1 General Idea of Compactification . . . . . . . . . . . . . . . . 105
3.11.2 Scalar Field Theory Compactified on S1 . . . . . . . . . . . . 105
3.11.3 Main Features of Field Theory Compactifications on S1 . . . . 107
3.11.4 String Theory Compactified on S1 . . . . . . . . . . . . . . . . 108
3.11.5 Features of String Theory on S1 . . . . . . . . . . . . . . . . . 112
3.12 T-duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.12.1 The T-duality Map . . . . . . . . . . . . . . . . . . . . . . . . 113
3.12.2 T-duality in Superstring Theory . . . . . . . . . . . . . . . . . 115
3.12.3 T-duality Action on Ramond States . . . . . . . . . . . . . . . 115
3.13 Gauge Symmetry Enhancement in Circle Compactification . . . . . . 117
3.13.1 Momentum and Winding as Abelian Charges . . . . . . . . . 118
3.14 Lattices and Torii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
3.15 Rectangular Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
3.15.1 Construction of a General Torus . . . . . . . . . . . . . . . . . 121
3.15.2 Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.15.3 “Integer” and “Even” Lattices . . . . . . . . . . . . . . . . . . 123
3.15.4 Dual Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
3.16 Heterotic String Theory . . . . . . . . . . . . . . . . . . . . . . . . . 124
3.16.1 Equations of Motion . . . . . . . . . . . . . . . . . . . . . . . 125
3.16.2 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . 125
3.16.3 Quantization . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
3.16.4 Mass Conditions (Spectrum) . . . . . . . . . . . . . . . . . . . 129
3.16.5 Extra Massless States . . . . . . . . . . . . . . . . . . . . . . . 130
3.16.6 Massless Sector of the Heterotic String Spectrum . . . . . . . 131
3.17 Type I Superstrings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
3
3.17.1 Chan-Paton Factors (Works for Bosonic and Superstrings) . . 135
3.18 D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.19 Multiple D-branes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.20 String Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
3.20.1 Type I - Heterotic SO(32) Duality . . . . . . . . . . . . . . . 149
3.20.2 Type IIA - IIB Duality . . . . . . . . . . . . . . . . . . . . . . 150
3.20.3 Heterotic SO(32) - Heterotic E8 × E8 Duality . . . . . . . . . 151
3.20.4 Type IIA - M-theory Duality . . . . . . . . . . . . . . . . . . 151
3.20.5 Heterotic E8 × E8 - M-theory Duality . . . . . . . . . . . . . . 156
4
1 Introduction
At the moment, string theory is the most promising candidate for a unified theory of
all fundamental particles and forces, including gravity. Furthermore, it unifies all the
known forces with the laws of quantum mechanics.
Traditional quantum gauge field theories have been a successful framework for
describing elementary particles up to currently testable energy scales. However, in
the big picture, they are to be considered as effective theories, approximations to the
“final” theory that lies somewhere underneath. For example, the Standard Model
contains 19 parameters, whose values are set by hand to agree with experiments.
String theory is based on the idea of replacing particles as fundamental con-
stituents by one-dimensional extended objects, strings. There is only one parameter
in the theory, the length of the string, ls. The string length is thought to be of the
order of Planck length 10−33 cm. The hope is that at larger scales, string theory
determines how the symmetries of Nature are selected and broken, and what is the
ladder of effective field theories that emerge, containing the Standard Model.
As a theory of quantum gravity, string theory is also hoped to teach us what
is superseded by space and time at truly small scales where quantum effects render
these concepts fuzzy. It is also hoped to answer some fundamental problems related
to quantum behavior of black holes. In this course we will see some examples of how
our usual concepts of spacetime are modified.
During the past years, string theory itself has developed rapidly. Our current
understanding of it is now much more rich. The term “string theory” no longer refers
to only strings and their interactions - it now refers to many kinds of different extended
objects and to a large web different but related techniques (including traditional gauge
field theories) and so on.
The hope is that this course will serve as an introduction to the basic concepts,
and enable and encourage to further study this fascinating field.
Editorial note. These lectures have been compiled from various sources (I will
provide a more detailed list of references when I remember them). Therefore I have
had to struggle with different conventions and notations – some confusion is bound
to remain even at this stage. Please let me know of any typos and mistakes that
you find. A portion of these notes was developed by Fawad Hassan, currently at
Stockholm University. The typesetting is mostly thanks to the efforts of Moundheur
Zarroug and Niko Jokela and I highly appreciate their work.
2 The bosonic string
For someone used to field theory, our starting point may perhaps look a bit peculiar
at first. In field theory, the particles appear as quanta of the field so one begins
5
with a “multiparticle” description. Here, in contrast, we will begin by investigating
the dynamics of a single string. Different oscillations of the string will turn out to
correspond to different particles. However, it will take a while before we will quantize
the string. We begin by studying its classical behavior.
2.1 The Nambu-Goto action
A string is simply a one-dimensional extended object moving and vibrating across
spacetime. As an introduction, it is useful first to consider the motion of relativistic
massive point particle.
As a particle moves through the spacetime, it sweeps out a curve, the worldline:
X
particle worldline
X0
i
Figure 1: Worldline of a point particle.
If we denote by τ the parameter along the worldline (the proper time that the
particle would measure if it carried a watch), we can parameterize the curve of the
worldline by
Xµ = Xµ(τ) , µ = 0, 1, . . . , D − 1 (2.1)
in a D-dimensional spacetime. If the spacetime is a flat Minkowski space, it has a
metric
ds2 = ηµν(X)dXµdXν = −(dX0)2 + (dX1)2 + (dX2)2 + · · ·+ (dXD−1)2. (2.2)
(I’m using the -++++...+ signature)
6
The action that controls the dynamics of the point particle is simply proportional
to the proper length of the worldline:
S = −m
∫ds . (2.3)
We can rewrite this using the embedding Xµ(τ) and the proper time τ :
ds2 = (−(X0)2 + (X1)2 + · · ·+ (XD−1)2)dτ 2 = ηµνXµXνdτ 2 (. ≡ d
dτ) (2.4)
⇒ S = −m
∫ √ηµνXµXνdτ . (2.5)
The classical equations of motion are minima of the action, so they are found by the
variational principle
δS = 0 (2.6)
Since the path integral contains c = 1
~ = 1(2.7)
we get thus [length] = [mass]−1 = [energy]−1. Since [dτ ] = [length], we must have
[mdτ ] = [length]0, so m has the units of a mass (of the particle). If the particle is
moving in a curved spacetime, we replace the flat metric ηµν by the curved metric
gµν(X). Then the action is
S = −m
∫ √gµν(X)XµXνdτ . (2.8)
You can check that the variational principle gives the geodesic equation as the (rela-
tivistic) equation of motion of the point particle. Recall that massive point particles
are supposed to follow timelike geodesics in curved spacetime.
2.1.1 Strings
Now consider a string moving in spacetime. It traces out a two-dimensional surface,
called the worldsheet. We have two natural choices for a string: an open string and
a closed string. Their worldsheets are depicted in Figure 2. The D-dimensional
spacetime where the string moves is often called the target space.
Just as the action for a relativistic point particle was proportional to the length
of the worldline, the logical guess for the action of a relativistic string is the area of
its worldsheet:
S = −T
∫“d2s” . (2.9)
7
0
open string closed string
X
X X
X
0
i i
Figure 2: Open and closed string worldsheets.
Now we just have to be a bit careful about what is the integration measure here,
i.e. what is meant by the infinitesimal area element “d2s”. First of all, we need two
parameters to parameterize the worldsheet. In addition to a proper time τ we need a
spacelike coordinate σ which parameterizes the string. For an open string we choose
σ ∈ [0, π] (σ = 0, π are the end points), for a closed string we choose σ ∈ [0, 2π] with
σ = 0 identified with σ = 2π (Fig. 3):
σ
π
2π
0
Figure 3: Parameterization of a closed string.
Note that the strings are oriented (imagine an arrow pointing to the direction of
increasing σ).
The worldsheet is characterized by its embedding to the spacetime:
Xµ = Xµ(τ, σ) . (2.10)
8
For a closed string we need a periodicity condition Xµ(τ, σ) = Xµ(τ, σ + 2π). If the
worldsheet was a flat strip or cylinder, the area would be simply
Area =
∫dτ
∫ π or 2π
0
dσ . (2.11)
However, the embedding to the spacetime allows the string to wiggle and bend (see
Fig. 2.), this induces a curved metric hαβ(X) to the worldsheet to characterize dis-
tances. To find it, we substitute the embedding (2.10) to the metric of the spacetime.
Let us take the spacetime to be a flat Minkowski space with metric ηµν . Then, by
substituting (2.10):
ds2 = ηµνdXµdXν = ηµνd(Xµ(τ, σ))d(Xν(τ, σ))
= ηµν
(∂Xµ
∂τdτ +
∂Xµ
∂σdσ
)(∂Xν
∂τdτ +
∂Xν
∂σdσ
)
= ηµν(∂αXµ)(∂βXν)dσαdσβ
≡ hαβ(X)dσαdσβ
(2.12)
where I used the notation
σα =
σ0 ≡ τ (α = 0)
σ1 ≡ σ (α = 1)(2.13)
for the worldsheet coordinates τ , σ, and ∂α ≡ ∂∂σα . So the string worldsheet is a
2-dimensional surface with curved metric
hαβ(X) = ηµν∂αXµ∂βXν .
What is its infinitesimal area element? In curved space, at every point we can intro-
duce (“local orthonormal”) coordinates ξα where the metric looks flat:
ds2WS = hαβdσαdσβ = −(dξ0)2 + (dξ1)2 . (2.14)
In these coordinates, the infinitesimal area element is simply dξ0dξ1. So we only need
to evaluate the Jacobian in transforming back to the original coordinates σα:
dξ0dξ1 = J(ξ, σ)dσ0dσ1 . (2.15)
You can convince yourself that the Jacobian is
J(ξ, σ) =√− det(hαβ) ≡
√−h . (2.16)
So now we are ready to write down the action of a relativistic string:
SNG = −T
∫dσ0dσ1
√− det(hαβ) = −T
∫dτdσ
√− det(ηµν∂αXµ∂βXν) . (2.17)
9
The action (2.17) is the Nambu-Goto action. For SNG to be a number (~ = 1),
T must scale like [mass]2 or [length]−2. One can check that T can be identified as the
tension of the string. The string is elastic. We can trade the tension to an another
parameter, the string length ls:
ls =1√πT
. (2.18)
Another equivalent parameterization uses the Regge slope α′ (pronounced “alpha
prime”):
T =1
2πα′. (2.19)
Note that the Nambu-Goto action (2.17) contains a square root. So the action is non-
polynomial, and it is no surprise that it would be very hard to proceed to quantize
it. We will therefore use a trick and move to an alternative description, the Polyakov
action, which is quadratic (in Xµ).
2.2 The Polyakov action
The action (2.17) can be thought as the action for a 1+1 dimensional field theory of
D scalar fields Xµ(τ, σ). Now we make a trick. We introduce a set of (a matrix of)
auxiliary fields hαβ(τ, σ) (to be identified with the worldsheet metric, but not yet)
and consider the action
Sp = −T
2
∫dτdσ
√−h hαβ∂αXµ∂βXνηµν . (2.20)
This is the Polyakov action. Since hαβ are nonpropagating degrees of freedom, their
equation of motion
δS
δhαβ
= 0 (2.21)
plays the role of a constraint. The equation of motion can be written as the equation
Tαβ ≡ − 1
T
1√−h
δS
δhαβ=
1
2(∂αXµ∂βXν − 1
2hαβhγδ∂γX
µ∂δXν)ηµν = 0 . (2.22)
We can use (2.22) to eliminate1 the auxiliary fields hαβ. Let us denote hαβ ≡∂αXµ∂βXνηµν and h = det(hαβ). From (2.22):
hαβ =1
2hαβhγδ
=hγδ︷ ︸︸ ︷∂γX
µ∂δXνηµν (2.23)
1Note that the equation (2.22) really introduces only 2 non-trivial constraints. Tαβ is symmetricso it has only 3 independent components. On the other hand it turns out to be automaticallytraceless (see next sections), so there is one linear relation between to components meaning thatonly 2 are independent.
10
⇒ h =1
4h(hγδhγδ)
2 . (2.24)
Substituting the square root of (2.24) to (2.20) yields
Sp = −T
2
∫dτdσ
√−h hαβhαβ (2.25)
= −T
∫dτdσ
√−h = SNG . (2.26)
Thus, by eliminating the auxiliary fields hαβ from the Polyakov action (2.20), we
recover the Nambu-Goto action (2.17).
***** END OF LECTURE 1 *****
2.3 Classical symmetries of the Polyakov action
The Polyakov action is invariant under the following three different groups of sym-
metry transformations:
(i) Global symmetries: The D-dimensional spacetime is invariant under Poincare
transformations. From the point of view of the 2-dimensional string action, these
transformations are global symmetry transformations acting on the D scalar fields
Xµ:
Xµ → Xµ + aµ (translations)
Xµ → Xµ + ωµνX
ν (ωµν = −ωνµ) (Lorentz)(2.27)
We will ignore these for the moment.
(ii) Worldsheet diffeomorphisms: These are general coordinate transformations
on the worldsheet,
σα → σα(σβ) (2.28)
or, infinitesimally:
σα = σα + εα(σβ) . (2.29)
These2 are also called reparametrizations of the world sheet. Under (2.28), the in-
finitesimal area element√−hdσ0dσ1 remains invariant3
d2σ√−h = d2σ
√−h . (2.30)
2Note that (2.29) is characterized by 2 local parameters εα.3They are both related to the simple area element in the local orthonormal frame.
11
To see how the worldsheet metric transforms, use
hαβ(σ)dσαdσβ = hαβ(σ)dσαdσβ
= hγδ(σ)dσγ
dσα
dσδ
dσβdσαdσβ
⇒ hαβ(σ) = hγδ(σ)dσγ
dσα
dσδ
dσβ. (2.31)
This transformation is compensated by the transformation of ∂αXµ∂βXνηµν , so that
hαβ∂αXµ∂βXνηµν is invariant. Thus SP is invariant under (2.28).
(iii) Weyl transformations: Weyl transformation means rescaling of the (world-
sheet) metric
hαβ(σ) → Λ(σ)hαβ(σ) (2.32)
by a local scale factor Λ(σ). (2.32) implies
√−h →
√− det
(Λh00 Λh01
Λh10 Λh11
)=√
Λ2√−h
hαβ → Λ−1hαβ . (2.33)
So the combination√−hhαβ is invariant4. Note that a Weyl transformation only
acts on hαβ, not on Xµ or ∂α. This is in contrast with the reparametrizations which
also transform ∂α. The Weyl transformation (2.32) is characterized by 1 local pa-
rameter. Thus, the different groups of transformations (i)-(iii) involve altogether 2+1
=3 local parameters. Recall that the Polyakov action involves 3 auxiliary local vari-
ables hαβ(τ, σ). We can use the symmetry transformations (ii)-(iii) to remove the 3
auxiliary variables and gauge fix hαβ.
2.3.1 Gauge fixing
1) First, any 2-dimensional metric can be related to a flat (Minkowski) metric by
a suitable coordinate transformation (reparametrization, symmetry (ii)), up to an
overall local scale factor. In other words, using (ii) we can write
(hαβ) = Λ(τ, σ)
( −1 0
0 1
). (2.34)
This is called the conformal gauge. For a proof, see e.g. Nakahara.
2) Second, we can perform a Weyl transformation (iii) to remove the overall scale
factor Λ(τ, σ). Then,
(hαβ) =
( −1 0
0 1
)≡ (ηαβ) . (2.35)
4Note that in general in n dimensions this is not true:√−hhαβ → Λn/2Λ−1
√−hhαβ .
12
This is called the covariant gauge. Thus, we have fixed the gauge by specifying to a
worldsheet coordinate system where hαβ takes the form of a flat Minkowski metric
ηαβ.
2.4 Conformal invariance:
Actually, the above procedure does not fix the gauge completely. In other words, we
did not completely specify the worldsheet coordinates by demanding (2.35). There
is a class of coordinates where (2.35) continues to hold, and these are related by
conformal transformations followed by Weyl transformations. Recall that a conformal
transformation is a special case of a reparametrization which satisfies
hαβ = ηαβ → hαβ = hγδ(σ)dσγ
dσα
dσδ
dσβ= Λ(σα)hαβ . (2.36)
Then, we can perform a Weyl transformation which cancels the scale factor:
hαβ → ˜hαβ = Λ−1(σα)hαβ = ηαβ (2.37)
to still stay in the covariant gauge hαβ = ηαβ. The combinations of (2.36) and (2.37)
σα → σα(σβ)
ηαβ → ηαβ(2.38)
are the residual gauge transformations.
It is useful to introduce light-cone coordinates on the worldsheet:
σ+ = σ0 + σ1 = τ + σ
σ− = σ0 − σ1 = τ − σ. (2.39)
Then
∂± ≡ ∂
∂σ±=
1
2(∂τ ± ∂σ) , (2.40)
and the flat metric becomes
ds2 = ηαβdσαdσβ = −dσ+dσ− (2.41)
so
(η++ η+−η−+ η−−
)=
(0 −1/2
−1/2 0
). (2.42)
Then the conformal transformations are reparametrizations
σ+ → σ+(σ+)
σ− → σ−(σ−)(2.43)
13
They satisfy
dσ+dσ− = σ′+σ′−dσ+dσ− ≡ Λ(σα)dσ+dσ− . (2.44)
[There is some abuse of language in the literature, sometimes the term “conformal
transformation” means the residual gauge transformation (2.38) which includes a
Weyl transformation.]
When we plug the covariant gauge worldsheet metric hαβ = ηαβ into the Polyakov
action (2.20), it takes a very simple form:
Sp = −T
2
∫d2σ(∂τX
µ∂τXν − ∂σX
µ∂σXν)ηµν . (2.45)
This looks like an action for D free massless scalar fields Xµ(τ, σ). However, X0 comes
with a “wrong sign” because of the Minkowski signature ηµν = (−++++...+). Since
hαβ has been gauged away, we have to remember the constraint
Tαβ = 0 (2.46)
which followed from the equation of motion. In light-cone coordinates (2.172), the
gauge fixed action is
Sp = 2T
∫d2σ∂+Xµ∂−Xνηµν (2.47)
and the constraint (2.46) is replaced by
T±± = 0 (2.48)
T+− = 0 . (2.49)
Actually, (2.49) is not a constraint. Recall that Tαβ is traceless, ηαβTαβ = 0. In an
exercise you will show that the tracelessness follows from the Weyl invariance (under
(2.32)) of the action. In light-cone coordinates the tracelessness reads
2η+−T+− = 0 ⇒ T+− = 0 . (2.50)
So (2.49) reflects the Weyl invariance. The two equations (2.48) are real constraint
equations. They are called the Virasoro constraints.
2.5 Equations of motion and boundary conditions
Consider now the field variation Xµ → Xµ + δXµ. The variation of the action (2.45)
is (integrating by parts)
14
δSP = T
∫ +∞
−∞dτ
∫ π or 2π
0
dσ[(∂2σ − ∂2
τ )Xµ]δXµ
−T
∫dτ [(∂σX
µ)δXµ] |σ=π or 2πσ=0
+T
∫dσ[(∂τX
µ)δXµ] |τ=+∞τ=−∞ (2.51)
where XµδXµ ≡ XµδXνηµν .
As usual, we take δXµ = 0 at τ = ±∞ so the last term = 0. The second term in
(2.51) is a surface term. We need to impose boundary conditions for Xµ.
1) For a closed string, we needed a periodicity condition Xµ(τ, σ) = Xµ(τ, σ + 2π)
(see after eqn (2.10)), so ∂τXµ(τ, 0) = ∂τX
µ(τ, 2π).
The variations δXµ must also be periodic in σ → σ + 2π. Thus the second term
in (2.51) vanishes (δXµ(τ, 2π)− δXµ(τ, 0) = 0).
2) For an open string we need a different boundary condition. Now the second
term in (2.51) is −T∂σXµ(τ, σ = π)δXµ(τ, σ = π)− ∂σX
µ(τ, σ = 0)δXµ(τ, σ = 0)This vanishes, if we impose the boundary conditions
∂σXµ(τ, σ = 0) = ∂σX
µ(τ, σ = π) = 0 . (2.52)
These are called Neumann boundary conditions. The end points of the open string
are free to vibrate:
Figure 4: Neumann boundary conditions for open string.
(There are other possible alternative open string boundary conditions, namely
keeping one or both endpoints fixed: δXµ = 0. These Dirichlet boundary conditions
will be discussed in the end of the course: they are associated with the existence of
other extended objects called D-branes in string theory.)
Then, what remains in (2.51) is the first term. Setting δSP = 0 gives the field
equations
[∂2τ − ∂2
σ]Xµ(τ, σ) = 0 , (2.53)
i.e., ¤Xµ = 0 where ¤ = ηαβ∂α∂β is the d’Alembertian. The eqn (2.53) is the good
old wave equation for a massless scalar field Xµ in 1+1 dimensions. Of course we
15
must remember the constraints Tαβ = 0. They are now written as
T00 = T11 =1
4(∂τX
µ∂τXµ + ∂σXµ∂σXµ) = 0 (2.54)
T01 = T10 =1
2∂τX
µ∂σXµ = 0 (2.55)
The tracelessness is
hαβTαβ = −T00 + T11 = 0 (2.56)
2.6 Mode expansion and quantization
Let us first consider the closed string. A general solution to the wave equation has
the schematic structure Xµ(τ, σ) = xµ + aµτ + bµσ+[superpositions of plane waves].
The periodic b.c. kills the term linear in σ. The general solutions consistent with the
periodic boundary condition can be written as
Xµ(τ, σ) = xµ +l2s2
pµτ + ils2
∑
n 6=0
1
nαµ
ne−in(τ−σ) +1
nαµ
ne−in(τ+σ)
. (2.57)
Since Xµ has the dimension of length, we have included the dimensional parameter
ls, the string length (2.20). The factors 1n
have been introduced by convention and
convenience into Fourier coefficients 1nαµ
n, 1nαµ
n .
An alternative way to write the general solution is to use the light-cone coordinates
σ±. The wave equation reads now
∂+∂−Xµ = 0 (2.58)
and the solution decomposes into
Xµ = XµL(σ−) + Xµ
R(σ+) , (2.59)
a superposition of a leftmoving wave XµL which depends only on σ− = τ − σ and a
rightmoving wave XµR which depends only on σ+ = τ + σ. Schematically:
The left and right movers can be expanded as
XµL(σ−) =
1
2xµ +
l2s2
pµLσ− +
∑
n 6=0
ilsn
αµne−inσ−
XµR(σ+) =
1
2xµ +
l2s2
pµRσ+ +
∑
n 6=0
ilsn
αµne−inσ+
, (2.60)
and the periodic b.c. requires
pµL = pµ
R ≡1
2pµ . (2.61)
16
LX
X R
Figure 5: Left- and rightmoving excitations on closed string.
Quantization. So far we discussed classical features. We now proceed to quantize
the fields Xµ using the standard canonical quantization procedure. We promote the
fields Xµ to operators, and impose canonical commutation relations. First, we need
to find the canonical momentum Pµ(τ, σ) conjugate to Xµ. Following the standard
definition of Pµ:
Pµ(τ, σ) =δL
δXµ= TXµ(τ, σ) . (2.62)
Then, we interpret Pµ(τ, σ) and Xµ(τ, σ) as Heisenberg operators, and impose the
equal time τ canonical commutation relations
[P µ(τ, σ), P ν(τ, σ′)] = [Xµ(τ, σ), Xν(τ, σ′)] = 0 (2.63)
and
[P µ(τ, σ), Xν(τ, σ′)] = T [Xµ(τ, σ), Xν(τ, σ′)] = −iηµνδ(σ − σ′) . (2.64)
In an exercise you will show that after substituting the mode expansion (2.57) into
(2.63) and (2.64), you recover the commutation relations
[pµ, xν ] = −iηµν (2.65)
[αµm, αν
n] = mηµνδm+n,0 (2.66)
[αµm, αν
n] = mηµνδm+n,0 (2.67)
17
for the mode coefficients. (Note: Bailin and Love have opposite signs for (2.65)-(2.67)
since they use ηµν = (+,−,−,−,−, ...,−).) We can make the following interpreta-
tions:
• xµ = (target space) center-of-mass coordinate of the string.
• pµ = (target space) center-of-mass momentum of the string. This can be justi-
fied as follows. To obtain the total momentum, integrate the momentum density
along the string: P µ ≡ ∫dσP µ(τ, σ)
(2.62)= T
∫ 2π
0Xµdσ = 2πT 1
2l2sp
µ = pµ.
• αµm, αµ
m create and annihilate oscillation degrees of freedom of the string.
Since Xµ is a Hermitean operator, (Xµ)† = Xµ, the center-of-mass coordinate xµ and
momentum pµ are also Hermitean. Furthermore, the oscillator coefficients αµm, αµ
m
must satisfy the following relations:
(αµm)† = αµ
−m (2.68)
(αµm)† = αµ
−m . (2.69)
Therefore, we can rescale them and denote
aµm ≡ 1√
| m |αµm ; aµ
m ≡ 1√| m | α
µm . (2.70)
Now the commutation relations (2.66), (2.67) become
[aµm, (aν
n)†] = ηµνδm,n (2.71)
[aµm, (aν
n)†] = ηµνδm,n (2.72)
These are the standard commutation relations for harmonic oscillator creation and
annihilation operators! Note however that because of ηµν , the µ = ν = 0 components
have a wrong sign. Let us define a vacuum state |0〉:
aµn|0〉 = 0 ∀ n > 0 . (2.73)
We can then create oscillation modes for the string by acting with a creation operator
(aνn)† = aν
−n. However, before moving forward, we make the following two observa-
tions:
18
1) Even if the string is not oscillating, it is moving in target space with some
center-of-mass momentum kµ. We need to take this into account in the definition of
the vacuum. So a more accurate notation for it is |0, kµ〉, with
pµ|0, kµ〉 = kµ|0, kµ〉 (2.74)
The vacuum |0, kµ〉 is an eigenstate of the c.o.m. momentum operator pµ which ap-
pears in the expansion of the field operators Xµ: Xµ(τ, σ) = xµ + 12l2sp
µτ + ....
2) Because of [a0n, (a
0m)†] = η00δn,m = −δn,m, the spectrum contains states with a
negative squared norm: consider e.g. (a0m)† | 0, kµ〉:
|| (a0m)† | 0, kµ〉 ||2= 〈0, kµ | a0
m(a0)†m | 0, kµ〉 = −1 .
Such states, called ghosts, are unphysical and we need to find a way to exclude them
from the spectrum. For this we need the Virasoro constraints.5
***** END OF LECTURE 2 *****
2.7 Constraints
We have seen that if we describe string dynamics with the Polyakov action, we had to
include (the Virasoro) constraints. When quantizing constrained systems, one has two
natural alternatives to proceed. Either one can first quantize all degrees of freedom,
and then apply the constraints to extract out the real physical states. Or, one can
first solve the constraints and find the real classical degrees of freedom, and then
quantize only those. For the string, we have been following the first road, called the
covariant quantization. So now we must take into account the Virasoro constraints.
In light-cone coordinates, they were (see (2.48))
T±±(σ±) =1
2∂±Xµ∂±Xµ = 0 . (2.75)
As with the Xµs, we will use Fourier expansions:
T−−(σ−) =l2s4
∞∑n=−∞
Lne−inσ− (2.76)
T++(σ+) =l2s4
∞∑n=−∞
Lne−inσ+
(2.77)
5An analogous situation exists in Quantum Electrodynamics: the timelike polarization of a pho-ton gives rise to ghost states, but they can be excluded from the spectrum by applying the GaussLaw constraint.
19
where
Ln =2
πl2s
∫ 2π
0
dσein(σ−)T−−(σ−) |τ=0 (2.78)
etc. If you substitute (2.60) and (2.75) into (2.78) (exercise)6, you can derive the
expressions for Ln s in terms of the oscillator coefficients αµn. The results are
Ln =1
2
∞∑m=−∞
αµn−mαν
mηµν (2.79)
Ln =1
2
∞∑m=−∞
αµn−mαν
mηµν (2.80)
where we have used a notation
αµ0 ≡
1
2lsp
µ ≡ αµ0 (2.81)
to express the results in a neat form. The Hamiltonian density H of the string is
obtained by a Legendre transformation:
H = PµXµ − L (2.45),(2.62)
=T
2XµXµ + X ′µX ′
µ (2.82)
(where . ≡ ∂τ ,′ ≡ ∂σ). Comparing with (2.54), you see that
H = 2T · T00 = 2TT++ + T−− . (2.83)
If we integrate H along the string, we obtain the Hamiltonian H:
H =
∫ 2π
0
dσH =2
πl2S∑
n
l2S4
Ln
∫dσe−in(τ−σ) + (2.84)
∑n
l2S4
Ln
∫dσe−in(τ+σ) = L0 + L0 . (2.85)
Substituting (2.79) and (2.80):
H =1
2
∞∑m=−∞
(αµ−mαµm + αµ
−mαµm) . (2.86)
6
T++ =12(T00 + T11)
T−− =12(T00 − T11)
20
Recall that the αµms with m < 0 were creation operators. It is standard to express
the Hamiltonian in the form where all the creation operators have been commuted
to the left of all the annihilation operators. Using the commutation relations (2.66),
(2.67) will then result to a constant term which is an infinite series. We write
H =1
2α2
0 +1
2α2
0 +∞∑
n=1
(αµ−nαµn + αµ
−nαµn)− 2a (2.87)
where −2a denotes the series. Formally, it is an infinite sum. In quantum field
theory, this is known as the zero-point or vacuum energy. Without gravity, this
would not be a problem, because experiments measure differences in energy (with
respect to vacuum). So there one could set a = 0. However, here a cannot be chosen
freely. In string theory, it is fixed by the requirement that unphysical states are
removed. We will return to this issue. The Virasoro constraints at classical level were
T++ = T−− = 0. At quantum level, these are replaced by conditions on expectation
values on the physical states:
〈phys|T++|phys〉 = 〈phys|T−−|phys〉 = 0 . (2.88)
This implies (using (2.76)):
Lm|phys〉 = Lm|phys〉 = 0 ∀n > 0 . (2.89)
For L0, L0, we have to be a bit more careful, because of the zero-point energy. First,
we redefine L0, L0 to denote only the normal ordered parts of the oscillator expansion:7
L0 =1
2αµ
0αµ0 +1
2:∑
n 6=0
αµ−nαµn :
=1
2αµ
0αµ0 +∞∑
n=1
αµ−nαµn
L0 =1
2αµ
0 αµ0 +∞∑
n=1
αµ−nαµn
(2.90)
Then,
H = L0 + L0 − 2a . (2.91)
7For Ln, n 6= 0, Ln =: Ln : trivially.
21
The constraint 〈T++〉 = 〈T−−〉 yields the level matching condition
L0|phys〉 = L0|phys〉 (2.92)
and the vanishing of 〈H〉 ∼ 〈T++〉+ 〈T−−〉 = 0 means that in total we get
(L0 − a)|phys〉 = (L0 − a)|phys〉 = 0
(2.93)
We can now check if the vacuum, as it was defined in (2.73), is a physical state:
Lm|0, kµ〉 = 0 for m > 0, but
(L0 − a)|0, kµ〉 = (1
2αµ
0αµ0 + 0− a)|0, kµ〉 (2.94)
= (l2s8
p2 − a)|0, kµ >= 0 (2.95)
This is satisfied if k2 = 8a/l2s . We will interpret this later. The operators Ln have a
special importance, and they have a name: they are called Virasoro operators. Using
the commutation relations (2.65)-(2.67), and the oscillator expansions (2.79),(2.80),
(2.90) we can derive commutation relations for the Lns. For n+m 6= 0, it is straight-
forward to derive
[Ln, Lm] = (n−m)Ln+m (n + m 6= 0) (2.96)
For the case n + m = 0, it is most convenient to first write an ansatz
[Ln, Lm] = (n−m)Ln+m + b(n)δn+m,0 (2.97)
where b(n) denotes the expected additional contribution, arising from the infinite
constant contributions in the normal ordering. The b(n) can then be evaluated by
considering the expectation value
b(n) = 〈0|[Ln, Ln]|0〉 = (exercise . . .) =D
12n(n2 − 1) . (2.98)
(See Bailin and Love.) So, all told, the commutation relations for Lns are
[Ln, Lm] = (n−m)Ln+m +D
12n(n2 − 1)δn+m,0
(2.99)
22
This is called the Virasoro algebra.
The Lms have similar commutation relations. Moreover, Ln, Lm commute with
each other:
[Ln, Lm] = 0 ∀n,m. (2.100)
So the closed string contains two Virasoro algebras, one for the left and one for the
right movers. Now let us return to the conditions (2.93). They give a mass formula
for the string excitations. The string moves through the target space with c.o.m
momentum kµ. Then, as seen in the target space, the string has a rest mass M , with
M2 = −kµkµ . (2.101)
From the point of view of the worldsheet, the energy of a string was a constraint
H|phys〉 = 0. That means
(L0 + L0 − 2a)|phys〉 = 0 . (2.102)
This gives a relation between the target space rest mass, c.o.m momentum, and
oscillations of the string. Recall:
L0 =1
2α2
0 +∞∑
n=1
α−nαn ≡ l2s8
p2 + NL (2.103)
and similarly L0 =l2S8p2 +NR. Then, for a physical state with momentum kµ, (2.102)
becomes
l2S4
k2 + NL + NR − 2a
|phys〉 = 0 . (2.104)
So we get the mass formula
M2 =4
l2S(NL + NR)− 8a
l2S(2.105)
where NL =∑∞
n=1 αµ−nαµn, NR =
∑∞n=1 αµ
−nαµn are called the level numbers. For a
physical state, level matching condition (2.92) requires NL = NR. The interpretation
is that different excitations of string will correspond to different sorts of particles in
the target space, with the rest mass given by (2.105). So far we have carried along
two parameters D, a, and we have not checked that the constraints (2.89) really
23
remove the unphysical states with negative squared norm, called the ghosts, from the
spectrum.
The proof of this “no-ghost theorem” is technical, and I will skip it. The idea is to
check that ghosts have a vanishing overlap with any physical state: 〈phys|ghosts〉 = 0.
Then they form a subspace of the Fock space which is orthogonal to that of physical
states:
H = (αµ1
−1)i1 · · · (αµn
−n)in(αν1−1)
j1 · · · (ανm−m)jm|0, kµ〉
= |phys〉 ⊕ |ghost〉 (2.106)
and hence can be ignored. However, this is possible only when D = 26 and a = 1 or
D ≤ 25 and a < 1. In the latter case the ghosts will show up again at one-loop level
as unphysical poles in string scattering amplitudes. Therefore we will focus on the
first case, critical string theory. The latter case is called the non-critical (bosonic)
string theory. In our case, the string then propagates in 26 spacetime dimensions!
2.8 Low-lying string states.
With a = 1, the mass formula (2.105) becomes M2 = 4l2S
(NL +NR)− 8l2S
. Let us check
the lowest mass states in the spectrum. Recall that the level matching condition
requires NL = NR.
1)NL = NR = 0 : This is just the vacuum |0, kµ〉 with M2 = −k2 = −8/l2S. This
was required for the vacuum to be a physical state. Thus, from the target space point
of view, a string which is not oscillating corresponds to a particle with a negative rest
mass. It is called the tachyon.
2) NL = NR = 1 : This is the first excited level,
αµ−1α
ν−1|0, kµ〉 (2.107)
with M2 = 4l2S
(1 + 1)− 8l2S
= 0. Any D×D matrix Aµν = αµ−1α
ν−1 can be decomposed
into three parts as follows:
Aµν =1
2(Aµν + Aνµ)− 2trA
Dηνµ
︸ ︷︷ ︸≡Gµν
symmetric traceless
+1
2(Aµν − Aνµ)
︸ ︷︷ ︸≡Bµν
antisymmetric traceless
+2trA
Dηµν trace (2.108)
These massless particles correspond to the graviton, the antisymmetric tensor, and
the dilaton.
24
3) NL = NR = 2 :
αµ1
−1αµ2
−1αν1−1α
ν2−1|0, kµ〉
αµ−2α
ν−2|0, kµ〉 (2.109)
These have M2 = 8l2S
. They correspond to very massive particles (M2 ∼ m2Pl).
Polarizations: Let us consider the effect of the constraints. Consider again the
massless level NL = NR = 1. We introduce a polarization tensor εµν(k) and write the
states as
εµν(k)αµ1
−1αµ2
−1|0, kµ〉 . (2.110)
[Compare with QED: quantize the U(1) gauge field Aµ and introduce a polarization
vector εµ(k) for the photon state εµ(k)aµk |0〉.] The momentum vector satisfies
k2 = −M2 = 0 . (2.111)
The constraint
L1εµναµ−1α
ν−1|0, kµ〉 = 0 (2.112)
implies:
1
2α−1α2 + α0α1 + α1α0 + α2α1 + · · · εα−1α−1|0〉
= εµνηγδαγ1α
δ0α
µ−1α
ν−1|0, kµ〉
=ls2
εµνηγδkδαν
−1αγ1α
µ−1|0, kµ〉
=ls2
εµνkµαν
−1|0, kµ〉 = 0 , (2.113)
where αγ1α
µ−1 = [αγ
1 , αµ−1] = ηγµ. So the polarization tensor and the momentum satisfy
k · ε ≡ kµεµν = 0 . (2.114)
[I suppressed the ν index on the RHS.]
Similarly, L1εµναµ−1α
ν−1|0, kµ〉 = 0 yields
ε · k = εµνkν = 0 . (2.115)
Let us choose a frame where (kµ) = (k0, k1, 0, . . . , 0). Then (2.111) requires k0 = k1 ≡k. Suppose that (2.111) corresponds to a graviton, then εµν must be symmetric and
traceless. Then, (2.114) and (2.115) mean that the polarization tensor can be reduced
into the form (after decoupling the longitudinal polarizations εµνkµkν) .
(εµν) =
0 0 0 0
0 0 εii εij
0 0 εji εjj
(2.116)
25
where ε is a symmetric, traceless (D−2)×(D−2) matrix. This is a higher dimensional
analogue of the transverse and traceless gauge for the graviton in four spacetime
dimensions. The polarization can be chosen to be transverse to the direction of
propagation. The number of true physical degrees of freedom is manifestly
(D − 2)(D − 1)
2− 1 .
[The QED analogue is, a photon has momentum kµ = (k0, k1, 0, . . . , 0). The con-
straint ∂µAµ = 0 gives the condition eµk
µ = 0 for the polarization vector. So the
photon has only two physical degrees of freedom, corresponding to the two transverse
polarizations.] So, we have seen that the constraints do reduce the number of degrees
of freedom to the real physical one. Now we will discuss an alternative to the covari-
ant quantization. We first solve the constraints and find the real physical degrees of
freedom, and quantize only them.
***** END OF LECTURE 3 *****
2.9 The light-cone gauge
Recall that the covariant gauge hαβ = ηαβ still allowed the freedom of residual gauge
transformations (2.38), which were combinations of conformal transformations (2.36)
and Weyl transformations (2.37). Such transformation
σ± → σ±(σ±) (2.117)
means in a Cartesian frame τ = (σ+ + σ−)/2, σ = (σ+ − σ−)/2 that the new time is
a superposition of an arbitrary function of σ+ and an arbitrary function of σ−:
τ = f(σ+) + g(σ−) . (2.118)
Hence it satisfies the 1 + 1 dimensional wave equation
(∂2τ − ∂2
σ)τ = 0 . (2.119)
Since all Xµ satisfy the same wave equation, we could pick one of them and set it to
be equal to τ by a suitable transformation (2.117). It would be natural to pick X0:
X0 = τ . (2.120)
This choice is sometimes used and known as the static gauge. However, for our
purposes there is a more useful pick. Define the light cone coordinates X± for the
string,
X± ≡ 1√2(X0 ±XD−1) . (2.121)
26
Since X± also satisfy the wave equation, we can pick e.g. X+ to be proportional to
τ (I drop the tilde now)
X+(τ, σ) = x+ +1
2πTp+τ , (2.122)
where we introduced the constants x+, p+ for convenience. This choice is called the
light-cone gauge. Recall the constraints (2.54):
T00 = T11 =1
2(∂τX
µ∂τXµ + ∂σXµ∂σXµ) = 0 (2.123)
T01 = T10 = ∂τXµ∂σXµ = 0 . (2.124)
In light-cone coordinates XµYµ = −X+Y − − X−Y + + X iY i. The equation (2.122)
becomes
1
2πTp+∂τX
− = ∂τX+∂τX
− + ∂σX+
︸ ︷︷ ︸=0
∂σX− =
1
2
∑i
(∂τXi)2 + (∂σX
i)2 (2.125)
and (2.123) becomes
1
2πTp+∂σX
− = ∂τX+∂σX
− + ∂τX− ∂σX
+
︸ ︷︷ ︸=0
=∑
i
∂τXi∂σX
i . (2.126)
Thus, if we substitute the mode solution for X i:
X i(τ, σ) = xi +l2s2
piτ + ils2
∑
n 6=0
1
nαi
ne−in(σ−) +
1
nαi
ne−in(σ+) ,
we can use (2.125) and (2.126) to solve for X− in terms of αins, pi, p+ and an integra-
tion constant x−.
So, in the light-cone gauge (2.122) we can solve the constraints, and find that the
real degrees of freedom are the X i (or the αin, α
in, x
i, pi)! Now recall that previously
we would have written X− as
X− = x− +l2s2
p−τ + · · · , (2.127)
but we can use (2.125) again to solve for p−. Again, commuting αi−ns to the left of
αins introduces a zero point contribution a . We find
2p+p− =∑
i
(pi)2 +4
l2s
[∑i
∞∑n=1
(αi−nα
in + αi
−nαin)− 2a
]. (2.128)
Then, the mass shell condition M2 = −p2 = 2p+p− − pipi becomes
27
M2 =4
l2S
[∑i
∞∑n=1
(αi−nαi
n + αi−nα
in)− 2a
].
(2.129)
Since we have solved the constraints, the Fock space now contains only physical states:
HLC = (αi1−1)
P1 · · · (αin−n)Pn(αi1
−1)q1 · · · (αjn
−n)qn|0, kµ > . (2.130)
Since we have already solved the constraints, now we need a different condition to de-
cide what are the allowed values of a and D. In solving the constraints we paid the fol-
lowing price. In the light cone gauge we have lost the manifest D-dimensional Lorentz
invariance of the target space. If we construct the generators for D-dimensional target
space Lorentz transformations and demand that they do satisfy the correct commuta-
tion relations of SO(1, D) (the D-dim. Lorentz algebra), we find that this only works
for
D = 26, a = 1 . (2.131)
In this case one can also check that the previous space of physical states is the same
as the state space in the light-cone gauge,
Hphys = |phys〉 | covariant gauge = HLC .
2.10 Lowest lying states
In the light-cone gauge, the lowest lying string excitations are again
1) NL = NR = 0: |0, kµ〉, tachyon
2) NL = NR = 1:
1
2(αi
−1αj−1 + αj
−1αi−1)−
2δklαk−1α
l−1
D − 2|0, kµ〉 graviton (2.132)
1
2(αi
−1αj−1 − αj
−1αi−1)|0, kµ〉 antisymm.tensor (2.133)
2δklαk−1α
l−1
D − 2δij|0, kµ〉 dilaton . (2.134)
Now the graviton has explicitly (D−2)(D−1)2
− 1 degrees of freedom.
28
2.11 Open strings
So far we have focused on the closed string. For open strings we can proceed in a
similar fashion. The difference is that the boundary conditions glue the left- and
rightmoving waves together to standing waves. Thus, in the covariant gauge
Xµ = xµ + l2spµτ + ils
∑
n 6=0
1
nαµ
ne−inτ cos(nτ) .
The quantization again yields
[pµ, xν ] = −iηµν
[αµm, αν
n] = mηµνδm+n,0 . (2.135)
There is only one set of oscillator coefficients αµn. The Virasoro generators Ln are
defined by
Ln = 2T
∫ π
0
dσein(τ+σ)T++ + ein(τ−σ)T−− . (2.136)
For n 6= 0 they are
Ln =1
2
∞∑m=∞
αµn−mαµm (2.137)
where we have now defined
αµ0 = lsp
µ . (2.138)
Note that (2.138) differs by a factor of 2 from the definition in the closed string case.
The L0 generator is again defined to be normal ordered:
L0 =1
2:
∞∑m=∞
αµ−mαµm :=
1
2αµ
0αµ0 +∞∑
m=∞αµ−mαµm .
29
The Hamiltonian is
H = L0 − a . (2.139)
and the physical state conditions are
Lm|phys〉 = 0 ∀ m > 0 (2.140a)
(L0 − a)|phys〉 = 0 . (2.140b)
Now there is no level matching condition. The no-ghost theorem again requires
D = 26, a = 1. The mass shell condition from (2.140a) becomes
M2 =2
l2S
∞∑n=1
αµ−nαµn − 2
l2Sa
(2.141)
The Virasoro generators (2.137), (2.139) satisfy the same Virasoro algebra commuta-
tion relations as in (2.99). The lowest lying states are
1) N = 0: the vacuum |0, kµ〉 with M2 = − 2l2S
. So it is again a tachyon.
2) N = 1: the first excited states eµαµ−1|0, kµ〉 are massless: M2 = 0. This corre-
sponds to a massless gauge particle. The physical state condition L1 = 0 gives the
polarization condition e · k = 0. So the physical degrees of freedom are (D − 2)
transverse polarizations, just like in QED. In the light cone gauge
M2 =2
l2S
( ∞∑n=1
∑i
αı−nαi
n − a
)(2.142)
and the real degrees of freedom are explicit.
2.12 Path integral quantization of the bosonic string
So far we have been discussing the “old fashioned” canonical quantization approach.
A more “modern” approach to quantize is via a path integral. In QFT, the idea is
to account for quantum fluctuations by considering all possible field configurations
30
(not only solutions of classical equations of motion) and weighting them by their
contribution to the action. The sum is the path integral,
Z =
∫Dφe
i~S(φ) . (2.143)
In string theory we sum over all possible worldsheets and their embeddings, so integral
runs over all worldsheet metrics hαβ and all embeddings Xµ:
Z =
∫DhDXeiSP (h,X) , (2.144)
where SP is the Polyakov action. The path integral gives vacuum-to-vacuum ampli-
tude, so in closed string theory that means that we are summing over all possible two
dimensional Riemann surfaces
...
...
...
...
Figure 6: Two-dimensional surfaces.
Figure 6 depicts that surfaces are not only deformed in shape, but one can also add
holes. The latter correspond to loop corrections. Note that in contrast to field theory,
the action is still a free theory. So the string interactions are introduced by different
surfaces, instead of adding nonlinear terms to the action! It turns out that there
are some important terms we can add into the Polyakov action, if we are considering
strings moving in more generic backgrounds than just an empty flat Minkowski space.
One important addition is
S = SP + λχ (2.145)
with (in the Euclidean signature8)
χ =1
4π
∫d2σ
√hR , (2.146)
8As usual, the path integral is best defined in the Euclidean signature and one then has tocontinue back to Minkowski signature.
31
where R is the Ricci scalar curvature of the worldsheet metric. Note that χ is a
two-dimensional version of the Einstein-Hilbert action. However, in two spacetime
dimensions χ does not really depend on the metric – it only depends on the topology
of the surface. The quantity χ is a topological invariant, the Euler number, equal to
2(1−g) (for manifolds without boundaries) where g is the number of holes or handles
(0 for sphere, 1 for torus, etc.). The factor λ looks like an arbitrary parameter. Yet
I said that in string theory theres only one, the string tension or length. In fact λ
depends on the background and is thought to be set dynamically. I will get back to
this later. The importance of the term λχ is that (in Euclidean continuation), the
path integral has the factor
e−SP−λχ = e−SP e−2λe2gλ . (2.147)
Thus, increasing the genus g by one, by adding a “handle”, the path integral picks
up an additional factor of e2λ. Now think of the addition of a handle as a sequence of
closed string interactions – a closed string is first emitted, then propagates along the
handle, and then is reabsorbed. The emission and absorption should be characterized
by the strength of the closed string interactions. In other words, they should be
associated with one power of a closed string coupling constant gc. Thus, adding a
handle corresponds to adding two powers of string coupling constant, and we are thus
lead to identify
g2c ≡ e2λ . (2.148)
For open strings the path integral is defined in a similar manner, only now the
surfaces must have a different topology. Recall that the (tree level) open string world
sheet looked like a strip of width π, so it looks like there are two boundaries associated
with the two open strin endpoints. However, taking into account the point at infinity,
the infinite strip has only one boundary and it can be conformally mapped to the
unit disk on a complex plane. For an open string, adding loops in the path integral
then corresponds to adding holes into surfaces that are topologically like the unit
disk. In other words, adding loops corresponds to adding boundaries. For example,
at one loop level the surfaces are topologically equivalent to the annulus, which has
two boundaries.
A more general formula for the Euler number, which also counts boundaries, is
χ = 2− 2g − b− c. (2.149)
Here b counts the number of boundaries, eg. b = 2 for the annulus, and c counts
something called crosscaps, you can forget that for now and consider c = 0. Now
you can see that adding a boundary to the open worldsheet introduces a factor eλ.
Thinking of this as an emission and reabsorption of an open string, we are lead to
identify
g2o = eλ , (2.150)
32
where go is the open string coupling constant. Alltogether then
g2o = gc = eλ . (2.151)
Gauge fixing. Recall that the action SP was invariant under worldsheet repa-
rameterizations (diffeomorphisms) and Weyl transformations. Denote the group of
such transformations by Diff × Weyl. Then the integral of surfaces induces a huge
overcounting: all surfaces (or hαβ and Xµs) related by Diff × Weyl are counted re-
dundantly, since they all contribute equally (to SP ). This is just like in gauge field
theory: if the action is invariant under a gauge group G (say, SU(2) of a non-abelian
field Aaµ), all gauge field configurations Aa
µ which are equivalent by gauge transfor-
mations contribute equally. So we have to remove the overcounting from the path
integral. The standard way is to use the Faddeev-Popov method. Let ξ symbolize a
Diff ×Weyl transformation:
h 7→ hξ : hξαβ(σ) = eρ(σ) ∂σγ
∂σα
∂σδ
∂σβhγδ(σ) . (2.152)
In particular, hαβ could be the worldsheet metric in the covariant gauge: (ηαβ) =
diag(−1, 1) and hξαβ anything else. The F-P trick is to insert a special “1” into the
path integral:
1 =
∫dξδ[F (hξ)] det
(δF (hξ)
δξ
)|ξ=0 (2.153)
where
F (hξ) ≡ hξαβ − hαβ . (2.154)
Let me denote det( δF (hξ)δξ
) |ξ=0≡ ∆(h). Insert (2.153) into the path integral (2.143):
Z =
∫dξ
∫DhDXδ(hξ − h)∆(h)eiS(X,h)
R Dh=
∫dξ
∫DX∆(hξ)eiS(X,hξ) (2.155)
Since the action is invariant under Diff ×Weyl, S(X, hξ) = S(X, η). One can show
that also ∆(hξ) is invariant, so
∆(hξ) = ∆(hξ′) = ∆(η) (2.156a)
Z =
∫dξ
∫DX∆(η)eiS(X,η)
= V ol(Diff ×Weyl)︸ ︷︷ ︸Rdξ
∫DX∆(η)eiS(X,η) . (2.156b)
33
The integral∫
dξ over Diff × Weyl transformations was trivial and just gave the
(infinite) volume of the group. It can be dropped from the path integral by a normal-
ization convention. Now we only need to evaluate the Jacobian ∆(η). I will simplify
this by taking a step back. Let us not fix the Weyl transformations: we only fix to
the covariant gauge h(c)αβ = eρηαβ. That means, we use S(X, hξ) = S(X, h(c)) and
∆(hξ) = ∆(h(c)). then instead of (2.156b),
Z = V ol(Diff)︸ ︷︷ ︸Drop
∫Dρ
∫DX∆(h(c))eiS(X,h(c)) . (2.157)
It is simpler to evaluate ∆(h(c)). Under an infinitesimal Diff transformation:
σα → σα + ξα , (2.158)
the metric transforms:
hαβ → hξαβ = hαβ + δhαβ , (2.159a)
where
δhαβ = −5α ξβ −5βξα (2.159b)
(See e.g. Nakahara for the mathematics or some gravity textbook.) In particular, in
the worldsheet light-cone coordinates,
δhξ++ = −25+ ξ+ (2.160a)
δhξ−− = −25− ξ− . (2.160b)
So
det(δF
δξ) = det(
δF++
δξ+
) det(δF−−δξ−−
) , (2.161)
where
det(δF±±(τ ′, σ′)δξ±(τ, σ)
) = det(25′± δ(τ ′ − τ)(σ′ − σ)) . (2.162)
The determinant can be exponentiated into the action by introducing 2 anticommut-
ing fields b, c callled Faddeev-Popov ghosts:
det(B) =
∫DcDb exp
i
π
∫dτ ′dσ′dτdσ c(τ ′, σ′)B(τ ′, σ′, τ, σ)b(τ, σ)
. (2.163)
So the path integral (??) can be written in the form
Z =
∫DρDXDc+Db++Dc−Db−−ei[SP (hc,X)+Sgh(b,c)] , (2.164)
34
where Sgh is ghost action
Sgh =1
π
∫d2σ(c−5+ b−− + c+ 5− b++) . (2.165)
The conventional labeling (c−, b−− etc.) for the ghost fields may look a bit strange,
it can be motivated by looking at their properties under conformal transformations.
Note: Since SP is invariant under Weyl transformations, the covariant derivatives in
Sgh reduce to ordinary derivatives, so it appears to be independent of ρ:
Sgh =1
π
∫d2σ(c−∂+b−− + c+∂−b++) . (2.166)
So the whole path integral looks independent of ρ. That would mean that Weyl
invariance is a symmetry at quantum as well as classical level9. However, a more
careful investigation of the measure Dρ shows that to be case only in D = 26. For
D 6= 26 Weyl invariance is broken at quantum level (“Weyl anomaly”) and ρ reappears
in the action. It is then called the Liouville field. D 6= 26 is the noncritical bosonic
string.
2.13 Conformal field theory (CFT)
In the discussion of the path integral, we mentioned the string interactions for the
first time. Before proceeding to discuss string interactions in more detail, it is useful
to go through some other issues which may seem slightly abstract at first. We need
to discuss some generic features of conformally invariant 2-dimensional field theories.
Conformal symmetry is a rather powerful feature. It does not appear only in string
theory, but it is also encountered in statistical mechanics and in some condensed
matter systems. For example, in statistical mechanics it arises in systems which have
a second order phase transition, at the critical point. The reason why conformal
invariance is so important in string theory is that it is a gauge symmetry. Often
in gauge theories, gauge transformations are used to eliminate unphysical degrees
of freedom (like the timelike photon in QED). Suppose that the gauge invariance is
then broken at quantum level. Then the unphysical degrees of freedom may return
and spoil the theory. So in general we like to preserve gauge symmetries at quantum
level too. Recall that string theory had reparametrization and Weyl invariance as
local (gauge) symmetries. Going to covariant gauge did not completely fix the gauge
symmetry. We still had the freedom to make conformal transformations combined
with Weyl transformations. We used this residual gauge freedom to go to the light-
cone gauge, where we eliminated all unphysical degrees of freedom and completely
fixed the gauge. Thus, conformal symmetry played a central role in eliminating the
unphysical degrees of freedom. Now suppose that the conformal symmetry is broken
9And it looks like we could do the remaining∫ Dρ functional integral in Z to pick up V ol(Weyl).
35
at quantum level. Then we can anticipate that the unphysical degrees of freedom will
return and string theory will be spoiled. So we want to preserve conformal symmetry
also at quantum level. Now let us move on to discuss conformal field theories. It is
standard and useful to use complex coordinates and Euclidean signature.
Complex coordinates. We continue the worldsheet time coordinate σ0 = τ to
Euclidean time τE ≡ σ2:
σ0 = τ = −iτE ≡ +iσ2 (2.167)
Then the metric on the (cylindrical) worldsheet becomes
ds2 = −(dσ0)2 + (dσ1)2 = (dσ1)2 + (dσ2)2 . (2.168)
The null coordinates σ± become
σ± = τ ± σ = ±(σ ∓ iτE) = ±(σ1 ± iσ2) . (2.169)
So we can introduce complex coordinates
w = σ1 + iσ2; w = σ1 − iσ2 (2.170)
and the Euclidean metric on the cylinder is written as
ds2 = dwdw . (2.171)
The closed string worldsheet, an infinite cylinder, can be mapped to the complex
plane by z = eiw = eτEeiσ
z = e−iw = eτEe−iσ ,
see Figure 7. This is clearly a conformal transformation:
dwdw︸ ︷︷ ︸cylinder
= d(ln z)d(ln z) =1
zzdzdz︸︷︷︸plane
. (2.172)
Since the action was invariant under conformal transformations, we could have equally
well written it using complex plane as a worldsheet (i.e., using complex coordinates
on a plane):
SP =T
2
∫dzdz∂zX∂zX . (2.173)
The solution to the field equation ∂z∂zX = 0 has again a factorizable form: X(z, z) =
XL(z) + XR(z). If a function depends only on z (only on z), we call it holomorphic
36
=const
τ
σ
=const
τ
τ-cylinder
=+ 8
=- 8
w
=constτ
=constσ
=- 8τ
8=+τ (maps to the boundary at infinity)
(origin)
z-p|ane
Figure 7: Mapping of cylinder to the plane. Here τ means the same as τE in the text.
(antiholomorphic). Holomorphic functions can be expanded as a Laurent series, a
series on zn’s. For example
−i∂zXL =∞∑
n=−∞αnz
−n−1 . (2.174)
The coefficients can be computed using the Cauchy formula, e.g.
αn =
∮dz
2πizn(−i)∂XL(z) , (2.175)
where∮
is along a closed contour around the origin. We will see that (2.174), (2.175)
are the complex plane versions of the Fourier series expansion and the Fourier coeffi-
cients. For the rightmovers, the expansion is
+i∂zXR =∞∑
n=−∞αnz−n−1 . (2.176)
37
Now let us investigate conformal transformations. An infinitesimal conformal trans-
formation looks like
z → z + ε(z)
z → z + ε(z) , (2.177)
where the ε(z), (ε(z)) is a small holomorphic (antiholomorphic) function. Laurent
expanding
ε(z) = −∑
n
εnz−n+1; ε(z) = −∑
n
εnz−n+1 (2.178)
and defining the holomorphic and antiholomorphic vector fields
ln(z) = −zn+1∂z;ˆln(z) = −zn+1∂z (2.179)
we can expand a generic holomorphic and antiholomorphic vector field as follows:
ε(z) = ε(z)∂z =∑
n
εnl−n(z)
ˆε(z) = ε(z)∂z =∑
n
εnˆl−n(z) . (2.180)
Therefore, the vector fields (2.179) form a basis in the spaces of holomorphic and
antiholomorphic vector fields. Infinitesimal conformal transformations (2.177) can be
thought to be generated by vector fields:
z → (1 + ε(z))z = z + ε(z)∂zz
z → (1 + ˆε(z))z = z + ˆε(z)∂z z . (2.181)
The basis vectors ln,ˆln satisfy the commutation relations
[ln, lm] = (n−m)ln+m (2.182a)
[ˆln, ˆlm] = (n−m)ˆln+m . (2.182b)
The algebra (2.182a) is called Witt algebra. Since [ln, lm] = 0, the basis vectors
ln, ˆln form two copies of Witt algebras. Witt algebra looks like the Virasoro algebra
(2.99) but without the δn+m,0 term. Hence we can anticipate that Virasoro algebra
is also related to conformal transformations. An important subalgebra of (2.182a) is
generated by l0, l±1. ([l0, l±1] = ∓l±1, [l+1, l−1] = 2l0 so the commutators close within
the subset). They generate the transformations summarized in Table 1.
Combinations of these transformations are conformal transformations of the gen-
eral form
z → az + b
cz + d, (2.183)
38
infinitesimal transf. finite transformation interpretation
l−1 z → z − ε z → z − b translations
l0 z → z − εz z → e−λz scaling
l+1 z → z − εz2 z → z1+cz
special conformal
transformations
Table 1: l0, l±1
with a, b, c, d ∈ C and ad − bc = 1. This is the SL(2, C)/Z2 group. (/Z2 since the
reflection (a, b, c, d) → −(a, b, c, d) does nothing.)
The transformations (2.183) are called global conformal transformations. They
are globally well defined on the Riemann sphere C ∪ ∞ ∼= S2. Another name for
the transformations (2.183) is Mobius transformations. The algebra
[l0, l±] = ∓l±1, [l+1, l−1] = 2l0 (2.184)
is the SL(2, C) Lie algebra. So far we have been discussing conformal transformations
on the worldsheet. Let us then move to discussing conformal field theories on the
worldsheet. The fields (operators, if we talk about a quantized theory) transform in
some way, when we perform a conformal transformation on the worldsheet. There is
a special set of operators that have a particularly simple (and useful) transformation
rule. These are the primary fields/operators. Consider a (not infinitesimal) conformal
transformation
z → z′ = f(z) ⇔ z′ → z = f−1(z′) (2.185)
and
z → z′ = f(z) ⇔ z′ → z = f−1(z′) . (2.186)
Consider an operator φ at point P = (z, z) or (z′, z′). See Figure 8. The operator
φ(z, z)P at point P maps to a transformed operator φ′ at point P , but with P labeled
by the transformed coordinates z′, z′:
φ(z, z)|P → φ′(z′, z′))|P . (2.187)
A primary field /operator of conformal weight (h, h) is an operator such that
φ′(z′, z′)|P =
(dz
dz′
)h
|P
(dz
dz′
)h
|Pφ(z(z′), z(z′))|P . (2.188)
39
P
(z coordinates)
(z’ coordinates)
Figure 8: Point P = (z, z) or (z′, z′).
Example: ∂zXL(z) is a primary field with conformal weight (1, 0):
(∂z′XL)(z′) =
(dz
dz′
)1
(∂zXL)z(z′) (2.189)
(using the chain rule of differentiation). An easy way to remember (2.188) is to think
of primaries as tensor fields:
φ′(z′, z′)(dz′)h ⊗ (dz′)h = φ(z, z)(dz)h ⊗ (dz)h . (2.190)
The expression above is invariant under z → z′, z → z′ provided that φ transforms
as in (2.188). Note: it can be shown that primaries of conformal weight (h, 0) (or
(0, h)) are holomorphic or (antiholomorphic):
φ(h,0)(z, z) = φh(z) . (2.191)
Consider a couple of special transformations:
(i) Scaling z′ = eλz, z′ = eλz:
φ′(z′, z′) = e−λ(h+h)φ(e−λz′, e−λz′) (2.192)
The sum ∆ ≡ (h + h) is the (mass) scaling dimension of the field/operator. For
example, ∂zX(z) has scaling dimension ∆ = 1.
(ii) Rotation of complex plane: z′ = eiθz, z′ = e−iθz:
φ′(z′, z′) = e−iθ(h−h)φ(e−iθz′, eiθz′) . (2.193)
The difference s ≡ h − h is the spin of the operator. [For example, let θ = 2π
so that you do a full rotation. If h = 1/2, h = 0 then s = 1/2, and you can see
40
that the operator is antiperiodic under the rotation.] Earlier I said that the Laurent
expansion is the complex plane analogue of the Fourier expansion. Let’s check this.
Consider for simplicity φ(z), a holomorphic primary field of conformal weight (h, 0).
The convenient way to write its Laurent expansion is
φ(z) =∑
n
φnz−n−h . (2.194)
Now let’s check what this corresponds to on the (Euclidean) cylinder. Using z = eiw,
z = e−iw:
φ(w) =
(dz
dw
)h
φ(z(w))
= (i)heihw∑
n
φne−inw−ihw
= ih∑
n
φne−inw . (2.195)
So the Laurent series (2.194) implies the usual Fourier series expansion (2.195). (check
conventions...?) So the φn are really just the usual Fourier modes.
***** END OF LECTURE 4 *****
The stress tensor. In any two-dimensional field theory we can define the stress
tensor Tµν . In the complex coordinates the components are Tzz, Tzz, Tzz = Tzz
(symmetry). Recall that Weyl invariance implied tracelessness T µµ = 0. In the two
dimensions, Weyl invariance implies conformal invariance and vice versa. So in CFT,
T µµ = ηzzTzz = 0 ⇒ Tzz = Tzz = 0. We also require energy and momentum to be
conserved, so Tµν satisfies the conservation law ∂µTµν = 0. In complex coordinates,
this implies
ηzz∂zTzz = 0 ⇒ Tzz ≡ T (z)
ηzz∂zTzz = 0 ⇒ Tzz ≡ T (z) (2.196)
In other words, in CFT Tzz (Tzz is a holomorphic (antiholomorphic) field, denoted for
short by T (z) (T (z)). So the only two nonvanishing components of the stress tensor
in CFT are T (z) and T (z). Now recall that in field theory, a continuous symmetry
implies the existence of a conserved current (Noethers theorem), let us denote it by
Jµ(X). We can then define an associated conserved charge Q by
Q(X0) =
∫dd~x J0(x0, ~x) (2.197)
by integrating the time component of the current over a space slice at fixed time x0.
The conservation law ∂µJµ = 0 implies the charge conservation, provided that Jµ
41
vanishes sufficiently rapidly at infinity:
∂0Q = −∫
V
ddX∂iJi = lim
r→∞
∫
∂V
dΣiJ i = 0 (2.198)
In our case, consider the infinite cylinder as the spacetime. fixed time means fixed τE
and the space integral is∫
dσ. On a complex plane τE =fixed,∫
dσ becomes∮
dz,
see Fig. 9.
τ fixed
|z| fixed
Figure 9: Constant time slices.
The stress-energy tensor is an example of a conserved current, it satisfies ∂µTµν =
0. In CFT, conformal symmetry also yields conserved currents: consider an infinites-
imal conformal transformation
xµ → xµ + εµ(x) (2.199)
The metric changes by
ηµν → ηµν + ∂µεν + ∂νεµ (2.200)
but on the other hand, we need
ηµν → ηµν + δ(x)ηµν (2.201)
because a conformal transformation just rescales the metric. One can show that (in
2 dimensions, with ηµν = (1, 1)) consistency requires
∂µεν + ∂νεµ = δ(x)ηµν = (ηαβ∂αεβ)ηµν (2.202)
Now, the current associated with (??) is of the form
Jµ = Tµνεν (2.203)
where Tµν is the stress-energy tensor. The current conservation law requires
0 = ∂µJµ = ∂µ (Tµνεν) = (∂µTµν) εν + Tµν∂
µεν =1
2Tµν (∂µεν + ∂νεµ) . (2.204)
42
But (2.202) gives
∂µJµ =1
2Tµνη
µν(ηαβ∂αεβ
)=
1
2T µ
µ (∂ · ε) = 0 . (2.205)
So the current is conserved because the stress-energy tensor is traceless.
Going into complex coordinates, and recalling that ε = ε(z) or ε(z) for confor-
mal transformations, we see that the conserved current associated with conformal
symmetry has components
Jz = Tzzεz i.e. J(z) = T (z)ε(z)
Jz = Tzzεz i.e. J(z) = T (z)ε(z) . (2.206)
We can then define the associated conserved charges. It is actually useful to define
them separately for the holomorphic and antiholomorphic sectors:
Qε =
∮dz
2πiε(z)T (z)
Qε =
∮dz
2πiε(z)T (z) . (2.207)
Note that there are infinitely many conserved charges, one for each conformal trans-
formation ε(z).
Now recall then from QFT (or from quantum mechanics) that the conserved charge
is actually the generator for the associated symmetry transformation. Thus, under
an infinitesimal conformal transformation z → z + ε(z), an operator transforms as10
φ → φ + δφ = φ + [Qε, φ]ETC . (2.208)
We can also view Qε to be associated with the holomorphic field ε(z) = ε(z)∂z. A
generic holomorphic vectorfield could be expanded in the basis e = zn+1∂z. The
corresponding charges are
Qn =
∮dz
2πizn+1T (z) . (2.209)
But if we substitute the Laurent expansion
T (z) =∑
n
Lnz−n−2 , (2.210)
we can see that the Qn’s are just the Virasoro generators:
Qn = Ln . (2.211)
So now we can see that the Virasoro generators are associated with conformal trans-
formations!11 Note that this time we did not specify what was the action of the CFT,
it could even have had interaction terms.10ETC = equal time commutator.11More precisely, they correspond to the representation of the generators of conformal transfor-
mations in the space of operators.
43
In a generic CFT, the Virasoro algebra commutation relations take the form
[Ln, Lm] = (n−m)Ln+m +c
12(n3 − n)δn+m,0 , (2.212)
where c is a real number, called the central charge. Its value depends on the specific
CFT in question, and it will play an important role in the future.
Note:
• L0, L±, generate again SL(2,C).
• if c = 0 then this is just like the Witt algebra of conformal transformations on
the worldsheet.
Now recall the transformation rule of primaries:
(dz′
dz
)h (dz′
dz
)h
φ(z′, z′) = φ(z, z) . (2.213)
Under an infinitesimal transformation z′ = z + ε(z), z′ = z + ε(z) then
δε,εφ(z, z) =(ε∂z + h(∂zε) + ε∂z + h(∂z ε)
)φ(z, z) . (2.214)
On the other hand, using (2.208)
δε,εφ(z, z) = [Qε, φ] + [Qε, φ] , (2.215)
so we can extract the commutator
[Qε, φ] = (ε∂z + h(∂zε)) φ(z, z) , (2.216)
and similarly for [Qε, φ]. In particular, for ε = zn+1 we obtain the commutator of a
Virasoro generator and a primary field:
[Ln, φ(z, z)] =zn+1∂z + h(n + 1)zn
φ(z, z) . (2.217)
Substituting the mode expansion φ(z) =∑
m φmz−m−h (for a holomorphic case) we
can extract out12
[Ln, φm] = n(n− 1)−mφm+n . (2.218)
12As an aside, note that all these followed from the transformation properties of the primaries, wedid not need to know any more detailed information about them!
44
2.13.1 Commutators in CFT and Radial Ordering
Actually, in the above I cheated a bit since I did not tell you what is meant by
the commutator in [Q, φ]. I need to be a bit more specific about operator normal
ordering. In QFT, products of operators contain an ordering ambiguity: a product
of two operators A(x), B(x′) depends on their order since in general they do not
commute:
A(x)B(x′) 6= B(x′)A(x) .
In QFT, of special importance are the time ordered products:
T (A(t, ~x)B(t′, ~x′)) =
A(t, ~x)B(t′, ~x′) , t > t′
B(t′, ~x′)A(t, ~x) , t′ > t, (2.219)
or in general
T (A1(t1, ~x1) · · ·An(tn, ~xn)) =
A1(t1, ~x1) · · ·An(tn, ~xn) , t1 > · · · > tn
....
(2.220)
In our case, the time coordinate on the cylinder becomes the radial coordinate on the
complex plane. So the time ordering of operators is replaced by a radial ordering,
denoted by R:
R (A(z)B(z′)) =
A(z)B(z′) , |z| > |z′|B(z′)A(z) , |z′| > |z| . (2.221)
Now consider the equal time commutator
[Q, φ(w)]ETC =
∮dz
2πi[J(z), φ(w)]ETC =
∮dz
2πiJ(z)φ(w)− φ(w)J(z) . (2.222)
Equal time means equal radius |z| = |w|. The integral is taken around a circle of
radius |z| around the origin.
Clearly we have to shift the contour a little bit to avoid crossing the point w. The
shift is depicted in Figure 10. So the contour becomes
[Q, φ(w)] =
(∮
|z|>|w|
dz
2πi−
∮
|z|<|w|
dz
2πi
)R (J(z)φ(w)) =
∮
Cw
dz
2πiR (J(z)φ(w)) ,
(2.223)
where Cw is the difference of the two contours: a circle around the point w (see Figure
11).
2.13.2 Operator Product Expansions
In the above, we ended up integrating around a contour wound tightly around the
point w. So in the integrand the operators J(z), φ(w) are evaluated at nearby points.
45
2nd term
ww
z) z)
1st term
Figure 10: Closed contours.
w
Cw
z)
Figure 11: Tiny contour Cw.
In QFT, products of local operators are typically singular at small separation.
The singular behavior can be isolated and the product of operators can be expressed
as a power series, a linear combination of local operators with singular coefficients.
This operator product expansion (OPE) was proposed by K. Wilson.
In 2-dimensional CFT on a plane, the OPE of two operators φi, φj is written as13
R(φi(z, z)φj(w, w)) ∼∑
k
Cijk
(z − w)hijk(z − w)hijkφk(w, w) , (2.224)
where hijk, hijk are exponents depending on the operators φi,j,k and Cijk are numbers.
The symbol “∼” means “= up to regular terms”.
13In these lectures we are always interested in R ordered products.
46
Earlier we evaluated the commutator
[Qε, φ(w)] =
∮dz
2πiR(ε(z)T (z)φ(w)) = h(∂wε(w))φ(w) + ε(w)∂wφ(w) . (2.225)
By using Cauchy’s theorem, we can deduce that the OPE of T (z) and φ(w) must be14
R(T (z)φ(w)) ∼ hφ(w)
(z − w)2+
∂wφ(w)
(z − w). (2.226)
The OPE (2.226) is characteristic for a primary field φ. Note in particular that h
appears as a coefficient in the first term. Further, the powers in the denominators
are fixed by scaling dimensions.
In an exercise, you will show that the OPE of T (z) with itself is
R(T (z)T (w)) ∼ c/2
(z − w)4+
2T (w)
(z − w)2+
∂wT (w)
(z − w). (2.227)
In particular, this means that T (z) is not a primary operator. Without the 1st term
on RHS, it would look like T (z) is a primary operator with conformal weight (2, 0).
In an exercise you can see by another way as well that T (z) is not a primary.
2.13.3 Correlation Functions
Correlation functions are vacuum expectation values of time ordered products of
operators:
〈vacuum|T (φ1(t1, ~x1) · · ·φn(tn, ~xn))|vacuum〉 . (2.228)
In CFT, we replace the time ordering by radial ordering and denote for short
〈φ1(z1, z1) · · ·φn(zn, zn)〉 = 〈0; 0|R(φ1(z1, z1) · · ·φn(zn, zn))|0; 0〉 . (2.229)
You have calculated some 2-point functions in a problem set. By using the Vi-
rasoro generators, and (2.217), one can also compute the 3- and 4-point functions of
generic primary operators:
〈φ1(z1, z1)φ2(z2, z2)φ3(z3, z3)〉 ∝∣∣∣∣
1
(z1 − z2)−h1−h2+h3(z2 − z3)−h2−h3+h1(z3 − z1)−h3−h1+h2
∣∣∣∣2
,
(2.230)
and
〈φ1(z1, z1) · · ·φ4(z4, z4)〉 = f(x, x)∏i<j
∣∣∣(zi − zj)−hi−hj+
Pk
13hk
∣∣∣2
, (2.231)
where f is an arbitrary function of the cross ratio
x =(z1 − z2)(z3 − z4)
(z1 − z3)(z2 − z4). (2.232)
14You can also replace φ(w) by φ(w, w), ∂wφ(w) by ∂wφ(w, w).
47
Note that for primaries, even the 3-point function is known exactly up to an overall
coefficient. Try to find a 3-point function to all loop orders in an interacting non-
conformal QFT15.
2.13.4 Wick’s Theorem
We will be interested in some specific OPE’s in bosonic string theory. As a warm-up,
consider a single bosonic scalar field X(z, z), with action
S = −T
2
∫dzdz ∂X(z, z)∂X(z, z) . (2.233)
The stress tensor has components
T (z) = −1
2:∂XL(z)∂XL(z):
T (z) = −1
2:∂XR(z)∂XR(z): (2.234)
where the normal ordering is defined as before by using mode expansions, or by
:∂XL(z)∂XL(w): = limz→w
(∂XL(z)∂XL(w) + (z − w)−2
). (2.235)
One can prove that these two definitions are equivalent.
Suppose we want to compute the OPE
R(T (z)∂X(w)) = −1
2R(:∂X(z)∂X(z):∂X(w)) (2.236)
to check that ∂X(w) is a primary field of weight (1, 0). We will use Wick’s theorem16:
R(A1A2A3 · · ·An) = :A1A2A3 · · ·An: + :A1A 2A3 · · ·An:+ (2.237)
+ all possible contractions inside the normal ordering, contractions of operators at
equal radius give zero. Further: normal orderings inside normal orderings can be
dropped.17
It would take several pages to give really precise definitions, let me instead choose
to illuminate this with examples, that will suffice for our purposes.
15Warning! Do not attempt this on your own. It is extremely hazardous and requires specialabilities.
16Works for free fields/operators.17..above still needs modifying...
48
Example 1.
T (∂X(z)∂X(w)) = :∂X(z)∂X(w): + ∂X(z)∂X(w) . (2.238)
Take VEV:
〈0; 0|T (∂X(z)∂X(w))|0; 0〉 = 〈0; 0|: · · · :|0; 0〉+ ∂X(z)∂X(w)〈0; 0|0; 0〉 , (2.239)
so the contraction is equivalent to 2-point function:
A1(z)A 2(w) = 〈A1(z)A2(w)〉 . (2.240)
For example,
∂X(z)∂X(w) = 〈∂X(z)∂X(w)〉 =−1
(z − w)2. (2.241)
Example 2.
R(T (z)∂X(w)) = −1
2R(:∂X(z)∂X(z):∂X(w))
= −1
2
::∂X(z)∂X(z):∂X(w): + ::∂X(z)∂X(z):∂X(w):
+ ::∂X(z)∂X(z):∂X(w):
= −1
2
:T (z)∂X(w): + 2∂X(z)
−1/(z−w)2︷ ︸︸ ︷〈∂X(z)∂X(w)〉
=∂X(z)
(z − w)2− 1
2:T (w)∂X(w):
− 1
2(z − w):∂T (w)∂X(w): + . . . (2.242)
z→w−→ ∂X(z)
(z − w)2+ regular or vanishing terms
=∂X(w) + (z − w)∂2X(w) + . . .
(z − w)2+ . . .
∼ ∂X(w)
(z − w)2+
∂∂X(w)
(z − w). (2.243)
In (2.242) we used T (z) = T (w) + (z−w)∂T (w) + . . .. This is the correct result, ∂X
is a primary with h = 1.
49
Example 3.
R(T (z)T (w)) =1
4R(:∂X(z)∂X(z)::∂X(w)∂X(w):)
=1
4
::∂X(z)∂X(z)::∂X(w)∂X(w):: + ::∂X(z)∂X(z)::∂X(w)∂X(w)::
+ 3 others with a single contraction
+ 2 · ::∂X(z)∂X(z)::∂X(w)∂X(w)::
= :T (z)T (w): + :∂X(z)∂X(w):〈∂X(z)∂X(w)〉+
1
2(〈∂X(z)∂X(w)〉)2 (2.244)
= :T (w)T (w): + (z − w):∂T (w)T (w): + . . .
− − 1
(z − w)2:∂X(w)∂X(w): + (z − w):∂∂X(w)∂X(w): + . . .
+1
2
1
(z − w)4
z→w−→ 1/2
(z − w)4+
2T (w)
(z − w)2− :∂∂X(w)∂X(w):
(z − w)+ regular
∼ 1/2
(z − w)4+
2T (w)
(z − w)2+
∂T (w)
(z − w). (2.245)
(Note that ∂T (w) = −12∂:∂X(w)∂X(w): = −:∂∂X(w)∂X(w):.) This is also the
correct result. Hence a single free boson has c = 1. T (w) is not a primary.
Another, important result (Problem set #3), denote Vp(w) = :eipX(w):. It satisfies
T (z)Vp(w) =p2/2Vp(w)
(z − w)2+
∂Vp(w)
(z − w). (2.246)
Hence Vp(w) is a primary, with h = p2/2.
2.13.5 Operator-state Correspondence
In particle physics, we are interested in scattering processes. The basic information
is contained in the probability amplitude, the inner product between initial and final
states
A = out〈φ|ψ〉in . (2.247)
In Quantum Field Theory, the probability amplitude is calculated by isolating inter-
actions and simple Fock space states describing non-interacting particles, using the
S-matrix, e.g.,
A = 〈~k1, ~k2|S|~k3, ~k4〉 = limT→∞
〈~k1, ~k2|UI(T1 − T )|~k3, ~k4〉 , (2.248)
where UI is the time-evolution operator in the interaction picture. UI contains the
higher order operators in the Lagrangian that correspond to interactions. So in QFT
it is useful to make a distinction between local operators and states of the theory.
50
In CFT, there is a useful one-to-one mapping between local primary operators and
states. As an introduction, consider the closed bosonic string theory as an example.
Suppose we are interested in an incoming string state and its overlap (amplitude)
between an outgoing string state (see Fig 12).
τ=+ 8 <- time 8
<φ | |ψ >out in
τ=−
Figure 12: Amplitude.
Since we have been working on the complex plane, now the initial state will be a
state at the origin, and the outgoing state will be a state at the point at infinity (Fig
13.)
|ψ >in
out
time
τ=const.
<φ |
Figure 13: Complex plane.
Let |0; 0〉 denote the vacuum state of a CFT (for a bosonic string, this is a string
which is neither oscillating nor moving: momentum is zero). For every primary
operator φ(z, z), we can define a corresponding incoming state as
|φ〉in = limτ→−∞
φ(τ, σ)|0; 0〉 = limz→0
φ(z, z)|0; 0〉 . (2.249)
For an outgoing state, rather than using the coordinates z, z to describe the point at
infinity, it is more well controlled to use the coordinates
w =1
z; w =
1
z. (2.250)
Now, in order to take a limit as in (2.249), it is easy to describe a small neighborhood
around the point at infinity: it is simply |w| = ε, say. In going from z to w, we have
51
to take into account how the primary operator transforms:
φ′(z, z) =
(dz
dw
)h (dz
dw
)h
φ(z, z)
=
(−1
w2
)h (−1
w2
)h
φ
(1
w,
1
w
). (2.251)
The outgoing state corresponding to the operator φ is then defined as
out〈φ| = limw→0
〈0; 0|φ′(w, w)
= limw→0
〈0; 0|(−w)−2h(w)−2hφ
(1
w,
1
w
). (2.252)
Motivated by this, we define the Hermitean conjugate of an operator as follows:
[φ(z, z)]† = φ
(1
z,1
z
)(−z)−2h(−z)−2h . (2.253)
Thus
out〈φ| = limz→0
〈0; 0|(−z)−2h(−z)−2hφ
(1
z,1
z
)(2.254)
= limz→0
〈0; 0| [φ(z, z)]†
= limz→0
(φ(z, z)|0; 0〉)† = |φ〉†in . (2.255)
(In the above, in taking the limit in (2.254), you can think that we simply used a
label z instead of w for the coordinate: like limz→0 f(z) = limw→0 f(w).)
Example 1. ∂zX(z) is a primary with weight (h, h) = (1, 0).
|∂zX〉in = limz→0
∂zX(z)|0; 0〉 = limz→0
∑n
αnz−n−1|0; 0〉
= α−1|0; 0〉 . (2.256)
In order for the limit (2.256) to be non-singular, we need αn|0; 0〉 ∀n ≥ 0. The only
term that survives is the n = −1 term.
Example 2. Vp(z) = :eikX(z): is a primary with weight (k2
2, 0).
|Vp〉in = limz→0
:eikX(z):|0; 0〉 = limz→0
: exp[ik(x + p ln z + osc.)]:|0; 0〉= eikx|0; 0〉 = |0; k〉 . (2.257)
In (2.257) we used p|0; 0〉 = 0, oscillators also give nothing due to normal ordering.
Recall from QM that eikx is an operator that creates a momentum eigenstate.
52
Thus :eikX(x): creates a momentum eigenstate |0; k〉. Generalizing to a bosonic
string we have
Vp = :eikµXµ(z,z): . (2.258)
If k2 = 2, the resulting state is a physical state: it is the tachyon |0; kµ〉 ! Then also
Vp has a conformal weight (1, 1). This is an example of a vertex operator. Vertex
operators create on-shell string states. The above operator is the tachyon vertex
operator.
Higher closed string excitations also have corresponding operators: e.g., at the
massless level
N = 0 : eµναµ−1α
ν−1|0; kρ〉 ↔ eµν:∂Xµ∂XνeikρXρ
:∣∣(z,z)=(0,0)
. (2.259)
To find the rules for finding the primary operator that creates a given state (so far we
have discussed the opposite: what is the state that corresponds to a given primary),
we proceed as follows (following the discussion in Polchinski’s book, “String Theory”,
vol. I chapter 2.8).
Consider states of the form
Qn|φ〉 , (2.260)
where φ is a primary which can be identified simply, so
|φ〉 = φ(0, 0)|0; 0〉 , (2.261)
and Qn is an operator that has an integral representation
Qn =
∮dz
2πizn+h−1J(z) . (2.262)
For example, Qn could be the nth coefficient in the Laurent expansion of some primary
of conformal weight (h, 0), e.g.,
Qn = αn =
∮dz
2πizn+1−1∂X(z) , (2.263)
or it could be a conserved charge associated with a current J(z) (of conformal weight
(h,0)). Since Q acts on the state from the left, the contour integral is taken around
the origin. Now, the rule to find the operator V (0, 0) (operator V (z, z) inserted at
the origin) which creates the state,
Q|φ〉 = V (0, 0)|0; 0〉 (2.264)
is the following:
Q|φ〉 ↔ V (0, 0) =
∮dz
2πizn+h−1J(z)φ(0, 0) . (2.265)
53
Example 1. Let φ = 1, then |1〉 = |0; 0〉 (rather obviously). Then take
Qm = αµ−m =
∮dz
2πiz−m∂Xµ(z) . (2.266)
Now, according to (2.265):
αµ−m ↔
∮dz
2πiz−m∂Xµ(z) , (2.267)
then Taylor expand inside the integral:
∂Xµ(z) = ∂Xµ(0) + z∂2Xµ(0) +1
2!z2∂3Xµ(0) + . . . . (2.268)
The only term that contributes (and yields a 1st order pole together with z−m) is the
(m-1)th term:
∮dz
2πiz−m
∂Xµ(0) + . . . +
1
(m− 1)!zm−1∂mXµ(0) + . . .
=
1
(m− 1)!∂mXµ(0) .
(2.269)
So we get V (0, 0) ∼ 1(m−1)!
∂mXµ(0), and
αµ−m|0; 0〉 =
1
(m− 1)!∂mXµ(0)|0; 0〉 . (2.270)
Example 2. Now let φ = :eikµXµ(0,0):, then18
αµ−m|0; kν〉 ↔
∮dz
2πiz−mR(∂Xµ(z):eikνXν(0,0):) . (2.271)
By Wick’s theorem:
R(∂Xµ(z):eikνXν(0,0):) = :∂Xµ(z)eikνXν(0,0): + contractions . (2.272)
In (2.272), the only non-zero contractions are between ∂Xµ(z) and Xµ(0, 0). They
can only produce terms ∼ 1z. So the only term that can produce a first order pole
comes from the Taylor expansion of ∂Xµ(z). Expanding as in (2.268),
R(∂Xµ(z):eikνXν(0,0):) = . . . +zm−1
(m− 1)!:∂mX(0)eikνXν(0,0): +
+ terms that do not contribute to (2.271) + . . . .(2.273)
Thus,
αµ−m|0; kν〉 ↔ 1
(m− 1)!:∂mX(0)eikνXν(0,0): . (2.274)
18I don’t understand the notation below.. How come kν on the LHS, since ν is a summing indexon RHS?
54
A similar calculation gives the graviton vertex operator,
eµναµ−1α
ν−1|0; kν〉 ↔ a · eµν:∂Xµ(0)∂Xν(0)eikρXρ(0,0): , (2.275)
with some numerical factor (normalization coefficient) a. Further, vertex operatos for
higher excited string states are found in similar manner. E.g., at level N = 2:
Cµνραµ−2α
ν−1α
ρ−1|0; kσ〉 ↔ Cµνρ:∂
2Xµ(0)∂Xν(0)∂Xρ(0)eik·X(0,0): . (2.276)
So now we have the technology to create incoming and outgoing on-shell string
states by vertex operators, and we are ready to begin discussing interactions of strings.
***** END OF LECTURE 5 *****
2.14 Tree-level Bosonic String Interactions
Recall that string theory differs from Quantum Field Theory in that interactions are
not introduced by including higher order local operators into the Lagrangian.
For example, consider what would happen if you were to, e.g., include a X4 term
into the Polyakov action:
S =T
2
∫dzdz
∂Xµ∂Xµ + λ(XµXµ)2
. (2.277)
This is not the way to go. The action still describes just a single string. The strings
are not quanta of the fields in the action. The quanta of the fields are the center-of-
mass momentum and the oscillators of a single string. So the X4 term only makes the
oscillations become coupled. But since we want to identify the different oscillatory
levels with particle states, we have only made the task of idenfication much harder!
We need something else to describe string interactions.
Consider now tree-level processes, e.g., the one where a single closed string splits
in two (Fig 14).
3
1
2
Figure 14: Three-string interaction.
55
The worldsheet depicted in the Figure 14 can be mapped by a conformal trans-
formation to a sphere with three little holes, or punctures as they are often called, see
Fig. 15.
3
1
2
Figure 15: A sphere with 3 punctures.
At each end of the branched cylinder of Fig. 14, or at each puncture of the sphere
of Fig. 15, we have an on-shell string state. On-shell string states are created by
vertex operators, so we insert a vertex operator for a graviton, say, if that’s what
we are scattering. We could also map the (Riemann) sphere with punctures to the
(compactified) complex plane, Fig. 16. By a global conformal map (SL(2,C)/Z2
3
1
2
Figure 16: The complex plane with 3 punctures.
transformation) we can further map three points to any specially chosen points, like
the origin, point at infinity, and z = z = 1, if we wish. Then the probability amplitude
for the three string scattering would be
A = 〈V (3)p |V (2)
p (1, 0)|V (1)p 〉 . (2.278)
We will make this more precise, of course. It is actually simpler (calculationally) to
first consider open string scattering.
56
For three open strings, the worldsheet is a branching strip, shown in Figure 17.
1
2 3
Figure 17: Tree-level diagram for scattering of 3 open strings.
This is conformal to a disk with three dents (like the punctures), depicted in
Figure 18.
3
1
2
Figure 18: Disk with 3 dents.
It is also conformal to the upper half-plane with three dents, Figure 19.
3 12
Figure 19: Upper half-plane with 3 dents.
57
Now, we insert an open string vertex operator to each dent to create an on-shell
string state. Let us consider scattering of open strings in more detail, starting with
the easiest example.
2.14.1 Scattering of Open String Tachyons
Consider a tree-level process involving M (on-shell) strings, Fig. 20.
M
1
2
3
4
5
Figure 20: Worldsheet of tree level scattering of M open strings.
By a suitable conformal transformation, we can map the worldsheet to a strip
with M − 2 dents, Fig 21. Let the (Euclidean) worldsheet time τE run from left to
right.
1
2 3 4 M-1
M
τ
σ
0
π
Figure 21: Worldsheet as a strip.
Let us use the complex coordinate
w = τE + iσ ; τE ∈ R , σ ∈ [0, π] (2.279)
to parametrize the strip. Note that all the string interactions happen at the boundary
at σ = 0 (which you can also think of including the points at infinity, τE = ±∞).
58
For each external string state at 1, . . . , M with momentum k1, . . . , kM there is a
corresponding vertex operator
Vi(ki, τEi)
which creates it. At first you might think that the scattering amplitude is
limτE1
→−∞τEM
→∞
〈0|VM(kM , τEM) · · ·V2(k2, τE2)V1(k1, τE1)|0〉 . (2.280)
However, this actually does not make sense. An experiment in the target space has
no way of measuring the points wi ≡ τEion the worldsheet where the strings come
from. Since the worldsheet points are not observable, we have to integrate over them:
A =
∫dτE1 · · · dτEM
〈0|VM(kM , τEM) · · ·V2(k2, τE2)V1(k1, τE1)|0〉 . (2.281)
Instead of using the strip as the worldsheet, we can use the upper half-plane (Fig.
22).
...
(z
1 2 3 4
Figure 22: Upper half-plane.
The map
z = ew (2.282)
does the job. On the strip, the open string coordinates Xµ(τE, σ) have the mode
expansion
Xµ = xµ − ilspµτE + ils
∑
n 6=0
1
nαµ
ne−nτE cos(nσ) . (2.283)
On the boundary at σ = 0 this becomes
Xµ = xµ − ilspµτE + ils
∑
n 6=0
1
nαµ
ne−nτE , (2.284)
and on the boundary of the upper half-plane, using z = eτE+iσ,
Xµ = xµ − ilspµ ln z + ils
∑
n 6=0
1
nαµ
nz−n (2.285)
59
with =z = 0. On the upper half-plane, we write the scattering amplitude (2.281) as
A =
∫dz1 · · · dzM〈0; 0|VM(kM , zM) · · ·V2(k2, z2)V1(k1, z1)|0; 0〉 (2.286)
where the integrals are evaluated on the boundary =z = 0, <z ≥ 0. To start with, we
assume that we are not rearranging the strings but keep them ordered: <z1 ≤ <z2 ≤· · · ≤ <zM
19.
As a specific example, consider M -tachyon scattering: all the vertex operators are
the holomorphic primaries
Vi(ki, zi) = :eikµXµ(z): , (2.287)
which you have considered before. As a homework problem20, you have already
evaluated the M -point function in (2.286). Thus,
A =
∫dz1 · · · dzM
∏i<j
eki·kj ln(zi−zj) . (2.288)
All the momenta ki satisfy the tachyon on-shell condition
k2i = 2 . (2.289)
This is crucial, because then the vertex operators Vi are of the holomorphic weight
h = k2/2 = 1, and the integrals∫
dzVi(ki, z) (2.290)
are invariant under conformal transformations z → f(z). This is an important re-
quirement. So, whatever the vertex operator V is, it must be of weight h = 1 as k is
on shell!
The expression (2.288) is not yet the final answer for the scattering amplitude as
it diverges - it contains a huge overcounting. The reason is that the upper half-plane
maps to itself (one-to-one) under all SL(2,R) transformations
z → az + b
cz + d(2.291)
with a, b, c, d ∈ R, det
(a b
c d
)= 1. These are the global conformal transformations
of the upper half-plane (open string). So we need to factor out the overcounting from
(2.288). This is done by a variant of the Faddeev-Popov trick. Insert into (2.288)
1 =
∫dαdβdγδ(z1 − z1)δ(zM−1 − zM−1)δ(zM − zM)
∣∣∣∣∂(z1, zM−1, zM)
∂(α, β, γ)
∣∣∣∣ . (2.292)
19For a complete discussion without this assumption, see page 63.20Really?
60
The δ-functions remove the z1, zM−1, zM integrals and fix z1, zM−1, zM to some
specific points z1, zM−1, zM on the positive real axis, which we can choose. The
Jacobian is calculated using the infinitesimal form of the SL(2,R) transformation:
z → z + δz = z + α + βz + γz2 . (2.293)
Thus the Jacobian is
∣∣∣∣∂(z1, zM−1, zM)
∂(α, β, γ)
∣∣∣∣ = det
1 z1 z21
1 zM−1 z2M−1
1 zM z2M
= (z1 − zM−1)(z1 − zM)(zM−1 − zM) .
(2.294)
For example, let’s consider 4-tachyon scattering, so M = 4. Then the amplitude
becomes21
A = (z1 − z3)(z1 − z4)(z3 − z4)
∫ z3
z1
dz2
∏i>j
(zi − zj)ki·kj . (2.295)
We want V4 to be in the far future, and V1 to be in the far past, so we fix
τE4 = ∞ → z4 = eτE4 = ∞τE1 = −∞ → z1 = eτE1 = 0 . (2.296)
The third point z3 is useful to fix to
z3 = 1 . (2.297)
Now, removing all the overall factors from A (and hence the infinite overcounting as
well), what remains is
A =
∫ 1
0
dz(1− z)k3·k2zk2·k1 . (2.298)
Note: we have used k1 + . . . + k4 = 0. In order to extract some physics from the
answer, let us introduce the Mandelstam variables
s = −(k1 + k2)2 = −2− 2− 2k1 · k2 = −4− 2k1 · k2
t = −(k2 + k3)2 = −(k1 + k4)
2 = −4− 2k2 · k3 (2.299)
so that
A =
∫ 1
0
dxx−s/2−2(1− x)−t/2−2 = B
(−s
2− 1,− t
2− 1
)=
Γ(− s
2− 1
)Γ
(− t2− 1
)
Γ(− s
2− t
2− 2
) ,
(2.300)
where B is the Beta function
B(a, b) =
∫ 1
0
dxxa−1(1− x)b−1 . (2.301)
The amplitude A (2.300) is the Veneziano amplitude.
21Check which i < or > j...
61
Two comments:
• To be complete, we have to assign an open string coupling constant go with
each vertex between the initial and final states, thus
AM ∼ gM−2o , (2.302)
and the 4-tachyon amplitude has a factor g2o .
• In the above, we have set ls = 1. In reality, we should have used
〈Xµ(z)Xν(z′)〉 = −l2sηµν ln(z − z′) (2.303)
in the contractions.
Restoring units and coupling constants, the answer for the Veneziano amplitude is
(but see also ...)
A = g2oB
(− l2ss
2− 1,− l2st
2− 1
). (2.304)
Now, finally, some physics. The amplitude has poles at
s =2
l2s· n , n = −1, 0, 1, 2, . . . (2.305)
corresponding to on-shell string states in the s-channel (Fig. 23)
41
2 3
Figure 23: s-channel.
and poles at
t =2
l2s· n , n = −1, 0, 1, 2, . . . (2.306)
corresponding to poles at the t-channel (Fig. 24). So the scattering tachyons are ex-
changing infinitely many particles corresponding to all the possible string excitations
(tachyon, photon,. . . ). We got all the intermediate particles “for free”.
Another sign of “stringiness” is displayed by the hard scattering limit s →∞ (very
energetic collision). Hard scattering is used to probe short distance substructure of
the scattering objects (consider, e.g., the deep inelastic scattering of electrons from
62
2 4
31
Figure 24: t-channel.
a nucleon which revealed the parton (or quark) substructure of the latter). For
scattering of point-like objects, the amplitude decays like a power law,
A ∼ s−a (2.307)
with a some positive number. The hard scattering limit of the Veneziano amplitude
produces
A ∼ sae−l2sf(θ)s , (2.308)
where f(θ) is a function of scattering angle. The exponential fall-off suggests that
the scattering objects are smooth, of finite size of the order of ls, as expected!
Note Added In page 60, at the beginning of the calculation I assumed for simplicity
the ordering <z1 ≤ <z2 ≤ · · · ≤ <zM for the vertices. This simplification you also
find often in the literature (like, e.g., in Bailin & Love). For identical particles (like
the tachyons), this in fact gives only a part of the result for the amplitude. If the
worldsheet is drawn as a disk, it corresponds to the ordering for the vertices (for
M = 4) depicted in Fig. 25.
4
a)3
1
2
Figure 25: Disk.
63
In fact there are five other inequivalent orderings to take into account, the per-
mutations depicted in Figure 26, with <z1 ≤ <z4 ≤ <z2 ≤ <z3 for b) etc.
1
f)
1 1 1 1
2 2
33
3
4
4
4
43
22
1
2 3
4
b) c) d) e)
Figure 26: More disks.
Adding the contributions b)-f) to the amplitude gives the full Veneziano amplitude
as a sum of 3 terms22:
A = #g2o
B
(− l2ss
2− 1,− l2st
2− 1
)+ B
(− l2st
2− 1,− l2su
2− 1
)
+ B
(− l2su
2− 1,− l2ss
2− 1
) (2.309)
with # a numerical coefficient. So the answer contains 3 Beta functions depending
on the three possible pairs of 2 Mandelstam variables out of the three s, t, u. In
particular, there are poles in each of the three channels. See Polchinski, vol. I, p.180.
2.14.2 Tree-level Scattering of Closed String Tachyons
The calculation goes like the open string calculation, but with some differences be-
cause now the left and right movers decouple, and the worldsheet is conformal to the
whole complex plane with punctures:
22Maybe u should be defined somewhere, here?
64
Now the starting point (2.286) for the amplitude is replaced by
A =
∫dz1dz1 · · · dzMdzM〈0; 0|VM(kM , zM , zM) · · ·V1(k1, z1, z1)|0; 0〉 , (2.310)
and the integrals go over the whole complex plane (with |zM | > · · · > |z1|). The
closed string (tachyon) vertex operator is
V (k, z, z) = :eikµXµ(z,z): = :eikµLXLµ(z)::eikµ
RXRµ(z): (2.311)
because X(z, z) = XL(z) + XR(z). So the vertex operator is like a product of the
previous vertex operators, one holomorphic and one antiholomorphic. Further, the
left and the right movers each carry 1/2 of the total center-of-mass momentum of the
closed string:
kµL = kµ
R =1
2kµ . (2.312)
In order to maintain conformal invariance, the vertex operator V (k, z, z) must have
conformal weight (h, h) = (1, 1) (because now we integrate with∫
dzdz). This means
that
k2L = k2
R =k2
4= 2 (2.313)
(in units where ls = 1) so that k2 = 8, corresponding to the on-shell tachyon with
mass
M2 = −8 . (2.314)
Now the two-point function is23
〈Xµ(z, z)Xν(z′, z′)〉 = − ln[(z − z′)(z − z′)] . (2.315)
The SL(2,R) invariance of the upper half-plane is replaced by the SL(2,C) invariance
of the full complex plane, the full global conformal group. This allows us to fix
z1, zM−1, zM and ˆz1, ˆzM−1, ˆzM . The four-tachyon amplitude becomes
A = |(z1 − z3)(z1 − z4)(z3 − z4)|2∫
dz2dz2
∏i>j
e2ki·kj ln |zi−zj | (2.316)
with z1 = 0, z2 = 1, z2 = ∞24. Dropping out the extra pieces, what remains is
A =
∫dzdz|1− z|2k3·k2|z|2k1·k2 . (2.317)
Again, we must include coupling constants, this time closed string coupling constants
gc. Restoring dimensions, and employing Mandelstam variables, the result is
A = g2c
Γ(− l2ss
8− 1
)Γ
(− l2st
8− 1
)Γ
(− l2su
8− 1
)
Γ(− l2ss
8+ 2
)Γ
(− l2st
8+ 2
)Γ
(− l2su
8+ 2
) (2.318)
23Missing ηµν?24What are the indices?
65
where u is the Mandelstam variable
u = −(k2 + k4)2 = −16− 2k2 · k4 . (2.319)
This is the Virasoro-Shapiro amplitude. Using the Gamma function identity
Γ(x)Γ(1− x) =π
sin πx, (2.320)
and the relation between the open string and closed string coupling constants
gc ∼ g2o (2.321)
(see section 1.5.6. of Green, Schwartz and Witten, vol. I), we can see that the closed
string amplitude factorizes into a product of open string amplitudes
Aclosed(s, t, u) ∼ sin(πt/8)Aopen(s/4, t/4)Aopen(t/4, u/4) . (2.322)
We might have expected something like this to happen, because the vertex operator
factorized to left and right movers. This is in fact an example of a more general
relation between closed and open string amplitudes.
We will now leave scattering amplitudes and move to a next case of interactions,
where a string interacts with a condensate of other strings.
***** END OF LECTURE 6 *****
2.15 Strings in Background Fields
So far we have been discussing the dynamics of a small number of strings moving in a
flat empty target space. However, in reality the strings move in a sea of other strings.
The other strings could be in any of the on-shell string states, giving rise (at long
distance) to various fields turned on. At long distances, of particular interest are the
massless string excitations. We consider first closed strings, so the massless states
are the graviton, antisymmetric tensor, and the dilaton. The associated fields are the
gravitational field, described by the (target space) metric Gµν(X), an antisymmetric
tensor field Bµν(X) and a scalar field φ(X).
A generic gravitational field means a generic curved target space. The Polyakov
action (before the gauge fixing) reads
SP =−T
2
∫d2σ
√−hhαβGµν(X)∂αXµ∂βXν . (2.323)
Now consider a small perturbation about the flat background like a gravitational wave
in flat space:
Gµν(X) = ηµν + χµν(X) . (2.324)
66
The integrand in the worldsheet path integral can then be expanded as a power series
in χ:
eiSP ' eiSP (ηµν ,X)
1− iT
2
∫d2σ
√−hhαβχµν(X)∂αXµ∂βXν
+1
2!
(−iT
2
)2 ∫d2σ
∫d2σ′
√−h√−h′hh′χχ∂X∂X∂X∂X + . . .
.(2.325)
The second term in the wavy brackets in (2.325) can be identified as the worldsheet
integral of the graviton vertex operator,
−iT
2
∫d2σ
√−hVgr(σ
α) , Vgr = eµνhαβ:∂αXµ∂βXνeik·X : (2.326)
if
χµν(X) = eµν(k)eik·X . (2.327)
A more general χµν(X) can be expressed as a Fourier integral (a superposition) of
the waves (2.327). So the path integral can be expressed as
Z =
∫DXDheiSP +iVgr ≡ 〈eiVgr〉 . (2.328)
If an operator A creates a single quantum from the vacuum, eA creates a coherent
state. So the action (2.328) can be thought to describe a string moving in the back-
ground corresponding to a coherent state of gravitons (from on-shell closed strings).
The vertex operators for the other massless closed string excitations are
Vast = aµνεαβ:∂αXµ∂βXνeik·X: (2.329)
= antisymmetric tensor
Vdil = φ:Rheik·X: (2.330)
= dilaton
where Rh is the Ricci scalar curvature of the worldsheet metric hαβ. By exponentiating
to generate coherent states, and generalizing to finite background fields, we find that
the Polyakov action becomes
SP =−1
4πα′
∫d2σ
√−h
hαβGµν(X)∂αXµ∂βXν + εαβBµν(X)∂αXµ∂βXν
+α′
2Rhφ(X)
,
(1
2πα′≡ T ≡ 1
πl2s
). (2.331)
The background fields Bµν(X), φ(X) are the antisymmetric tensor and the dilaton.
The action is invariant under the gauge transformation
Bµν(X) → Bµν(X) + ∂µξν(X)− ∂νξµ(X) (2.332)
67
or
BµνdXµdXν → BµνdXµdXν + d(ξµdXµ) (2.333)
using differential forms in target space.
The Bµν field is a generalization of a gauge field Aµ; or in other words, if Aµ is a
gauge field which couples to 0-dimensional objects (point particles), Bµν is a gauge
field which couples to 1-dimensional objects (strings).
Coupling of Aµ to a point particle:
S(0)coupling = −e
∫dτAµ∂τX
µ = −e
∫AµdXµ . (2.334)
[Aµ∂τXµdτ is the pull-back of the target space 1-form AµdXµ to the particle world-
line.] Coupling of Bµν to a string:
S(1)coupling = − 1
4πα′
∫d2σBµνε
αβ∂αXµ∂βXν = − 1
4πα′
∫Bµν∂αXµ∂βXνdσα ∧ dσβ
= − 1
4πα′
∫BµνdXµ ∧ dXν . (2.335)
[So now there’s a pull-back of 2-form B = BµνdXµ∧dXν to the worldsheet (assumed
flat for simplicity in the above).] By comparing the equations (2.334) and (2.335) you
can conclude that the string carries a charge
”est” =1
4πα′.
The sign of the charge depends on the orientation of the string. Reversing the orien-
tation flips the sign. Note also that the string can carry a charge because it is oriented.
The field strength corresponding to B is the three-form
H = dB or Hµνλ = ∂µBνλ + ∂νBλµ + ∂λBµν . (2.336)
It is invariant under the gauge transformation (2.332) or (2.333).
Since the string couples to Bµν like a charged point particle couples to Aµ, it means
that the string carries a charge which acts as a source for Bµν . For an electrically
charged point particle which creates a field configuration Aµ, the charge is obtained
by integrating the field strength around a sphere which surrounds it. If the particle
is in 4 dimensions, the charge q is found by
q =
∫
S2
∗F =
∫
S2
dxk ∧ dxlε0iklF0i =
∫
S2
d~s · ~E . (2.337)
The generalization to D dimensions is
q =
∫
SD−2
∗F (2.338)
68
where SD−2 is a (D−2)-sphere around the point particle. The Hodge dual of a 2-form
F is a (D − 2)-form ∗F . The analogous formula for a string in D dimensions with
the field strength 3-form H is (∗H = (D − 3)-form)
q =
∫
SD−3
∗H . (2.339)
Example geometries are shown in D = 3 + 1 = 4 in Fig. 27.
a)
1
b)
S2 S
Figure 27: Figure a) depicts a two-sphere around a point particle in D = 4. Figure
b) depicts a one-sphere around a string in D = 4.
Note that since Gµν , Bµν , φ all depend on X, in general the action is no longer
quadratic in X but becomes non-linear. For Bµν = φ = 0 the action with Gµν(X)
describes a field theory where the field space is a curved manifold. These are known as
non-linear σ-models25. If we expand the action around a classical solution Xµ(σα) =
xµ0 :
Xµ(σα) = xµ0 + Y µ(σα) , (2.340)
the action has a power series expression
S = − 1
4πα′
∫d2σ
√−hhαβ
Gµν(x
µ0)∂αY µ∂βY ν + Gµν,λ(x
µ0)Y λ∂αY µ∂βY ν
+ Gµν,λρ(xµ0 )Y λY ρ∂αY µ∂βY ν + higher
. (2.341)
The first term is the kinetic term, the rest are various interaction terms. Now the
oscillations of the string are coupled. If the curvature radius of the target space is of
the order ∼ R0, then derivatives of the metric are of the order ∼ R−10 . So the effective
dimensionless coupling constant for the interaction terms is
λ ∼ lsR0
∼√
α′
R0
. (2.342)
25You may encounter these also when studying effective theories for pion fields.
69
If the target space curves only very slightly over a string scale,
R0 À ls , (2.343)
the coupling constant is small and one can study the model perturbatively.
Note also that at long wavelengths R0 the corresponding energy scale 1/R0 ¿ 1/ls.
Since the massive string excitations are ∼ 1/ls, at the low-energy scale they do not
contribute. This is why we have restricted our attention to massless background
fields.
Another feature of low energies or length scales À ls is that a string appears to
be point-like. In such a regime one can ignore the internal structure of the string
and use field theory, or more precisely, low-energy effective field theory. That will be
our next subject. It is connected with the issue of Weyl invariance, so we start from
there.
2.16 Weyl Invariance and the Weyl Anomaly
Recall that at the classical level, the Polyakov action was invariant under Weyl trans-
formations hαβ → Λ(σ)hαβ
Xµ , ∂αXµ unchanged. (2.344)
A global Weyl transformation
Λ(σ) = Λ ≡ a2 (2.345)
is equivalent to rescaling all lengths on the worldsheet by a, σα → aσα. Consider a
familiar φ4 theory in 3 + 1 dimensions
L =1
2∂µφ∂µφ− λφ4 . (2.346)
The field φ has mass dimension 1, and the coupling constant λ is dimensionless. So
the classical action is invariant under rescalings26
xµ → axµ ≡ x′
φ(x) → a−1φ(x/a) ≡ φ′(x) . (2.348)
However, at quantum level there are divergencies, which need first to be regular-
ized and renormalized. The regularization process (e.g., by dimensional regulariza-
tion) introduces dimensionful parameters (e.q., λ becomes dimensionful) which breaks
26E.g.,
∫d4x∂φ(x)∂φ(x) →
∫d4x′∂′φ′(x′)∂′φ′(x′) =
∫d4xa4
(∂
∂(ax)aφ
(ax
a
))2
=∫
d4xa4a−4∂xφ∂xφ .
(2.347)
70
the scale invariance. At the end of the day, when the regularization is removed, it may
be that the scale invariance is not restored. Then the physical value of the coupling
constant depends on the energy (or length) scale, and the dependence is described by
a Beta function.
For string theory, the story with the Weyl invariance is similar. The key question
is if the path integral is independent of choices of metrics related by Weyl transfor-
mations:
Z[Λ(σ)hαβ]?= Z[hαβ] . (2.349)
We are also interested in path integrals with additional operator insertions, or in
other words, operator expectation values:
〈· · ·〉h =
∫DXDbDce−iSP [X,b,c,h] · · · . (2.350)
Then Weyl invariance would require
〈· · ·〉Λh = 〈· · ·〉h . (2.351)
In evaluating the path integral, we need to introduce some regularization prescrip-
tion27. That in turn will break the Weyl invariance.
The energy momentum tensor Tαβ is defined as the infinitesimal variation of the
path integral with respect to the metric:
δ〈· · ·〉h = − 1
4π
∫d2σ
√−hδhαβ〈Tαβ · · ·〉h . (2.352)
Weyl invariance requires Tαα = 0. However, it turns out that if the central charge
c 6= 0, the Weyl invariance is broken at quantum level, due to regularization effects.
The stress tensor then has a non-vanishing trace
T αα = − c
12Rh , (2.353)
where Rh is the worldsheet Ricci scalar28. (2.353) is known as the Weyl anomaly.
2.17 The Bosonic String Beta Functions and the Effective
Action
Now consider a string moving in the coherent background of massless on-shell strings.
For small background perturbations about the empty flat target space, the path
integral was
Z =
∫DhDXe−i(SP +Vmassless) '
∫DhDXe−iSP 1− iVmassless + . . . . (2.354)
27You should review the path integral description of a QM harmonic oscillator in the Pauli-Villarsscheme.
28So far a flat worldsheet, Rh = 0 and there’s no anomaly.
71
Consider now an infinitesimal Weyl variation δΛ, we get
δΛ〈1〉h = δΛZ =
∫DhDXe−iSP
− c
12Rh − iδΛV + . . .
. (2.355)
In addition to the Weyl anomaly that we already discussed, we get additional con-
tributions from the Weyl variation of the vertex operator. Adding together linear
combinations (Fourier sums) of vertex operators, and contributions from higher order
terms, after a long and involved analysis, the result can finally be written as
δΛ〈1〉h = 〈− 1
2α′βG
µνhαβ∂αXµ∂βXν − i
2α′βB
µνεαβ∂αXµ∂βXν − 1
2βφRh〉h ≡ 〈Tα
α 〉h(2.356)
so the trace of the stress tensor is
T αα = − 1
2α′βG
µνhαβ∂αXµ∂βXν − i
2α′βB
µνεαβ∂αXµ∂βXν − 1
2βφRh , (2.357)
and the coefficients βG, βB, βφ are, up to two spacetime derivatives,
βGµν = α′Rµν + 2α′∇µ∇νφ− α′
4HµλρH
λρν +O(α′2)
βBµν = −α′
2∇λHλµν + α′∇λφHλµν +O(α′2)
βφ = − c6− α′
2∇2φ + α′∇λφ∇λφ +O(α′2)
. (2.358)
For the Weyl invariance to be restored at the quantum level (no anomaly), the trace
must vanish, T αα = 0, which requires
βGµν = βB
µν = βφ = 0 . (2.359)
The coefficients (or functions) β govern the dependence on the physics on worldsheet
scale. The expressions (2.358) encode the leading order effects at long (target space)
distances, at shorter distances there are higher (O(α′2)) stringy corrections. The
equations (2.359) tell that the β-functions vanish and these equations look like field
theory equations of motion. The equation βGµν = 0 resembles the Einstein equation
with matter field sources, and βφ = 0 resembles a scalar field equation. The equation
βBµν = 0 is a 3-form generalization of Maxwell’s equation.
Indeed, the field equations (2.358), (2.359) can be derived from the target space
action
SEFT =1
2κ20
∫dDx
√−Ge−2φ
− 2c
3α′+ RG − 1
12HµνλH
µνλ + 4∂µφ∂µφ +O(α′)
,
(2.360)
where c = D − 26 for the bosonic string in D dimensions.
The action (2.360) is written in a specific target space coordinate system, known
as the “string frame”. It contains terms that are almost like the Einstein-Hilbert
action for gravity and kinetic term for a scalar in D dimensions, except that there
72
is an overall e−2φ factor so that the terms are not in the standard form. If φ is a
constant (or has a constant factor, φ = φ0 + φ), then e−2φ0 is an overall coupling
constant factor g−2.
We can express the action in another form, because we have a freedom to make
field redefinitions. By defining
Gµν = exp
(4(φ0 − φ)
D − 2
)Gµν (2.361)
φ = φ− φ0 (2.362)
the action can be shown to take the form
SEFT =1
2κ2
∫dDx
√−G
RG −
1
12e−8φ/(D−2)HµνλH
µνλ
− 4
D − 2∂µφ∂µφ− 2c
3α′e4φ/(D−2) +O(α′)
. (2.363)
Now the action contains standard expressions for the Einstein-Hilbert term and the
scalar kinetic term. The overall factor
κ = κ0eφ0 = (8πGN)1/2 =
(8π)1/2
MPl
(2.364)
is the observed gravitational constant in D dimensions, it also gives the D-dimensional
Planck mass MPl. The coordinate frame where the metric components are Gµν is
known as the Einstein frame. Note that string frame and Einstein frame are related
by a Weyl transformation in D dimensions.
The important part about the vanishing of the β-functions was that it gives a con-
sistency condition for allowed string backgrounds. We have said before that Lorentz
invariance and unitarity (the decoupling of ghosts) requires that conformal/Weyl sym-
metry of string theory is preserved at quantum level. Without background fields, in
flat target space that required only that total central charge c = 0 (i.e., D = 26 for
the bosonic string).
In more general backgrounds, the condition c = 0 is replaced by a more compli-
cated set of consistency equations β = 0. So the allowed backgrounds contain all
kinds of curved spacetimes with suitable dilaton and antisymmetric field configura-
tions. From the point of view of 4-dimensional physics, the hope is that the back-
ground consistency conditions will contain solutions where 6 dimensions are compact
and of very small size.
***** END OF LECTURE 7 *****
73
2.18 An Example of a One-loop Amplitude: the Vacuum-to-
vacuum Amplitude, i.e., the Partition Function
We have discussed some simple bosonic tree-level scattering processes. At one-loop
level, the simplest process to consider is the vacuum-to-vacuum scattering. It illus-
trates nicely some physics and technical aspects of string theory, so we will discuss it
next. (Added “bonus”: good material examwise .)
Recall that for a point particle, a one-loop vacuum amplitude involves a circle
diagram (Fig. 28 a)). So the closed string analogue is a torus T 2 (Fig. 28 b)). In
k
a) b)
Figure 28: a) Point particle loop. b) Closed string loop.
the path integral formulation, the one-loop vacuum amplitude would be the partition
function
Z =
∫
T 2
DXDh
Vol(Diff)× Vol(Weyl)e−SP , (2.365)
where we integrate over all torii which are not related by Diff×Weyl transformations
(since we have already removed Diff×Weyl transformations from the integral/sum by
gauge fixing).
Specifically, in the path integral we must integrate over the moduli space of the
torus,
M1 =metrics
Weyl × Diff , (2.366)
the space of all metrics on the toroidal worldsheet, not related by Weyl transfor-
mations or diffeomorphisms. The parameters τi ∈ M1 are called moduli (modular
parameters). What are the moduli of the torus?
We can define a (complex) torus by periodic identifications of the complex plane.
Pick two (linearly inequivalent) complex numbers λ1, λ2 ∈ C and consider them as
vectors on the complex plane. Then identify all points shifted by an integer amount
of λ1 or λ2:
z ' z + mλ1 + nλ2 . (2.367)
Then the complex plane is tiled by cells, see Fig. 29. Since complex rescalings
z → λz, λ ∈ C are one way to relate equivalent torii, in order to count just the
inequivalent one we can rescale by 1/λ1 so that the ratio λ2/λ1 is the only relevant
74
2
Re z
Im z
λ
λ1
Figure 29: Construction of a torus by tiling of the complex plane.
complex parameter. In other words, we can set λ1 = 1, λ2 ≡ τ is then the only
complex parameter which characterizes the torus:
z ' z + m + nτ . (2.368)
We can also restrict to =τ > 0 because of the freedom of interchanging λ1 ↔ λ2. The
parameter τ is called a Teichmuller parameter, and it describes a point in Teichmuller
space:
Teichmuller = conformally inequivalent complex torii .
However, there is a further restriction on τ . Torii are also related by so called global
diffeomorphisms which are not smoothly connected to the identity transformation.
In the case of the torus, these turn out to be generated by the transformations
τ → τ + 1 (2.369)
τ → −1/τ (2.370)
(corresponding to so called Dehn twists on the two homology cycles of the torus).
They generate the group SL(2,Z)/Z2:
τ → aτ + b
cτ + d, a, b, c, d ∈ Z , ad− bc = 1 (2.371)
(mod out by Z2: a, b, c, d → −a,−b,−c,−d does nothing).
The group of global diffeomorphisms which leaves the Riemann surface invariant
is called its modular group. So SL(2,Z)/Z2 is the modular group of the torus. The
transformations τ → aτ+bcτ+d
are called modular transformations of the torus. The
moduli space of the torus is then
M1 =Teichmuller
modular group=τ ∈ C|=τ > 0
SL(2,Z)/Z2
=
τ ∈ C∣∣∣|<τ | < 1
2, |τ | > 1
. (2.372)
75
-1
τ
-1/2 11/2
Re τ
Im
Figure 30: The moduli space M1 of the torus is the semi-infinite vertical strip.
This is also called the fundamental domain of the torus. See Fig. 30. The path
integral for the torus amplitude will then contain an explicit integral over the moduli
space:
ZT 2 =
∫
M1
dτdτZ(τ) . (2.373)
Let us then calculate the torus (= one-loop vacuum) amplitude. Rather than evalu-
ating the path integral, it is simpler to do the calculation using the Hamiltonian for-
mulation. Using the relation between the path integral and Hamiltonian formalism,
we can consider the partition function/one-loop amplitude for a torus with modular
parameter τ = τ1 + iτ2 ∈ C as a trace
Z(τ) = Tre−2πτ2H+2πiτ1P
. (2.374)
The Hamiltonian
H = L0 + L0 − 2a (2.375)
generates translations in (Euclidean worldsheet) time σ2, and in the above we perform
a time evolution for a time 2πτ2. The momentum
P = L0 − L0 (2.376)
generates translations in σ1, in the above we translate σ1 by 2πτ1. We then identify
the ends, which in the operator language corresponds to taking a trace over the string
states. Using (2.375), (2.376), the partition function (2.374) takes the form
Z(τ) = (qq)−(D−2)/24 Tr(qL0 qL0
)(2.377)
where29 we used a = (D − 2)/24. I am cutting some corners and will use the light-
cone guage. (In the textbooks, the calculation is most often done in the covariant
29q ≡ e2πiτ , q ≡ e−2πiτ
76
formulation, and then one must also include the b, c ghosts and take the trace trace
over them as well - that involves some subtleties). In the light-cone gauge, I will use
L0 =1
2α2
0 +∞∑
n=1
αi−nα
in =
l2s8
p2 +D−2∑i=1
∞∑n=1
n ai−nai
n︸ ︷︷ ︸≡Nin
≡ α′
4p2 +
D−2∑i=1
∞∑n=1
nNin . (2.378)
Similarly,
L0 =α′
4p2 +
D−2∑i=1
∞∑n=1
nNin . (2.379)
The trace in (2.377) then breaks up into an integral over the center-of-mass momenta
kµ and a sum over the occupation numbers Nin, Nin. Note that
qL0 qL0 = e2πi(τ1+iτ2)α′4
p2+...e−2πi(τ1−iτ2)α′4
p2+... = e−πτ2α′p2+... . (2.380)
We then get
Z(τ) = (qq)−(D−2)/24VD
∫dDk
(2π)De−πτ2k2α′
D−2∏i=1
∞∏n=1
∞∑Nin=0
∞∑
Nin=0
qnNin qnNin . (2.381)
The D-dimensional target spacetime volume factor comes from the usual conversion
of the latticized momentum sum to an integral:∑
k → VD
∫dDk
(2π)D .
Note that in the above the metric signature in the target space is Minkowski, we
must rotate to Euclidean signature to get a convergent momentum integral:
k0 → ikD
dDk = dk0dk1 · · · dkD−1 → idk1dk2 · · · dkD ≡ idDkE
k2 = −(k0)2 + (~k)2 → (k1)2 + . . . + (kD)2 .
The occupation number sums involve∑∞
N=0(qn)N = (1−qn)−1. Thus, (2.381) becomes
Z(τ) = iVD
[∫ ∞
−∞
dk
2πe−πτ2α′k2
]D
(qq)−(D−2)/24
D−2∏i=1
∞∏n=1
(1− qn)−1(1− qn)−1
= iVD
[1
2π
√π
πτ2α′
]D[q−1/24
∞∏n=1
(1− qn)−1q−1/24
∞∏n=1
(1− qn)−1
]D−2
.(2.382)
Introducing the Dedekind eta function
η(τ) = q1/24
∞∏n=1
(1− qn) (2.383)
the result becomes (with D = 26 for the bosonic string)
Z(τ) = iV26[4π2τ2α
′]−13|η(τ)|−48 . (2.384)
77
To get the full one-loop vacuum amplitude, we must include the integral over the
moduli space. An additional subtlety is that we need to factor out reflections σa →−σa and translations σa → σa + aa which leave the metric of the torus, ds2 =
2((dσ1)2 + (dσ2)2), invariant. The former introduces an overall factor 1/2 and the
latter means that we have to divide the measure by the area of the torus, proportional
to τ2. The correct result then turns out to be
ZT 2 =
∫dτdτ
4τ2
Z(τ) = iV26
∫dτdτ
4τ2
[4π2τ2α′]−13|η(τ)|−48 . (2.385)
Now an important consistency check is that the one-loop amplitude indeed be
modular invariant. I will leave it as an exercise to check that
dτdτ
τ 22
(2.386)
is modular invariant, and so is
τ2|η(τ)|4−12 . (2.387)
In order to understand the physics content of the amplitude, it is useful to compare it
with the corresponding quantity in field theory, the sum over all particle paths with
the topology of a circle. The latter is given by
ZS1(m2) = VD
∫dDk
(2π)D
∫ ∞
0
dl
2le−(k2+m2)l/2 = VD
∫ ∞
0
dl
2l(2πl)−D/2e−m2l/2 . (2.388)
In the above, m is the mass of a point particle, 12(k2+m2) is the worldline Hamiltonian,
l is the modulus of the circle, and 2l in the denominator removes the overcounting
from reversal and translation of the worldline coordinate.
Now, we take the point particle result and sum over all states of the string which
we interpret as different point particles (in the field theory approximation). Recall
the mass formula (2.105) (again using α′ ≡ l2s/2 and a = 1)
m2 =2
α′(NL + NR − 2) , (2.389)
and the level matching condition NL = NR. It is useful to write that condition in the
integral form
δNL,NR=
∫ π
−π
dθ
2πei(NL−NR)θ . (2.390)
Let us then sum over the physical string spectrum (again in light-cone gauge so the
Hilbert space includes just the transverse oscillations):
∑i∈H⊥
ZS1(m2i ) = iVD
∫ ∞
0
dl
2l
∫ π
−π
dθ
2π(2πl)−D/2
∑i∈H⊥
e−(NLi+NRi−2)l/α′−i(NLi−NRi)θ
= iVD
∫
R
dτdτ
4τ2
(4π2α′τ2)−D/2
∑i∈H⊥
qNLi−1qNRi−1 (2.391)
78
where we introduced τ = θ+il/α′2π
. The region of integration R is
R : τ2 > 0 , |τ1| < 1
2. (2.392)
Note that the one-loop amplitude (2.388) for a single particle diverges, with the
divergence arising from the short-distance (ultraviolet) limit l → 0. Summing over
the string spectrum only makes this worse as all states contribute with the same
sign - (2.391) diverges in the τ2 → 0 limit. However, (2.388) is not the actual string
amplitude - it is similar to (2.385), but the latter actual string amplitude has a
different integration region: it was
M1 : |τ | > 1 , |τ1| < 1
2. (2.393)
Thus, while the point particle one-loop amplitude suffers from the usual UV diver-
gence of quantum field theory, the UV divergent region is absent from the correspond-
ing string amplitude!
Another possible divergence comes from the long-distance (infrared) limit τ2 →∞ where torus becomes very long. In this region, the asymptotic behavior of the
integrand is controlled by the lightest string states (the massive ones are exponentially
suppressed), and the amplitude has the expansion
Z ' iV26
∫ ∞ dτ2
2τ2
(4π2α′τ2)−13[e4πτ2 + 242 + . . .] . (2.394)
The series is in the increasing order of mass2. The first term diverges due to the
positive exponential, and arises from the tachyon. The other terms converge. Hence
the divergence is an artifact of the tachyon in the bosonic string spectrum. Other
more realistic string theories (superstring, heterotic string) do not have tachyons and
do not suffer from the IR divergence either.
So we have seen a general feature of string theory, which holds for all string
amplitudes: there is no UV region of moduli space that might give rise to high-
energy divergences! More over, all limits are in fact controlled by the lightest states
- the long-distance physics.
2.18.1 The Vacuum Energy
Another important physics aspect of the vacuum amplitude is its relation to the
vacuum energy, or cosmological constant.
In point particle theory, vacuum paths consist of any number of disconnected
circles:
©+©©+©©©+ . . . (2.395)
Including a factor of 1/n! for permutation symmetry and summing in n gives
Zvac(m2) = ©+
1
2!©©+
1
3!©©©+ . . . = e© = eZS1 (m2) . (2.396)
79
On the other hand, in field theory
Zvac(m2) = 〈0|e−iHT |0〉 = e−iρ0VD (2.397)
where ρ0 is the vacuum energy density
ρ0 =i
VD
ln Zvac(m2) =
i
VD
ZS1(m2) . (2.398)
The l-integral in ZS1(m2) diverges as l → 0, but we can get some insight by
inserting a regulator by cutting off the integral at l = ε, then drop the divergent
terms and take ε → 0. This gives
limε→0
∫ ∞
ε
dl
2le−(k2+m2)l/2 = −1
2ln(k2 + m2) + div. , (2.399)
and for the vacuum energy:
ρ0 =i
VD
ZS1(m2) = i
∫dk0dD−1~k
2π(2π)D
∫ ∞
0
dl
2le−(k2+m2)l/2 →
∫dD−1~k
(2π)D−1
ω~k
2(2.400)
where ω~k =√
~k2 + m2. The result (??) is just the usual sum of zero point energies.
To make contact with the field theory calculation, compare with the path integral for
a massive scalar field:
ρ0 =i
VD
ln Zvac(m2) = − i
2VD
Tr ln(−∂2 + m2) = −iVD
2
∫dDk
(2π)Dln(k2 + m2) .
(2.401)
By (2.399), (2.401) is the same as the point particle result (2.400).
The generalization of the vacuum energy calculation to a theory with particles of
arbitrary spin is
ρ0 =i
VD
∑i
(−1)FiZS1(m2i ) (2.402)
where the sum runs over all physical particle states. The Fi is the spacetime fermion
number (1 for fermions, 0 for bosons) so fermions contribute with the opposite sign.
The vacuum energy gives a source term in the Einstein’s equation, the cosmo-
logical constant. Observations indicate that spacetime is nearly flat and thus the
cosmological constant is very small,
|ρ0| . 10−44 (GeV)4 . (2.403)
In contrast, if one evaluates (2.402) from field theory including vacuum fluctuations
of all the known particles up to the currently explored energy ∼ mEW (electroweak
scale) one obtains a large vacuum energy
|ρ0| ∼ m4EW ∼ 108 (GeV)4 . (2.404)
80
This is 52 orders of magnitude too large!
In bosonic string theory, the one-loop vacuum energy was also non-zero. On
dimensional grounds it must be of the order of string scale (or Planck scale). In four
dimensions this would be even more enormous:
|ρ0| ∼ 1072 (GeV)4 . (2.405)
Note that when we considered strings in a background, the beta functions were in-
terpreted as equations of motion of the effective field theory
SEFT =1
2κ2
∫dDx
√−G
RG − 2c
3α′e4φ/(D−2) + . . .
. (2.406)
For a constant dilaton background, the second term (which arises from tree-level) is
a cosmological constant term. However, c = 26−D for bosonic string, so at tree-level
the cosmological constant is zero.
But we just found that at one-loop level there is an enormous contribution (2.405)
to the cosmological constant. So at 1-loop level Minkowski space with constant dilaton
is most definitely not a solution of the β-function equations, i.e., not a good string
background, unlike what was implicitely assumed when we started to quantize the
string!
(In supersymmetric theories the cosmological constant is zero and Minkowski
space is a consistent string background).
***** END OF LECTURE 8 *****
3 Superstrings (“Where It Begins Again”)
The bosonic string is an interesting model, but unsatisfactory because of two obvious
shortcomings. There are no fermions in the theory, and the theory contains a tachyon
which is a sign of troubles (like instability of the theory, or causal issues like sending
messages backwards in time containing a message asking someone to kill your mother,
etc.). The tachyon problem in principle could have a solution within bosonic string
theory. It could be that what we have been calling the vacuum is not the true
vacuum (when all perturbative and non-perturbative effects have been included).
Reformulating the theory (and the action) around the true vacuum could then lead
to a spectrum without the tachyon. But no one has succeeded to do that (yet).
It is then more fruitful to move forward and search for a new theory. We will
now add fermionic excitations to the model. This is done in a special way, there is a
new worldsheet symmetry that relates the fermions with the bosons. It is called the
worldsheet supersymmetry. Supersymmetry is hoped to be a yet unraveled symmetry
of Nature. It is hoped that it will be found experimentally in the next generation of
81
accelerators. Supersymmetry is a topic of a separate course, so I’m assuming that
you will use that as a source for more details. Here we will not need that much of the
machinery, for the most part you can just consider the superstring as a model which
includes bosonic and fermionic excitations.
I will also use some concepts from part I of this course without motivating them
again, to keep things concise.
3.1 The Superstring Action
The superstring action is a jazzed up Polyakov action,
S = − 1
2πα′
∫d2σ
∂αXµ∂
αXµ − iψµρα∂αψµ
(3.1)
where we have added D Majorana fermions ψµ = (ψµα), µ = 0, 1, 2, . . . , D − 1. The
index µ is a target space vector index, so ψµ transforms in the vector representation
of SO(1, D − 1). The index α is a worldsheet (spinor) index.
The notation ρα refers to Dirac matrices in 1+1 dimensions, satisfying the Clifford
algebra ρα, ρβ
= −2ηαβ . (3.2)
A representation that you see often is that ρα are imaginary,
ρ0 =
(0 −i
i 0
), ρ1 =
(0 i
i 0
), ρ3 ≡ ρ0ρ1 =
(1 0
0 −1
). (3.3)
Then the Dirac operator iρα∂α is real, and the spinors ψµ can have real components.
They are built from 1-component Weyl spinors ψµ+, ψµ
−:
(ψµα) =
(ψµ−
ψµ+
), (3.4)
both components are real. The symbol ψ means
ψ = ψ†ρ0 = ψT ρ0 . (3.5)
Note that spinors are anticommuting variables, ψαχβ = −χβψα.
Now, in addition to the symmetries of the bosonic action, there is a new symmetry,
δXµ = εψµ
δψµ = −iρα∂αXµε (3.6)
where the parameter ε is global (independent of worldsheet coordinates σα), and
also a two-component real Majorana spinor. This transformation obviously maps the
82
worldsheet bosons to fermions and vice versa. It is known as a N = 130 worldsheet
supersymmetry31.
Let us again use the null coordinates (as in (??),(??)):
σ± = τ ± σ , ∂± =1
2(∂τ ± ∂σ) . (3.7)
Now we can rewrite the action (3.1) as
S =1
πα′
∫d2σ∂+Xµ∂−Xµ +
i
2πα′
∫d2σ (ψµ
+∂−ψ+µ + ψµ−∂+ψ−µ) . (3.8)
Using the notation ε =(
ε−ε+
)we can divide the SUSY32 transformations in left- and
rightmoving pieces:
δXµ = iε+ψµ
−δψµ
− = −2∂−Xµε+,
δXµ = −iε−ψµ
+
δψµ+ = 2∂+Xµε−
. (3.9)
Note that the field equation for ψ± have become
∂+ψ− = 0 → ψ− = ψ−(σ−)
∂−ψ+ = 0 → ψ+ = ψ+(σ+) . (3.10)
You may have noticed that the bosonic sector of the actions (3.1),(3.8) is the gauge
fixed form of the Polyakov action, in the covariant gauge hαβ = δαβ. Gauge fixing has
eliminated the variable hαβ and its equation of motion is now a constraint Tαβ = 0.
Similarly, the fermionic sector also involves a gauge fixing. You should read the
details from Bailin & Love, p. 176-178. The real starting point should be an action
that includes the superpartner of hαβ, the gravitino χα. The correct action can
be found by promoting the SUSY transformations (3.6) to local transformations,
ε = ε(σα). The action that is invariant under local SUSY, contains the gravitino χα.
That action has the bosonic reparameterization & Weyl symmetries, but also a local
fermionic symmetry, the superconformal symmetry. The gauge fixing then involves
hαβ = δαβ, but also χα = 0, leading to constraints
Tαβ = ∂αXµ∂βXµ +i
4ψµ(ρα∂β + ρβ∂α)ψµ
− 1
2ηαβ
(∂γXµ∂γXµ − i
2ψµργ∂γψµ
)= 0 (3.11)
Jα =1
2ρβραψµ∂βXµ = 0 . (3.12)
30Or N = 1?31N = 1, since only 1 spinorial parameter ε.32Told already SUSY = supersymmetry?
83
The equation (3.11) is the constraint which replaces the equation of motion of hαβ,
with fermions now included in the stress tensor Tαβ.
The equation (3.12) replaces the equation of motion of χα. The Jα is a new
current, the worldsheet supercurrent.
These are the constraints that need to be included when using the gauge fixed
action (3.1) or (3.8).
3.2 Equations of Motion and Boundary Conditions
The bosonic part of the action (3.8) is the same as before, and we have already
discussed its equations of motion etc. The new part is the fermions, so we focus on
them:
δSF =i
2πα′
∫d2σ δψ+∂−ψ+ + ψ+∂−(δψ+) + (. . .)
=i
2πα′
∫d2σ δψ+∂−ψ+ + ∂−(ψ+δψ+)− (∂−ψ+)δψ+ + (. . .)
=i
2πα′
∫d2σ2δψ+(∂−ψ+) + 2δψ−(∂+ψ−)
+ ∂−(ψ+δψ+) + ∂+(ψ−δψ−) = 0 (3.13)
leads to the equations of motion
∂+ψ− = ∂−ψ+ = 0 (3.14)
(which I already mentioned) provided that the boundary term vanishes:
∫ ∞
−∞dτ
∫ 2π
0
dσ∂+(ψ−δψ−) + ∂−(ψ+δψ+) = 0 . (3.15)
Using ∂± = 12(∂τ ± ∂σ) and assuming that
limτ→±∞
(δψ±) = 0 , (3.16)
we need ∫ 2π
0
dσ∂σ(ψ−δψ−)− ∂σ(ψ+δψ+) = 0 (3.17)
for a closed string. Since left and right movers are independent, the boundary terms
for ψ∓ must vanish independently:
ψ∓δψ∓∣∣∣σ=2π
= ψ∓δψ∓∣∣∣σ=0
. (3.18)
This is possible, if we choose the functions to be either periodic or antiperiodic:
ψµ−(2π) = ±ψµ
−(0)
ψµ+(2π) = ±ψµ
+(0) (3.19)
84
(±: + periodic, − antiperiodic). Then δψ± are also periodic or antiperiodic, and the
boundary terms vanish. So now we have two kinds of boundary conditions, with the
following names associated with them
Ramond (R) boundary cond. : ψµα(σ = 2π) = ψµ
α(σ = 0) (3.20)
Neveu− Schwarz (N) boundary cond. : ψµα(σ = 2π) = −ψµ
α(σ = 0) . (3.21)
Memorizing rule: “Neveu-Schwarz = Anti-Periodic”. The right movers could satisfy
either one of the two b.c.’s and the left movers could satisfy either one, so alltogether
there are 4 possibilities, and the Fock space is divided into 4 sectors:
NS-NS, NS-R, R-NS, R-R .
3.3 Mode Expansions and Quantization
I’ll just list the relevant formulas:
NS− sector : ψµ− =
∑
r∈Z+ 12
bµr e−ir(τ−σ) (3.22)
and
ψµ+ =
∑
r∈Z+ 12
bµr e−ir(τ+σ) . (3.23)
Quantization:
bµr , b
νs = bµ
r , bνs = ηµνδr+s,0 (3.24)
bµr , b
νs = 0 . (3.25)
Number operators:
N(b)− =
∑
r∈Z+ 12
rbµ−rbrµ (3.26)
N(b)+ =
∑
r∈Z+ 12
rbµ−rbrµ (3.27)
(you can derive these by substituting the mode expansions into the Hamiltonian, see
Bailin & Love).
R− sector : ψµ− =
∑
n∈Zdµ
ne−in(τ−σ) (3.28)
and
ψµ+ =
∑
n∈Zdµ
ne−in(τ+σ) . (3.29)
85
Quantization:
dµn, d
νm = dµ
n, dνm = ηµνδn+m,0 (3.30)
dµn, d
νm = 0 . (3.31)
Number operators:
N(d)− =
∞∑n=1
ndµ−ndnµ (3.32)
N(d)+ =
∞∑n=1
ndµ−ndnµ . (3.33)
Note that the number operators N(d)∓ do not contain the zero mode operator dµ
0 . It
will have a special role, to be discussed later.
Again, we can make the following interpretations:
b−r, b−r, d−n, d−n = creation operators (3.34)
br, br, dn, dn = annihilation operators , (3.35)
for r, n > 0. The NS vacua in the left- and rightmoving sectors are defined as follows:
|0〉−NS : αµn|0〉−NS = bµ
r |0〉−NS = 0 , (3.36)
∀n ≥ 0, r > 0, where αµn are the bosonic annihilation operators, and
|0〉+NS : αµn|0〉+NS = bµ
r |0〉+NS = 0 , (3.37)
respectively. The R vacua are defined in a similar fashion:
|0〉−R : αµn|0〉−R = dµ
m|0〉−NS = 0 (3.38)
|0〉+R : αµn|0〉+R = dµ
m|0〉+NS = 0 , (3.39)
for all n ≥ 0,m > 0. In order to analyze the spectrum, we will again need the
mass-shell formula, which was derived from the Virasoro constraints. We could again
calculate T++ and T−−, substitute the mode expansions of XµL,R, ψµ
± and derive ex-
pressions for the Virasoro generators Ln, Ln in terms of the oscillators. For the
mass-shell formula, we will need L0, L0. Again, there we define them to be normal
ordered and isolate the constant contributions arising from the commutators when
creation operators are moved to the left of annihilation operators. We will only quote
the results:
LNS0 =
1
2αµ
0α0µ +∞∑
n=1
αµ−nαnµ +
∞∑
r= 12
rbµ−rbrµ (3.40)
LNS0 =
1
2αµ
0 α0µ +∞∑
n=1
αµ−nαnµ +
∞∑
r= 12
rbµ−rbrµ , (3.41)
86
where αµ0α0µ = l2s
4p2 as before, and
LR0 =
1
2αµ
0α0µ +∞∑
n=1
αµ−nαnµ +
∞∑m=1
mdµ−mdmµ (3.42)
LR0 =
1
2αµ
0 α0µ +∞∑
n=1
αµ−nαnµ +
∞∑m=1
mdµ−mdmµ , (3.43)
with α20 = l2s
4p2.
Again, there is a level matching condition
L0|phys〉 = L0|phys〉 (3.44)
and the mass-shell formula comes from the constraint
(L0 − a)|phys〉 = (L0 − a)|phys〉 = 0 , (3.45)
where L0 = LNS0 or LR
0 and a = aNS or aR and similarly for L0. Alltogether there are
4 combinations, corresponding to the NS-NS, NS-R, R-NS and R-R sectors.
A generic state in the Fock space has the form
|N〉 ⊗ |N〉 =∏
i
∏j
(αµj
−nj)aj(f νi
−ki)bi|0〉 ⊗
∏p
∏q
(αµp
−np)ap(f
νq
−kq)bq |0〉 , (3.46)
where |0〉 is either |0〉−NS or |0〉−R and |0〉 is either |0〉+NS or |0〉+R; and
f νi−ki
=
bνi−ri
(NS sector) or
dνi−mi(R sector)
(3.47)
with r = integer + 12, m = integer, depending on whether we are building states on
|0〉NS or |0〉R. (Similarly for f .) The total levels are
N =∑
j
ajnj +∑
i
biki , ki = ri or mi (3.48)
N =∑
p
apnp +∑
q
bqkq , (3.49)
level matching requires N = N . It can be shown that the zero point energies are
aR = 0 (3.50)
aNS =1
2. (3.51)
The mass-shell conditions (with p2 = −M2) give
M2 =4
l2sNbos + Nbos + Nfer + Nfer − 4a
l2s(3.52)
87
with
Nbos =∞∑
n=1
αµ−nαnµ (3.53)
Nfer =∞∑
r= 12
rbµ−rbrµ or
∞∑m=1
mdµ−mdmµ (3.54)
(similarly for Nbos, Nfer); and
a = aL + aR =
aNS
aR
+
aNS
aR
. (3.55)
3.4 Constraints on Physical States
Because of the indefinite signature of the metric, the Fock space again contains un-
physical states, to be removed by the constraint conditions.
In light-cone coordinates, the constraint equations are
T++ = ∂+XµL∂+XLµ +
i
2ψµ
+∂+ψµ+ = 0 (3.56)
T−− = ∂−XµR∂−XRµ +
i
2ψµ−∂−ψµ
− = 0 (3.57)
J+ = ψµ+∂+XLµ = 0 (3.58)
J− = ψµ−∂−XRµ = 0 (3.59)
(c.f. eqn (2.75)). We will substitute the mode expansions and work with the Fourier
components(∼)
Ln =1
4πl2s
∫ 2π
0
dσ±einσ±T±±(σ+) . (3.60)
For the superconformal current, in the NS sector we define
NS :(∼)
Gr ≡ 1
4πl2s
∫ 2π
0
dσ±eirσ±J±(σ±) (3.61)
where r = integer + 12; in the R sector we define
R :(∼)
Fm ≡ 1
4πl2s
∫ 2π
0
dσ±eimσ±J±(σ±) (3.62)
where m = integer. In terms of the mode operators, the results are
Ln = L(α)n + L(b)
n (NS) (3.63)
Ln = L(α)n + L(d)
n (R) (3.64)
88
where L(α)n are the bosonic components
L(α)n =
1
2:
∞∑m=−∞
α−m · αn+m: (3.65)
as before, and
L(b)n =
1
2:
∑
r∈Z+ 12
(r +
n
2
)b−r · bn+r: (3.66)
L(b)n =
1
2:∑
m∈Z
(m +
n
2
)d−m · dn+m: . (3.67)
From the superconformal current, we obtain fermionic generators (Ln are bosonic):
Gr =∞∑
n=−∞α−n · br+n (3.68)
Fm =∞∑
n=−∞α−n · dm+n . (3.69)
The super-Virasoro algebra of the generators in the NS sector is
[Lm, Ln] = (m− n)Lm+n +1
8Dm(m2 − 1)δm+n,0 (3.70)
NS : [Lm, Gr] =(m
2− r
)Gm+r (3.71)
Gr, Gs = 2Lr+s +1
2D
(r2 − 1
4
)δr+s,0 , (3.72)
where D is the target spacetime dimension. The anomaly coefficients (the last terms
in (3.70), (3.72)) are obtained by considering vacuum expectation values. Note that
the central charge is now different from the purely bosonic string case.
Note also that L±1,0 and G± 12
generate a closed superalgebra, known as OSp(1|2).
It is a N = 1 supersymmetric extension of SL(2,R).
In the R sector, the super-Virasoro algebra has m3 instead of m(m2 − 1), but
this could be cured by shifting L0 by a constant. The above choice is a convenient
convention. Now, if you try to add F0 into the algebra of L0, L±1, you will generate
all the other generators! So in the R sector there is no extension of SL(2,R).
Physical states must satisfy the constraints
Lm|phys〉 = Lm|phys〉 = 0 , m > 0 (3.73)
(L0 − a)|phys〉 = (L0 − a)|phys〉 = 0 (3.74)
NS :(∼)
Gr|phys〉 = 0 , r > 0 (3.75)
R :(∼)
Fm|phys〉 = 0 , m > 0 . (3.76)
89
Note: now because a 6= a in the NS-R, R-NS sectors, the level matching condition is
not simply L0|phys〉 = L0|phys〉, but a, a need to be included as well. So forget the
eqn (3.44)!
3.5 Emergence of Spacetime Spinors
So far we have discussed worldsheet bosons Xµ and spinors ψµα. Now we want to
interpret the Fock space states as being bosons or fermions from the target space
point of view. We start from Ramond sector.
The Ramond sector contained the zero mode operators dµ0 which were not required
to annihilate the Ramond vacuum. They are neither creation or annihilation opera-
tors, and they commute with LR0 = L
(α)0 +L
(d)0 . Thus, any eigenstate of LR
0 is mapped
to another eigenstate by acting by dµ0 . So they must form a representation of the
algebra of dµ0 ’s:
dµ0 , d
ν0 = ηµν . (3.77)
If we define Γµ = i√
2dµ0 , the algebra can be written as
Γµ, Γν = −2ηµν (3.78)
which is the 1+9 dim. Clifford algebra. Its irreducible representations correspond
to spinors of SO(1, 9), i.e., spinors in the D = 10 dimensional target space. Thus
every state in the R sector is a spinor, and hence fermionic in the target space. The
Γµ can be represented by Dirac matrices in 1+9 dimensions. These have 210/2 = 32
components.
In particular, the Ramond vacuum is a 32-component spinor. So we use the
notation |0〉aR for it, where a is the spacetime spinor index a = 1, . . . , 32. The Ramond
vacuum can be shown to be a Majorana spinor (following from ψµ± = real). So it has
32 real degrees of freedom.
Let’s see how we can construct |0〉aR explicitly. Define first the operators (for
simplicity, we rotate to Euclidean signature so we are considering SO(10) instead of
SO(1, 9))
ek = Γk + iΓ5+k
e†k = Γk − iΓ5+k , k = 1, . . . , 5 . (3.79)
Then the Clifford algebra takes the form of fermionic oscillator algebra:
ek, el = e†k, e†l = 0
ek, e†l = δkl . (3.80)
Then we define a “ground state” |0〉 annihilated by all ek: ek|0〉 = 0. Now the
full representation, i.e., all components of the Ramond vacuum, is constructed as
summarized in Table 2.
90
states multiplicity total
|0〉 1 1
e†k|0〉 5 5
e†ke†l |0〉
(52
)10
e†ke†l e†m|0〉
(53
)10
e†ke†l e†me†n|0〉
(54
)5
e†1e†2e†3e†4e†5|0〉 1 1
total = 32
Table 2: State multiplicities
All these states have the L0 = L(α)0 +
∑∞m=1 mdµ
−mdµm eigenvalue 0, since the
construction only involved the dµ0 modes. So they are all components of the Ramond
vacuum |0〉aR.
The vacuum has zero center-of-mass momentum kµ. Consider now the states
|0, kµ〉aR (3.81)
with non-zero momentum. In order for it to be a physical state, it must satisfy the
constraint
F0|0, kµ〉aR =∑
n∈Zαµ−ndnµ|0, kµ〉aR
= αµ0d0µ|0, kµ〉aR =
−ils
2√
2(pµΓµ|0, kµ〉R)a
= 0 . (3.82)
In other words,
pµΓµab|0, kµ〉bR = kµΓµ
ab|0, kµ〉bR = 0 . (3.83)
This is the massless Dirac equation in momentum space in 10 dimensions. So these
states are indeed target space fermions. The massive states must be spinors too, since
the spinor index comes only from dµ0 .
3.5.1 Chirality
It is possible to consider spinors of definite chirality (Weyl spinors) in 10 dimensions.
The chirality operator is
Γ11 ≡ Γ0Γ1 · · ·Γ9 . (3.84)
Then, we can split the 32-component Majorana spinor |0〉aR into two 16-component
Majorana-Weyl spinors |0,±〉aR:
Γ11|0, +〉R = +1|0, +〉RΓ11|0,−〉R = −1|0,−〉R , (3.85)
91
Fock space Target space nature
|NS〉 ⊗ |NS〉 boson
|R〉 ⊗ |R〉 boson (bispinors!)
|R〉 ⊗ |NS〉 fermion
|NS〉 ⊗ |R〉 fermion
Table 3: State interpretations
|0〉R decomposes as
|0〉R = |0, +〉R ⊕ |0,−〉R . (3.86)
Using the creation operators e†k, |0, +〉R is created by an even # of them, |0,−〉R with
an odd #33.
We have been focusing on the left moving sector, the right moving sector has an
identical structure.
The NS sector has no spinor indices. All NS sector states are spacetime bosons.
So when we put the left- and rightmovers together, we obtain the Table 3 for the
target space interpretations of the states.
An added freedom is that we can choose Γ11 = ±1 independently in the right
moving and left moving sector. We will return to that later. Before we move to
discuss the Fock space spectrum in more detail, I’d like to divert to one more issue:
3.6 The Spin Field
We can actually relate the R vacuum to the NS vacuum. Recall from over CFT
discussion that there was a natural mapping between states and operators,
|φ〉 = φ(0)|0〉 . (3.87)
So we expect that we could create the R vacuum from the NS vacuum by some
operator Sa:
|0〉aR = Sa(0)|0〉NS . (3.88)
Such an operator should be a 32-component (Majorana) spinor constructed from
the variables of the theory. Since |0〉aR was constructed using the zero modes of ψµ+
(consider again the leftmovers), we expect Sa to be constructed from the fields ψµ+.
An explicit construction is indeed possible. I will not present it here, because
it involves something called bosonization (of fermions). I will only note that since
Sa links target space bosons with fermions, it is like a target space supercharge.
33Note: from the worldsheet point of view e†k are fermionic. So |0, +〉R is a worldsheet boson,|0,−〉R is a worldsheet fermion. But both are target space fermions.
92
Moreover, to construct vertex operators for target space fermions, one need the spin
field.
Now we can finally move on to analyze the Fock space states.
3.7 Lowest Lying Excitations of Closed Superstrings
Let’s proceed sector by sector:
3.7.1 NS-NS Sector
Mass formula (now a = a = 12):
M2 =4
l2s
∞∑n=1
(αµ−nαnµ + αµ
−nαnµ) +∞∑
r= 12
(rbµ−rbrµ + rbµ
−rbrµ)− 1
(3.89)
• |0, k〉NS ⊗ |0, k〉NS:
M2 = − 4
l2s(3.90)
This is a tachyon. It is a target space scalar.
However, I will show that it can be removed from the spectrum.
• eµν(k)bµ
− 12
|0, k〉NS ⊗ bν− 1
2
|0, k〉NS:
M2 =4
l2s(1
2+
1
2− 1) = 0 (3.91)
These contain the graviton hµν , the antisymmetric tensor field Bµν , and the
dilaton φ.
• fµν(k)αµ−1|0, k〉NS ⊗ αν
−1|0, k〉NS:
M2 =4
l2s(1 + 1− 1) =
4
l2s(3.92)
Now these are massive states.
3.7.2 R-NS Sector
Mass formula (a = 0, a = 12):
M2 =4
l2s
∞∑n=1
(αµ−nαnµ + αµ
−nαnµ) +∞∑
m=1
(mdµ−mdmµ +
∞∑
r= 12
(rbµ−rbrµ)− 1
2
(3.93)
93
• |0, k〉R ⊗ |0, k〉NS:
M2 = − 2
l2s(3.94)
This can again be removed from the spectrum.
• |0, k〉R ⊗ ξµ(k)bµ
− 12
|0, k〉NS:
M2 = 0 (3.95)
This is spin 12⊗ 1 = 3
2⊕ 1
2. In other words, it decomposes into a spin 3
2field,
the gravitino ψµa and a spin 12
field χa. They are superpartners of the massless
bosons hµν , Bµν .
• massive fermions. . .
3.7.3 NS-R Sector
• The states are identical to those from R-NS sector.
3.7.4 R-R Sector
Mass formula (a = a = 0):
M2 =4
l2s
∞∑n=1
(αµ−nαnµ + αµ
−nαnµ) +∞∑
m=1
m(dµ−mdmµ + dµ
−mdmµ)
(3.96)
• |0, k〉R ⊗ |0, k〉R:
M2 = 0 (3.97)
These are bispinors, decomposing into various bosons. We will look at them in
more detail after showing how to deal with tachyons (the GSO projection).
• massive bosons. . .
3.7.5 Problems with the Spectrum
1. Tachyonic states
2. Inequal number of bosons and fermions at each mass level
3. Target space fermions correspond to both worldsheet bosons and fermions, like-
wise for target space bosons
The cure is to project out certain states from the spectrum. This is known as the
GSO projection. In the way I will present it, it will appear rather ad hoc. But it
arises naturally from some more advanced features of the theory, as a consequence of
the “1-loop modular invariance” (and 2-loop modular invariance).
***** END OF LECTURE 9 *****
94
3.8 The GSO projection (GSO=Gliozzi-Scherk-Olive)
The GSO projection acts independently on the right- and leftmovers. Consider left-
movers:
3.8.1 NS Sector
• GSO projection rule: throw away all states with even number of bµ−n’s.
Mathematically: define fermion number operator F
F ≡∞∑
r= 12
bµ−rbrµ (3.98)
→(−1)F |even # of b〉 = |even〉(−1)F |odd # of b〉 = −|odd〉 (3.99)
Then demand NS-GSO: (−1)F |phys〉 = −|phys〉 for all physical states.
Examples. The tachyonic vacuum |0〉NS is eliminated. If the projection acts iden-
tically on the rightmovers, then in the closed string spectrum |0〉NS ⊗ |0〉NS (the
tachyon!) is eliminated, and so is fµναµ−1|0〉NS ⊗ αν
−1|0〉NS.
However eµνbµ
− 12
|0〉NS ⊗ bν− 1
2
|0〉NS (the graviton etc.) are preserved.
3.8.2 R Sector
The GSO projection acts similarly in the R sector, but now there’s some more detail:
F ≡∞∑
m=1
dµ−mdmµ (3.100)
counts the # of dµm’s with m > 0. In order to count the dµ
0 ’s (modulo 2), recall
|0〉R = |0, +〉R ⊕ |0,−〉R with Γ11|0,±〉R = ±|0,±〉R . (3.101)
The |0, +〉R (|0,−〉R) contain an even (odd) # of dµ0 ’s. Thus
Γ11|φ〉 = +|φ〉 → even # of dµ0′s
−|φ〉 → odd # of dµ0′s . (3.102)
In total, to differentiate between states with even or odd # of dµm’s, m ≥ 0, we define
the projection operator
Γ ≡ Γ11(−1)F . (3.103)
95
The GSO projection then corresponds to throwing away states with Γ = +1 or −1. In
the R sector there is no natural preference for either choice. Both lead to consistent
string theories. Thus R-GSO: Γ11(−1)F |phys〉 = ±|phys〉.Note that in particular the R-GSO tells us that the vacuum is either |0, +〉R or
|0,−〉R. So the massless fermions are Majorana-Weyl spinors of definite chirality.
3.9 Type IIA and Type IIB Superstrings
When we put the left- and rightmovers together, we have two choices for the R-GSO
projection, independently on the left and right. (The NS sector has no freedom of
choice.) It turns out that this gives two consistent closed superstring theories, called
Type IIA and Type IIB theories:
Type IIA Theory Choose the left and right moving R-states to have opposite GSO
projections: in particular then
Γ11|0〉leftR = −Γ11|0〉rightR . (3.104)
So the massless (target) spacetime fermions from the NS-R & R-NS sectors,
bµ
− 12
|0〉NS ⊗ |0〉R , |0〉R ⊗ bµ
− 12
|0〉NS (3.105)
have opposite chiralities. Further, the theory has two spacetime supersymmetries (∼S(0), S(0)) of opposite chirality. Hence the “II” in IIA. Note also that the tachyonic
states |0〉NS⊗|0〉R, |0〉R⊗|0〉NS in the NS-R and R-NS sectors are eliminated already
due to the GSO projection in the NS sector (these states contain an even number of
fermionic NS creation operators: zero).
Type IIB Theory Choose the left and right moving R-states to have equal GSO
projections, then:
Γ11|0〉leftR = Γ11|0〉rightR . (3.106)
Now the massless spacetime fermions from NS-R, R-NS sectors have the same chiral-
ity. There are also two spacetime supersymmetries of the same chirality. The IIB is
a chiral theory.
In the section 3.7.4 I mentioned that the R-R sector gives rise to various bosonic
fields without being more specific. Next we will construct the fields explicitly. The
field content will be different in IIA and IIB theories.
We will need to look at first the spin field and how the GSO projection affects it.
Since
Γ11|0〉R = Γ11S(0)|0〉NS = ±|0〉R = ±S(0)|0〉NS , (3.107)
under the GSO projection the left and right spin fields transform as:
Γ11S = ±S , Γ11S = ±S (3.108)
96
and thus
Γ11S = −Γ11S , for IIA
Γ11S = +Γ11S , for IIB . (3.109)
Before the projection, S and S have 32 real components each. The projection leaves
16 real components for each. The equations of motion leave 8 on-shell degrees of
freedom out of 16.
3.9.1 Gamma Matrix Conventions
We will need SO(1, 9) gamma matrices in calculations. Below are some conventions
that we use:
Algebra : Γµ, Γν = −2ηµν = −2diag(−, +, . . . , +) (3.110)
Chirality projection : Γ11 = Γ0Γ1 · · ·Γ9 (3.111)
Γµ chosen to be : purely imaginary → (Γµ)∗ = −Γµ (3.112)
Symmetry :(Γi
)T= Γi , i = 1, . . . , 9 (3.113)
(Γ0
)T= −Γ0 (3.114)
Γ0Γ†µΓ0 = Γµ → Γ0ΓµΓ0 = Γ†µ = −ΓTµ (3.115)
(Γ11)2 = 1 , Γ11, Γ
µ = 0 (3.116)
Γ11Γ0 = −Γ0Γ11 , ΓT
11 = Γ11 (3.117)
Antisymmetrization : Γµ1···µk ≡ 1
k!Γ[µ1···Γµk]
=1
k!Γµ1Γµ2 · · ·Γµk ± permutations (3.118)
The antisymmetrized product is antisymmetric under the interchange of any two
indices, e.g.,
Γ1234 = −Γ2134 = −Γ4231 = −Γ3214 . (3.119)
As an explicit example of the definition, consider
Γµ1µ2µ3 =1
3!
Γµ1Γµ2Γµ3 + Γµ2Γµ3Γµ1 + Γµ3Γµ1Γµ2
− Γµ2Γµ1Γµ3 − Γµ3Γµ2Γµ1 − Γµ1Γµ3Γµ2
, (3.120)
even permutations come with a + sign, odd permutations with - sign. Note also:
k = 0 in (3.118) ↔ Γµ1···µk ≡ 1 . (3.121)
Recall that Γµ are pure imaginary. Consider the antisymm. products
1 , iΓµ , i2Γµ1Γµ2 , i3Γµ1Γµ2Γµ3 , . . . , i10Γ0...9 . (3.122)
97
These are all real, and can be shown to be linearly independent. There are
1 +
(10
1
)+
(10
2
)+
(10
3
)+ . . . +
(10
10
)= (1 + 1)10 = (25)2 = 32× 32 (3.123)
combinations. Thus they form a basis in the vector space of 32 × 32 real matrices:
we can expand a matrix (Mab)
Mab =9∑
k=0
Mµ1...µk
ik
k!(Γµ1...µk)ab (3.124)
with coefficients Mµ1...µk. Now let us consider the massless states from the R-R sector,
the R-R ground states.
3.9.2 R-R Ground States
The ground states were |0〉leftR ⊗ |0〉rightR = S(0)|0〉NS⊗ S(0)|0〉NS. The vertex operator
that creates such a bispinor (center-of-mass momentum included) is
V RR = F ab(k):Sa(0)(iΓ0
)bc
Sc(0)eik·X(0): . (3.125)
Let us denote the spin field part of the vertex operator by
Σab = Sa
(iΓ0
)bc
Sc . (3.126)
Why the iΓ0? Well, to form a tensor from two vectors ~A, ~B, take M = ~A~BT . Similarly,
from two spinors ψ, χ take M = ψχT = ψχ†Γ0. Then recall that S is real. The i is
inserted since we want the field F to be real (Γ0 is imaginary).
More importantly, now the trace
Tr(Σab) = iSaΓ0abSb = iSbΓ
0abSa = i ¯SS (3.127)
is Lorentz invariant and real. Hence also the polarization bispinor Fab(k). Note that
it is actually the polarization tensor (or vector, or bispinor as Fab(k) here) that will
become the target space field34. By (3.124) we can expand
Fab =9∑
k=0
ik
k!Fµ1...µk
(Γµ1...µk)ab , (3.128)
where Fµ1...µkare antisymmetric and real tensors of rank k.
Recall then the equation of motion (3.83) which followed from the constraint
Jµ = 0 (or the zero mode constraint F0 = 0 in the R sector). Combining the left and
right sectors, for the bispinor Fab we find the equations
kµΓµF = FΓµkµ = 0 . (3.129)
34Can’t read this line!!!
98
For the antisymmetric tensors Fµ1...µk, they imply
k[µF ν1...νk] = kµFµν2...νk = 0 . (3.130)
In coordinate space these become
dF = 0 , d ∗ F = 0 (3.131)
interpreting F as a k-form Fµ1...µkdxµ1 ∧ · · · ∧ dxµk . The first equation in (3.131) is
an equation of motion, the second one is a Bianchi identity. The equation of motion
tells us that F is a closed form, so there is a potential C = Cµ1...µk−1dxµ1∧· · ·∧dxµk−1
so that
F = dC . (3.132)
Or, in the component form,
Fµ1...µk=
1
(k − 1)!∂[µ1Cµ2...µk] . (3.133)
The Fµ1...µkcan be interpreted as field strengths of massless bosonic gauge fields
Cµ2...µk.
Now we need to take into account the GSO projection to see which gauge fields
survive in IIA and IIB theories. We will need the following identities (I’ll skip the
proof):
Γ11Γµ1...µk =
(−1)[k/2]
(10− k)!εµ1...µ10Γµk+1...µ10 (3.134)
Γµ1...µkΓ11 =(−1)[(k+1)/2]
(10− k)!εµ1...µ10Γµk+1...µ10 , (3.135)
where [k/2] = the integer part of k/2.
Type IIA Theory The GSO projection Γ11S = S, Γ11S = −S implies
FabSa(iΓ0S
)b
= Fab (Γ11S)a(iΓ0S
)b
= (Γ11S)T F(iΓ0S
)= ST ΓT
11F(iΓ0S
)
= (Γ11F )ab Sa(iΓ0S
)b
→ Γ11F = F (3.136)
and
ST F(iΓ0S
)= −ST F
(iΓ0Γ11S
)= ST F
(iΓ11Γ
0S)
= (FΓ11)ab Sa(iΓ0S
)b
→ FΓ11 = F (3.137)
99
so
Γ11F =10∑
k=1
(i)k
k!Fµ1...µk
Γ11Γµ1...µk
=10∑
k=1
(i)k
k!Fµ1...µk
(−1)[k/2]
(10− k)!εµ1...µ10Γµk+1...µ10
= FΓ11 =10∑
k=1
(i)k
k!Fµ1...µk
(−1)[(k+1)/2]
(10− k)!εµ1...µ10Γµk+1...µ10 . (3.138)
This is possible only for those terms in the sums which satisfy [k/2] = [(k + 1)/2],
i.e., when k is even. Other terms must vanish.
Thus IIA theory only contains Fµ1...µkfor k = even.
Furthermore,
Γ11F =10∑
k=1
(i)k
k!Fµ1...µk
(−1)[k/2]
(10− k)!εµ1...µ10Γµk+1...µ10
= F =10∑
k=1
(i)k
k!Fµ1...µk
Γµ1...µk . (3.139)
For the differential forms Fk ≡ Fµ1...µkdxµ1 ∧ · · · ∧ dxµk this implies the relation
Fk = ∗F10−k . (3.140)
So in IIA theory the independent forms are F0, F2, F4. The zero form F0 turns out
to be constant, hence non-propagating. The propagating degrees of freedom then
correspond to the gauge potentials Cµ and Cµ1µ2µ3 .
Type IIB Theory Now the GSO projection Γ11S = S, Γ11S = −S implies that
Γ11F = −FΓ11 = F . (3.141)
Then the equation (3.138) will receive an additional minus sign, so that the equa-
tion can be satisfied only when
(−1)[k/2] = −(−1)[(k+1)/2] , (3.142)
i.e., when k is odd. The other condition (3.139) remains unchanged, so that the higher
forms are related to the lower forms by the Hodge duality condition (3.140) as before.
Interestingly, the five form F5 is a special case: it is selfdual:
F5 = ∗F5 . (3.143)
So in Type IIB the independent forms are F1, F3, and the selfdual F5. The corre-
sponding gauge potentials are C,Cµν , Cµ1µ2µ3µ4 .
100
3.10 Type IIA and IIB Supergravity
As was the case for the bosonic string, at low energies/long distances the superstring
theories will reduce to low-energy effective field theories. They will be the Type IIA
and IIB supergravity theories in 10D.
Type IIA Supergravity Field content from superstring:
Sector Massless fields # of physical degrees of freedom
NS-NS φ dilaton 1
Bµν antisymm.tensor 28
Gµν graviton 35
R-R Cµ vector 8
Cµνλ antisymm.tensor of rank 3 56
Table 4: Bosons with total of 128 degrees of freedom
Sector Massless fields # of physical degrees of freedom
NS-R χa spin 12
fermion 8
ψµa spin 3
2fermion (gravitino) 56
R-NS χ′a another spin 12
fermion 8
ψ′µa another gravitino 56
Table 5: Fermions with total of 128 degrees of freedom
So there are an equal number of fermionic and bosonic degrees of freedom (=
physical polarizations allowed by the constraints), as required for unbroken super-
symmetry.
Note that the spinors χa, ψµa have an opposite chirality with respect to χ′a, ψ
′µa .
The low-energy effective action (only the bosonic sector shown here) in the Ein-
stein frame looks like
SIIA =1
2κ2
∫d10x
√−G
R− 1
2(∇φ)2 − 1
12e−φH2
− 1
4e3φ/2F 2
2 −1
48eφ/2F 2
4
− 1
2304
1√−Gεµ0...µ9Bµ0µ1(F4)µ2...µ5(F4)µ6...µ9
+ . . . , (3.144)
where
(∇φ)2 = Gµν∂µφ∂νφ (3.145)
H2 = HµνλHµνλ , indices raised with Gµν (3.146)
Hµνλ = ∂µBνλ + ∂νBλµ + ∂λBµν (3.147)
101
and
(F2)µν = ∂µCν − ∂νCµ (3.148)
(F4)µνλκ = field strength constructed from the R− R 3− form Cµνλ(3.149)
and1
2κ2=
1
16πG(10)N
, (3.150)
where G(10)N = Newton’s constant in 10D.
A puzzle: until 1995, there was a puzzle associated with the R-R gauge fields
Cµ, Cµνλ. The NS-NS gauge field couples to a charged object, namely the string,
as we discussed in section 2.15. Now it would be natural to expect that also the
R-R gauge fields also couple to charged objects (carrying R-R charges, as opposed to
NS-NS charge). But we would expect these objects to be pointlike (coupling to Cµ)
or two-dimensional (coupling to Cµνλ). What are these mystery objects in Type IIA
theory? J. Polchinski showed in 1995 that nonperturbative objects called D-branes
(constructed but ill-understood already in 1989) carry the R-R charges; they are
p-dimensional objects, with p even in Type IIA theory.
Type IIB Supergravity Field content from superstring:
Sector Massless fields # of physical degrees of freedom
NS-NS φ dilaton 1
Bµν antisymm.tensor 28
Gµν graviton 35
R-R C0 scalar 1
Cµν antisymm.tensor 28
Cµνλκ selfdual rank 4 35
Table 6: Bosons with total of 128 degrees of freedom
Sector Massless fields # of physical degrees of freedom
NS-R χa spin 12
fermion 8
ψµa spin 3
2fermion (gravitino) 56
R-NS χ′a another spin 12
fermion 8
ψ′µa another gravitino 56
Table 7: Fermions with total of 128 degrees of freedom
Again, 128 bosons + 128 fermions. The NS-NS sector is as in Type IIA SUGRA.
Now the fermions χa, ψµa and χ′a, ψ
′µa have the same chirality.
102
Now it is trickier to write down a low-energy effective action, because of the
selfdual Cµνλκ. It is generally difficult to find Lorentz covariant actions for selfdual
tensors. The following action comes close (from Polchinski’s book). Note that it is
written in the string frame:
SIIB =1
2κ2
∫d10x
√−G
e−2φ
(R + 4(∇φ)2 − 1
2H2
)
− 1
2
(F 2
1 + F 23 + F 2
5
)
− 1
4κ2
∫C4 ∧H ∧ F3 (+ fermionic sector) , (3.151)
where (using the differential form notation):
H = dB2 NS− NS form (3.152)
F1 = dC0 1− form from R− R scalar C0 (3.153)
F3 = dC2 R− R 3− form (3.154)
F5 = dC4... (3.155)
F3 = F3 − C0 ∧H3 (3.156)
F5 = F5 − 1
2C2 ∧H3 +
1
2B2 ∧ F3 . (3.157)
In particular, the equation of motion and the Bianchi identity for F5 which result
from the action, are
d ∗ F5 = dF5 = H3 ∧ F5 . (3.158)
These are consistent with the selfduality condition
∗F5 = F5 (3.159)
but do not imply it. Thus the selfduality condition must be imposed separately, as a
constraint. This formulation is enough for classical level. At quantum level it is not
satisfactory.
Moving to Einstein frame, it is useful to introduce the following notations (GEµν
is the metric in the E. frame):
GEµν = e−φ/2Gµν (3.160)
τ ≡ C0 + ie−φ (3.161)
(Mij) ≡ 1
=τ
( |τ |2 −<τ
−<τ 1
)(3.162)
(F i
3
)=
(H
F3
)(3.163)
103
and write the action as follows:
SIIB,Einstein =1
2κ2
∫d10x
√−G
RE − 1
2(=τ)2∂µτ ∂µτ
− 1
2MijF
i3 · F j
3 −1
2F 2
5
− εij
4κ2
∫C4 ∧ F i
3 ∧ F j3 . (3.164)
The benefit of this complicated looking form of the action is that one can recognize
a new symmetry, an SL(2,R) symmetry of the following form:
τ → aτ + b
cτ + d= τ ′ (3.165)
F i3 → Λi
jFj3 = F ′i
3 , with(Λi
j
)=
(d c
b a
)(3.166)
F5, GEµν unchanged . (3.167)
The invariance of the F3 kinetic term follows from
M→ (Λ−1)TMΛ−1 = M′ . (3.168)
The a, b, c, d are real numbers with det
(a b
c d
)= 1.
There are two interesting features of this symmetry that are easy to observe:
1. In Type IIB string theory there will be nonperturbative objects, D-branes, like
in IIA theory. They couple to C0, C2, C4. The SL(2,R) symmetry mixes C2
with B. If the symmetry reflects the properties of IIB string theory, it suggests
that the D1-brane (D-string) maps to the “fundamental” string.
2. The mapping also mixes e−φ with eφ. Since the closed string coupling constant
was gs = eφ. The suggested symmetry in string theory seems to relate weak
coupling to strong coupling.
The observations 1) & 2) do reflect a symmetry of the underlying Type IIB su-
perstring theory, the S-duality (charge quantization will break SL(2,R) to SL(2,Z)).
***** END OF LECTURE 10 *****
3.11 Toroidal Compactification and T-duality
History (Kaluza 1921) 35 Consider gravity in 5 dimensions (x1, . . . , x5). Denote
the metric by gµν . Consider the coordinate x5 to be a coordinate on a circle of radius
35Interestingly, the idea of combining EM with gravity by introducing an extra dimension wasalready presented by a Finn, Gunnar Nordstrom, in 1914. However, he used his own scalar theoryof gravity.
104
R. Then x5 ' x5 + 2πR. Denote the remaining coordinates by xµ (µ = 1, 2, 3, 4).
Then the 5-dimensional metric splits into a 4-dimensional metric, a gauge filed and
a scalar:
gµν ∼ gµν , Aµ ∼ gµ5 , φ ∼ g55 (3.169)
(the actual field redefinitions are slightly more complicated, but we do not need them
here).
Thus, gravity in 5 dimensions leads to gravity + electromagnetism + scalar field
in 4 dimensions. In the simplest case, we assume that all fields above are independent
of the coordinate x5. In general, fields could depend on x5 and the consequence of
this will be discussed below.
3.11.1 General Idea of Compactification
• Consider a theory in D + d dimensions.
• Replace the d-dimensional part of the space by a compact space (e.g., a d-
dimensional torus).
• Rewrite the (D + d)-dimensional theory on the d-dimensional compact space as
a D-dimensional theory.
Clearly, one needs some kind of compactification to get a 4-dimensional theory out of
string theory in 10 dimensions. We will not discuss these phenomenologically more
interesting compactifications here.
Below, we discuss field theory and string theory compactification on a circle (S1)
and discuss the notion of T-duality. Then, we briefly describe compactification on a
d-dimensional torus (T d).
3.11.2 Scalar Field Theory Compactified on S1
φ(x): scalar field in D + 1 dimensions with coordinates xµ, µ = 0, 1 . . . , D. Equation
of motion:
(¤(D+1) − M2)φ(x) = 0 , (3.170)
where
¤(D+1) = −(
∂
∂x0
)2
+D∑
µ=1
(∂
∂xµ
)2
(3.171)
is the d’Alembertian and M is the mass of the scalar field φ in D + 1 dimensions.
Let:
xµ = xµ , µ = µ = 0, . . . , D − 1 (3.172)
xD = y , µ = D . (3.173)
105
Now
¤(D) = −(
∂
∂x0
)2
+D−1∑µ=1
(∂
∂xµ
)2
(3.174)
¤(D+1) = ¤(D) +∂2
∂y2. (3.175)
Let y take values on a circle: y ' y +2πR. Fourier expand φ(x) = φ(xµ, y) on S1:
φ(xµ, y) =∑
n
φ(n)(xµ)einy/R , (3.176)
where φ(n)(xµ) are the Fourier modes of φ(xµ, y). Then, the equation of motion in
D + 1 dimensions implies:
∑n
(¤(D) +
∂2
∂y2− M2
)φ(n)(x)einy/R = 0 →
(¤(D) − (M2 +
n2
R2)
)φ(n)(x) = 0 .
(3.177)
φ(n) are scalar fields in D dimensions with masses36
M2n = M2 +
n2
R2, n = 0, 1, . . . ,∞ (3.178)
(Kaluza-Klein modes of φ on S1).
Essentially, this is pure kinematics: in D +1 dimensions, for a particle of momen-
tum pµ,
−p2 = M2 or − pµpµ − p2
y = M2 . (3.179)
D-dimensional mass is M2 = −pµpµ. Thus
M2 = M2 + p2y . (3.180)
If y is a coordinate on a circle, then py is quantized as py = n/R. Hence
M2 = M2 +n2
R2. (3.181)
For simplicity, let us focus on the case when M = 0 (massless D-dimensional scalar
field). Then
M2n =
n2
R2. (3.182)
36Should n take negative values as well? In addition, shouldn’t this also be true for the windingnumber introduced below?
106
3.11.3 Main Features of Field Theory Compactifications on S1
• A single massless field in D + 1 dimensions leads to a single massless field in
D dimensions (i.e., φ(n=0)(x)) + a tower of massive modes φ(n)(x) with M2n =
n2/R2, n > 0.
• The massive modes get heavier as R gets smaller and disappear as R → 0,
leaving only φ(0) behind.
• As R → ∞, Mn become lighter and the spacing between masses also reduces:
the discrete mass spectrum tends to a continuous spectrum. Thus, the appear-
ance of a continuous mass spectrum in D dimensions signals the opening up of
a new dimension (R →∞) resulting in a (D + 1)-dimensional theory.
String theory compactified on S1 could differ from field theory in all these aspects
this is because close strings could move on S1 just as particle, but could also wind
around S1, something particles cannot do. See Fig 31.
particle closed string winding closed string
R R
p=n/R winding length = m2 πRp=n/R
Figure 31: Particle vs closed strings moving along a compact circle.
One can also consider other fields instead of scalars, e.g., Aµ, gµν etc. in D + 1
dimensions.
In all cases, the mass spectrum has the same features as for scalars. For example,
consider Aµ(x) in D + 1 dimensions. Let:
Aµ(x) = Aµ(x, y) , µ = µ = 0, 1, . . . , D − 1 (3.183)
AD(x) = φ(x, y) . (3.184)
Fourier expand Aµ(x, y) and φ(x, y) on the circle as before. Then the D+1-dimensional
equation of motion in Lorentz gauge yields
¤(D+1)Aµ = 0 →
(¤(D) −M2n)A
(n)µ = 0
(¤(D) −M2n)φ(n) = 0
, (3.185)
where M2n = n2/R2.
For gµν , some field redefinitions are also needed but the main features remain the
same.
107
3.11.4 String Theory Compactified on S1
The procedure for studying string theory on a circle (or T d) is a slight generalization
of the uncompactified case. Below, we follow the standard steps. Consider string in
D + 1 dimensions (D + 1 = 10 for superstrings and 26 for bosonic strings). 37
Step 1. Solving equation of motion:
(∂2τ − ∂2
σ)XM = 0 or ∂+∂−XM = 0 . (3.186)
In superstring theory, we also have the worldsheet fermions satisfying ∂+ψM− = 0,
∂−ψM+ = 0. This sector is not affected by compactification on S1, so we will not write
it explicitly (ψM± are not target space coordinates).
General solution:
XM = xM + α′pMτ + LMσ +∑
n
cMn ein(τ−σ) +
∑n
cMn ein(τ+σ) . (3.187)
Step 2. Boundary conditions: it is here that the S1 compactification differs from
the uncompactified case. Split XM , M = 0, 1, . . . , D, as follows:
Xµ , µ = 0, 1, . . . , D − 1 : uncompactified coordinates (3.188)
XD : compactified on a circle of radius R . (3.189)
Then the closed string boundary conditions are:
Xµ(σ + 2π) = Xµ(σ) (3.190)
XD(σ + 2π) = XD(σ) + 2πmR , (m ∈ Z no. of windings) , (3.191)
yielding
Xµ(σ, τ) = xµ + α′pµτ + i
√α′
2
∑
n6=0
αµn
ne−in(τ−σ) + i
√α′
2
∑
n 6=0
αµn
ne−in(τ+σ) , (3.192)
and for the compact coordinate:
XD(σ, τ) = xD + α′pDτ + mRσ + i
√α′
2
∑
n6=0
αDn
ne−in(τ−σ) + i
√α′
2
∑
n 6=0
αDn
ne−in(τ+σ) .
(3.193)
Note: the oscillators are the same as in the uncompactified case. The only new
feature is the “mRσ” term in XD. Clearly, “m” is the winding number. What is
pM = (pµ, pD)?
37Note that from now on we have switched µ, ν to M,N . . .
108
Translations in XM , XM → XM +aM , are generated by the momentum operators
ΠM :
ΠM =∂L
∂(∂τXM)=
1
2πα′∂τX
M : momentum density along the string (3.194)
(the index is raised and lowered by the flat metric ηMN).
The total linear momentum of the closed string (= center-of-mass momentum) is
given by:
PM =
∫ 2π
0
dσΠM(σ) = pM , pM = ηMNpN . (3.195)
pM is thus the center-of-mass momentum of the closed string.
Since XD is compact, singlevaluedness of the wave function eipDXDimplies that
pD =n
R. (3.196)
Hence:
Xµ = xµ + α′pµτ + oscillators (3.197)
XD = xD + α′pDτ + LDσ + oscillators (3.198)
pD =n
R(3.199)
LD = mR (3.200)
(in superstring theory, for fermions we get the usual NS and R sectors. These are not
affected by the boundary condition on S1).
Since the theory naturally splits into left (+) and right (-) moving parts it is
convenient to rewrite XM making this explicit:
XM(σ, τ) = XML (τ + σ) + XM
R (τ − σ) , (3.201)
with
XML =
1
2xM
L +α′
2pM
L (τ + σ) + (oscillators α) (3.202)
XMR =
1
2xM
R +α′
2pM
R (τ − σ) + (oscillators α) (3.203)
where
M = µ : pML = pM
R = pµ (3.204)
M = D :1
2(pD
L + pDR) = pD ,
α′
2(pD
L − pDR) = LD , (3.205)
and
pDL = pD +
LD
α′(3.206)
pDR = pD − LD
α′. (3.207)
109
Substituting for pD and LD,
pDL =
n
R+
mR
α′(3.208)
pDR =
n
R− mR
α′. (3.209)
Step 3. Quantization: This proceeds as in the uncompactified case by imposing
[XM(σ), ΠN(σ′)] = iηMNδ(σ − σ′) (3.210)
which yields38
[xM , pN ] = iηMN (3.211)[αM
m , αNn
]=
[αM
m , αNn
]= mδm+n,0η
MN . (3.212)
Similarly, for the fermions, one gets the usual Neveu-Schwarz and Ramond (NS
and R) sectors. We can now define the “number” operators NR, NL,
NR =∞∑
n=1
αM−nα
Nn ηMN + fermionic contribution (NS or R) (3.213)
NR =∞∑
n=1
αM−nα
Nn ηMN + fermionic contribution (NS or R) . (3.214)
Building up the Fock space:
αM−n, α
M−n : raising operators (n > 0) (3.215)
αMn , αM
n : lowering operators (n > 0) (3.216)
Fock ground state : αMn |ground〉 = αM
n |ground〉 = 0 , (n > 0) . (3.217)
Note: pM commutes with α’s and NL,R implying that in the uncompactified theory,
the ground state can carry a momentum kM :
pM |k〉 = kM |k〉 . (3.218)
In compactified theory the Fock space ground state can carry both momenta and
windings: |m,n〉.The Fock space constructed in this way have negative norm states, which are
unphysical and should not exist. These unphysical states disappear once the Virasoro
constraints are imposed.
38Do not confuse “m” and “n” here with the momentum and winding quanta!
110
Step 4. Constraints and mass formula Virasoro constraints:
T++ = 0 , T−− = 0 , (3.219)
where Tαβ is the energy-momentum tensor on the worldsheet.
We are interested only in the constraints that lead to the mass formula and hence
will also ignore the super Virasoro constraints related to worldsheet supersymmetry.
T++ =1
2∂+XM∂+XM + (fermions) (3.220)
T−− =1
2∂−XM∂−XM + (fermions) (3.221)
Virasoro generators Ln, Ln:
T++ =∑
n
Lne−in(τ+σ) (3.222)
T−− =∑
n
Lnein(τ−σ) (3.223)
Physical states are defined by imposing the Virasoro constraints on the Fock space:
Lm|phys〉 = 0 (3.224)
Lm|phys〉 = 0 (3.225)
+ super Virasoro (will not be discussed any further) (3.226)
for m > 0: these eliminate the negative norm states.
(L0 − a)phys = (L0 − a)phys = 0 : (3.227)
these define the mass spectrum of string excitations. (a, a): normal ordering con-
stants, the values of which depend on the theory. The two also imply (σ-translation
invariance):
(L0 − L0 + (a− a))phys = 0 . (3.228)
|phys〉 has the structure: |phys〉 = |〉L|〉R. The above equation implies that |〉L and
|〉R are not independent.
One can easily evaluate L0 and L0 for the string theory on S1. The answer is:
L0 =α′
4pM
R pRM + NR (3.229)
L0 =α′
4pM
L pLM + NL (3.230)
which yield
L0 =α′
4pµpµ +
α′
4(pD
R)2 + NR (3.231)
L0 =α′
4pµpµ +
α′
4(pD
L )2 + NL (3.232)
111
or
L0 =α′
4pµpµ +
α′
4
(n
R− mR
α′
)2
+ NR (3.233)
L0 =α′
4pµpµ +
α′
4
(n
R+
mR
α′
)2
+ NL . (3.234)
D-dimensional mass (excluding S1): M2 = −pµpµ. (3.227) yield
α′M2 = α′(
n
R− mR
α′
)2
+ 4(NR − a) (3.235)
α′M2 = α′(
n
R+
mR
α′
)2
+ 4(NL − a) (3.236)
These two are consistent provided:
α′(
n
R− mR
α′
)2
+ 4(NR − a) = α′(
n
R+
mR
α′
)2
+ 4(NL − a) (3.237)
→ 4(NR −NL + a− a) = 4α′(nm
α′
)(3.238)
NR −NL + a− a = nm . (3.239)
Adding (3.235) and (3.236) we get a more symmetric expression for the mass:
α′M2 = α′(
n2
R2+
m2R2
α′2
)+ 2(NR + NL − a− a) . (3.240)
Bosonic: a = a = 1, superstrings: a = 0, a = 12
(depending on the sector). Compare
this with the field theory formula (note that 2(NR + NL − a− a) replaces M2 in the
field theory case).
3.11.5 Features of String Theory on S1
• For a given massless states in D + 1 dimensions one may get more than one
massless state in D dimensions, for the right combination of R and (a, a) (verify
that this can happen for bosonic strings but not for superstrings).
• As R → 0, momentum modes get heavier, but winding modes get lighter.
• As R → ∞, winding modes get heavy and disappear in the limit. Momentum
modes get lighter.
Note: from D-dimensional point of view one gets a continuous mass spectrum even
as R → 0 due to winding modes getting lighter. Thus from the D-dimensional point
of view, R → 0 looks similar to R →∞. In particular, even when R → 0, it appears
as if a new dimension has opened up leading to a D+1-dimensional theory! (There is
no way of totally getting rid of XD by shrinking R to zero.) This feature is formalized
in a property of string theory called T-duality.
***** END OF LECTURE 11 *****
112
3.12 T-duality
Consider a string theory in D + 1 dimensions (D + 1 = 26 for bosonic string and
10 for superstring, or heterotic string). Denote the spacetime coordinates by XM ,
(X0, X1, . . . , XD), and assume that the coordinate XD is compactified on a circle of
radius R, (XD ' XD + 2πR). We have seen that the mass spectrum in this theory is
given by
α′M2 = α′(
m2
R2+
n2R2
α′2
)+ 2NR + 2NL − 4 , (3.241)
where M is the D-dimensional mass and m and n denote the closed string momentum
quantum number and winding along XD. Let us rewrite the above mass formula in
terms of a new variable R,
R =α′
R. (3.242)
Then one gets
α′M2 = α′(
m2R2
α′2+
n2
R2
)+ 2NR + 2NL − 4 . (3.243)
Clearly, this looks like the mass formula for a string theory compactified on a circle
of radius R = α′/R with momentum n and winding m !
In other words, a string theory on a circle of radius R has the same mass spec-
trum as one on a circle of radius R = α′/R with momentum and winding modes
interchanged, M(m,n, R) = M(n,m, R). This relationship between a theory with
radius R and one with radius R = α′/R is called T-duality.
So far, we have only mapped the spectra in the two theories. However, this can
easily be extended to the full theory, including states and interactions.
3.12.1 The T-duality Map
Let us consider the coordinate XD. We have seen that,
∂+XD =α′
2
(n
R+
mR
α′
)+
√α′
2
∑n
αDn e−in(τ+σ) (3.244)
∂−XD =α′
2
(n
R− mR
α′
)+
√α′
2
∑n
αDn e−in(τ−σ) (3.245)
and
XD = xD + α′n
Rτ + mRσ + i
√α′
2
∑
n 6=0
1
n
(αD
n e−in(τ+σ) + αDn e−in(τ−σ)
). (3.246)
Clearly,
XD(σ + 2π) = XD(σ) + m(2πR) , (circle of radius R) . (3.247)
113
Let us now consider another string theory, now compactified on a circle of radius
R (for the time being, we do not relate R and R). Denoting the corresponding
coordinate by XD, we obviously have
∂+XD =α′
2
(n
R+
mR
α′
)+
√α′
2
∑n
˜αDn e−in(τ+σ) (3.248)
∂−XD =α′
2
(n
R− mR
α′
)+
√α′
2
∑n
αDn e−in(τ−σ) (3.249)
and
XD = xD + α′n
Rτ + mRσ + (oscillators) (3.250)
XD(σ + 2π) = XD(σ) + m(2πR) , (circle of radius R) . (3.251)
Let us now impose a relationship between the two theories, such that:
∂+XD = ∂+XD , ∂−XD = −∂−XD , (3.252)
(Xµ = Xµ, (µ = 0, 1, . . . , D − 1)). Then, using the expansions above, we see that
R =α′
R, m = n , n = m . (3.253)
Equation (3.252) above is the basic T-duality map that relates not only the spectra
but also the vertex operators in the two theories.
For the oscillators αMn , αM
n the map implies
˜αMn = αM
n , αµn = αµ
n , αDn = −αD
n . (3.254)
The number operators NL, NR are quadratic in oscillators and remain unchanged39.
Since ∂+ = ∂τ + ∂σ and ∂− = ∂τ − ∂σ, the T-duality map corresponds to
∂τXD = ∂σX
D (3.255)
∂σXD = ∂τX
D (3.256)
Note that at the special radius R0 =√
α′, the theory becomes selfdual, i.e.,
R0 = R0. A theory with radius R < R0 =√
α′
is equivalent, by T-duality, to a
theory with radius R > R0. In this sense it is unnecessary to consider radii smaller
than R0 (which is of the string scale).
39Note: we use “−” to denote leftmovers and “∼” to denote T-dual. In other places whereT-duality does not appear, we have used ∼ to denote leftmovers
114
3.12.2 T-duality in Superstring Theory
In superstring theory, besides the bosonic fields XM , we also have worldsheet fermions
ψM± . The action of T-duality on ψM
± can be deduced very easily by using worldsheet
supersymmetry transformations. Recall that the supersymmetry transformations in-
clude
δψM+ = 2∂+XMε− (3.257)
δψM− = −2∂−XMε+ . (3.258)
We can write similar equations for ∂±XM and ψM± in the dual theory. Now, using the
T-duality map between XM and XM we can easily see that
ψD+ = ψD
+ , ψD− = −ψD
− , ψµ± = ψµ
± . (3.259)
This gives the T-duality map for the worldsheet fermions.
3.12.3 T-duality Action on Ramond States
Recall that with any periodic boundary condition for ψM (ψM(σ+2π) = ψM(σ)), i.e.,
in the Ramond sector, the fermion zero modes give rise to Dirac Gamma matrices in
10 dimension40:
ψM+,0 ∼ ΓM , ψM
−,0 ∼ ΓM , ΓM , ΓN = 2ηMN . (3.260)
The ground state in the Ramond sector is a Majorana-Weyl spinor of the Lorentz
group SO(1, 9), say, |0〉αR, where α is a spinor index.
One can construct a spinor field Sα(σ, τ), such that, acting on the Neveu-Schwarz
ground state |0〉NS, it produces the Ramond ground state
Sα(0)|0〉NS = |0〉αR . (3.261)
Clearly, Sα is also a Majorana-Weyl spinor. We can construct Sα for both the left
and right moving sectors of the worldsheet theory. Let us denote these by
SαL , Sα
R . (3.262)
(Note that “R” here refers to right moving and not Ramond sector.) SαL,R are the basic
spinorial objects in the theory in the same way that ∂±XM are the basic vectorial
objects.
The fact that SL,R are Weyl spinors means that they have definite 10-dimensional
chirality, i.e.,
Γ11SL,R = ±SL,R . (3.263)
40Not the same anticommutator as we used earlier for SO(1, 9)!
115
The GSO projection implies that not all chiralities appear in the same theory.
If SL and SR have opposite chiralities, we get Type IIA theory. If they have the
same chirality, we get Type IIB theory.
The action of T-duality on SL,R can be found as follows: SR is a spinor associated
with the Dirac algebra ΓM , ΓN = 2ηMN . The Dirac matrices are the zero modes of
ψM− , ΓM ∼ ψM
−,0.
In the dual theory we denote the corresponding objects by SR, ΓM , ψM− . We have
seen that ψ9− = −ψ9
− (we take D + 1 = 10). This implies that Γ9 = −Γ9, Γµ = Γµ.
We can explicitly construct a transformation that changes the sign of Γ9, keeping all
other Γ’s invariant:
Γ9 = Ω†Γ9Ω = −Γ9 , Γµ = Ω†ΓµΩ = Γµ . (3.264)
One can check that
Ω = Γ11Γ9 . (3.265)
This is the spinor representation of ∂−X9 → −∂−X9.
It is now clear that on the associated spinors SR, one has the action
SR = ΩSR = Γ11Γ9SR . (3.266)
Since ψµ+ = ψµ
+, we conclude that
SL = SL . (3.267)
Consequence. SL and SL are equal and hence have the same chirality. However,
Γ11, Γ9 = 0, Γ211 = 1 so that
Γ11SR = Γ11ΩSR = −Ω(Γ11SR) (3.268)
or
Γ11SR = ±S± → Γ11SR = ∓SR . (3.269)
Hence, T-duality reverses the chirality of SR and therefore interchanges IIA and IIB
theories41!
Comment. Before T-duality, the Dirac matrices ΓM acted on the spinors SR. To
discuss T-duality we introduced ΓM = Ω†ΓMΩ. The transformation of the spinors is
obtained by absorbing Ω into SR: ΩSR = SR. So at the end of the day we are left
with the Dirac matrices ΓM and spinors SR (and NOT with ΓM and SR!). ΓM do not
appear in the final theory and were simply used to obtain the transformation of SR.
41IIA on a circle of radius R is dual to IIB on a circle of radius α′/R with momenta and windingsinterchanged.
116
3.13 Gauge Symmetry Enhancement in Circle Compactifica-
tion
Consider bosonic string theory with X25 compactified on a circle (X25 ' X25 +2πR).
We have already obtained the mass spectrum for this theory as:
α′M2 = α′(
n2
R2+
m2R2
α′2
)+ 2(NR + NL − 2) (3.270)
NR −NL = nm (3.271)
with,
NR =24∑
µ=0
∑n>0
αµ−nα
νnηµν +
∑n>0
α25−nα
25n (3.272)
NL =24∑
µ=0
∑n>0
αµ−nα
νnηµν +
∑n>0
α25−nα
25n . (3.273)
The compactified theory has two obviously massless abelian gauge fields,
Aµ ∼ Gµ25 , A′µ ∼ Bµ25 (3.274)
and hence a gauge group U(1)× U(1). They are associated with states
Aµ → 1
2(αµ
−1α25−1 + α25
−1αµ−1)|0〉 (3.275)
A′µ → 1
2(αµ
−1α25−1 − α25
−1αµ−1)|0〉 , (3.276)
where NR = NL = 1 and n = m = 0. At a very special value of the radius, the mass
formula gives 4 more massless vector states. The extra massless states are obtained
for R =√
α′based on a ground state that carries nonzero winding and momentum
n = m = ±1. Let us denote the ground state by |n,m〉. Then the extra massless
vector states are
αµ−1|1, 1〉 , n = m = 1 , NR = 1 , NL = 0 (3.277)
αµ−1|−1,−1〉 , n = m = −1 , NR = 1 , NL = 0 (3.278)
αµ−1|1,−1〉 , n = −m = 1 , NR = 0 , NL = 1 (3.279)
αµ−1|−1, 1〉 , n = −m = −1 , NR = 0 , NL = 1 . (3.280)
Note: R =√
α′, is the T-duality invariant radius.
These extra vector fields combine with the abelian ones to enhance the gauge
symmetry SU(2)× SU(2). We will not carry out the explicit construction here, but
will look at the charge assignments to make this plausible.
117
3.13.1 Momentum and Winding as Abelian Charges
Abelian gauge fields couple to abelian charges. In field theory, the coupling appears in
the action as∫
dDxeJµAµ, where e is the charge and Jµ is the current. For example,
in charged scalar field theory the coupling is∫
dDxeAµ(φ∗∂µφ− φ∂µφ∗) . (3.281)
A point particle carrying charge “e” couples to a gauge field through the term
e
∫dxµAµ = e
∫dτ
(∂xµ
∂τAµ
), (3.282)
where the integral is along the worldline and τ is the proper time.
We use these expressions to identify the charges that couple to Aµ ∼ Gµ25 and
A′µ ∼ Bµ25.
For Aµ, consider the effective low-energy theory which, for example, has a term∫d26xGµν∂
µφ∗∂ νφ (we use a “∧” for quantities in 26 dimensions). Compactifying x25
on a circle, φ(xµ, x25) =∑
k φ(k)(xµ)eikx25
. Then we have
∫d25xdx25Gµ25∂
µφ∂25φ =∑
k>0
∫d25xkAµ(φ∗∂µφ− φ∂µφ∗) (3.283)
(only the zero mode of Gµ25 is retained to perform the x25 integral).
This shows that the quantized momenta k25 = n/R are the charges to which
Aµ ∼ Gµ25 couples. The effective low-energy theory does not lead to such couplings
for A′µ ∼ Bµ25, i.e., its charge does not show up in the field theory limit. This is a
hint that the A′µ charge is characteristically string theoretic. Note that the worldsheet
action contains a term ∼ 12πα′
∫dτdσBµν∂τX
µ∂σXν which inturn contains
1
2πα′
∫dτdσBµ25∂τX
µ∂σX25 . (3.284)
Use
X25 = x25 + α′p25 + L25σ +∑
n
An(τ)einσ , (3.285)
and assume that as a background, Bµ25 depends only on the zero modes of Xµ, so
that it is independent of σ. Substituting for X25 and performing the σ integral one
gets1
2πα′
∫dτLBµ25∂τX
µ(2π) =L
α′
∫dτA′
µ∂τXµ , L = mR . (3.286)
Thus, the winding no. on S1 appears as A′µ electric charge in the uncompactified
dimensions42.
42Note: by integrating over σ, we are effectively treating the closed string as a point particle. Thestring winding becomes the electric charge for the corresponding particle.
118
To summarize,
Aµ = Gµ25 couples to charges k =n
R(3.287)
A′µ = Bµ25 couples to charges
L
α′=
mR
α′. (3.288)
The vector index on Aµ and A′µ has contributions from both left and right moving
oscillators. We can define net gauge fields that do not mix between the left and right
moving sectors in this way. Let:
ARµ =
Aµ + A′µ√
α′ , AL
µ =Aµ − A′
µ√α′ (3.289)
(√
α′for latter convenience). In terms of states:
ARµ :
[12(αµ
−1α25−1 + α25
−1αµ−1) + 1
2(αµ
−1α25−1 − α25
−1αµ−1)
] |0〉 = αµ−1α
25−1|0〉 (3.290)
ALµ :
[12(αµ
−1α25−1 + α25
−1αµ−1)− 1
2(αµ
−1α25−1 − α25
−1αµ−1)
] |0〉 = αµ−1α
25−1|0〉 , (3.291)
so
ARµ : αµ
−1α25−1|0〉 (3.292)
ALµ : αµ
−1α25−1|0〉 , (3.293)
U(1)R × U(1)L. Hence, we can interpret AR and AL as the gauge fields associated
with the right and left moving sectors, respectively. The corresponding charges can
be found easily. Consider a state that carries charges p and L/α′ under Aµ and A′µ,
respectively. Then, the coupling is given by∫
dxµ(pAµ + L/α′A′µ) =
∫dxµ
[1
2(p + L/α′)(Aµ + A′
µ) +1
2(p− L/α′)(Aµ − A′
µ)
]
=
∫dxµ
[√α′
2pR
Aµ + A′µ√
α′ +
√α′
2pL
Aµ − A′µ√
α′
]
=
∫dxµ
(qRAR
µ + qLALµ
)(3.294)
so the ARµ and AL
µ charges are given by
qR =
√α′
2pR =
√α′
2(p + L/α′) =
√α′
2
(n
R+
mR
α′
)(3.295)
qL =
√α′
2pL =
√α′
2(p− L/α′) =
√α′
2
(n
R− mR
α′
). (3.296)
Let us now look at the charges of the extra massless vector states that appear at
R =√
α′. At this value of the radius,
qR =1
2(n + m) , qL =
1
2(n−m) , (at R =
√α′) . (3.297)
119
Then, for the right moving states,
AR(+)µ ↔ αµ
−1|1, 1〉 : qR = +1 , qL = 0 (3.298)
AR(−)µ ↔ αµ
−1|−1,−1〉 : qR = −1 , qL = 0 (3.299)
and for the left moving states,
AL(+)µ ↔ αµ
−1|1,−1〉 : qR = 0 , qL = +1 (3.300)
AL(−)µ ↔ αµ
−1|−1, 1〉 : qR = 0 , qL = −1 . (3.301)
In other words, the vector fields AR(±)µ carry charges ±1 under AR
µ and AL(±)µ carry
charges ±1 under ALµ . Compare this with the structure of the SU(2) group, where
[T 3, T±] = ±T±. Identifying AL,Rµ with A
L,R(3)µ , we see that the extra massless vectors
AL(±)µ , A
R(±)µ combine with the U(1)L,R gauge fields AL
µ , ARµ to enhance the gauge
group from U(1)L × U(1)R to SU(2)L × SU(2)R.
***** END OF LECTURE 12 *****
3.14 Lattices and Torii
A d-dimensional torus (T d) can be constructed by modding out the d-dimensional
space Rd by a lattice Γd.
Example. Circle (S1): consider a basic vector ~e of length l on R1 (~e = lx, for unit
vector x). Clearly, the points n~e, for all integer n, form a lattice Γ1 on R1. Every
R0 l 2l-l-2l
e
Figure 32: Construction of a circle of radius R = l/2π.
point on R1 can be obtained by translating a point 0 ≤ x < l by an appropriate lattice
vector (x+nl for some n). Conversely, modding out R1 by Γ1 will project every point
to some x, 0 ≤ x < l. In particular, x = l is identified with x = 0. This gives a
circle of radius R = l/(2π) (Fig. 32). Moving by n cells on the lattice corresponds
to going around the circle n times. This is how a circle S1 can be regarded as R1/Γ1
(x ' x + 2πR).
A torus is obtained by generalizing this construction to d dimensions.
120
3.15 Rectangular Torus
In Rd, consider unit vectors xI along the coordinate axes and construct basis vectors
~e1 = e1x1, . . . , ~ed = edxd of length eI , I = 1, . . . d. Then the vectors∑d
i=1 ni~ei define
2
e = e x
e = e x
11
2 2
1
2
x
x
1
Figure 33: Construction of a rectangular torus.
a lattice (Fig. 33). The entire Rd can be obtained by translating a single cell by
the lattice. Conversely, Rd modded out by this d-dimensional lattice Γ(d), defines the
d-dimensional torus, T d. In terms of the coordinates, for any vector ~x in Rd, we make
the identification
~x ' ~x +∑
i
ni~ei → xI ' xI + nIeI , no summation over I . (3.302)
This gives a simple rectangular torus (based on a rectangular lattice). In general, a
torus need not be rectangular.
3.15.1 Construction of a General Torus
In Rd, consider a set of d linearly independent vectors ~ei, i = 1, . . . , d. In the coordi-
nate basis they are given by
~ei =∑
I
eI) x
I (3.303)
121
2e
e1
Figure 34: Construction of a general torus.
Consider a vector∑
i ni~e(i) for integer ni. The set of all these vectors, which corre-
spond to taking all possible sets of integers n1, n2, . . . , nd define a lattice Γ(d) (Fig.
34). Note that lattices are also imporant in condensed matter physics in discussions
of crystals. In that context they are called Bravais lattices. We mod out Rd by Γ(d)
by identifying a vector ~x with its translations by lattice vectors:
~x ' ~x +∑
i
ni~ei , ∀ n1, n2, . . . , nd (3.304)
or ∑I
xI xI '∑
I
xI xI +∑
I
∑i
nieIi x
I (3.305)
which yields
xI ' xI +∑
i
nieIi . (3.306)
This defines a torus T d = Rd/Γ(d).
122
3.15.2 Metric
Since the lattice vectors e(i) are not parallel to the coordinate vectors xI , one can
describe the lattice in terms of a metric gij:
gij =∑
I
eIi e
Ij = ~ei · ~ej (3.307)
(note that eIi are nothing but the vielbeins for the metric gij). The volume of the unit
cell of the lattice is given by the metric,
vol(unit cell) =√
det(gij) . (3.308)
This metric defines both the size, as well as the shape of the torus. However, it is
sometimes useful to use a scale factor (say, c) to specify the overall size of the torus
instead of including this information in the metric. In that case we define the torus
by
xI ' xI + c∑
i
nieIi (3.309)
and define the metric as before. For example, in string theory compactifications we
may need torii of string size√
α′. This information will be included in c and not in
the metric. Now the volume of the d-dimensional torus is
vol(Torus) = cd√
det(gij) . (3.310)
3.15.3 “Integer” and “Even” Lattices
Consider a lattice vector v = ni~ei. This has a length squared
(v, v) = ninj(ei, e)) = ninjeIi e
Jj
δIJ︷ ︸︸ ︷(xI , xJ) = ninjgij . (3.311)
Here, (, ) denotes the scalar product on Rd. Γ(d) is an integer lattice if the (length)2
of every lattice vector is an integer. Clearly, this is a property of the metric gij. If
for every lattice vector v, (v, v) is an even integer, then the lattice is said to be even.
Note that
(v, v) = 2∑i<j
ninjgij +∑
i
n2i gii , (3.312)
thus for an even lattice, gii will be even integers.
3.15.4 Dual Lattice
For every lattice Γ(d) defined by lattice vectors ~ei, we can define a dual lattice Γ∗(d)
with lattice vectors ~e∗i such that
(~e∗i , ~ej) =d∑
I=1
e∗Ii eIj = δij . (3.313)
123
The notion of a dual lattice is useful for describing the notion of momentum
quantization for a particle on a torus. The wave function has a factor ei~k·~x which has
to be single valued under ~x → ~x + ni~ei. If ~k is on the dual lattice, ~k = 2πmj~e∗j , then
the single valuedness of the wave function is assured since
2πmj~e∗j · ni~ei = 2πmini = 2π(integer) . (3.314)
In condensed matter physics, dual lattices are called reciprocal lattices.
3.16 Heterotic String Theory
We have seen that when bosonic string theory is compactified on a circle, at an special
value of the radius (R =√
α′) extra massless states appear that enhance the gauge
symmetry from U(1)L × U(1)R to SU(2)L × SU(2)R. However, the bosonic theory
has a tachyon and does not contain fermions. Superstring theory has fermions but an
investigation of the mass formula shows that extra massless states DO NOT appear
in this case (this is due to the fact that the normal ordering constants a, a are now
0, or 12
as opposed to +1 for the bosonic theory). Both these problems are solved in
heterotic string theory which also has many other nice features.
Basic idea: the critical dimension (D = 10 for superstring and D = 26 for bosonic
string) was needed to cancel the conformal anomaly in the theory insuring consistent
quantization. The conformal field theory splits into left and right moving sectors,
each having its own conformal anomaly. Hence, it is not necessary to have the same
conformal anomaly in both left and right sectors.
Therefore, we can construct a theory where the left moving sector comes from
bosonic string theory and the right moving sector from superstring theory:
Bosonic string Superstring
µ = 0, . . . , 9 :
Xµ = XµL(τ + σ) + Xµ
R(τ − σ) Xµ = XµL(τ + σ) + Xµ
R(τ − σ)
no fermions ψµL(τ + σ) , ψµ
R(τ − σ)
I = 1, . . . , 16 :
XI = XIL(τ + σ) + XI
R(τ − σ) no extra coordinates
Combine then the leftmovers from the bosonic string and the rightmovers from the
superstring into:
Heterotic string
124
µ = 0, . . . , 9 : Xµ = XµL(τ + σ) + Xµ
R(τ − σ) (3.315)
ψµR = ψR(τ − σ) (no ψµ
L) (3.316)
I = 1, . . . , 16 : XI = XIL(τ + σ) (Xµ
R(τ − σ) = 0) . (3.317)
Thus, Xµ(σ, τ) are the usual 10-dimensional coordinates. Worldsheet fermions are
only right moving (ψµR). (Since in superstring theory, ψµ
R and ψµL are totally indepen-
dent, setting ψµL = 0 has no non-trivial consequence). The “internal” coordinates XI
(I = 1, . . . , 16) are compactified on a torus T 16. This can be done by first compactify-
ing the unconstrained XI (i.e., XI = XIL+XI
R) on T 16 and then setting XIR = 0. Both
XIL and XI
R contain zero mode parts and oscillator modes. The oscillator modes are
independent and the XIR oscillators can be set to zero without further consequence.
But the XIL and XI
R zero modes are not independent. Hence setting (XIR)zero mode = 0
imposes a constraint on XIL which affects its quantization non-trivially.
To study heterotic strings, we follow the standard procedure:
• Equations of motion (worldsheet)
• Boundary conditions
• Quantization (commutation relations)
• Virasoro constraints and mass formula.
3.16.1 Equations of Motion
∂+∂−Xµ = 0 , µ = 0, . . . , 9 → Xµ = XµR(τ − σ) + Xµ
L(τ + σ) (3.318)
∂+ψµR = 0 , → ψµ
R = ψµR(τ − σ) (3.319)
∂+∂−XI = 0 , I = 1, . . . , 16 → XI = XIR(τ − σ) + XI
L(τ + σ) (3.320)
Constraint:
XIR = 0 (3.321)
3.16.2 Boundary Conditions
Xµ(σ + 2π) = Xµ(σ) (3.322)
ψµR(σ + 2π) = ±ψµ
R(σ) ,
+1 : Ramond b.c.
−1 : Neveu− Schwarz b.c.(3.323)
XI(σ + 2π) = Xµ(σ) + c∑
i
nieIi , (c =
√2α′π) , (3.324)
125
i.e., XI are compactified on T 16. The scale factor c determines the overall size of the
torus, along with the metric gij.
Note: the shape and size of the torus (depending on c and gij) have not yet
been specified. As in the circle compactification, for a specific shape and size, one
gets enhanced gauge symmetry. However, unlike circle compactification, this shape
and size need not be fixed by hand. These parameters are fixed by requirement of
consistency of string 1-loop amplitudes (the, so called, modular invariance). The
resulting torus is precisely the one that enhances the gauge symmetry from U(1)16 to
E8 × E8 or SO(32).
The equations of motion can now be solved subject to the boundary conditions:
Xµ : Xµ = xµ + α′pµ + i
√α′
2
∑
n 6=0
1
n
(αµ
ne−in(τ−σ) + αµne−in(τ+σ)
). (3.325)
Once again, pµ can be identified with the string center-of-mass momentum in 10
dimensions, hence it is related to the mass operator M2 = −pµpµ.
ψµR : ψµ
R =∑
n
dµne−in(τ−σ) Ramond sector (3.326)
=∑
n
bµ
n+ 12
e−i(n+ 12)(τ−σ) Neveu− Schwarz sector (3.327)
For XI , solving the equation of motion gives
XI = xI + α′pIτ + LIσ + i
√α′
2
∑n
1
n
(aI
ne−in(τ−σ) + aI
ne−in(τ+σ))
. (3.328)
The torus boundary condition
XI(σ + 2π) = XI(σ) +√
2α′π∑
i
nieIi (3.329)
gives43
LI =
√α′
2
∑i
nieIi . (3.330)
Now,
XI = XIL(σ + τ) + XI
R(σ − τ) , (3.331)
with
XIL = xI
L +1
2α′(pI + LI/α′)(τ + σ) + i
√α′
2
∑
n6=0
aIn
ne−in(τ+σ) (3.332)
XIR = xI
R +1
2α′(pI − LI/α′)(τ − σ) + i
√α′
2
∑
n 6=0
aIn
ne−in(τ−σ) . (3.333)
43Now, the metric gij gives the torus size in units of the string length√
α′.
126
Then XIR = 0 implies:
xIR = 0 , aI
n = 0 , (for n 6= 0) (3.334)
(these can be set to zero with further conditions on XIL).
pIR = pI − LI/α′ = 0 → pI = LI/α′ . (3.335)
This affects XIL which also contains pI and LI .
In general,
XIR = 0 → ∂−XI = 0 (3.336)
(since ∂−XIR = ∂−XI) or
∂τXI = ∂σX
I . (3.337)
This is a phase space constraint which has to be taken into account while quantizing
the theory.
***** END OF LECTURE 13 *****
3.16.3 Quantization
Xµ and ψµR are quantized as in superstrings, leading to
[αµm, αν
n] = [αµm, αν
n] = mδm+n,0ηµν (3.338)
[xµ, pν ] = iηµν (3.339)
dµm, dν
n = ηµνδm+n,0 : Ramond sector (3.340)
bµr , b
νs = ηµνδr+s,0 : Neveu− Schwarz sector (r, s :
1
2−odd integer) .(3.341)
We can define the corresponding number operators:
N(α)R =
∑n>0
α−n · αn (3.342)
N(α)L =
∑n>0
α−n · αn (3.343)
N(d)R =
∑n>0
nd−n · dn (3.344)
N(b)R =
∞∑
r= 12
rb−r · br (3.345)
(clearly, N(d)L = N
(b)L = 0).
Then for XI , due to constraint ∂σXI = ∂τX
I , these should be quantized using
the Dirac bracket, instead of the Poisson bracket.
127
Let us write:
XI = XIL = xI
L + α′p′IL(τ + σ) + i
√α′
2
∑
n 6=0
aIn
ne−in(τ+σ) . (3.346)
For aIn one gets the usual commutation relation:
[aI
n, aJm
]= nδm+n,0δ
IJ (3.347)
with the corresponding number operator
N(a)L =
∑n>0
aI−naJ
nδIJ . (3.348)
For xIL and p′IL one gets: [
xIL, p′JL
]=
1
2iδIJ (3.349)
or, [xI
L, 2p′JL]
= iδIJ (3.350)
(notation: p′IL = 12pI
L = 12(pI + LI/apr)). This implies that
p′IL ∼ − i
2
∂
∂xIL
. (3.351)
Since xIR = 0, a translation in XI is the same as a translation in XI
L44. However,
the generator of this translation is 2p′IL and not p′IL . An eigenfunction of p′IL with
eigenvalue kI is e2ikIxIL :
p′ILe2ikIxIL = kIe2ikIxI
L . (3.352)
This is the function that should be single valued on a torus, implying
e2ikIxIL = e2ikI(xI
L+√
2α′πnieI(i)
) (3.353)
or √2α′kIeI
(i)ni = integer (∀ integer ni) . (3.354)
Thus, kI are on the dual lattice:
kI =1√2α′
mie∗I(i) . (3.355)
The considerations so far have not assumed any specific shape for the lattice/torus.
But there is already a constraint:
XIR = 0 → pI
R = pI − LI/α′ = 0 → pI = LI/α′ . (3.356)
44Check capitalization all around.
128
On the other hand,
p′IL =1
2(pI + LI/α′) = LI/α′ . (3.357)
p′IL has eigenvalues kI = 1√2α′
∑i mie
∗I(i) and LI =
√α′2nie
I(i).
Therefore, pI = LI/α′ implies
∑i
mie∗I(i) =
∑i
nieI(i) , ∀ integer (mi, ni) . (3.358)
Taking the scalar product with∑
j njeI(j) gives:
ninjgij = nimi = integer . (3.359)
Thus, Γ(16) is an integral lattice, if we demand the constraint (3.358) to have
solutions for nonzero mi ja ni. In general, Γ(16) could be a sublattice of its dual Γ∗(16).
Then the momenta kI could be any element Γ(16). A special case is when the lattice
is selfdual, Γ(16) ≈ Γ∗(16).
In fact the modular invariance of the 1-loop amplitude requires T 16 to be “even”
and “selfdual”.
It is known that in 16 dimensions, there are only two even-selfdual lattices: Γ(16)
corresponding to the root lattice of the group SO(32) and Γ(8) × Γ(8) corresponding
to the root lattice of the group E8 × E8 (will not be discussed in detail).
3.16.4 Mass Conditions (Spectrum)
The Virasoro conditions leading to the mass equation are follows.
Left moving sector:
(L0 − 1)|phys〉 = 0 (3.360)
with
L0 =α′
4pµp
µ +α′
4pI
LpIL + N
(a)L + N
(α)L (3.361)
and
pIL = pI + LI/α′ = 2LI/α′ = 2p′IL . (3.362)
This leads to the mass equation:
α′
4M2 =
α′
4pI
LpIL + NL − 1 ≡ α′p′ILp′IL + NL − 1 (3.363)
where, NL = N(a)L + N
(α)L .
For the right moving sector,
(L0 − a)|phys〉 = 0 ,
a = 1
2(NS)
a = 0 (R)(3.364)
129
with
L0 =α′
4pµp
µ + NR − a ,
NR = N
(α)R + N
(b)R (NS)
NR = N(α)R + N
(d)R (R)
. (3.365)
Mass shell condition:
α′
4M2 = (N
(α)R + N
(b)R )− 1
2, (NS) (3.366)
α′
4M2 = N
(α)R + N
(d)R , (R) (3.367)
The left and right moving masses should be equal, leading to the level matching
conditions:
α′
4pI
LpIL + N
(α)L + N
(a)L = N
(α)R + N
(b)R +
1
2, (NS) (3.368)
α′
4pI
LpIL + N
(α)L + N
(a)L = N
(α)R + N
(d)R + 1 , (R) . (3.369)
Note: as such, both sectors have tachyons. However, the usual GSO projection of
superstring theory, acts on the right moving sector which eliminates the NS tachyon
corresponding to N(b)R = 0. Thus, the lowest value of N
(b)R is 1/2. Then the level
matching condition in the NS sector eliminates the bosonic string tachyon.
One can easily check that for pILpI
L = 0, the massless spectrum contains
Gµν , Bµν , φ and their superpartners ψαµ , λα
in the gravity sector, as well as abelian gauge fields
AIµ (I = 1, . . . , 16) and the gauginos λI
α
in the gauge sector.
3.16.5 Extra Massless States
From the mass formula,α′
4M2 =
α′
4pI
LpIL + NL − 1 (3.370)
it is evident that extra massless states appear for special torii (with NL = 0) when
α′
4pI
LpIL = 1 (3.371)
or
α′p′ILp′IL = 1 , (3.372)
since, p′IL = 1√2α′
∑i mie
I(i), (using selfduality) this implies
mimjgij = 2 . (3.373)
130
Thus, on an even, selfdual lattice, p′IL are associated with lattice vectors of length
squared 2. Since, the lattice is even, this is the minimum nonzero length possible.
On any even lattice, vectors of (length)2 = 2 correspond to nonzero roots of some
Lie algebra. Denoting the abelian generators by HI , and the remaining by Eα (in the
Cartan basis), the roots αI appear as
[HI , Eα
]= αIEα . (3.374)
For the Lie algebra associated with our (length)2 lattice vectors we clearly get
αI = kI . (3.375)
On the other hand, from the generalization of circle compactification, it is clear that
kI = p′IL are the charges to which the abelian gauge fields AIµ couple (note that AI
µ
now arise only in the left moving sector). Therefore, associating the abelian gauge
fields AIµ to the Cartan generators HI , we see that the extra massless states are
naturally associated with the remaining generators Eα. The symmetry group U(1)16
is thus enhanced to the non-abelian group G with U(1)16 as its Cartan subalgebra.
As mentioned earlier, modular invariance restricts G to either E8 × E8 or SO(32).
With (α′/4pILpI
L = 1, NL = 0), the level matching conditions give N(α)R = 0,
N(b)R = 1
2in the NS sector (non-abelian vector fields) or N
(α)R = 0, N
(d)R = 0 in the
Ramond sector (corresponding to the non-abelian components of the gaugino).
It should be emphasized that in heterotic string theory, the shape of the torus and,
hence, the emergence of the non-abelian gauge group is entirely dictaded by modular
invariance (that we have not studied here) and no parameter is adjusted by hand to
get extra massless states.
3.16.6 Massless Sector of the Heterotic String Spectrum
Let us summarize what we know about the massless sector of the heterotic string and
count all the physical degrees of freedom.
The mass equation for the left moving bosonic excitations was
α′
4M2
L =α′
4
∑I
pILpI
L + NL − 1 (3.376)
where
NL = N(α)L + N
(a)L (3.377)
and the mass equation for the right moving superstring excitations was
α′
4M2
R =
N
(α)R + N
(b)R − 1
2(NS sector)
N(α)R + N
(d)R (R sector)
. (3.378)
131
Level matching requires
M2L = M2
R (3.379)
and for the superstring, we also need to take into account the GSO projection
(−1)F |phys〉 = −|phys〉 (NS sector) (3.380)
Γ11(−1)F |phys〉 = ±|phys〉 (R sector) . (3.381)
Again, for NS sector this means that |0〉NS is eliminated, but states with an odd
number of b’s are kept. For R sector, there is the usual choice of chirality for the
vacuum. It does not matter which choice is made, both are equivalent.
So let’s then list the left moving states with M2L = 0. I’ll list only the the physical
polarizations, so I’m using the light-cone gauge:
αi−1|0〉 i = 1, . . . , 8 SO(8) vector (3.382)
αI−1|0〉 I = 1, . . . , 16 internal degrees of freedom (3.383)
|pIL〉 vacuum with nonzero internal momentum
satisfyingα′
4
∑I
pILpI
L = 1 . (3.384)
In the right moving sector, the states with M2R = 0 are
|0〉R SO(8) spinor vacuum with Γ11|0〉R = ±|0〉R(one choice for sign) from R sector (3.385)
bi12|0〉NS i = 1, . . . , 8 SO(8) vector from NS sector . (3.386)
Now let’s put these together:
States Spacetime fields # of physical degrees of freedom
B1: traceless symm. 10D graviton 35
eijαi−1|0〉bj
12
|0〉NS antisymm.tensor 28
dilaton 1
B2: ξi(k)αI−1|0〉bi
12
|0〉NS 16 U(1) vector fields 16× 8
(I labels them)
B3: additional massless gauge vectors 480× 8
ξi(k)|pIL〉bi
12
|0〉NS of E8 × E8 or SO(32) (480 = # of roots
of E8 × E8 or SO(32))
Table 8: Bosons with total of 8× 8 + 16× 8 + 480× 8 degrees of freedom
So there are an equal number of physical degrees of freedom in the bosonic and
fermionic sectors: (8 + 16 + 480) × 8. This is in agreement of supersymmetry. The
132
States Spacetime fields # of physical degrees of freedom
F1: ψia(k)αi−1|0〉|0〉aR spin 3
210D gravitino ψi
a 56
spin 12
fermion χa 8
F2: Ψa(k)αI−1|0〉|0〉aR 16 gauginos of U(1) 16× 8
(spin 12)
F3: λa(k)|pIL〉|0〉aR gauginos of E8 × E8 or SO(32) 480× 8
Table 9: Fermions with total of 8× 8 + 16× 8 + 480× 8 degrees of freedom
16+480 massless vector fields combine into the 496 components of a vector field in
the adjoint representation of E8 × E8 or SO(32).
We can group the massless fields into N = 1 supermultiplets:
B1 & F1 = graviton multiplet of N = 1 supergravity
B2-3 & F2-3 = gauge multiplet of N = 1 super Yang-Mills.
In other words, the low-energy effective field theory of the heterotic string is
N = 1 sugra + N = 1 E8 × E8 or SO(32) super Yang-Mills. The gauge vectors and
the gauginos transform in the adjoint representation of E8 ×E8 or SO(32). The fact
that it is N = 1 SUSY is reflected by a single gravitino. (Type IIA, IIB theories have
two gravitinos, one from R-NS and one from NS-R sector.)
***** END OF LECTURE 14 *****
3.17 Type I Superstrings
There are five perturbatively defined supersymmetric string theories. So far we have
discussed the four theories based on closed strings only, the Type IIA and IIB and
the heterotic E8 × E8 and SO(32) theories. The remaining one includes also open
strings. We start by going back to the classical superstring action. When we derived
the boundary conditions (??), we overlooked the possibility of open strings.
Let’s go back to the vanishing of the boundary term, eqn. (??):∫ ∞
−∞dτ
∫ π
0
dσ∂+(ψ−δψ−) + ∂−(ψ+δψ+) = 0 (3.387)
(for open string, the worldsheet coordinate σ ∈ [0, π]). Doing the σ-integral, we get∫ ∞
−∞[ψµ
+δψ+µ − ψµ−δψ−µ]
∣∣∣σ=π
σ=0= 0 . (3.388)
For open strings, the left and right moving excitations are not decoupled, so we
cannot require ψ+δψ+ and ψ−δψ− to vanish independently. Instead, for open strings
we impose
ψµ+δψ+µ = ψµ
−δψ−µ (3.389)
133
at both endpoints σ = 0, π. This is equivalent to
δ((ψ+µ)2
)= δ
((ψ−µ)2
)(3.390)
so we can have ψµ+ = ±ψµ
− at both ends σ = 0, π, and choose the sign independently
at each end. The overall sign does not matter (recall the GSO projection of the closed
string), so there are 2 independent choices:
R : ψµ+(τ, σ = 0) = ψµ
−(τ, σ = 0) and ψµ+(τ, σ = π) = ψµ
−(τ, σ = π) (3.391)
NS : ψµ+(τ, σ = 0) = ψµ
−(τ, σ = 0) and ψµ+(τ, σ = π) = −ψµ
−(τ, σ = π) .(3.392)
The two choices are the Ramond sector and the Neveu-Schwarz sector of open super-
strings.
The mode expansions are:
R sector : ψµ+ =
1√2
∞∑n=−∞
dµne−in(τ+σ) (3.393)
ψµ− =
1√2
∞∑n=−∞
dµne−in(τ−σ) (3.394)
NS sector : ψµ+ =
1√2
∑
r∈Z+ 12
bµr e−ir(τ+σ) (3.395)
ψµ− =
1√2
∑
r∈Z+ 12
bµr e−ir(τ−σ) (3.396)
(the 1/√
2 is a convention due to σ ∈ [0, π] for open, σ ∈ [0, 2π] for closed). Note
that there is only one set of oscillators (no d, b). The commutation relations of b, d
are as in (??), (??). The mass shell equation for the open superstring is
M2 =2
l2sNbos + Nfer − 2
l2s(3.397)
where
Nbos =∞∑
n=1
αµ−nαµn (3.398)
Nfer =
∑∞r= 1
2rbµ−rbµr (NS)∑∞
m=1 mdµ−rdµm (R)
(3.399)
and
a =
aNS = 1
2
aR = 0. (3.400)
134
Again, the spectrum would include a tachyon coming from the NS vacuum. So
once again we need the GSO projection
NS : (−1)F |phys〉 = −|phys〉 (3.401)
R : Γ11(−1)F |phys〉 = ±|phys〉 . (3.402)
Now the sign choice in R sector is an overall sign, so it does not matter. The GSO pro-
jection eliminates states with even # of b’s in the NS sector, including the tachyonic
vacuum. The massless excitations are the lowest physical states. They are:
bµ
− 12
|0〉NS ← massless U(1) vector (3.403)
|0〉aR ← massless spin1
2fermion . (3.404)
The low-energy effective field theory is a N = 1 super Yang-Mills with U(1) gauge
group. However, we can enrich the gauge symmetry by introducing Chan-Paton
factors.
3.17.1 Chan-Paton Factors (Works for Bosonic and Superstrings)
One can show that the end points of the open string behave like point charges (of
opposite charge). More precisely, they are U(1) charges. When an open string splits,
-e
+e
-e
+e
-e
+e
Figure 35: Splitting of an open string and pair creation of point charges.
a pair of charges is created.
However, one can promote the U(1) charges into non-abelian charges. This is
analogous to replacing ±e by something like quarks - point charges transforming
under some representations R and its complex conjugate R of a non-abelian gauge
group G. Let us denote the degrees of freedom of the representations R, R (analogous
to the color of a quark) by labels i and j. This is additional structure which is simply
tensored with each open string state:
|state〉 → |state〉 ⊗ |ij〉 ≡ |state; ij〉 (3.405)
135
ji
Figure 36: Oriented open string with Chan-Paton factors.
The degrees of freedom i, j are called Chan-Paton factors. For example, R and R
could be the fundamental and antifundamental representations n and n of a gauge
group U(n). Then we could use the defining representation matrices λaji to transform
to a new basis
|state; a〉 =n∑
ij=1
λaji|state; ij〉 . (3.406)
When the representation R is complex, so that R 6= R, there is a clear distinction
between the endpoints of the string. However, some gauge groups like SO(n) allow
for real representations R = R. In that casem the two endpoints are identical. Then,
it does not make sense to require the open string to be oriented anymore. So we
consider it to be an unoriented string, with a symmetry condition on every physical
state
Ω|state〉 = |state〉 . (3.407)
where Ω is an operator which changes the direction of the string:
Ω : σ → π − σ . (3.408)
Another feature of open strings is that the ends can join so that a closed string is
created:
Figure 37: Open string becomes a closed string. Worldsheet and snapshots of the
string.
Therefore, open string theories (with Neumann boundary conditions) must be
coupled to closed string theories.
136
If you make an unoriented closed string from a Type IIB theory by projecting out
all states with Ω|phys〉 6= |phys〉, you will project out one of the two gravitini and
other superpartners so that the supersymmetry gets broken from N = 2 to N = 1.
It turns out that the only consistent superstring theory including open superstrings
with a non-abelian gauge group G, is the unoriented open superstring theory with
G = SO(32) coupled to unoriented closed superstrings (IIB→ N = 1 SUSY). This
theory is the Type I superstring.
Interestingly, the low-energy effective theory is the same N = 1 supergravity +
N = 1 SO(32) super Yang-Mills theory as for the SO(32) heterotic string!
It has been discovered that in fact the two theories are dual descriptions of the
same underlying string theory. This is one link in the web of dualities that connects
all 5 superstring theories to a single underlying theory.
***** END OF THE OFFICIAL PART OF THE COURSE - EXAM MATERIAL
UP TO HERE *****
3.18 D-branes
You may have noticed three gaps in what we have discussed so far:
• In discussing bosonic open string boundary conditions (the equation just before
()), I only consider the possibility that the endpoints vibrate freely:
∂σXµ(τ, σ = 0) = ∂σX
µ(τ, σ = π) = 0 , (3.409)
the Neumann boundary conditions. In the class I mentioned that there is an-
other possibility: that the endpoints are fixed
δXµ(τ, σ = 0) = δXµ(τ, σ = π) = 0 , (3.410)
these are the Dirichlet boundary conditions. But I did not use that possibility
so far.
• The low-energy effective field theories of the Type II theories contained RR
gauge fields. I said that they couple to nonperturbative extended objects called
D-branes but I did not elaborate on that.
• We have discussed the T-duality map only in the context of closed strings. What
happens to open strings, when one target space direction is compactified on a
circle and we perform T-duality?
Let us start from the last question. Consider an oriented bosonic string moving
in 25 + 1 dimensions Xµ, µ = 0, . . . , 25. We compactify one direction, say X25, on
a circle of radius R. In closed string theory, T-duality would invert the radius to
137
α′/R and momentum/winding modes would become winding/momentum modes. Is
there any reason consider T-duality in open string theory? Yes, because the open
string ends can interact and close to form a closed string. Because open string theory
will contain closed strings as well, we need to understand how the former map under
T-duality.
The open strings carry momentum, but there are no obvious winding modes,
because of topological reasons. You can always unwind an open string: Instead,
25X
Figure 38: Open string unwinds due to its tension.
recall that the T-duality map for closed string mapped
X25 → X25 (3.411)
∂+X25 = ∂+X25 (3.412)
∂−X25 = −∂−X25 (3.413)
or, if we separate the left movers from right movers
X(τ, σ) = XL(σ+) + XR(σ−) , σ± = τ ± σ (3.414)
the T-duality map is
XL = XL , XR = −XR . (3.415)
Recall then the mode expansion of an open string:
Xµ = xµ + l2spµτ + ils
∑
n6=0
1
nαµ
n cos(nσ)e−inτ . (3.416)
Let us break this to left and right movers (they are coupled, having the same oscillator
coefficients). Write
Xµ = XµL(τ + σ) + Xµ
R(τ − σ) (3.417)
XµL(τ + σ) =
xµ + cµ
2+
1
2l2sp
µ(τ + σ) +ils2
∑
n 6=0
αµn
ne−in(τ+σ) (3.418)
XµR(τ − σ) =
xµ − cµ
2+
1
2l2sp
µ(τ − σ) +ils2
∑
n 6=0
αµn
ne−in(τ−σ) . (3.419)
Now, we use the T-duality map and flip the sign of X25R :
X25L = X25
L , X25R = −X25
R . (3.420)
138
Then, in the T-dual target space (of radius l2s/R45):
X25 = X25L + X25
R = X25L −X25
R = c25 + l2spµσ + ls
∑
n 6=0
α25n
nsin(nσ)e−inτ . (3.421)
This obviously satisfies
δX25(τ, σ = 0) = δX25(τ, σ = π) = 0 (3.422)
(i.e., X25(τ, σ = 0), X25(τ, σ = π) are constants) at the endpoints of the open string,
as sin(0) = sin nπ = 0. Thus, in the T-dual circle the open string satisfies Dirichlet
boundary conditions at the endpoints.
Note also that X25, X25 satisfy
∂τX25 = ∂σX
25 = −ils∑
n
sin(nσ)e−inτα25n (3.423)
∂σX25 = ∂τX
25 = ls∑
n
cos(nσ)e−inτα25n + lsp
25 , (3.424)
i.e., T-duality interchanges the worldsheet coordinates τ, σ as in the closed string
case.
In the original picture, in the mode expansion of X25, p25 was the (25-component)
of the center-of-mass momentum of the open string. Because X25 was a circle of
radius R, p25 has to be quantized:
p25 =m
R, m ∈ Z . (3.425)
Now in the T-dual circle,
X25 = c25 + l2sp25σ + . . . . (3.426)
Because p25 multiplies σ, instead of τ , it no longer corresponds to momentum. Now
it actually does correspond to winding: now the σ = 0 endpoint of the string is stuck
at X25 = c25 (we can set c25 = 0 by a choice of origin here) and the σ = π endpoint
of the open string is stuck at
X25 = c25 + l2smπ
R= 2πm
l2s/2
R= 2πmR , α′ ≡ l2s
2. (3.427)
Thus, the open string winds around m times before ending stuck to the same position
where it started from: Note that in the other (noncompact) directions Xµ=0,...,24
the open string satisfies Neumann b.c.’s so the endpoints are free to move. That
means that the open string is stuck on a 24-dimensional hyperplane in the 25 space
dimensions:
45Check the definition of α′...
139
m=1
R~
m=0
m=1
Figure 39: Now the open string can wind because its endpoints are stuck.
The 24-dimensional hyperplane is a higher-dimensional analogue of a membrane.
It is clear that instead of compactifying just one dimension (X25), we can compactify
25 − p of the 25 spacelike dimensions and perform T-duality on all of them - this
produces a 25 − (25 − p) = p-dimensional hyperplane, called a Dirichlet p-brane
or Dp-brane for short. So the figure above depicts a D24-brane. To sum up, the
boundary conditions for a Dp-brane are
X1, . . . , Xp : Neumann (string can move) (3.428)
Xp+1, . . . , X25 : Dirichlet (string stuck) . (3.429)
Note that when we shrink the original circle (or torus) to zero size (R → 0 or V25−p →0), the dual target space decompactifies (R → ∞ or V25−p → ∞). Then we have D-
branes in an infinite Minkowski space. Of course we could as well then skip the
T-duality procedure altogether, there is simply a choice of Neumann or Dirichlet
b.c.’s for open strings, corresponding to open strings propagating on Dp-branes of
different p. (In particular, a D25-brane is the 26-dimensional spacetime itself.)
3.19 Multiple D-branes
Now that we know how to introduce a single D-brane, we can ask if there can be
several of them. For several D-branes, the natural starting place is to equip the open
string with Chan-Paton factors.
140
m=1
−π πR R
identify
X
X
25
µ=25
D-brane
m=0
m=1
Figure 40: Open strings become attached to a hyperplane.
Consider an oriented bosonic open string with endpoints carrying charges that
transform under the fundamental representation N and the antifundamental repre-
sentation N of the gauge group U(N): The states of the open string have the form
_i j
Figure 41: Chan-Paton factors.
|ψ, ij〉 = |ψ〉 ⊗ |ij〉 ≡ |ψ〉λij (3.430)
where (λij) is an U(N) matrix. In string scattering diagrams, the Chan-Paton factors
cannot change in one part of the boundary of the diagram. Consider, e.g., a tree-level
open string diagram: The amplitude picks up a contribution
i
i
j
j
2
kk
1
23
i j
k
1
3
Figure 42: Open string disc diagram with Chan-Paton factors.
141
λ2ijλ
2jkλ
3ki (3.431)
from the Chan-Paton factors. We must also sum over all possible values of i, j, k, so
we get ∑
ijk
λ1ijλ
2jkλ
3ki =
∑i
(λ1λ2λ3)ii = Tr(λ1λ2λ3) . (3.432)
All open string amplitudes will contain traces over products of U(N) matrices like in
the above. Such traces are invariant under U(N) transformations
λa → ΛλaΛ† , ∀Λ ∈ U(N) (3.433)
reflecting the underlying U(N) gauge symmetry.
Now, compactify X25 on a circle of radius R. Again, the dual spacetime is a circle
of radius R, in other words we identify
X25 = X25 + 2πnR . (3.434)
Now, as X25 → X25 + 2πnR (the center-of-mass of the string is rotated around the
dual circle), in the group space the string state can transform in a more complicated
way. The Chan-Paton factors need not come back to their original values, but the
|ij〉 ≡ λij part of the string state can come back to itself up to a U(N) transformation:
(λij) ≡ λ → MλM † . (3.435)
This is called gauge holonomy, and M is called a Wilson line. Let us choose M to be
diagonal:
M = diag(e−iθ1 , . . . , e−iθN ) . (3.436)
Then, upon transporting around a circle, a string state with momentum p25 and CP
factor λij acquires a phase factor
|ψ〉λij → ei(θj−θi)|ψ〉λij . (3.437)
This means that the momentum p25 does not need to be quantized in units of 1/R.
Now we require
eip252πR = ei2πn+i(θj−θi) → p25 =n
R+
θj − θi
2πR. (3.438)
Now the string coordinates Xµ are also promoted to N ×N matrices, to capture the
N ×N CP degrees of freedom. Substituting p25:
X25ij = c25
ij + l2s
(n
Rδij +
θj − θi
2πR
)σ + . . .
= c25ij + 2R
(nδij +
θj − θi
2π
)σ + . . . . (3.439)
142
Now it is convenient to choose c25ij = Rθi: then
X25ij (τ, σ = 0) = Rθi , open string starts at angle θi (3.440)
X25ij (τ, σ = π) = Rθj + 2πRnδij , and end at angle θj . (3.441)
Diagonal entries i = j correspond to open strings that can wind around the circle and
then end at the same angle as they started from. Off-diagonal entries correspond to
strings that go from θi to θj. The interpretation is that now there are N D-branes,
located at angles θi, with N ×N open strings interpolating between them. Below is
a picture for N = 3: or, drawing hyperplanes:
θ
θ
~
1
θ2
3
1
2
3
1-2
1-3
winding 3-3
winding 3-3
winding 3-3
2-3
1-1R
Figure 43: N = 3 leads to 3 fixed points.
0
1-1
1-2 2-3
1-3
winding 3-3
winding 3-3
θ1 θ2
θ3
R
R
R
~
~ ~
1 2 3
2πR~winding 3-3
Figure 44: N = 3 becomes the number of D-branes.
143
The mass formula for the open string excitations (between D24-branes) is
M2 = (p25)2 +1
α′(N − 1)
=
[2πn + θj − θi
2πR
]2
+1
α′(N − 1)
=
[2πn + θj − θi
2πα′R
]2
+1
α′(N − 1) . (3.442)
For the vector states at level N = 1 with zero winding (n = 0) we get
M =θj − θi
2πα′R ≡ TLji (3.443)
where T = 1/(2πα′) is the tension of the string and Lji = (θj − θi)R is the length of
the open string between branes i, j.
Now, if all the branes are at different positions, θi 6= θj, the only massless vector
states come from open strings which start and end on the same brane. The gauge
group is U(1)N .
If we put M of the N branes to the same position, there are M(M −1) additional
massless states. The gauge group becomes
U(M)× U(1)N−M . (3.444)
We can also introduce D-branes in superstring theories. Then one can have closed
(Type IIA or IIB) superstrings between the branes (in the “bulk”) and open su-
perstrings trapped on or interpolating between the branes. The low-energy the-
ory of the open strings is a supersymmetric Yang-Mills theory. By moving the
branes with respect to one another, we can break the gauge symmetry from U(N) to
U(M)× U(1)N−M . This corresponds to the Higgs mechanism!
One can show (by supersymmetry analysis) that only Dp-branes with p odd are
possible in Type IIB theory and p even in Type IIA theory. Thus IIB theory contains
D1,Dp,. . . ,D9-branes and IIA theory D0,D2,. . . ,D9-branes.
One can also simultaneously have D-branes of different dimension. They can also
form bound states. Thus, e.g., IIB theory can have (marginally) bound states of N1
D1-branes and N5 D5-branes: Then one has open strings that interpolate between
different kinds of D-branes: thus one end of the open string can have a Dirichlet b.c.
in one direction while the other end has a Neumann b.c. in the same direction:
Different brane configurations give arise to multitude of gauge symmetries and
matter contents. This gives a geometrical way to look at (supersymmetric) gauge
theories.
Another feature of D-branes is that at weak string coupling they are more massive
than the fundamental strings. The D-brane masses scale like
M ∼ 1/gst (3.445)
144
D1-branes
N5 D5-branes
N1
Figure 45: Stack of N1 D1-branes and N5 D5-branes.
whereas Mstrings ∼ g0st. The D-branes are nonperturbative objects in string theory.
One can also show that Dp-branes in Type II theories indeed carry charges that
couple to the RR gauge fields. This calculation is beyond the scope of these lectures.
The basic idea is to consider closed string exchange at tree level between parallell
D-branes:
One can perform a string theory calculation for the amplitude for the closed string
exchange, isolate the contribution from RR sector and compare with a field theory
calculation. Agreement of the calculations shows that D-branes carry RR charge and
also gives the value for the tension (mass density) of the D-branes.
In the low-energy supergravity, one can also find how the D-branes deform the
spacetime. The metric of a Dp-brane looks like
ds2 = f−1/2p (−dt2 + dx2
1 + . . . + dx2p) + f 1/2
p (dx2p+1 + . . . + dx2
9) (3.446)
in the string frame, with a dilaton dependence
e−2φ = f (p−3)/2p (3.447)
and a RR gauge field
A0···p = −1
2(f−1
p − 1) (3.448)
145
D(free to move)
(fixed)
N
Figure 46: Open string with D and N boundary conditions.
Figure 47: Tree-level closed string exchange between parallel D-branes.
where...46 with Qp the charge of the brane.
By suitable brane configurations, one can generate all kinds of interesting space-
time metrics. E.g., by taking a stack of D5-branes and D1-branes and compactifying
them on a 5-torus, one can generate a charged black hole in 4 + 1 dimensions. More
complicated configurations can even yield “ordinary” electrically (or magnetically)
charged Reissner-Nordstrom black holes in 3 + 1 dimensions.
The use of doing that is that one can use the underlying D-brane/string picture
to investigate deep issues associated with thermodynamics and quantum mechanics
of black holes. This is one case where string theory has allowed us to solve problems
which have been mystifying without using string theory.
There is also an increasing interest of using brane dynamics in developing cosmo-
logical models. There one considers our world as a brane living in a higher dimensional
spacetime.
46My last line is not visible...
146
3.20 String Dualities
We have covered all the known perturbative formulations of consistent string theories:
Type IIA, Type IIB, Het SO(32), Het E8 × E8, Type I .
However, you have already seen an example of a relation between two theories. The
IIA theory compactified on a circle is T-dual to IIB theory on a dual circle. By
T-duality and shrinking the radius of the circle to zero, you can go from IIA in 10D
to IIB in 10D and vice versa. The T-duality is an example of a perturbative duality
(mapping defined at the level of perturbation theory).
I have also mentioned two other dualities. The low-energy EFT of the Type
IIB theory was dual to itself under SL(2,R) transformations. In particular, they
changed weak coupling to strong coupling. The SL(2,R) breaks to a SL(2,Z) group
by requiring that the NSNS and RR charges are quantized. The SL(2,Z) turns out to
be a self-duality of the IIB string theory. The duality maps weak coupling to strong
coupling and vice versa, such a duality is called S-duality.
Another relation was found between the low-energy EFT’s of Type I and Het
SO(32) theory, both were N = 1 SUGRA + SO(32) N = 1 SYM theories47. It turns
out that the string theories are also dual to each other, however the strong coupling
limit of one is the weak coupling limit of the other. This is S-duality, but not a
self-duality, since it relates two different looking theories.
So why can we have dualities between different looking theories? Let’s step back
for a moment and think about quantum field theories.
In quantum field theory, we generally start with a set of fields and an action S
which depends on several parameters λ1, λ2, . . .: the masses and the coupling con-
stants. All the information at the quantum level is contained in the path integral
Z = Dφiei~S(φi,λi) . (3.449)
However, because the action is non-linear48 (and there are subtleties with the mea-
sure), we cannot evaluate the path integral. Instead, what we do is that in the space
of all possible values of the parameters λi, called the moduli space M, we pick a
particular point P = λc1, λ
c2, . . . ∈ M corresponding to the classical values of the
parameters. Then we consider a small neighborhood of P , and use perturbation
techniques for calculating, e.g., scattering amplitudes.
Suppose that we can use perturbation theory reliably in the neighborhood U of
P . We have no understanding on the behavior of the theory in the neighborhood of
U ′ of P ′, which is in the “strong coupling” regime.
What can happen that around point P ′ there is actually another way to write the
theory. There can be a complicated mapping from the original set of fields φi to a
47Again, N or N ...and have you told that SUGRA=supergravity and SYM=super Yang-Mills?48Was it non-linear or nonlinear...
147
U’
P
P’
M
U
Figure 48: Two perturbative neighborhoods in the full moduli space.
new set of fields ψi and a new parameterization κi of the moduli space M around
the point P ′, with a different looking action S(ψi, κi) such that we can again use
perturbation theory around P ′ = κc1, κ
c2, . . . to evaluate
Z =
∫Dψe
i~ S(ψi,κi) (3.450)
(the same path integral expressed in a different way), this time using δκ1 = κ1−κc1, . . .
as the small parameters and using the Feynman rules obtained from S(ψi, κi).
Then we say that the actions S(φi, λi) and S(ψi, κi) are dual perturbative for-
mulations of the same theory, but valid (or useful) around different points in moduli
space.
Recall that there are a lot of simple functions that do not have a Taylor expan-
sion around a point P . Example: suppose that we compute a scattering amplitude
perturbatively around λ = 0. Then we expect that A(λ) has a Taylor expansion
A(λ) = A(0) + A′(0)λ + A′′(0)λ2
2+ . . . . (3.451)
However, they may be other effects in the theory whose contribution to A(λ) goes
like (for example)
Anp(λ) ∼ e−1/λ . (3.452)
Clearly, this cannot be expanded around λ = 0. Such effects are then nonperturbative
effects. However, it may be that the theory has a dual formulation with = 1/λ49 as a
coupling constant. Then the dual formulation can be used perturbatively as λ →∞.
So the effects which were nonperturbative in the original formulation may become
perturbative in the dual formulation (with ???).
49What symbol?
148
A simple QFT example: the Sine-Gordon model
LSG =1
2(∂φ)2 − 1
λ
[cos(
√λφ)− 1
](3.453)
and the massive Thirring model
LT = iψ/∂ψ −mψψ − λD(ψψ)2 (3.454)
are dual to each other with the coupling constants related ny
λ =1
1 + λD
(3.455)
and the fields related by
ψ = :eiφ: (3.456)
(“bosonization”)50.
A similar story is expected to hold for string theory. The current understanding
(with gaps. . . ) is that there is only one theory, with a big moduli space, with the
five different “theories” corresponding to different perturbative definitions valid at
different points (neighborhoods) of the moduli space.
Let’s consider first the Type I - Het SO(32) duality in some more detail.
3.20.1 Type I - Heterotic SO(32) Duality
The low-energy effective field theory of Type I has the action (bosons only):
SIeff ∼
∫d10x
√g
R− 1
2(∂φ)2 − 1
4eφ/2 Tr(FµνF
µν)− 1
12eφH2
(3.457)
where F aµν is the YM field strength of the SO(32) gauge field Aa
µ and Hµνλ is the field
strength of the antisymmetric tensor field Bµν (≡ Cµν) from the RR sector. φ is the
dilaton, the action is written in the Einstein frame.
The heterotic SO(32) theory has the low-energy EFT action
SHOeff ∼∫
d10x√
g
R− 1
2(∂φ)2 − 1
4e−φ/2 Tr(FµνF
µν)− 1
12e−φH2
(3.458)
where φ is the (heterotic string) dilaton, Hµνλ = ∂µBνλ + . . . the field strength of
the (bosonic sector) antisymmetric tensor field, F aµν the field strength of the SO(32)
vector field Aaµ.
The actions are identical up to the sign of the dilaton. Note another subtlety.
The actions depend on the string coupling gs, but in the above the gs-dependence
50S. Coleman, PRD 11, 2788 (1975)
149
has been absorbed into fields by a suitable rescaling. It turns out that the effective
actions are invariant under
φ → φ− c (3.459)
gs → ecgs (3.460)
where c is an arbitrary constant. Thus, by setting ec = g−1s we can remove gs from
the action; this has been done in the above. If we then map the two actions into each
other by
φ → −φ (3.461)
we must also invert the string coupling
gs → g−1s (3.462)
in order for
eφ−c = gseφ (3.463)
to remain invariant.
Thus the weak coupling limit of the Type I theory corresponds to the strong
coupling limit of Het SO(32) and vice versa. This is why the two can look so different.
The S-duality of Type I and Het SO(32) theories has not been proven rigorously.
There are nonperturbative tests of the duality, that can be done, and so far the duality
conjecture has not failed.
3.20.2 Type IIA - IIB Duality
The T-duality of the theories was already discussed, but let me mention a few com-
ments on the relation of the parameters: the low-energy EFT actions of IIA on S1 of
radius R and IIB on radius R are of the form
IIA : S ∼ 1
g2s
∫d10x. . . =
2πR
g2s
∫d9x. . . (3.464)
IIB : S ∼ 1
g2s
∫d10x. . . =
2πR
g2s
∫d9x. . . . (3.465)
Since the radii are related by
R =α′
R, (3.466)
the coupling constants must be related by
gs = gs
√α′
R. (3.467)
So the coupling constants are proportional to each other, as expected since T-
duality is a perturbative duality.
150
3.20.3 Heterotic SO(32) - Heterotic E8 × E8 Duality
The two heterotic theories are also T-dual to each other when compactified on a circle.
Although the E8×E8 and SO(32) lattices (corresponding to the 10-dimensional torii
of the internal dimensions) are inequivalent, it can be shown that the larger lattices,
that result after adding one more compact dimension, are equivalent. For details, see
the textbooks. The parameters are related by
R =α′
R(3.468)
gs = gs
√α′
R. (3.469)
3.20.4 Type IIA - M-theory Duality
We have seen that the strong coupling limit of Type IIB theory is dual to its weak
coupling limit. What is the strong coupling limit of IIA theory? It turns out to be a
new theory that we have not discussed yet!
The low-energy EFT of IIA is the Type IIA supergravity in ten dimensions. It
was known before that it can be obtained from dimensional reduction of the highest-
dimensional supergravity theory, the 11-dimensional supergravity theory. What is
the 11-dimensional SUGRA a low-energy theory of?
It is illuminating to consider the D0-branes of IIA theory. They are the lightest
nonperturbative objects in Type IIA theory, with a mass
T0 =1
gs
√α′
. (3.470)
Dp-branes have tension (mass for pointlike D0-branes)
Tp =1
gs(2π)p√
(α′)p+1. (3.471)
A bound state of N D0-branes has a mass
m =N
gs
√α′
, (3.472)
and their RR-charge is equal to the mass. Thus there is an infinite tower of evenly
spaced masses from the bound states of D0’s.
Now suppose that we compactify the 11-dimensional supergravity on a circle. The
momentum of the massless graviton satisfies
−m211 = p2 = −(p0)2 + (p1)2 + . . . + (p9)2 + (p11)2 = 0 (3.473)
(we label the 11th dimension by 11 instead of 10, as in the literature). Now, the p11
component must be quantized,
p11 =N
R11
(3.474)
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(R11 denotes the radius of the circle). Hence, in 10 dimensions there is a tower of
massive objects:
−m210 = −(p0)2 + . . . + (p9)2 = −
(N
R11
)2
→ m10 =N
R11
= p11 . (3.475)
Now, on the other hand the 11-dimensional metric GMN reduces in 10 dimensions
to the 10-dimensional metric gµν , a vector field Aµ = Gµ,11, and the dilaton φ =
G11,11. The vector field of the IIA SUGRA is the RR 1-form Cµ, which couples to
the D0-brane. It should correspond to Aµ. The charge coupling to Aµ = Gµ,11 is the
momentum p11. So the dimensional reduction gives a tower of particles with mass
equal to the charge,
m = p11 =N
R11
(3.476)
just like the D0-branes. So we would like to identify the two. Then we should have
the relation
R11 =√
α′gs . (3.477)
This fits, because as R11 → 0 we recover the perturbative IIA theory, gs → 0, ???
excitations become massive just like the D0-branes. On the other hand, as gs →∞,
the IIA theory will grow another dimension and becomes 11-dimensional, and the
low-energy ??? theory is the 11-dimensional supergravity. The full 11-dimensional
theory is called M-theory. Many of its features are known, but most of it remains
Mysterious. At this point, there is no perturbative formulation for it, and we do not
know even good perturbative degrees of freedom. M-theory would be a topic for a
separate course.
There are some parameter relations that are easy to obtain. The fundamental
unit of M-theory is the 11-dimensional Planck length lPl. 11-dimensional supergravity
action looks like
S11 ∼ 1
l9Pl
∫d11x
√gR + . . . (3.478)
(l9Pl for dimensional reasons). Reducing to 10D on a circle, we obtain for 10-dimensional
SUGRA
S10 ∼ R11
l9Pl
∫d10xe−2φ√gR + . . . . (3.479)
On the other hand, the low-energy action of IIA (in string frame) looks like
SIIA10 ∼ 1
l8sg2s
∫d10xe−2φ√gR + . . . (3.480)
so we must identify (using R11 = lsgs)
lPl = g1/3s ls . (3.481)
To summarize the chain of dualities so far, we have:
To discuss the missing link, we need some more machinery. We’ll divert first to
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IIB IIA Het SO(32) IS
TT
M
S
HetE8xE8
S1
Figure 49: String-string dualities.
Orbifolds An orbifold is an example of a slightly more complicated possible com-
pact manifold on which to compactify string theory. Consider a space X (e.g., Rn or
T n (torus)) and its symmetry (isometry) group G. Consider a discrete subgroup H of
G. Example: X = R2, H = Z2 acting by x → −x (reflection). The coset space X/H
is an orbifold. It is obtained by identifying the points x, gx for all x ∈ X, g ∈ H. In
general there will be orbifold fixed points where x = gx.
Example 1. R2/Z2 The tip of the cone (x = (0, 0)) is the fixed point. As above,
a cone-x
x
0
0
identify
0
Figure 50: The orbifold R2/Z2.
the orbifold fixed points are usually singular points.
Example 2. S1/Z2
R0 πR
0 π
Figure 51: The orbifold S1/Z2.
Now there are two fixed points, 0 and πR.
Consider the Hilbert space H of a theory defined on the original space containing
X. We can gauge the action of the symmetry group H by projecting to a H-invariant
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subspace of H: keep all H-invariant states:
HH =|ψ〉 ∈ H
∣∣g|ψ〉 = |ψ〉 ∀g ∈ H
. (3.482)
Consider a closed string theory on an orbifold. Apart from closed string states,
where the closed string is periodic in the usual manner, called the untwisted sector
states (see Fig. 49 for an example)
Figure 52: Untwisted string in R2/Z2.
with X(τ, σ + 2π) = X(τ, σ), we have to consider additional string states where
the string is periodic up to the action of H: X(τ, σ+2π) = gX(τ, σ), for some g ∈ H.
Such states are called twisted sector states. An example of these is depicted in Fig.
50.
closed string
closed string
2π/3
Figure 53: Twisted string in R2/Z3.
In the original plane before orbifolding, this would be an open string. So the
twisted sector states are new, i.e., not present in the closed string theory before
orbifolding.
Then we consider another construction:
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Orientifolds From the discussion of Type I theory, recall the worldsheet parity
reversal operation Ω:
Ω : σ →
π − σ (open strings)
2π − σ (closed strings). (3.483)
Mapping the (Euclidenized) worldsheet to plane by z = eτ−iσ, Ω acts by
Ω : z → z . (3.484)
So for bosonic fields, Ω interchanges the left and right movers:
X = XL(z) + XR(z)Ω→ X ′ = XR(z) + XL(z) . (3.485)
Now suppose that we compactify one direction and perform T-duality. T-duality
changes the sign of XR:
X = XL(z) + XR(z)T→ X = XL(z)−XR(z) . (3.486)
Suppose that we then apply Ω (interchanging XL ↔ XR):
XΩ→ X ′ = XR(z)−XL(z) . (3.487)
This is the same as first mapping X → −X and then doing parity on the worldsheet
z ↔ z. The combined operation is thus a Z2 reflection followed by a worldsheet
parity reversal. Recall that X ∈ S1. If we would perform a Z2 reflection only, we
would obtain a S1/Z2 orbifold (from the dual circle). However, because of the addi-
tional worldsheet parity reversal, we get something new. This is called an orientifold.
There are still two fixed points, or including the 8 noncompact dimensions, 2 8 + 1-
dimensional hyperplanes called the orientifold planes:
155
In the bulk, we still have oriented closed strings because the orientifold construc-
tion relates an oriented closed string to its mirror image in spacetime with an opposite
orientation. The string and its mirror image meet at the fixed plane, so in the orien-
tifold planes the strings will be unoriented.
Recall that Type I theory can be obtained from IIB by projecting with Ω. Now,
adding T-duality, we obtain IIA theory in the bulk and unoriented closed strings in
the orientifold planes. The full theory is called Type I’ theory.
Now we are ready to present our last duality:
3.20.5 Heterotic E8 × E8 - M-theory Duality
What is the strong coupling limit of heterotic E8 × E8 theory? (c.f., IIB→ IIB, I→HO, HO→ I, IIA→ M)
Compactify first E8 × E8 heterotic theory on S1 (direction X9 with radius R9)
and apply T-duality. We obtain heterotic SO(32), with
R′9 =
α′
R9
(3.488)
g′s = gs
√α′
R9
. (3.489)
Then, S-duality maps it to Type I theory, with
R′′9 =
R′9√g′s
=α′3/4
√R9gs
(3.490)
g′′s =1
gs
=R9√α′gs
. (3.491)
Then, T-duality maps Type I to Type I’ theory = IIA with 2 orientifold planes at
the boundary.
The parameters are
R′′′9 =
α′
R′′9
=√
R9gsα′1/4 (3.492)
g′′′s = g′′s
√α′
R′′9
= R3/29 α′−3/4g−1/2
s . (3.493)
Now we want to let R9 →∞ corresponding to Het E8×E8 in 10D. Since we ended
up with I’=IIA in the bulk, with g′′′s →∞, we know that we obatain a 11-dimensional
theory with 2 orientifold 9-branes, M-theory in the bulk. One can show that the E8
gauge groups reside on the orientifold 9-branes at the boundaries of 11 dimensions –
we obtain the E8 × E8 M-theory on a segment, depicted in Fig. 52.
This is the Horava-Witten model. The final chain of dualities of these lectures is
then depicted in Fig. 53.
For a refined discussion, see the literature. It must be evident for you that we
have just scratched the surface of string dualities (and string theory in general)!
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8
X11
ME8 E
Figure 54: M -theory compacified on a line segment.
IIB IIA Het SO(32) IS
TT S
2
HetE8xE8
S1M
S1/Z
Figure 55: The chain of string-string dualities.
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